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+Caustics in the sine-Gordon model from quenches in coupled 1D Bose gases
+Aman Agarwal,1, 2, 3, 4, 5, 6, ∗ Manas Kulkarni,3, † and D. H. J. O’Dell1, ‡
+1Department of Physics and Astronomy, McMaster University,
+1280 Main St.
+W., Hamilton, Ontario, Canada L8S 4M1
+2BITS-Pilani, K. K. Birla Goa Campus, NH17B, Bypass Road, Zuarinagar, Goa 403726, India
+3International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru – 560089, India
+4Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5
+5Department of Physics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1
+6Institute of Physics, University of Greifswald, 17489 Greifswald, Germany
+(Dated: January 23, 2023)
+Caustics are singularities that occur naturally in optical, hydrodynamic and quantum waves,
+giving rise to high amplitude patterns that can be described using catastrophe theory.
+In this
+paper we study caustics in a statistical field theory setting in the form of the sine-Gordon model
+that describes a variety of physical systems including coupled 1D superfluids. Specifically, we use
+classical field simulations to study the dynamics of two ultracold 1D Bose gases (quasi-condensates)
+that are suddenly coupled to each other and find that the resulting non-equilibrium dynamics are
+dominated by caustics. Thermal noise is included by sampling the initial states from a Boltzmann
+distribution for phononic excitations. We find that caustics pile up over time in both the number and
+phase difference observables leading to a characteristic non-thermal ‘circus tent’ shaped probability
+distribution at long times.
+I.
+INTRODUCTION
+Wave focusing is ubiquitous in nature and leads to
+localized regions of high amplitude called caustics that
+dominate wavefields.
+Everyday examples are provided
+by rainbows and also the bright lines on the bottom of
+water pools which are caused by the focusing of sunlight
+by raindrops and surface water waves, respectively [1].
+Caustics also occur in water waves themselves as ship
+wakes [2] and more dramatically as tsunamis (focused by
+the topography of the seabed [3–5]) and tidal bores (fo-
+cused by v-shaped bays [6]). Astrophysical examples in-
+clude gravitational lensing by matter and the twinkling of
+starlight due to time-dependent fluctuations in the den-
+sity of Earth’s atmosphere. Natural focusing also leads
+to the phenomenon of branched flow [7] and is speculated
+to have given rise to the filamented nature of the large
+scale structure of the universe [8–11]. In all these systems
+caustics give rise to extreme amplitude fluctuations that
+occur more frequently than those predicted by gaussian
+statistics [12].
+A remarkable property of caustics is that they com-
+monly take on particular characteristic shapes. This is
+because caustics are singularities of the ray description,
+i.e. they are places where two or more rays coalesce lead-
+ing to a diverging intensity in the short wavelength limit
+[13]. Such singularities are described by Thom’s catas-
+trophe theory which rigorously shows that only certain
+shapes of singularity are structurally stable against per-
+turbations and hence occur under ‘natural’ or generic
+∗ aagarw03@uoguelph.ca
+† manas.kulkarni@icts.res.in
+‡ dodell@mcmaster.ca
+conditions [14–16].
+These special shapes or catastro-
+phes form a hierarchy organized by dimension where the
+higher ones contain the lower ones. Each member of the
+hierarchy represents a class of equivalent shapes that can
+be smoothly transformed into each other, but each class
+is distinct and cannot be smoothly transformed into any
+of the others. In two dimensions the only structurally
+stable shape is the cusp and we shall see it appear fre-
+quently when we plot quantities such as number fluctu-
+ations versus time.
+It is worth noting in this context
+that the humble point focus that we associate with lens-
+ing is structurally unstable and unfolds into an extended
+caustic in the presence of perturbations (aberrations).
+Natural lenses are of course never perfect and so typi-
+cally produce the shapes predicted by catastrophe theory.
+The upshot of all this is that caustics represent a form of
+universality in nonequilibrium wave dynamics: they fall
+into equivalence classes each with their own shapes and
+scaling properties analogous to, but a generalization of,
+equilibrium phase transitions [13, 17].
+Caustics should equally be present in quantum waves
+where, due to the probabilistic interpretation, they cor-
+respond to regions of high probability density. Quantum
+matter wave caustics have been seen in experiments with
+cold neutrons [18, 19], electron microscopes [20], atom op-
+tics [21–23], and most recently in atom lasers [24]. The-
+oretical works on such matter wave caustics have also
+considered their ‘fine structure’ [13] which features a lat-
+tice of vortices [25–27]. Quantum fields are another area
+where caustics are expected to form naturally during dy-
+namics. Early work centred on the electromagnetic field
+[28, 29], including an interpretation of Hawking radiation
+as a ‘quantum catastrophe’ [30], and more recently this
+idea has been extended to quantum many-particle sys-
+tems including bosonic Josephson junctions [26, 31, 32],
+the XY model with long-range interactions (Hamiltonian
+arXiv:2301.08410v1  [cond-mat.quant-gas]  20 Jan 2023
+
+2
+Figure 1. Schematic of the setup we consider. The top fig-
+ure shows two quasi one-dimensional gases that are prepared
+independently and then suddenly coupled together. We call
+this process of sudden coupling a “J-quench”. ρ1(z) and ρ2(z)
+represent the density (red) in the first and second conden-
+sates, respectively. Similarly, φ1(z) and φ2(z) represent the
+phases (black) of the two condensates. Prior to the J-quench,
+these fields in the two condensates are independent and con-
+tain thermal fluctuations. The bottom figure shows how a J-
+quench could be implemented by suddenly reducing the tun-
+neling barrier height in a double well potential from a higher
+to a lower value.
+mean field model) [33], quantum spin chains [27] and the
+Bose-Hubbard model [34].
+One point to appreciate is
+that the caustics in many-body systems can occur in the
+wavefunction associated with an entire N-body configu-
+ration. Quantum many-particle caustics therefore live in
+Fock space which can have a large number of dimensions
+and hence lead to very complicated catastrophes [34].
+However, catastrophes obey projection identities which
+means that when projected down to lower dimensions one
+obtains either the same catastrophe or one lower down
+the hierarchy [35]. Thus, low order correlation functions
+obtained by integrating out most of the degrees of free-
+dom will also generically contain caustics [27].
+In this paper we study caustics in the sine-Gordon (SG)
+model.
+The (classical) SG model obeys the nonlinear
+wave equation
+∂2φ
+∂t2 − c2
+0
+∂2φ
+∂z2 + ω2
+0 sin φ = 0
+(1)
+where φ = φ(z, t) is a one dimensional field, and c0 and
+ω0 represent a characteristic speed and frequency, respec-
+tively. If c0 is taken to be the speed of light then Eq. (1) is
+relativistically covariant, being a nonlinear version of the
+Klein-Gordon equation and reducing to it when φ ≪ 1
+such that sin φ ≈ φ. The SG model received attention
+from the high energy physics community in the 1970s due
+its soliton solutions [36–39], but also describes the low en-
+ergy physics of a considerable range of condensed matter
+systems including crystal dislocations [40], domain walls
+in magnetic [41] and binary superfluid [42] systems, the
+Heisenberg spin chain with a field induced gap [43–45],
+one-dimensional Bose gases in periodic potentials (that
+can capture the Mott-insulator to superfluid transition
+in one dimension) [46, 47], two-dimensional Bose gases
+realizing the XY model [48], trapped ions [49], and two
+tunnel-coupled one-dimensional Bose gases [50–57]. The
+fact that the SG model is both nonlinear and integrable
+means that attention is often focused on its soliton so-
+lutions, but part of our mission in this paper is to point
+out that these same properties also imply that caustics
+(which are associated with the existence of tori in phase
+space [58]) are expected to occur generically, and we are
+aware of only one previous study of caustics in this model
+[59].
+The particular physical realization we have in mind
+for this paper is a system composed of two elongated
+quasi-one dimensional Bose gases coupled by tunneling
+along their length; the field φ(z, t) in Eq. (1) gives the
+relative phase between the two quantum gases.
+Quasi
+one-dimensional Bose gases have been created in a num-
+ber of experiments over the last two decades using tightly
+trapped ultracold atoms, and the remarkable tunability
+of these systems allows the strongly interacting Tonks-
+Girardeau regime [60, 61], the weakly interacting quasi-
+condensate regime [62–65], and also the crossover be-
+tween the two [66, 67], to be reached. It is important to
+note that, in accordance with the Mermin-Wagner theo-
+rem [68], one-dimensional Bose gases do not undergo true
+Bose-Einstein condensation at low temperature, unlike
+three dimensional gases. Instead, they can form quasi-
+condensates where density fluctuations are suppressed
+but phase fluctuations remain [69, 70]. In this paper we
+shall work in the weakly interacting regime and assume a
+state of the system consisting of a quasi-condensate plus
+small thermal fluctuations.
+A system comprised of two coupled quasi-one dimen-
+sional gases can be made by taking a single gas and
+splitting it in two along its long axis by switching on an
+elongated double well potential. This is the experimen-
+tal protocol typically adopted in a series of experiments
+conducted by the Vienna group [63, 71–77]. The com-
+bination of almost complete isolation from the environ-
+ment, long relaxation times and spatially resolved mea-
+surements of phase and number difference make these
+experiments ideal for investigating many-particle quan-
+tum dynamics, including fundamental questions such as
+whether and how closed quantum systems reach equi-
+librium. The gas can be split slowly so that it always
+remains close to equilibrium leading to number squeezed
+states [78, 79] or it can be split rapidly, leading to a so-
+called quantum quench which launches the system into a
+nonequilibrium state.
+In this paper we shall consider the opposite quench
+
+pi(2)
+(2)d
+P2(2)
+p1(z)3
+where two one-dimensional gases are suddenly connected
+together (see schematic representation in Figure 1). This
+touches on rather fundamental considerations in quan-
+tum mechanics since it describes the build-up of coher-
+ence between two initially independent systems, and is
+therefore related to the double-slit experiment for many-
+particle systems [53, 80–83]. We shall refer to this as a
+“J-quench” because J is often used to denote the cou-
+pling strength between the two wells. In a simple two-
+mode description of a bosonic Josephson junction, i.e.
+one that assumes a single mode in each well without the
+quasi-continuum of low energy longitudinal modes that
+are present in highly elongated traps, such a quench is
+predicted to result in a periodic collapse and revival of
+the atom number distribution between the two wells [84–
+86]. Essentially the same behavior, but π/2 out of phase,
+occurs in the relative phase which is the conjugate vari-
+able to number difference. In Refs. 26, 31, and 32 these
+revivals are shown to be examples of quantum caustics
+in a many-particle system. One of our main aims here
+is to investigate what happens to these caustics in the
+presence of the dispersive longitudinal modes present in
+the SG model, and is part of a wider program attempt-
+ing to understand the role of caustics in quantum many
+particle dynamics [17, 26, 27, 31–34].
+Due to the difficulty of solving the fully quantum SG
+model we take a semiclassical-style approach based on
+classical field configurations which are solutions of Eq.
+(1).
+Each configuration is analogous to a single geo-
+metric ray in optics and we include fluctuations by sum-
+ming many configurations. The initial conditions for each
+field configuration are randomly sampled from a Boltz-
+mann distribution.
+This approach is similar in spirit
+to the truncated Wigner approximation (TWA) [87–92]
+which includes quantum fluctuations around the classi-
+cal field by summing many rays sampled from a quan-
+tum probability distribution. The TWA has previously
+been applied to one-dimensional Bose gases by Martin
+and Ruostekoski [93, 94] who studied dark solitons, and
+also to the connection problem of two zero temperature
+one-dimensional Bose gases by Dalla Torre, Demler and
+Polkovnikov [53], who proposed a universal scaling form
+for the phase dynamics after the quench. More recently,
+the TWA has been used by Horváth et al. [95] to study
+the surprisingly sudden relaxation of the phase seen in
+the Vienna BEC splitting experiments [77]. In this paper,
+we include both the quantum fluctuations arising from
+coupling two independent systems and thermal fluctua-
+tions arising from thermal phonons in the longitudinal
+modes and compare the time evolution of macroscopic
+variables (the total number difference and phase differ-
+ence) in the SG system against the simpler two mode
+system [17, 26, 31].
+We find that following a quench
+caustics dominate the dynamics of the macroscopic vari-
+ables of both systems, even in the presence of thermal
+fluctuations. Due to the singular nature of caustics, and
+combined with their structural stability, we therefore pro-
+pose that strong nongaussian fluctuations are a generic
+phenomenon following a quench in the SG model (and
+indeed, in integrable or moderately chaotic many-body
+systems in general).
+The caustics we discuss in this paper also have implica-
+tions for the question of relaxation towards equilibrium at
+long times in many particle systems. While chaotic (non-
+integrable) and open quantum systems should thermalize
+(although a complete description is still the subject of ac-
+tive research [96–103]), closed integrable models do not
+reach a conventional Gibbs state. We show here that in
+the SG model there is a pile-up of caustics leading to a
+singular shape for the long time probability distribution
+for the macroscopic variables that resembles the shape of
+a circus tent and is quite distinct from the thermal equi-
+librium prediction. We find that an analytic approxima-
+tion to the singular distribution based on an ergodic pen-
+dulum (assuming a microcanonical or ‘equal-probability’
+distribution) provides a good fit to the numerical data.
+The plan for the rest of this paper is as follows. We
+start in Sec. II by deriving the SG hamiltonian from the
+many-body description of two coupled 1D Bose gases. In
+Sec. III we describe the natural length and time scales
+and use them to write the SG hamiltonian and equa-
+tions of motion in convenient dimensionless forms. Sub-
+sequently, in Sec. IV we develop a method for finding the
+initial conditions for the SG equations of motion.
+We
+assume that prior to the quench the two Bose gases are
+independent and at thermal equilibrium with a bath at
+temperature T. The initial conditions are obtained by
+stochastically sampling the Fourier modes of a 1D quasi-
+condensate obeying the Tomonaga-Luttinger liquid the-
+ory. With the initial conditions in hand, in Sec. V we
+give the main results of this paper which are the dy-
+namics of the macroscopic number and phase difference
+variables obtained by solving the equations of motion
+numerically. In Sec. VI we consider the bigger picture
+and examine the universal aspects of our results includ-
+ing the influence of caustics on the coherence as well as
+the long time dynamics and the establishment of (non-
+thermal / non-Gaussian) equilibrium.
+We conclude in
+Sec. VII. There are also six appendices where we give
+the details of the calculations as well as bench marking
+our numerical method.
+II.
+FROM TWO COUPLED CONDENSATES TO
+THE SINE-GORDON PLUS MODEL
+We begin by deriving the SG model as an effective low
+energy description for two coupled one-dimensional Bose
+gases. For the sake of clarity, we list the main simplifica-
+tions employed in this work:
+• the treatment of a quantum many body problem
+by a semiclassical method (TWA).
+• the neglect of a weak harmonic trap along the
+long axis which would otherwise lead to a non-
+uniform longitudinal density (this can be avoided
+
+4
+in box traps which, although rarer, can be realized
+[76, 104])
+• the assumption of a constant value for the tunnel
+coupling J along the entire length of the gases
+• the neglect of coupling to symmetric and higher
+transverse modes. Some more involved theoretical
+models do include these effects [56, 57].
+These simplifications are not expected to qualitatively
+alter the main results of this work due to the structural
+stability of caustics. In other words, caustics are known
+to be robust to perturbations in both the Hamiltonian
+and initial conditions.
+A theoretical description of two ultracold quasi-one di-
+mensional gases made up of bosonic atoms of mass m,
+and held parallel to each other so that the atoms can
+tunnel between them at rate J, can be obtained from the
+following microscopic Hamiltonian [50, 51, 74]
+ˆH =
+�
+j=1,2
+� L/2
+−L/2
+dz
+�
+− ℏ2
+2m
+ˆψ†
+j(z) ∂2
+∂z2 ˆψj(z) + U(z) ˆψ†
+j(z) ˆψj(z) + g1D
+2
+ˆψ†
+j(z) ˆψ†
+j(z) ˆψj(z) ˆψj(z)
+�
+−
+� L/2
+−L/2
+dz ℏJ
+�
+ˆψ†
+1(z) ˆψ2(z) + ˆψ†
+2(z) ˆψ1(z)
+�
+.
+(2)
+The indices j = 1, 2 label the two gases and each is as-
+sumed to be tightly trapped in the x and y directions
+so that those degrees of freedom are frozen into their
+ground states. Only the longitudinal degree of freedom
+z in each gas is taken to be active. In experiments there
+will usually be a weak longitudinal trapping potential
+U(z), although as mentioned above for simplicity we set
+it to zero and hence consider a uniform system of length
+L with periodic boundary conditions. The quantum field
+operator ˆψj(z) annihilates a particle at point z and to-
+gether with its hermitian conjugate obeys bosonic com-
+mutation relations [ ˆψj(z), ˆψ†
+j′(z′)] = δjj′δ(z − z′). The
+interaction constant g1D characterizes the effect of atom-
+atom scattering within each gas on the longitudinal de-
+gree of freedom and can be controlled both in magnitude
+and sign either through Feshbach or confinement-induced
+scattering resonances [105]. We note in passing that a
+possible alternative physical realization of this problem
+could be a spinor Bose gas in a single quasi-one dimen-
+sional trap [106]. In fact, bosonic Josephson junctions
+where the atoms are held in a single trap and two atomic
+spin states are used for the two states have already been
+realized experimentally [107].
+A weakly interacting three-dimensional Bose gas at ul-
+tracold temperatures will undergo Bose-Einstein conden-
+sation and can be described to high accuracy by a clas-
+sical field approximation (Gross-Pitaevskii theory [108]).
+In a quasi-one dimensional geometry quantum fluctua-
+tions can still be small if the density is not too low, and
+under these circumstances the gas can be treated as a
+quasi-condensate where the quantum field operators are
+replaced by classical fields [69, 109, 110]
+ˆψj(z) → ψj(z) =
+�
+n1D + ρj(z) exp[iφj(z)] .
+(3)
+Here n1D = N/L is the background density where N is
+the number of atoms in each gas (for simplicity we as-
+sume an equal number of atoms N in each gas; the struc-
+tural stability of caustics means that they are stable to
+small differences in n1D between the two gases). ρj(z)
+and φj(z) are the atom number density and phase fluc-
+tuations at each point z, respectively. These are canon-
+ically conjugate variables and can even be quantized in
+a semiclassical regime such that they obey the commu-
+tation relations [ˆρj(z), ˆφj′ (z′)] ≈ δjj′δ(z − z′) in a coarse
+grained sense [110]. However, in the present paper ρj(z)
+and φj(z) will be purely classical fields subject only to
+thermal fluctuations.
+We can further decompose the fields into their sym-
+metric and antisymmetric components
+ρs(z) = ρ1(z) + ρ2(z)
+2
+,
+ρa(z) = ρ1(z) − ρ2(z)
+2
+φs(z) = φ1(z) + φ2(z),
+φa(z) = φ1(z) − φ2(z) . (4)
+If the fluctuations are small ρa will be small whereas ρs
+will be comparatively large. The particle-particle inter-
+action energy will then typically cause the dynamics of
+the symmetric modes to occur at higher energy than the
+antisymmetric ones, and consequently we can ignore the
+symmetric degrees of freedom as long as we restrict at-
+tention to low energies [50, 55, 72, 75]. The Hamiltonian
+purely describing the antisymmetric variables is (see Ap-
+pendix A for details)
+HSG+ =
+� L/2
+−L/2
+dz
+�
+g1D ρ2
+a(z) + ℏ2n1D
+4m
+�∂φa
+∂z
+�2
++
+ℏ2
+4mn1D
+�∂ρa
+∂z
+�2
+− 2ℏJn1D cos φa(z)
+�
+.
+(5)
+We refer to this as the “sine-Gordon plus” (SG+) Hamil-
+tonian because it includes an extra term (the third term)
+in comparison to the standard SG Hamiltonian.
+This
+term involves gradients of density fluctuations and results
+
+5
+in an energy cost which automatically suppresses den-
+sity fluctuations at small length scales. It is also worth
+noting that including this term means that the density
+and phase fluctuations [the second term in Eq. (5)] are
+incorporated on an equal footing.
+This is also in ac-
+cordance with Gross-Pitaevskii theory which suppresses
+density fluctuations with wavelengths below the healing
+length [95]
+ξh =
+ℏ
+√mg1Dn1D
+.
+(6)
+However, when n1D is relatively large the third term is
+naturally suppressed in comparison to the others and can
+be dropped as long as the density gradients are small
+leading to the SG Hamiltonian [55, 74]
+HSG =
+� L/2
+−L/2
+dz
+�
+g1D ρa(z)2 + ℏ2n1D
+4m
+�∂φa
+∂z
+�2
+− 2ℏJn1D cos φa(z)
+�
+.
+(7)
+The nonlinear piece in both Hamiltonians is the cosine
+term which originates from tunneling between the two
+wells and occurs in all Josephson junction type prob-
+lems.
+It provides an effective potential well for phase
+configurations φ(z, t) that play the role of rays. In fact,
+as we shall see in Section V, it acts as an (imperfect) lens
+that focuses rays excited by the quench to form caustics.
+For the sake of brevity, and when we deem no confusion
+can arise, we will omit the ‘a’ subscript on antisymmet-
+ric variables since we will not be dealing with symmetric
+degrees of freedom.
+The fact that the two fields φ(z) and ρ(z) form a conju-
+gate pair means that their equations of motion are given
+by Hamilton’s equations
+˙φ = 1
+ℏ
+δH
+δρ(z)
+˙ρ = −1
+ℏ
+δH
+δφ(z)
+(8)
+where H is the Hamiltonian density defined via
+H =
+� L/2
+−L/2
+H dz.
+(9)
+Applying these equations to the SG+ Hamiltonian given
+in Eq. (5) we find the following of equations of motion
+dφ(z, t)
+dt
+= 2 g1D
+ℏ ρ(z, t) + 2
+ℏ
+4mn1D
+∂2ρ(z, t)
+∂z2
+dρ(z, t)
+dt
+= 2 ℏn1D
+4m
+∂2φ(z, t)
+∂z2
+− 2Jn1D sin[φ(z, t)] .
+(10)
+These are the key equations we use to solve for the dy-
+namics of the field configurations. They have the form of
+Josephson’s equations [111] augmented by second order
+spatial derivatives ∂2φ/∂z2 and ∂2ρ/∂z2 which account
+for phase and density fluctuations along the longitudi-
+nal direction. Combined with the sine term, they will
+cause wavepackets to disperse along z. In the absence of
+these terms we have exactly the equations of motion for
+a pendulum where φ is the angular displacement from
+equilibrium and ρ plays the role of angular momentum.
+The dependence on z suggests an interpretation in terms
+of a continuous chain of many pendula each coupled to its
+neighbors by the spatial derivative terms and is reminis-
+cent of the Fermi-Pasta-Ulam-Tsingou problem [50, 112].
+In this paper the coupled equations of motion given in
+Eq. (10) will be solved numerically for a system of length
+L. To perform the numerical computations we discretize
+the system on a spatial grid with NL + 1 points which
+makes the grid spacing a = L/NL. The positions of the
+grid points are given by z = ra where r is an integer
+r = −NL
+2 , . . . , NL
+2
+(11)
+and NL is chosen to be an even integer.
+There is in fact a physical limitation on the grid size.
+Eq. (10) is classical and valid only on length scales greater
+that healing length ξh [51, 95]. Therefore, any numerics
+performed on Eq. (10) are meaningful only when the lat-
+tice grid size a is greater than ξh.
+In particular, NL
+should be such that a > ξh which implies
+N 2
+L < mg1Dn1DL2
+ℏ2
+.
+(12)
+We fulfil the condition given in Eq. (12) in our numerics.
+III.
+NATURAL SCALES
+Let us express the SG/SG+ Hamiltonians and equa-
+tions of motion in terms of the natural scales for a one-
+dimensional quantum fluid. For a length scale we chose
+the healing length ξh given in Eq. (6). The ratio of the
+healing length to the mean interparticle spacing 1/n1D
+motivates the definition of the Luttinger parameter
+K =
+�
+n1D(ℏπ)2
+4g1Dm .
+(13)
+This dimensionless quantity measures how strongly in-
+teracting the system is - when K ≫ 1 the healing length
+is much greater than the interparticle spacing and the
+system is in the weakly interacting (quasi-condensate)
+regime. Another key physical quantity is the speed of
+sound
+c =
+�g1Dn1D
+m
+.
+(14)
+This can be used to define a characteristic energy, namely
+that associated with phonons (quanta of sound)
+E = ℏω = ℏc
+ξh
+(15)
+
+6
+where we have set the natural frequency ω to be the ratio
+of the speed of sound to the healing length.
+We therefore transform to the following dimensionless
+variables
+z −→ ˜z = z
+ξh
+,
+t −→ ˜t = c
+ξh
+t
+ρ −→ ˜ρ = ρ ξh
+,
+φ −→ ˜φ = φ
+(16)
+and defining ˜HSG = HSG/E and likewise for ˜HSG+ we
+obtain the two Hamiltonians in dimensionless form
+˜HSG =
+� L/2
+−L/2
+d˜z
+�
+Γ ˜ρ2 + ϵ
+�
+∂ ˜φ
+∂˜z
+�2
+− 2J cos ˜φ
+�
+(17)
+and
+˜HSG+ =
+� L/2
+−L/2
+d˜z
+�
+Γ ˜ρ2 + ϵ
+�
+∂ ˜φ
+∂˜z
+�2
++ Γ
+4
+�∂˜ρ
+∂˜z
+�2
+− 2J cos ˜φ
+�
+(18)
+where the coefficients are given by
+Γ =
+π
+2K , ϵ = K
+2π , J = K
+2π
+ξ2
+h
+ξ2s
+.
+(19)
+In the last term we have introduced the spin healing
+length
+ξs =
+�
+ℏ
+4mJ
+(20)
+which provides a measure for the distance over which
+coherence between the two gases is restored due to the
+tunnel coupling J [55]. At finite temperatures another
+useful length scale is the thermal phase coherence length
+λT = 2ℏ2n1D
+mkBT .
+(21)
+The dimensionless form of the equations of motion can
+now be given. For the SG model we find
+d˜φ
+d˜t = 2Γ˜ρ
+d˜ρ
+d˜t = 2ϵ∂2 ˜φ
+∂˜z2 − 2J sin ˜φ
+(22)
+and for the SG+ model we obtain
+d˜φ
+d˜t = 2Γ˜ρ − Γ
+2
+∂2˜ρ
+∂˜z2
+d˜ρ
+d˜t = 2ϵ∂2 ˜φ
+∂˜z2 − 2J sin ˜φ .
+(23)
+IV.
+INITIAL CONDITIONS
+The dynamics we seek to study in this paper start from
+a J-quench where two independent one-dimensional gases
+at thermal equilibrium are suddenly coupled. In order
+to obtain the initial density and phase fluctuations of
+these gases we use the Tomonaga-Luttinger (TL) model
+that provides the universal low energy effective theory for
+one-dimensional systems (low energy limit of the Lieb-
+Lininger theory, for example) [53].
+A.
+Tomonaga-Luttinger (TL) liquid
+In our notation the TL Hamiltonian reads
+HTL =
+� L/2
+−L/2
+dz
+�
+g1Dρj(z)2 + ℏ2n1D
+4m
+�∂φj
+∂z
+�2�
+(24)
+where j labels either of the two gases. We henceforth,
+omit this label for the sake of brevity with the under-
+standing that in this section the density and phase fields
+refer to just one of the two gases. Eq. (24) has the same
+mathematical structure as the SG model but without the
+tunnelling term.
+If we include density fluctuations we
+find
+HTL+ =
+� L/2
+−L/2
+dz
+�
+g1Dρ(z)2 + ℏ2n1D
+4m
+�∂φ
+∂z
+�2
++
+ℏ2
+4mn1D
+�∂ρ
+∂z
+�2 �
+.
+(25)
+The TL model is quadratic and hence its thermal fluc-
+tuations can be treated exactly. To this end it is useful
+to work in Fourier space and we apply discrete Fourier
+transforms defined on the numerical grid with NL points
+as discussed at the end of Section II. The phase field φ
+and its Fourier transform ϕ are related by
+φr =
+1
+√NL + 1
+NL/2
+�
+k=−NL/2
+ϕk exp
+�
+i 2πkr
+NL + 1
+�
+ϕk =
+1
+√NL + 1
+NL/2
+�
+r=−NL/2
+φr exp
+�
+−i 2πkr
+NL + 1
+�
+.
+(26)
+The discrete data {φr} = {φ−NL/2, . . . , φ0, . . . , φNL/2}
+and its transform are located symmetrically about r = 0
+and k = 0, respectively. Since the value φr of the field
+at each coordinate space grid point is a real number the
+condition
+ϕ−k = ϕ∗
+k must hold. Similarly the density
+fluctuation field ρ and its Fourier transform ϱ are related
+by
+ρr =
+1
+√NL + 1
+NL/2
+�
+k=−NL/2
+ϱk exp
+�
+i 2πkr
+NL + 1
+�
+ϱk =
+1
+√NL + 1
+NL/2
+�
+r=−NL/2
+ρr exp
+�
+−i 2πkr
+NL + 1
+�
+(27)
+
+7
+where again the reality of the field in coordinate space
+requires that ϱ−k = ϱ∗
+k. Inserting these transformations
+in Eq. (25) we obtain (see Appendix B for details)
+HTL+ = a g1D
+NL/2
+�
+k=−NL/2
+|ϱk|2
++ a ℏ n1D
+NL/2
+�
+k=−NL/2
+ℏπ2k2
+mL2 |ϕk|2
++ a
+ℏ2
+4mn1D
+NL/2
+�
+k=−NL/2
+4π2k2
+L2
+|ϱk|2 .
+(28)
+Before proceeding with further analysis of Eq. (28), it is
+worth noting that it can be recast in a standard Luttinger
+liquid form
+HTL+ = acℏ
+2
+NL/2
+�
+k=−NL/2
+�K
+π
+4π2k2
+L2
+|ϕk|2 + π
+K |ϱk|2
++ K
+π
+4π2k2
+N 2 |ϱk|2
+�
+(29)
+where the strength of the terms depends either on K or
+1/K.
+Applying the transformations given in Eq. (16), the
+Fourier space variables can be written in dimensionless
+form as
+ϱk −→ ˜ϱk = ξhϱk
+,
+ϕk −→ ˜ϕk = ϕk
+(30)
+and the TL+ Hamiltonian given in Eq. (28) scaled by the
+energy E = ℏc/ξh is given by
+˜HTL+ =
+˜L
+NL
+NL/2
+�
+k=−NL/2
+�ϵ 4π2k2
+˜L2
+| ˜ϕk|2 + Γ|˜ϱk|2
++ Γ π2k2
+˜L2
+|˜ϱk|2
+�
+(31)
+where ˜L = L/ξh is the ratio of the system size to the
+healing length. Comparison with the spatial version of
+HTL+ given in Eq. (25) shows where this factor comes
+from: as the size is increased the range of the integration
+increases linearly and this is accounted for by ˜L in the
+Fourier transformed version. Note that all parameters
+and variables in Eq. (31) are dimensionless.
+B.
+Thermal equilibrium
+To find the initial conditions on the fields ρj(z) and
+φj(z) we assume that each gas is at thermal equilibrium
+such that the excitation (phonon) modes of the TL+
+Hamiltonian are populated with a probability given by
+the Boltzmann distribution. The range of temperatures
+we simulate is listed in Table I along with the values
+of all the other key parameters, and is chosen so as to
+correspond to realistic experimental conditions (the tem-
+perature must be low enough that the quasi-condensate
+description is valid).
+In the canonical ensemble of statistical mechanics the
+probability that a system at thermal equilibrium has
+the phase space configuration s = q1, p1, q2, p2...qN, pN
+is proportional to the Boltzmann weight exp[−βH(s)],
+where β = 1/kBT and H = �
+i p2
+i /2m + V (qi).
+The
+Hamiltonian in Eq. (31) is quadratic and hence the Boltz-
+mann weight becomes that of a series of independent har-
+monic oscillators
+e− ˜β ˜
+HTL+ =
+�
+k
+e−P 2
+k /2σ2
+ρ+ e−Q2
+k/2σ2
+φ+(k)
+(32)
+where ˜β = (ℏc/ξh)/kBT is the appropriately scaled tem-
+perature parameter and we have introduced the real vari-
+ables Qk and Pk which are related to the old variables
+by
+˜ϕk = Qkeiαk,
+˜ϱk = Pkeiβk.
+(33)
+The phases αk and βk allow for the fact that ˜ϕk and ˜ϱk
+can be complex numbers. The variances in Eq. (32) are
+given by
+σ2
+ρ+(k) = NL
+2˜β
+1
+Γ˜L(1 + π2k2/˜L2)
+(34)
+σ2
+φ+(k) = NL
+2˜β
+˜L
+4π2k2ϵ .
+(35)
+The partition function can now be written down as
+Z =
+�
+k
+� ∞
+−∞
+e− ˜β ˜
+HTL+ dPkdQk
+=
+�
+k
+�
+σρ+
+√
+2π
+� �
+σφ+(k)
+√
+2π
+�
+(36)
+and hence the probability P of a particular configuration
+(Q1, Q2, ...., P1, P2, ....) is
+P =
+�
+k
+�
+e−P 2
+k /2σ2
+ρ+
+σρ+
+√
+2π
+� �
+e−Q2
+k/2σ2
+φ+(k)
+σφ+(k)
+√
+2π
+�
+.
+(37)
+This is seen to be the total probability distribution for
+independent random variables Pk and Qk drawn from
+normal distributions. Thus, the absolute values of the
+Fourier coefficients ˜ϱk and ˜ϕk are normally distributed
+random variables with zero mean and variances given by
+Eqns. (34) and (35). We sample these numerically from
+normal distributions to generate the initial system con-
+figuration. The phases αk and βk given in Eq. (33) do
+not appear in the Boltzmann weight and are chosen ran-
+domly from the range [0, 2π). In fact, for both the phases
+and the amplitudes we only need to choose the values for
+
+8
+terms with k ≥ 0 because the reality conditions imply
+that we can put
+Qk = Q−k ,
+Pk = P−k,
+αk = −α−k ,
+βk = −β−k .
+(38)
+So far we have only considered the initial state of a sin-
+gle gas. By subtracting the results for two gases we can
+obtain the initial values of the antisymmetric variables
+ρa(z) and φa(z) defined in Eq. (4). Actually, due to the
+fact that the SG+ Hamiltonian with J = 0 and expressed
+in terms of antisymmetric variables as given in Eq. (5)
+formally has the same structure as the TL+ Hamiltonian
+given in Eq. (25), sampling initial data for two gases is
+unnecessary and one can obtain ρa(z) and φa(z) directly
+by sampling them as though they were from one gas de-
+scribed by the TL+ Hamiltonian. However, in doing so,
+consideration needs to be given to the average value of
+relative phase φa(z) because both the SG+ and TL+
+Hamiltonians only contain the spatial derivative of the
+phase but not the phase itself. Its average value is there-
+fore not determined by energy considerations and is left
+to float freely. This is also apparent in the Fourier trans-
+formed version of the TL Hamiltonian given in Eq. (31)
+where the k = 0 term involving ˜ϕ0 is absent due to the
+vanishing of its coefficient which is proportional to k2.
+To take into account the random phase difference be-
+tween the two gases one can chose ˜ϕ0 to be a random
+number in the range [−π . . . π) but multiplied by a factor
+of √NL + 1 in order to respect the normalization in Eq.
+(26). This gives values of the average value of φa(z) in
+the desired range −π and +π.
+The random value of the initial phase difference is ac-
+tually a key feature of the J-quench. It populates the
+cosine potential landscape in the Hamiltonian with uni-
+form probability. As the trajectories roll back and forth
+in this potential they form caustics.
+In effect, the co-
+sine potential acts as an imperfect lens that focuses an
+initially flat ‘wavefront’ over time.
+C.
+Choice of parameters
+There are three constraints which must be satisfied in
+order to have a quasi-one dimensional condensate [55].
+To ensure minimal scattering into the transverse modes
+we need the interaction to be sufficiently weak which im-
+plies µ = g1Dn1D ≪ ℏω⊥ where µ is the chemical poten-
+tial and ω⊥ is the transverse trapping frequency. More-
+over, the temperature needs to be low enough such that
+transverse modes are not thermally excited leading to the
+inequality kBT ≪ ℏω⊥. Finally, in order to have a quasi-
+condensate which permits a semiclassical approach we
+need weak interactions in comparison to the zero-point
+kinetic energy associated with the density of the parti-
+cles. This implies n1Dg1D ≪ ℏ2n2
+1D/m which means the
+Symbol
+Parameter
+Value
+ω⊥
+trapping frequency
+2π × 3 kHz
+m
+mass of Rb atom
+1.41 × 10−25 kg
+as
+scattering length
+98 × 0.52 Å
+N
+number of atoms
+1200
+L
+system length
+18 µm
+n1D
+average density
+6.7 × 107m−1
+g1D
+2 ℏascatω⊥
+2 × 10−38 Jm
+K
+Luttinger parameter
+25
+T
+temperature
+2 - 20 nK
+J
+J-quench
+0 - 30 Hz
+NL
+number of grid points
+50
+c
+speed of sound
+3 × 10−3 m s−1
+a
+grid spacing
+0.36 µm.
+ξh
+healing length
+0.24 µm
+λT
+phase coherence length
+38 − 380 µm
+ξs
+spin healing length
+2.5 µm
+Table I. Table containing important parameters and their val-
+ues. The parameters are chosen to be experimentally feasible
+and correspond roughly to those reported in references [72–
+77].
+Luttinger parameter should obey K ≫ 1. All the param-
+eter values we use satisfy these three inequalities.
+In quasi-one dimensional gases the interatomic inter-
+action parameter g1D is related to the scattering length
+as and transverse trapping frequency as g1D = 2ℏasω⊥.
+For 87Rb atoms we have as ≈ 98 × 0.52 Å[113] and we
+will assume ω⊥ = 2 π×3 kHz [77]. The full list of pa-
+rameters used in our simulations is given in Table I and
+roughly corresponds to those used in the experiments by
+the Vienna group [72–77].
+For our numerical simulations we choose a grid size
+that slightly exceeds the healing length because, as ex-
+plained above, this cuts off unphysical density fluctua-
+tions [51, 95]. This condition is given in Eq. (12) but can
+be expressed succinctly in terms of Γ as N 2
+L < ΓN 2. The
+magnitudes of ˜ρ and ˜φ also need to be considered. The
+phase difference can take the full range +π to −π, but
+the number difference is limited by the condition that
+the total number difference (integrated over the entire
+system) cannot exceed the total number of particles. In
+fact, due to the random nature of sampled thermal fluc-
+tuations, the integral of ˜ρ is always approximately zero.
+However, the validity of the SG/SG+ model requires that
+local density fluctuations be small in comparison to the
+background density n1D, see Appendix A. Translated
+into the scaled variables this means that at any point
+˜ρ(˜z) ≪ n1Dξh. In practice we choose ˜ρ(˜z) ≤ 1.6 so that
+the fluctuations are an order of magnitude smaller than
+the background density.
+
+9
+D.
+Examples of Initial conditions
+In Figure 2 we present typical spatial profiles of the
+initial number difference field ˜ρ (upper row) and phase
+difference field ˜φ (lower row). Each profile provides the
+initial conditions for a single classical field trajectory and
+is obtained by summing up thermally activated phonons
+(Fourier modes) using the Tomonaga-Luttinger model.
+The different columns show the effect of changing tem-
+perature T or Luttinger parameter K.
+As expected,
+when T is increased the fluctuations in both ˜ρ and ˜φ
+increase. By contrast, if K is increased the maximum
+magnitude and jaggedness of ˜ρ increases but the jagged-
+ness of ˜φ decreases. Referring to Eq. (19) we can see that
+this is because the coefficient multiplying the density fluc-
+tuation term in the Hamiltonian is Γ = π/2K which de-
+creases as K increases leading to increased variance of ϱk
+modes according to Eq. (34). The phase fluctuation term
+shows the opposite behavior because its coefficient in the
+Hamiltonian (which only appears as the spatial gradient
+of ˜φ) is ϵ = K/2π which increases as K increases and
+this reduces the variance of the ϕk modes according to
+Eq. (35), thereby making the ˜φ profiles smoother.
+V.
+NUMERICAL SIMULATIONS OF THE
+DYNAMICS
+In this section we explore the dynamics following a J-
+quench.
+Our approach is inspired by the TWA where
+multiple classical field configurations are propagated in
+time using the classical equations of motion, although in
+our case the initial conditions are sampled from a ther-
+mal distribution as described in Section IV rather than
+a quantum distribution as in the standard TWA.
+J-quench dynamics have previously been explored for
+the simpler case of a two-mode zero temperature bosonic
+Josephson junction where it was found that caustics dom-
+inate the number and phase difference probability distri-
+butions [17, 26, 31]. In the two-mode case it is possible
+to compute the exact quantum dynamics for some thou-
+sands of particles and compare them against the TWA.
+The results (see Figure 1 in [31]) show good qualitative
+agreement and give us confidence that the TWA can cap-
+ture the main features of the quantum dynamics. Fur-
+thermore, the inevitable presence of decoherence due to
+the environment will tend to reduce the quantum dy-
+namics to their classical limit (this has been investigated
+in the two-mode case for a J-quench in [32]) increasing
+the relevance of semiclassical calculations. In the present
+work we are interested in whether the phonons along the
+long axis disrupt or sustain these caustics. We will start
+by reproducing the caustics presented in Ref. 31 for the
+two-mode case and then add in the longitudinal modes
+after that.
+A.
+Numerical Methods
+The initial conditions are generated via random sam-
+pling from Gaussian distributions. We then evolve the
+equations of motion (Eq. 23 for the case of the full SG+
+model) using a Runge-Kutta solver with a user-defined
+time step [114]. The endpoints of our system are treated
+by imposing periodic boundary conditions. In Appendix
+C we demonstrate the numerical convergence of the solver
+by varying the temporal and spatial steps by tracking the
+time evolution of the total energy (hamiltonian) which
+should be a constant of the motion and obtain the fidu-
+cial time and space resolution for all our calculations.
+B.
+Special case: two-mode approximation
+In the two-mode approximation only a single mode in
+each well is taken into account. This description is rele-
+vant to the SG/SG+ model in the limit where the entire
+length of each quasicondensate is perfectly synchronized
+so that the fields ˜ρ(˜z) and ˜φ(˜z) do not depend on ˜z.
+In this case the spatial derivative terms vanish and the
+equations of motion in Eq. (23) reduce to
+d˜φ
+d˜t = 2Γ˜ρ
+,
+d˜ρ
+d˜t = −2J sin ˜φ .
+(39)
+These are the standard Josephson equations of motion
+and also correspond to those of a classical pendulum
+[115]. Such synchronization can occur at very low tem-
+peratures or when the coefficients ϵ and Γ are large
+enough that they suppress spatial fluctuations in the ini-
+tial conditions.
+In Figure 3 we display the post-quench dynamics in
+the two-mode approximation.
+The left hand and cen-
+tral panels show the time dependence of 150 indepen-
+dent solutions of Eq. (39) which give the trajectories for
+the number difference and phase difference, respectively.
+Note that in this paper we use the color blue for tra-
+jectories calculated within the two mode approximation
+and reserve red for the trajectories of the full many mode
+model. In accordance with our assumption that the two
+wells start with an equal number of atoms, each solution
+starts with ˜ρ = 0.
+And as discussed in Section IV B,
+the initial value of ˜φ is randomly chosen from the range
+[−π, π) because the two condensates are independent be-
+fore the J-quench.
+The most striking feature of Figure 3 is the series of
+cusp-shaped caustics that form in both variables. In or-
+der to guide eye, we have have outlined the first cusp
+caustic in the number difference variable using a black
+curve (the calculation for this curve is given in Appendix
+D). Like in optics, caustics are regions of high intensity
+formed by the envelopes of families of rays (trajectories).
+Each caustic is born at the centre of the distribution at
+the tip of a cusp before spreading out in two arms that
+move towards the edges of the distribution.
+The fact
+
+10
+Figure 2.
+Examples of initial spatial profiles of the number difference ˜ρ (top row) and phase difference ˜φ (bottom row). Each
+profile is obtained by randomly sampling a thermal distribution using the method described in Section IV B, and each panel
+includes ten different profiles. The parameter values common to all panels include the number of computational lattice points
+NL = 50, grid spacing a = 0.36µm, and healing length ξh = 0.24 µm (the remaining parameters are listed in Table I). The
+difference between the columns is as follows. The left column has the Luttinger parameter K = 25, and temperature T = 2
+nK giving a phase coherence length of λT = 380 µm. In the middle column K = 25, but the temperature is increased to 20 nK,
+giving λT = 38 µm. In the right column, the value of K is artificially increased (without changing any other parameters)
+to K = 250 and T = 2 nK. Increases in temperature excite stronger fluctuations in the profiles as expected. Increases in
+the Luttinger parameter have opposite effects on ˜ρ and ˜φ. The maximum value and jaggedness of ˜ρ is increased whereas the
+jaggedness of ˜φ is reduced. An explanation of this behavior is given in the main text.
+that they are cusp shaped is in agreement with the pre-
+diction of catastrophe theory that in two dimensions the
+only structurally stable and hence generic singularities
+are cusps.
+Each trajectory represents a single experimental run.
+The idea behind the TWA is that the number of tra-
+jectories reaching a point ˜ρ at time ˜t is proportional to
+the probability that a measurement of the true quantum
+system would yield that value of ˜ρ. An equivalent inter-
+pretation holds for the ˜φ trajectories. The caustics have
+the highest probability density and hence give the values
+most likely to be observed. Of course, if we only con-
+sider the average values of ˜ρ or ˜φ we would get zero in
+both cases due to the symmetry of the distributions and
+hence miss the caustics. Many experimental runs must
+be performed in order to obtain the probability distribu-
+tion where these patterns live.
+The mechanism underlying caustics can be understood
+from a phase space perspective, as shown in the right
+hand panel of Figure 3. Each dot gives the number and
+phase difference at a particular time for a different ini-
+tial condition. The red dots are the initial values which
+lie in a horizontal line because at ˜t = 0 all trajectories
+have ˜ρ = 0. As time evolves the dots rotate around the
+origin: the green and blue dots show two successively
+later times. However, the nonlinearity of the Josephson
+equations means dots further from the origin rotate more
+slowly and this leads to the formation of a spiral or whorl.
+At places where the whorl has a vertical segment a range
+of different solutions all have the same value of ˜φ and this
+stationarity of the distribution with respect to changes in
+the initial conditions is what generates a caustic, in this
+case a ˜φ-caustic.
+Conversely, horizontal segments give
+rise to ˜ρ-caustics.
+In the absence of nonlinearity the equations reduce to
+those of a harmonic oscillator
+d˜φ
+d˜t = 2Γ˜ρ
+,
+d˜ρ
+d˜t = −2J ˜φ
+(40)
+giving rise to rigid rotation in phase space and the forma-
+tion of perfect focal points in the number and phase dif-
+ference variables, as shown in Figure 4. However, these
+perfect revivals of the initial state are not stable: any
+nonlinearity will cause the focal points to evolve into the
+extended cusp caustics shown in Figure 3.
+The frequency of the linearized motion is known in
+Josephson junction terminology as the plasma frequency.
+
+3
+2 
+1
+Initial 
+0
+-1
+-2 
+-3
+-20
+-10
+0
+10
+20
+Grid points (r)3
+2
+1
+Initial pr
+0
+-1
+-2
+-3 
+-20
+-10
+0
+10
+20
+Grid points (r)3
+2
+1
+Initial 
+0
+-1
+-2
+-3 
+-20
+-10
+0
+10
+20
+Grid points (r)3
+2
+Initial $r
+1
+0
+-1
+-2
+-3
+-20
+-10
+0
+10
+20
+Grid points (r)3
+2
+1
+Initial $r
+0
+-1
+-2
+-3
+-20
+-10
+0
+10
+20
+Grid points (r)3
+2
+1
+0
+Initi:
+-1
+-2
+-3
+-20
+-10
+10
+10
+20
+Grid points (r)11
+Figure 3.
+Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and phase space distribution (right) following
+a J-quench from J = 0 to J = 30 Hz in the two mode approximation governed by the Josephson equations given in Eq. (39).
+The other parameter values are given in Table I. Each panel contains 150 trajectories: each trajectory starts with ˜ρ = 0 at time
+˜t = 0 but has an initial phase randomly sampled from [−π, π). Both number and phase difference variables display a series of
+cusp shaped caustics given by the envelopes of families of trajectories; to guide the eye we have outlined the first cusp caustic
+in the ˜ρ variable with a black curve. In the right panel three different time slices of the results are plotted in phase space (˜ρ
+versus ˜φ). Each dot corresponds to a different initial condition (trajectory) and the colors indicate the time: ˜t=0 (red), ˜t=50
+(green), ˜t=100 (blue). During time evolution the initial horizontal line winds into a whorl and the caustics in the ˜ρ and ˜φ plots
+occur due to horizontal and vertical segments of a whorl, respectively.
+Figure 4.
+Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and the phase space distribution (right) in
+the linearized version of the two-mode approximation [Eq. (40)] following a J-quench from J = 0 to J = 30 Hz. Like in Figure
+3, there are 150 trajectories shown in each panel corresponding to different values of the initial value of ˜φ. However, in this
+linearized case we obtain a series of perfect focus points (revivals of the initial state). This is because linearization gives rise to
+rigid rotation in phase space without whorls. Unlike the extended cusp caustics seen in Figure 3 (which will be qualitatively
+robust to details of the nonlinearity) perfect focus points are nongeneric because they are unstable to perturbations such as the
+effects of nonlinearity. All parameter values and color labels are the same as Figure 3.
+In our notation it reads
+ωp =
+√
+4ΓJ
+(41)
+and the period of the motion is therefore given by 2π/ωp.
+For the case shown in Figure 4 we have Γ = 0.063 and
+J = 0.037 giving a period ≈ 65. In fact, the tips of the
+cusps in the nonlinear case also occur with this period
+since they are formed from small amplitude trajectories
+that only experience the quadratic bottom of the cosine
+potential.
+C.
+General case: many-mode SG+ model
+Simulations of the full SG+ model are shown in Figure
+5, which represents one of the main results of this paper.
+The trajectories in the left panel give the spatially av-
+eraged number difference ⟨˜ρ(˜t)⟩z as a function of time
+obtained by solving the equations of motion given in Eq.
+(23) for the many-mode system and then averaging over
+its length. The trajectories in the middle panel of Figure
+5 give the equivalent spatial average of the phase differ-
+ence ⟨˜φ(˜t)⟩z, and the right-hand panel is the phase space
+picture.
+Each trajectory is evolved from a single ran-
+domly sampled field configuration (describing thermally
+activated phonons) such as those shown in the top row
+of Figure 2 and for the parameters given in Table I. We
+observe that despite the inclusion of longitudinal modes
+and the randomness of the initial conditions, the caustics
+survive and are quite similar to those of the two-mode
+approximation shown in Figure 3.
+This suggests that
+caustics are a generic feature of many particle dynamics
+
+1.5
+1.0
+0.5
+2Q
+0.0
+-0.5
+-1.0
+-1.5
+.3
+-1
+0
+1
+2
+32
+1
+2Q
+0
+-1
+-2
+0
+25
+50
+75
+5100 125 150 175 200
+2+3
+2
+1
+20
+0
+-1
+-2
+.3
+0
+25
+50
+75
+100 125 150 175 200
+2+2
+1
+2Q
+0
+-1
+-2
+-3
+-1
+0
+1
+2
+w
+iΦ2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5 
+-2.0
+10
+25
+50
+75
+100 125 150 175 2003
+2
+1
+20
+0
+-1
+-2
+-3
+0
+25
+50
+75
+100125 150.175 20012
+Figure 5.
+Dynamics of the spatially averaged number difference ⟨˜ρ⟩z (left), phase difference ⟨˜φ⟩z (middle), and phase space
+distribution (right) for the full many-mode SG+ model following a J-quench from J = 0 to J = 30 Hz. Each panel contains
+150 trajectories which are solutions of Eq. (23). The initial conditions are randomly sampled thermal phonons with the same
+parameter values as those shown in the top row of Figure 2 and described in Table I. In particular, the number of numerical
+lattice points is NL = 50 separated by a grid spacing of a = 0.36 µm, and the temperature is T = 2 nK. The healing length is
+ξh = 0.24 µm, the spin healing length is ξs = 2.5 µm and the phase coherence length is λT = 380 µm. The different colors on
+the phase space plot correspond to the same time slices as in the previous phase space plots.
+following quenches, at least for systems whose underlying
+physics is based on coupled nonlinear oscillators. Each
+oscillator starts with a random phase and a noisy momen-
+tum but the quench acts so as to give all the oscillators
+a momentum kick at the same time ˜t = 0 leading to an
+initial partial synchronization. As the system evolves in
+time after the kick the different periods of nonlinear os-
+cillators leads to cusp catastrophes in the distribution of
+trajectories. If we had instead calculated only the expec-
+tation values of the number and phase differences then
+this underlying structure would not have been visible be-
+cause it lives in the probability distribution rather than
+the mean values.
+A slice at fixed time through the probability distri-
+bution for the spatially averaged phase variable ⟨˜φ⟩z is
+shown in Figure 6. This is obtained by sorting the ⟨˜φ⟩z(˜t)
+trajectories into bins each of which covers a small range
+of ⟨˜φ⟩z and counting the number of trajectories in each
+bin. The result is noisy due to the thermal fluctuations
+but the caustics are clearly visible as strong peaks. These
+peaks display the characteristic ‘square root’ divergence
+of fold caustics [1]
+P(⟨˜φ⟩z) ∝
+1
+�
+˜φc − ⟨˜φ⟩z
+(42)
+where P(⟨˜φ⟩z) is the probability density and ˜φc is the
+location of the caustic. The blue dashed lines in Figure 6
+are fits of Eq. (42) to the numerical data and we see that
+the agreement is good. Although the height of the singu-
+larities predicted by Eq. (42) is infinite at the caustic, this
+function is integrable so that a probability distribution
+with caustics is still normalizable (of course, the peaks in
+the numerical data are of finite height because the num-
+ber of trajectories is finite). A very similar pattern of
+square root singularities at each caustic is obtained for a
+time slice through the probability density for the number
+difference variable so we shall not show it here.
+Figure 6.
+The probability density (red curve) as a func-
+tion of ⟨˜φ⟩z obtained from the density of trajectories at time
+˜t = 162 for the SG+ model. This corresponds to a slice at
+fixed time through the middle panel of Figure 5, although
+calculated using 10000 trajectories to improve the statistics
+and averaged over a short time window of ∆˜t = 1 to remove
+rapid time fluctuations. The red curve has been drawn with
+a bin width d˜φ = 0.04 and is normalised such that the area
+under the graph is 1. The caustics are clearly visible as di-
+verging peaks and are well fitted (blue dashed curves) by the
+inverse square root form given in Eq. (42) that is expected
+for fold catastrophes [1] (the satellite caustics also have this
+shape but the fit is not shown to avoid obscuring the data).
+A very similar profile is obtained for the probability density
+in the ⟨˜ρ⟩z variable (not shown).
+D.
+Effect of dispersion on the caustics
+The double derivative terms in the SG+ equations of
+motion given in Eq. 23 are responsible for transmitting
+wave disturbances along the longitudinal axis and are not
+present in the simpler two-mode case discussed in Section
+
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+0
+25
+50
+75
+1001251501752003
+2
+1
+0
+-1
+-2
+-3
+0
+25
+50
+75
+100 125 150 175 200
+2+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+0
+1
+2
+3
+()z1.0
+0.8
+M 0.4
+0.2
+0.0
+-3
+-1
+0
+1
+2
+3
+(0)z13
+V B. Initial thermal fluctuations in the SG+ model will
+therefore disperse in z over time and it is interesting to
+see what difference this makes to the caustics; comparison
+of Figures 3 and 5 suggests it makes little difference to
+spatially averaged variables. However, this observation is
+for only one choice of the parameters ϵ and Γ that govern
+the size of the derivative terms and also for relatively
+short times. In particular, in Figure 5 the parameters
+are ϵ ≈ 4 and Γ ≈ 0.06 which were chosen to match
+experimental values [72–77]. In Figure 7 we compare the
+long time dynamics of the two-mode approximation and
+the SG+ model for the case where ϵ in the SG+ model
+has been artificially increased by a factor of 10 (without
+changing any other parameters), thereby increasing the
+effect of spatial dispersion. Apart from this change, the
+initial conditions and J-quench are similar to those used
+in Figure 5. Note that we only use this increased value
+of ϵ for the time propagation and not for the generation
+of the thermal initial conditions. This avoids changing
+the starting phase fluctuations from those used in Figure
+5 which would otherwise be energetically suppressed and
+would also lead to significantly different dynamics but is
+not the comparison we would like to make here. From
+Figure 7 we see that the strong coupling of neighboring
+‘pendula’ does seem to largely wash out the caustics at
+long times in comparison to the dispersionless two-mode
+case, although some faint structure is still present which
+underlines the structural stability of caustics. The long
+time behavior will be further analyzed in Section VI.
+E.
+Effect of J on the caustics
+Another parameter that affects the dynamics is the
+tunnel coupling strength J [or its dimensionless version
+J which is defined in Eq. (19)] that becomes non-zero
+after the quench. The quench itself creates a strongly
+nonequilibrium phase difference where all values of ˜φ are
+equally probable independently of the value of J by virtue
+of the fact that before the quench there is no phase co-
+herence between the two quasicondensates. However, J
+does control the post-quench dynamics. One way it does
+this is via the frequency of the Josephson oscillations.
+The cusps occur with a frequency given by the plasma
+frequency in Eq. (41) which goes as
+√
+J.
+In Figure 8 we examine the effect of quenching to dif-
+ferent J values, with the value of J increasing from left
+to right. We can see the expected increase in frequency.
+The amplitude of the motion also increases with J be-
+cause immediately after the quench each trajectory finds
+itself at a random point on the cosine potential energy
+surface whose depth between valley top and valley bot-
+tom is 2J . The initial potential energy of a field config-
+uration is therefore −2J ⟨cos ˜φ0⟩z, where ˜φ0 is the phase
+field ˜φ(˜z, ˜t) at the initial time. This configuration evolves
+under the full Hamiltonian and upon spatial averaging is
+seen to execute oscillations about the potential minimum.
+The upper row in Figure 8 plots the spatially averaged
+Figure 7.
+Comparison of the long-time behavior of the phase
+difference in the two-mode approximation (upper) and many-
+mode SG+ model (lower). Both panels contain 150 different
+runs and the initial conditions and J-quench are similar to
+those of Figure 5 except that ϵ has been artificially multiplied
+by 10 (without changing any other parameters) in the lower
+panel. This enhances the effect of the spatial derivative term
+in φ in the SG+ model (this term does not appear in the two
+mode model). We see that in the upper panel the caustics
+are still visible. By contrast, the stronger spatial interaction
+causes dispersion and makes the caustics much less visible in
+the lower panel.
+number difference and according to Eq. (18) the maxi-
+mum amplitude this can have is
+⟨˜ρ⟩max
+z
+=
+�
+2J (1 − ⟨cos ˜φ0⟩z)
+Γ
+(43)
+where we have ignored the effects of spatial coupling (sec-
+ond order derivative terms). Thus, ⟨˜ρ⟩max
+z
+also scales as
+√
+J, and this is in correspondence with Figure 8.
+The lower row of Figure 8 shows the behavior in phase
+space. In these figures we have also included the unaver-
+aged data, i.e. the ˜ρ and ˜φ values of each grid point at
+the three selected times. This gives a sense of the size
+of the statistical fluctuations due to the spatial degrees
+
+3
+2
+1
+0
+-1
+-2
+-3
+600625650675700725 750775800
+2+3
+2
+1
+0
+-1
+-2
+-3
+600 625 650 675 700 725 750 775 800
+2+14
+Figure 8.
+Effect of quench strength J for J = 0 Hz, 3 Hz, and 30 Hz (from left to right). The top row shows the dynamics
+of ⟨˜ρ⟩z with initial conditions sampled in the same way as in Figure 5. The bottom row plots the corresponding phase space
+distributions. Like in previous figures, the different colors give different time instants: ˜t=0 (red), ˜t=50 (green), ˜t=100 (blue).
+The dots with intense colors are the spatially averaged values. We have also included the raw data (without spatial averaging)
+as faint dots. This gives an idea of the size of the statistical fluctuations due to the thermal initial conditions and is the same
+for all values of J. In the left column there is no coupling between the two quasicondensates and hence no time evolution of
+the spatially averaged data (the intense red, green, and blue dots sit on top of each other) although there can be evolution
+of unaveraged data due to intrawell dynamics, i.e. without the J term in Eq. (10). As we increase the magnitude of J time
+evolution leads to whorls with a greater vertical extent because more energy can be extracted from the cosine potential in Eq.
+(18) giving larger values of ⟨˜ρ⟩max
+z
+.
+of freedom. In the left hand column J remains zero for
+all time and the only dynamics that can occur is along
+the long-axis of each quasicondensate individually. The
+middle and right hand panels, which have J = 3 and
+J = 30 Hz, respectively, have the same initial statistical
+fluctuations as the left hand one because, as mentioned
+above, the initial distribution is set by the pre-quench
+thermal fluctuations in the two quasicondensates and is
+independent of J. However, as time evolves the effects of
+J described by Eq. (43) become apparent because larger
+J allows a greater value of ⟨˜ρ⟩max
+z
+and this stretches the
+distribution along the vertical direction in comparison to
+a smaller value of J. For a whorl to become apparent
+⟨˜ρ⟩max
+z
+should at least exceed the width of the statistical
+fluctuations and becomes better and better defined as J
+is increased.
+VI.
+UNIVERSALITY AND CAUSTICS
+We have already discussed the relationship between
+nonlinearity and caustics in the preceding section.
+As
+motivated earlier, and expounded in Refs. 17, 26, 31, 33,
+and 34, caustics also have implications for the universal
+dynamics of quantum systems. We explore a few of these
+effects in this section.
+A.
+Long time distribution: the circus tent
+The quench generates collective excitations that lead
+to caustics as shown in Figures 3 and 5 for the two non-
+linear models (two mode and SG+) discussed above. The
+caustics are born at the center of the probability distri-
+bution (in either the ˜ρ or the ˜φ variable) at intervals of
+the plasma period and move out to the edges over time.
+Figure 6 plots the probability distribution for the SG+
+model as a function of ⟨˜φ⟩z at an intermediate time where
+four pairs of fold caustics are discernible and shows how
+they diminish in strength but are still present as they
+move to the edges. The question then naturally arises
+as to what happens at long times ˜t → ∞ when the dis-
+tribution comprises of a large number of caustics and
+whether it tends to a characteristic shape? The answer
+is yes, and is shown in Figure 9 which is made in the
+same way as Figure 6 but this time by calculating the
+density of ⟨˜ρ⟩z trajectories and averaging over a time
+window extending between ˜t = 800 and ˜t = 980 in order
+
+2.0
+1.5
+1.0
+0.5
+z(g)
+0.0
+-0.5
+-1.0 
+-1.5 
+-2.0
+0
+25
+50
+75 100 125 150 175 200
+t2.0
+1.5
+1.0
+0.5
+z(d)
+0.0
+-0.5 
+-1.0 
+-1.5
+-2.0
+0
+25
+50
+75 100 125 150 175 200
+t2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+0
+25
+50
+75
+100 125 150 175 2002.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+-3
+-2
+-1
+0
+1
+2
+3
+(0)z2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+-3
+-2
+-1
+0
+1
+2
+3
+(0)z2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+-3
+-2
+-1
+0
+1
+2
+3
+(0)z15
+to remove rapid fluctuations. The probability distribu-
+tion takes a shape reminiscent of a ‘circus tent’ or ‘big
+top’ and can be understood as follows.
+The strongest
+singularities present are the cusp tips born at the center
+of the distribution which leads to this being the highest
+point. Each cusp then splits into two fold arms (which
+according to catastrophe theory are lower singularities)
+that move outwards, reducing in height as they go, before
+accumulating at the edges where there is a sharp drop to
+zero. The position of the outer edge is set by the maxi-
+mum energy that can be extracted from the quench and
+is given by Eq. 43.
+An analytic expression for the circus tent distribution
+is given by the integral
+PCT(˜ρ) =
+1
+2πB
+� 1
+˜ρ2/B2
+U(m, ˜ρ)
+K(m)
+dm
+(44)
+where
+U(m, ˜ρ) =
+1
+�
+m(1 − m)(m − ˜ρ2/B2)(1 + ˜ρ2/B2 − m)
+,
+(45)
+K(m) is the complete elliptic integral of the first kind,
+and B = 2
+�
+J /Γ. This expression is plotted in Figure
+9 as the dashed line and is derived in Appendix E un-
+der the assumption that at long times we can model the
+system by an ensemble of independent pendulua where
+each pendulum is ergodic. In other words, each pendu-
+lum obeys a microcanonical distribution where there is
+equal probability for it to be found anywhere on its en-
+ergy shell. The nature of the J-quench is such that it
+leads to an ensemble with an equal probability for any
+starting angle (this is different to an equal probability
+for each energy due to the dependence of the density of
+states on angle). As can be seen from Figure 9, PCT(˜ρ)
+gives a good fit to the numerical data generated by both
+the SG+ and two-mode models considered in this paper.
+In Figure 9 we also include the thermal probability
+distribution
+PT (˜ρ) = 1
+Z
+� ∞
+0
+PE(˜ρ) e−E/T D(E) dE
+(46)
+describing an ensemble of pendula at thermal equilibrium
+at temperature T where PE(˜ρ) is the probability distri-
+bution at fixed energy E, D(E) is the density of states
+and Z is a normalizing factor. The details of our cal-
+culation of PT (˜ρ) are given in Appendix F, where, for
+example, PE(˜ρ) is given in Eq. (F2). The temperature
+of this distribution is chosen such that the mean energy
+of the thermal distribution ⟨E⟩T is equal to the mean
+energy of the states excited by the quench. For a quench
+to J = 30 Hz we show in Appendix F that the effective
+temperature is 5.4 nK.
+Clearly, the thermal distribution is very different to the
+circus tent distribution: the thermal distribution takes
+the form of a smooth gaussian with wings that extend
+beyond ⟨˜ρ⟩max
+z
+because the thermal Boltzmann factor
+Figure 9.
+The long time probability distribution for the
+number difference ˜ρ. The data points are from the different
+nonlinear models considered in this paper averaged over the
+spatial coordinate z and also over a time window ranging from
+˜t = 800 to ˜t = 980 to remove fluctuations. The pink dashed
+line is the circus tent distribution PCT given in Eq. (44) and
+derived in Appendix E under the assumption of ergodicity;
+the circus tent shape is due to the proliferation of caustics
+at long times and gives a good fit to the data.
+The solid
+black curve is the thermal distribution PT with a temperature
+chosen so that the expectation value of the energy matches
+that provided by the quench.
+allows for excitations with any energy (albeit with ex-
+ponentially small probability) including those involving
+pendula undergoing rotation as well as libration, whereas
+the J-quench only excites librational motion. The proba-
+bility distribution for a thermal pendulum is in fact quite
+delicate to compute because of the singularity in the den-
+sity of states between libration and rotation but the com-
+bined result is smooth; see Appendix F for more details.
+B.
+Structural stability of caustics
+The defining characteristic of the singularities de-
+scribed by catastrophe theory is structural stability
+against perturbations and this ensures that they occur
+generically. The same is not true of isolated singularities
+as can be seen by comparing Figures 3 and 4 where it
+is shown that point foci do not survive the introduction
+of nonlinearity. In two dimensions cusps are the unique
+structurally stable catastrophe and from Figures 3 and 5
+we see that cusp-shaped caustics are indeed stable against
+random thermal fluctuations. However, thus far we have
+imposed the symmetrical starting condition that the ini-
+tial number difference between the two quasicondensates
+is zero. One may therefore wonder whether the caustics
+we see are a consequence of this symmetry. To check that
+this is not the case we show in Figure 10 the dynamics
+for the case where the initial background density n1D in
+the two quasicondensates differs by 10%. We see that
+
+1.0
+PcT
+Thermal
+two-mode
+0.8
+many-mode SG+
+t
+0.6
+0.4
+0.2
+0.0
+-2.0-1.5-1.0-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(p)z,t16
+Figure 10.
+Structural stability of caustics: here we investigate the effect of unbalanced densities on caustics by tracking the
+same SG+ model dynamics as those shown in Figure 5 except for an initial density imbalance of 0.1 in the background of ˜ρ
+at each point z. We see that the cusp caustics in the plots of ⟨˜ρ⟩z and ⟨˜φ⟩z versus time are distorted but still maintain their
+basic structure. This is because the whorl in phase space is left intact despite having a displaced centre. Caustics are resilient
+against imperfections and perturbations and we expect them to be present under realistic experimental conditions.
+although the caustics in both ⟨˜ρ⟩z and ⟨˜φ⟩z are distorted
+they maintain their basic cusp shape. Furthermore, the
+phase space whorls still occur and this guarantees the
+existence of caustics.
+C.
+Coherence factor and relaxation towards
+equilibrium
+Cold atom experiments have the ability to measure cor-
+relation functions in nonequilibrium many-body states
+[74, 116–118]. As a simple example let us consider the
+coherence factor
+C(˜t) =
+�
+⟨cos ˜φ⟩z
+�
+(47)
+which depends on the spatial average of the phase dif-
+ference field ˜φ(˜z, ˜t) between points along the two qua-
+sicondensates. The outer brackets indicate an ensemble
+average which means averaging over many trajectories
+each sampled from the thermal distribution discussed in
+Sec. IV. In the Vienna experiments, where one quasicon-
+densate is suddenly split into two, the coherence starts
+near unity and decays over time as the two quasiconden-
+sates decohere [76, 77]. In the opposite case, where two
+independent quasicondensates are suddenly coupled, one
+expects the converse where the coherence starts at zero
+and grows. This situation has been previously modelled
+by Horváth et al. using both the TWA and a truncated
+conformal space approach [95].
+They found that C(˜t)
+initially grows and then undergoes damped oscillations
+as it settles down towards a finite constant value. The
+coherence factor therefore provides a measure of how the
+system reaches equilibrium. In this context we note that
+C(˜t) actually corresponds to an ensemble average of the
+cosine term in the SG/SG+ Hamiltonian and thus gives
+information on the exchange of energy between the dif-
+ferent parts. In other words, since the total energy is a
+constant of the motion, if the ‘potential’ part of the en-
+ergy settles down to a constant this suggests the ‘kinetic’
+parts of the energy are also constant, at least from an
+ensemble averaged point of view. Our aim in this sec-
+tion is to see if the dynamics of C(˜t) is connected to the
+caustics.
+In Figure 11 we plot C(˜t) for two models: the full
+SG+ model which is many-mode and nonlinear and a
+linearized version which obeys the equations of motion
+d˜φ
+d˜t = 2Γ˜ρ − Γ
+2
+∂2˜ρ
+∂˜z2
+d˜ρ
+d˜t = 2ϵ∂2 ˜φ
+∂˜z2 − 2J ˜φ.
+(48)
+This differs from the linearized two-mode approximation
+defined by Eq. (40) because it describes an elongated
+multi-mode system. From Figure 11 we see that C(˜t) for
+the SG+ model (dark blue curve) does indeed initially
+grow, undergo damped oscillations and settle down to a
+non-zero value (the fact that C(˜t) ̸= 0 at ˜t = 0 is due
+to random fluctuations in the initial conditions: as we
+include more trajectories we find that the initial value
+gets smaller). Meanwhile, C(˜t) for the linear model (red
+dashed curve) executes undamped oscillations and hence
+does not settle down to equilibrium. Both models agree
+during the first oscillation but strongly differ after that.
+It is clear that nonlinearity is important for reaching
+equilibrium at least as far as global quantities such as
+C(˜t) are concerned.
+We can understand this by inter-
+preting the SG+ model as describing a chain of coupled
+pendula. The nonlinearity of each pendulum means that
+its period depends on the amplitude of its motion and
+hence an ensemble of pendula whose motion is initiated
+together by the quench, but all with different degrees of
+excitation, will dephase from one another over time so
+that collective oscillations are damped out. By contrast,
+linear oscillators have a period independent of their am-
+plitudes of motion and hence remain in phase.
+Apart from the ensemble averages shown by the darker
+curves in Figure 11, we have also included the individ-
+ual trajectories for ⟨cos ˜φ⟩z as fainter curves. The linear
+
+2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5 
+-2.0
+0
+25
+50
+75
+100 125 150 175 2003
+2
+1
+0
+-1
+-2
+-3
+0
+25
+50
+75
+100 125 150 175 200
+2t2.0
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+-3
+-2
+-1
+0
+1
+2
+3
+(0)z17
+Figure 11.
+The two dark lines give the time evolution of the
+coherence factor C(˜t) defined in Eq. (47) for a linear model
+(dashed-dotted red) and the SG+ model (solid blue). Both
+models are multi-mode (many longitudinal modes along ˜z)
+but the SG+ model is nonlinear. Also included as faint lines
+are the raw trajectories ⟨cos ˜φ⟩z from which C(˜t) is composed.
+As everywhere in this paper, ⟨. . .⟩z indicates a spatial aver-
+age.
+This figure highlights that recurrences present in the
+linear case are suppressed by nonlinearity in the SG+ sys-
+tem.
+The ensemble average over trajectories with different
+periods causes C(˜t) to relax towards an equilibrium value in
+the case of the SG+ model in line with previous experimental
+observations [76, 77] and theory [95].
+model displays harmonic motion and hence perfect re-
+vivals whereas the trajectories in the nonlinear model
+give rise to half-cusp caustics.
+These caustics overlap
+in time such that averaging over them causes the coher-
+ence to strongly relax after a single period. It is not so
+much that the caustics cause the relaxation, but rather
+that both have a common origin in the nonlinearity of
+the model and hence are generic features of dynamics in
+complex systems.
+VII.
+SUMMARY AND CONCLUSIONS
+The sine-Gordon (SG) model is a nonlinear integrable
+field theory that can be used to describe a wide range
+of systems from high energy physics to condensed matter
+physics. A series of landmark experiments using two cou-
+pled 1D atomic quasicondensates [63, 71–77] have real-
+ized the SG model in a controllable quantum many body
+environment. The key parameters can be varied in time
+allowing the implementation of sudden quenches that ex-
+cite many modes leading to nonequilibrium dynamics.
+This is the setting we adopt for the current paper where
+we use experimentally realistic parameters and compute
+the dynamics of the number and phase difference fields.
+However, in contrast to the usual experimental protocol
+where the tunnel coupling J is suddenly switched off,
+we consider quenches where it is suddenly switched on.
+While the former case is adapted to studying dephasing,
+decay and thermalization between the two subsystems,
+the many body dynamics is governed by the Tomonaga-
+Luttinger Hamiltonian describing independent 1D quasi-
+condensates. If instead J is suddenly switched on then
+the dynamics is that of the full SG model.
+Our calculations employ a thermal version of the
+semiclassical truncated Wigner approximation (TWA)
+method. More specifically, we propagate a large num-
+ber of classical field configurations over time with initial
+conditions sampled from a distribution at thermal equi-
+librium. The time evolved configurations (trajectories)
+can be summed to obtain the probability distributions
+for the observables and we find that these are dominated
+by singular caustic patterns. The natural mathematical
+description of caustics is catastrophe theory that predicts
+a hierarchy of structurally stable singularities with char-
+acteristic shapes that depend on dimension. In two di-
+mensions (e.g. number or phase difference versus time)
+the structurally stable catastrophes are fold lines that
+meet at cusps. This is exactly what we find in both the
+number and phase differences following a J-quench, see
+Figure 5.
+The probability distributions develop trains
+of caustics that are born periodically as cusp points (lo-
+cated at the center of the distribution if there is no tilt) at
+each plasma period and evolve into pairs of fold lines that
+gradually move out to the wings where they accumulate.
+Fold catastrophes manifest as strong non-gaussian fluc-
+tuations in the form of inverse square root divergences in
+the intensity (probability density), as shown in Figure 6.
+A special case is provided by the dynamics of a two
+mode system as shown in Figure 3.
+Here the equa-
+tions of motion are the Josephson equations given in
+Eq. (39). The only fluctuations we include in this ex-
+ample are the quantum fluctuations in the initial rela-
+tive phase between the two condensates as mandated by
+the uncertainty principle applied to systems in relative
+number eigenstates.
+The two-mode case is relevant to
+small systems where the higher modes are well above
+the temperature scale and so any spatial fluctuations are
+suppressed. By contrast, the many-mode case shown in
+all the other figures includes both quantum fluctuations
+and thermal fluctuations in the longitudinal modes, i.e.
+thermal occupation of phonon modes in the 1D quasicon-
+densates. Despite the presence of the many longitudinal
+modes (typically 50 in our calculations, as set by the pa-
+rameter NL) which give rise to highly random looking
+phase and density profiles as seen in Figure 2, we find
+that number and phase caustics survive for experimen-
+tally realistic parameters. Furthermore, the qualitative
+features of the caustics are stable against variations in
+quench strength and density imbalance, as seen in Fig-
+ures 8 and 10, respectively, and also against the details
+of the model (in this paper we use the SG+ model which
+augments the SG model by including longitudinal den-
+sity gradients). All of these different examples confirm
+the structural stability of caustics which is the reason why
+they occur universally without the need for fine tuning.
+
+1.00
+0.75
+0.50
+(cos((z))z
+0.25
+0.00
+-0.25
+-0.50
+-0.75
+Linearmany-mode
+-1.00
+Non-linear many-mode SG+
+0
+50 100 150 200 250 300 350 400
+t18
+The proliferation of caustics over time combined with
+their migration to the edge of the probability distribution
+has important consequences for the long time probability
+distribution. It takes on the shape of a circus tent featur-
+ing a strong central peak due to the cusp tips which are
+the most singular part of a caustic, flatter intermediate
+regions, and rapidly decaying edges where the caustics
+pile up, see Figure 9. This shape is quite distinct from a
+gaussian thermal distribution and can be derived assum-
+ing an ergodic hypothesis in which individual pendula
+have equal probability to be anywhere on their energy
+shell (see Appendix E). The approach to this equilibrium
+distribution can be tracked over time using the coherence
+factor (Figure 11) which is a spatial and ensemble average
+over the phase field and corresponds to the cosine term in
+the Hamiltonian if the latter is ensemble averaged. The
+attainment of equilibrium relies on the nonlinearity of the
+system to dephase itself when ensemble averaged. The
+caustics also rely on the nonlinearity without which they
+would reduce to nongeneric perfect revivals (point foci).
+In this sense caustics are mutually exclusive to recur-
+rences, at least in the statistical sense in which caustics
+appear in this paper.
+Caustics in the SG model could be observed experi-
+mentally by measuring the probability density for either
+the phase difference or the number difference. For ex-
+ample, the phase difference can be obtained by releasing
+the two quasicondensates from their double well potential
+and letting them overlap [80–82]. This process must be
+repeated many times and for as near identical initial con-
+ditions and time evolution as possible in order to build
+up a probability distribution, although due to the struc-
+tural stability of caustics they will not be particularly
+sensitive to differences in the experimental setup from
+run to run. If the probability distribution is obtained for
+a single time then we expect to see something like that
+shown in Figure 6. In order to observe the time evolu-
+tion of a caustic, one must then repeat the whole process
+for a range of different evolution times. This is laborious
+but technically possible, and since the first cusp caustic
+appears at half the plasma period the experiment does
+not need to run for long.
+The singular nature of caustics means that they dom-
+inate wave fields and are well known in hydrodynam-
+ics and optics through phenomena such as tsunamis and
+gravitational lensing. The results of this paper show that
+they also occur in the nonequilibrium dynamics of 1D su-
+perfluids where a quench plays an analogous role to an
+underwater earthquake by generating strong excitations
+beyond the linear regime that are focused in this case by
+the cosine term in the SG Hamiltonian. The universal
+properties of catastrophes imply caustics likely also occur
+in the post-quench dynamics of other condensed matter
+systems too: systems with more degrees of freedom will
+display higher catastrophes beyond folds and cusps such
+as hyperbolic and elliptic umbilics [34]. However, a spe-
+cial feature of the SG model is that it is integrable and
+so one may ask if that property plays a crucial role in
+the existence of caustics. In this context, we note that
+in classical mechanics caustics are closely associated with
+the existence of tori in phase space upon which trajec-
+tories live [58]. Tori are broken up by chaos, and thus
+caustics are not expected to survive for long in systems
+which are deep in the chaotic regime. Despite this, the
+Kolmogorov-Arnold-Moser (KAM) theorem shows that
+some tori survive in moderately chaotic systems [119],
+which suggests caustics may also survive in cases where
+the classical phase-space is mixed, which is the typical
+case. Indeed, they survive in the three site Bose-Hubbard
+model [34] which is known to be chaotic [120]. The im-
+portant problem of extending the KAM theorem to quan-
+tum mechanics [121] is thus intertwined with the analysis
+of caustics in quantum systems and provides an interest-
+ing direction for extending the present work.
+ACKNOWLEDGEMENTS
+We thank Ryan Plestid for contributions on ther-
+mal field sampling in the early stages of this project,
+Josh Hainge for suggesting the term ‘circus tent’, and
+Igor Mazets for correspondence and advice about ex-
+periments.
+This work was supported by the Mitacs
+Globalink research internship, by the Natural Sciences
+and Engineering Research Council of Canada (NSERC),
+and Research at the Perimeter Institute is supported
+in part by the Government of Canada, through the
+Department of Innovation, Science and Economic De-
+velopment Canada, and by the Province of Ontario,
+through the Ministry of Colleges and Universities. M.K.
+would like to acknowledge support from the project
+6004-1 of the Indo-French Centre for the Promotion of
+Advanced Research (IFCPAR), Ramanujan Fellowship
+(SB/S2/RJN-114/2016), SERB Early Career Research
+Award (ECR/2018/002085) and SERB Matrics Grant
+(MTR/2019/001101) from the Science and Engineering
+Research Board (SERB), Department of Science and
+Technology (DST), Government of India. M.K. acknowl-
+edges support from the Infosys Foundation International
+Exchange Program at ICTS. M.K acknowledges support
+of the Department of Atomic Energy, Government of In-
+dia, under Project No. 19P1112R&D.
+Appendix A: Derivation of the sine-Gordon
+Hamiltonian
+In this appendix we derive the Hamiltonian HSG as
+the effective low energy description of two cigar shaped
+tunnel-coupled quasicondensates [50, 74] within a clas-
+sical field description (Gross-Pitaevskii theory). Along
+the way we also obtain a slightly enhanced Hamiltonian
+HSG+ that includes contributions from the gradient of
+density fluctuations that are not included in the sine-
+Gordon (SG) Hamiltonian. These contributions are not
+very important for our parameters but play an impor-
+
+19
+tant conceptual role by introducing an energetic price
+for a rapidly varying density and hence effectively cut off
+these fluctuations.
+Assuming tight radial trapping such that each quasi-
+condensate is in its radial ground state, meaning that
+only longitudinal excitations are taken into account, the
+second quantized Hamiltonian for the total system be
+written
+H =
+� ∞
+−∞
+dz
+� �
+j=1,2
+�
+− ℏ2
+2m
+ˆψ†
+j(z)∂2 ˆψj(z)
+∂z2
++
+U(z) ˆψ†
+j(z) ˆψj(z) + g1D
+2
+ˆψ†
+j(z) ˆψ†
+j(z) ˆψj(z) ˆψj(z)
+�
+− ℏJ
+�
+ˆψ†
+1(z) ˆψ2(z) + ˆψ†
+2(z) ˆψ1(z)
+��
+.
+(A1)
+The quantum field operator ˆψj(z) annihilates a particle
+at the point z in the jth well, where z is the coordinate
+along the longitudinal direction (long axis of the system).
+m is the mass of the particles, U(z) is a possible external
+potential (in this paper it will be set to zero), g1D con-
+trols the interparticle interaction strength, and J is the
+tunneling frequency between the two wells. In the classi-
+cal field approximation we replace the field operators by
+complex functions
+ˆψj(z) → ψj(z) = eiφj(z)�
+n1D + ρj(z) .
+(A2)
+Note that φj and ρj are the phase and density variables
+for each well rather than their antisymmetric versions
+which are used extensively in the main text.
+Let us start by manipulating the kinetic energy term
+−
+�
+j=1,2
+� ∞
+−∞
+dz ℏ2
+2m
+ˆψ†
+j(z)∂2 ˆψj(z)
+∂z2
+(A3)
+=
+� ∞
+−∞
+dz
+�
+j=1,2
+ℏ2
+2m
+� � ∂
+∂z e−iφj(z)�
+n1D + ρj(z)
+�
+×
+� ∂
+∂z e+iφj(z)�
+n1D + ρj(z)
+� �
+=
+� ∞
+−∞
+dz
+�
+j=1,2
+ℏ2
+2m
+�
+− i∂φj
+∂z
+ˆψ†
+j +
+e−iφj ∂ρj
+∂z
+2√n1D + ρj
+�
+×
+�
+i∂φj
+∂z
+ˆψj +
+eiφj ∂ρj
+∂z
+2√n1D + ρj
+�
+=
+� ∞
+−∞
+dz
+�
+j=1,2
+ℏ2
+2m
+�
+ˆψ†
+j ˆψj
+�∂φj
+∂z
+�2
++
+( ∂ρj
+∂z )2
+4(n1D + ρj)
++ i
+∂ρj
+∂z
+∂φj
+∂z
+2√n1D + ρj
+[ ˆψje−iφj − ˆψ†
+jeiφj]
+�
+=
+� ∞
+−∞
+dz
+�
+j=1,2
+ℏ2
+2m
+�
+ˆψ†
+j ˆψj
+�∂φj
+∂z
+�2
++
+( ∂ρj
+∂z )2
+4(n1D + ρj)
+�
+≈
+� ∞
+−∞
+dz ℏ2
+2m
+�
+n1D
+2
+��∂φs
+∂z
+�2
++
+�∂φa
+∂z
+�2�
++
+1
+2n1D
+��∂ρs
+∂z
+�2
++
+�∂ρa
+∂z
+�2� �
+(A4)
+where
+φa = φ1 − φ2,
+φs = φ1 + φ2
+(A5)
+ρa = ρ1 − ρ2
+2
+,
+ρs = ρ1 + ρ2
+2
+,
+(A6)
+and we assume that n1D ≫ ρj. Next we consider the
+interactions
+�
+j=1,2
+g1D
+2 ψ†
+jψ†
+jψjψj =
+�
+j=1,2
+g1D
+2 [n1D + ρj(z)]2
+=
+�
+j=1,2
+�
+g1Dn2
+1D
+2
++ g1Dρ2
+j
+2
++ g1Dn1Dρj
+�
+=g1Dn2
+1D + g1D(ρ2
+s + ρ2
+a) + 2g1Dn1Dρs .
+(A7)
+Finally, we consider the tunneling term
+−ℏJ
+�
+ψ†
+1(z)ψ2(z) + ψ†
+2(z)ψ1(z)
+�
+(A8)
+= − ℏJ
+�
+(e−i(φ1−φ2) + e−i(φ2−φ1))√n1D + ρ1
+√n1D + ρ2
+�
+= − 2ℏJ cos(φa)√n1D + ρ1
+√n1D + ρ2
+= − 2ℏJ cos(φa)
+�
+n2
+1D + 2n1Dρs + ρ2s − ρ2a
+≈ − 2ℏJ cos(φa)(n1D + ρs) ≈ −2ℏn1DJ cos(φa) .
+(A9)
+
+20
+At very low temperatures the symmetric and antisym-
+metric components decouple and hence can be treated
+separately. The lower energy terms are the antisymmet-
+ric ones and we obtain the following Hamiltonian
+HSG+ =
+� ∞
+−∞
+dz
+�
+g1D ρa(z)2 + ℏ2n1D
+4m
+�∂φa
+∂z
+�2
++
+ℏ2
+4mn1D
+�∂ρa
+∂z
+�2 �
+−
+� ∞
+−∞
+dz 2ℏJn1D cos [φa(z)] .
+(A10)
+When the higher wavelength ρ modes are suppressed this
+reduces to the sine-Gordon model
+HSG =
+� ∞
+−∞
+dz
+�
+g1D ρa(z)2 + ℏ2n1D
+4m
+�∂φa
+∂z
+�2
+− 2ℏJ n1D cos [φa(z)]
+�
+.
+(A11)
+Eq. (A11) is the finally obtained SG Hamiltonian HSG
+which is the low energy description of two cigar shaped
+tunnel-coupled quasicondensates [50, 74].
+Appendix B: Derivation of the Tomonaga-Luttinger
+(TL) Hamiltonian in Fourier space
+In this appendix we derive the Fourier space version
+of the Tomonaga-Luttinger (TL) Hamiltonian. Starting
+from Eq. (25), and applying the discrete Fourier decom-
+positions given in Eq. (26) and Eq. (27), we have
+HTL+(ra) =
+� ∞
+−∞
+dz
+g1D
+NL + 1
+�
+�
+NL/2
+�
+k=−NL/2
+ϱkei 2πkr
+NL+1
+�
+� ×
+�
+�
+NL/2
+�
+l=−NL/2
+ϱlei 2πlr
+NL+1
+�
+�
++
+� ∞
+−∞
+dz
+ℏ2n1D
+4ma2(NL + 1)
+∂
+∂r
+�
+�
+NL/2
+�
+k=−NL/2
+ϕkei 2πkr
+NL+1
+�
+� × ∂
+∂r
+�
+�
+NL/2
+�
+l=−NL/2
+ϕlei 2πlr
+NL+1
+�
+�
++
+� ∞
+−∞
+dz
+ℏ2
+4mn1Da2(NL + 1)
+∂
+∂r
+�
+�
+NL/2
+�
+k=−NL/2
+ϱkei 2πkr
+NL+1
+�
+� × ∂
+∂r
+�
+�
+NL/2
+�
+l=−NL/2
+ϱlei 2πlr
+NL+1
+�
+�
+= a
+NL/2
+�
+r=−NL/2
+NL/2
+�
+k=−NL/2
+NL/2
+�
+l=−NL/2
+�
+�g1Dϱkϱlei 2π(k+l)r
+NL+1
+NL + 1
+�
+�
+− a
+NL/2
+�
+r=−NL/2
+NL/2
+�
+k=−NL/2
+NL/2
+�
+l=−NL/2
+ℏ2n1D
+4ma2(NL + 1) ×
+�
+2π
+NL + 1
+�2
+klϕkϕlei 2π(k+l)r
+NL+1
+− a
+NL/2
+�
+r=−NL/2
+NL/2
+�
+k=−NL/2
+NL/2
+�
+l=−NL/2
+ℏ2
+4mn1Da2(NL + 1) ×
+�
+2π
+NL + 1
+�2
+klϱkϱlei 2π(k+l)r
+NL+1
+(B1)
+where we have split the z coordinate into NL + 1 grid
+points separated by distance a so that z = r a where r
+in an integer lying in the range specified by Eq. (11).
+Using the fact that NLa = L, and applying the identity
+�NL/2
+r=−NL/2 ei 2π(k+l)r
+NL+1
+= (NL + 1)δk,−l we obtain
+HTL+ ≈a
+�
+k
+�
+l
+g1Dϱkϱlδk,−l
+− a
+�
+k
+�
+l
+�ℏ2n1Dπ2
+mL2
+�
+klϕkϕlδk,−l
+− a
+�
+k
+�
+l
+�
+ℏ2π2
+mn1DL2
+�
+klϱkϱlδk,−l
+(B2)
+where in the second term we have also replaced a2(NL +
+1)2 by L2 which holds when NL ≫ 1. The limits of the
+summation in Eq. (B2) has been omitted for the sake of
+
+21
+brevity. We therefore find
+HTL+ ≈
+�
+k
+�
+ag1Dϱkϱ−k+aℏ2n1Dπ2k2
+mL2
+ϕkϕ−k
++ aℏ2π2k2
+mn1DL2 ϱkϱ−k
+�
+=
+�
+k
+�
+ag1D|ϱk|2+aℏ2n1Dπ2k2
+mL2
+|ϕk|2
++ aℏ2π2k2
+mn1DL2 |ϱk|2
+�
+(B3)
+where we used the property of real fields that
+ϕ−k = ϕ⋆
+k,
+and
+ϱ−k = ϱ⋆
+k .
+(B4)
+Hence the Hamiltonian takes the form given in Eq. (28)
+of the main text.
+Appendix C: Bench marking of the numerical
+method
+The results given in this paper rely on numerically
+evolving the equations of motion over time for various
+models [e.g. for the full SG+ model the equations of mo-
+tion are given in Eq. (22)], which we accomplish using
+the Julia package DifferentialEquations.jl [114]. This im-
+plements a Runge-Kutta solver with a user-defined time
+step.
+As a measure of the accuracy of our numerical
+method we use the deviation of the Hamiltonian from
+its initial value. Since the Hamiltonian should be a con-
+stant of motion this gives an indication of the size of the
+numerical errors.
+In Figures 12 and 13 we plot the relative error in the
+SG+ Hamiltonian given in Eq. (18) for different time
+and spatial resolutions. More precisely, Figure 12 shows
+the effect of varying the time step d˜t, whereas Figure 13
+shows the effect of varying the number of grid points NL
+which sets the spatial step d˜z. In both cases we have
+evolved the system for a total elapsed time of ˜t = 1000
+which corresponds to the longest times we use in this
+paper (for the calculation of the long-term distribution
+shown in Figure 9), and also taken an ensemble average
+over 100 different trajectories similar to those in Figure 5.
+Furthermore, we also performed a moving time average
+of 30-time steps around ˜t = 1000 to average out the effect
+of fast oscillations.
+As expected, the relative error decreases as d˜t and d˜z
+decrease. For all the calculations in this paper we chose
+d˜t = 0.2 and NL = 50 because this keeps the relative
+error below 10 % and does not significantly slow down
+the simulations.
+Figure 12.
+The relative error in the SG+ Hamiltonian is
+plotted here as a function of the time step d˜t. The definition
+of the SG+ Hamiltonian is given in Eq. 18 and should be
+a constant of the motion were it not for numerical errors.
+The moving time average of relative error is evaluated after
+propagating the equations of motion for a total elapsed time
+of ˜t = 1000. All parameter values are the same as in Figure
+5 including NL = 50.
+Figure 13.
+The relative error in the SG+ Hamiltonian is
+plotted here as a function of the number of lattice points NL
+on the numerical spatial lattice. Like in Figure 12, the Hamil-
+tonian is evaluated after evolving the equations of motion for
+a total elapsed time of ˜t = 1000. The moving time average
+of the relative error fluctuates (at around 10 %) but does de-
+crease as d˜z decreases (or NL increases). All other parameter
+values are the same as in Figure 5 with d˜t = 0.2
+Appendix D: Caustic curve
+In this appendix we use the exact solution for the mo-
+tion of a pendulum to calculate the caustic curve plotted
+as the solid black line in Figure 3. The caustic is in fact
+the envelope of a whole family of trajectories. To begin,
+we take the equations of motion for the SG model given in
+
+0.6
+0.5
+(0) + 9SH  (1) 
+0.4
+0.3
+0.2
+0.1
+10-1
+100
+101
+dt0.16
+0.14
+0.12
+0.10
+0.08
+0.06
+0.04
+20
+40
+60
+80
+100
+NL22
+Eq. (22) and drop the second order derivative term pro-
+portional to ϵ which couples the different pendula. Next,
+we make the change of variables
+˜t = At,
+˜ρ = Bp,
+˜φ = 2y
+(D1)
+where
+A = 1
+2
+1
+√J Γ
+,
+B = 2
+�
+J
+Γ
+(D2)
+so the equations of motion simplify to
+dy
+dt = p
+(D3)
+dp
+dt = −1
+2 sin 2y .
+(D4)
+These equations are Hamilton’s equations obtained from
+a standard pendulum hamiltonian of the form
+H(y, p) = p2
+2 + 1
+2 sin2 y .
+(D5)
+The equations of motion given in Eqns. (D3) and (D4)
+have exact solutions in terms of the Jacobi elliptic func-
+tions sn[u|m] and cn[u|m] [122]. For the case relevant
+to us where the pendulum starts at angle y0, with zero
+initial angular momentum, they are
+y(t, y0) = arcsin{sin y0 sn[t + K(sin y0)| sin y0]}(D6)
+p(t, y0) = sin(y0) cn[t + K(sin y0)| sin y0]
+(D7)
+where K(m) =
+� π/2
+0
+dθ/
+�
+1 − m2 sin2 θ is the complete
+elliptic integral of the first kind [122] (we caution the
+reader that some computer packages such as Mathematica
+use the syntax K(m2) for this integral).
+Caustics occur when trajectories are focused, in other
+words they are the places where the trajectory does not
+change (to first order) when the initial conditions are
+varied. Thus, caustics in the momentum variable p oc-
+cur when dp/dy0 = 0 since the initial condition here is
+specified by y0. By differentiating Eq. (D7) an implicit
+expression for the position of the caustics can be found
+[123]
+sn(u|m)dn(u|m)
+�E(am(−t|m) |m)
+cos(y0)
++ t cos(y0)
+�
+− cos(y0)cn(u|m) = 0
+(D8)
+where u = t+K(sin y0), m = sin y0, E(u|m) is an elliptic
+integral of the second kind, dn(u|m) is another Jacobi
+elliptic function, and am(u|m) = arcsin[sin(φ)/m] is the
+Jacobi amplitude [122]. Finding the roots y0 of Eq. (D8)
+numerically at each value of the time gives pairs of values
+(y0, t) that can then be put back into Eq. (D7) to yield
+the black curve for the caustic shown in Figure 3. The
+match to the numerics is very good.
+Appendix E: Derivation of ergodic (“circus tent”)
+probability distribution at long times
+In this appendix we outline the derivation of an an-
+alytic approximation to the probability distribution for
+the number difference at long times, as shown in Figure
+9. This derivation is based upon a calculation given in
+Ref. 124 and assumes that the average behaviour of a con-
+tinuous chain of coupled pendula (the mechanical system
+that underlies the sine-Gordon model) can be described
+by a suitably ‘ergodized’ single pendulum.
+To keep the calculation general we use the pendulum
+Hamiltonian in standard form as given in Eq. (D5). With
+this hamiltonian we define a microcanonical probability
+density in phase space:
+dm(y, p; y0) =
+δ[H(y, p) − H(y0, p)]
+� �
+dy dp δ[H(y, p) − H(y0, p)]
+(E1)
+where y0 is the initial angle of the pendulum which fixes
+its total energy to be E = (1/2) sin2 y0 if the the initial
+angular momentum is zero (this is the appropriate ini-
+tial condition for the tunneling quench considered in this
+paper where the initial number difference is taken to be
+zero), and the denominator ensures that dm is normalized
+to unity. A microcanonical distribution has equal prob-
+ability to be anywhere on its energy shell (in this case
+a closed curve in y, p phase space) and thus by adopt-
+ing Eq. (E1) we are making an ergodic hypothesis. This
+does not hold for a single pendulum starting at position
+y0 since it will spend the most time at its turning points
+y = ±y0, but when averaged over y0 and y (see below)
+it gives a very good approximation at long times, as can
+be seen in Figure 9.
+The normalization integral can be evaluated exactly
+by re-expressing the delta function using the relation
+δ[g(x)] = �
+i δ(x − xi)/|g′(xi)|, where xi are the roots
+of g(x). In the present case this gives
+δ[(p2 + sin2 y − sin2 y0)/2] =δ[p − p1]
+|p1|
++ δ[p − p2]
+|p2|
+=2δ[p − p1]
+|p1|
+(E2)
+where |p1| = |p2| =
+�
+sin2 y0 − sin2 y. In obtaining this
+expression we have used the fact that for values of y
+within the range accessed by the pendulum, there are
+two values of p where the integral crosses the energy shell.
+The integral over p is now trivial due to the delta func-
+tion and the integral over y can be performed by putting
+sin y = sin y0 sin ζ so that
+
+23
+2
+� y0
+−y0
+dy
+|p(y, y0)| = 2
+� y0
+−y0
+dy
+�
+sin2 y0 − sin2 y
+= 2
+� π/2
+−π/2
+dζ
+�
+1 − sin2 y0 sin2 ζ
+= 4
+� π/2
+0
+dζ
+�
+1 − sin2 y0 sin2 ζ
+= 4K(sin y0) .
+(E3)
+Therefore, the normalized microcanonical probability
+density can be written as
+dm(y, p; y0) =
+1
+4K(sin y0)δ[(p2 + sin2 y − sin2 y0)/2]
+=
+1
+2K(sin y0)δ(p2 + sin2 y − sin2 y0)
+(E4)
+where we have used the property of delta functions that
+δ(αx) = (1/α)δ(x).
+The initial condition for our dynamics is such that the
+number difference is well defined but the phase differ-
+ence is completely undefined. We must therefore average
+the microcanonical probability density over all y0. This
+gives the phase space probability density relevant to J-
+quenches as being
+W(y, p) = 1
+π
+� π/2
+−π/2
+dy0 dm(y, p; y0)
+(E5)
+where we employ the notation W to indicate that this is
+a classical version of the Wigner function. The properties
+of the delta function can once more be used to write
+δ(p2 + sin2 y − sin2 y0) =
+�
+i
+δ(y0 − y0i)θ(cos y − |p|)
+2
+�
+p2 + sin2 y
+�
+cos2 y − p2
+(E6)
+where θ(x) is the Heaviside step function. The integral
+over y0 can now be evaluated exactly to give
+W(y, p) = 2
+4π
+θ(cos y − |p|)
+K(
+�
+p2 + sin2 y)
+�
+p2 + sin2 y
+�
+cos2 y − p2 .
+(E7)
+The final step is to integrate out the y coordinate to
+obtain the probability distribution PCT(p) for p alone
+PCT(p) =
+� π/2
+−π/2
+dy W(y, p) ,
+(E8)
+where “CT” stands for circus tent. Although this integral
+cannot be done analytically, it can be put in a form which
+is convenient to evaluate numerically.
+Denoting m =
+sin y0 =
+�
+p2 + sin2 y, one finds that
+PCT(˜ρ) =
+1
+2πB
+� 1
+˜ρ2/B2
+dm
+K(m)
+�
+m(1 − m)(m − ˜ρ2/B2)(1 + ˜ρ2/B2 − m)
+(E9)
+where we have also converted back from angular momen-
+tum p to number difference ˜ρ using Eq. (D1). This equa-
+tion is given in the main text as Eq. (44) and is plotted
+in Figure 9 where it is compared against the long-time
+spatially and temporally averaged numerical data for the
+various nonlinear models considered in this paper.
+As
+can be seen in Figure 9, PCT is characterized by a di-
+verging (yet normalizable) peak at the center and then
+relatively flat wings until it drops sharply to zero at the
+edges. In Ref. 124 it is shown that PCT(˜ρ) diverges log-
+arithmically at the origin ˜ρ = 0 and also tends suddenly
+to zero with logarithmic singularities at ˜ρ = ±B. Both
+these non-thermal features can be attributed to the pres-
+ence of caustics.
+Appendix F: Pendulum at thermal equilibrium
+In Figure 9 the long time probability distribution for
+the number difference is compared against the ergodic
+prediction derived in Appendix E, and also against the
+thermal equilibrium prediction. In this Appendix we ex-
+plain how to calculate the latter case. In order to make
+the calculation tractable we make the assumption that
+the SG+ model can be approximated by a thermal en-
+semble of independent pendula. We also adopt the same
+notation as Appendix E and hence work with a pendulum
+Hamiltonian in the standard form H = (1/2)(p2+sin2 y).
+This is related to the two mode Hamiltonian H2M =
+Γ˜ρ2 − 2J cos φ by H = H2M/8J + 1/4.
+We proceed in two steps: we first calculate the prob-
+ability distribution PE(p) for the momentum variable p
+(that here plays the role of the number difference) for a
+fixed energy E. Secondly, we assume our system is at
+thermal equilibrium with a bath at temperature T such
+that the relative probability of any energy is given by the
+Boltzmann factor exp[−E/T]. Thus the thermal proba-
+bility distribution is
+PT (p) = 1
+Z
+� ∞
+0
+PE(p) e−E/T D(E) dE
+(F1)
+where Z is a normalizing factor (found numerically) and
+
+24
+D(E) is the density of states.
+The probability distribution PE(p) at fixed E is pro-
+portional to 1/ ˙p as this determines how long the pendu-
+lum spends at each value of p. According to Hamilton’s
+equation ˙p = −∂H/∂x = −(1/2) sin 2y, and using the
+fact that sin y =
+�
+2E − p2, we find that this probability
+distribution for a fixed value of E is
+PE(p) =
+N
+(1/2) sin(2 arcsin
+�
+2E − p2)
+,
+(F2)
+where N is a normalization factor given by the period
+of the motion.
+Two cases must be distinguished: for
+E < 1/2 the energy is less than the separatrix and the
+pendulum undergoes vibrational motion (also known as
+librational motion in some literature). Conversely, when
+E > 1/2 the energy is above the separatrix and the pen-
+dulum undergoes rotational motion.
+For motion below the separatrix we have |p| < pmax =
+√
+2E. We must therefore supplement the expression for
+PE(p) with the condition that it is zero if |p| > pmax and
+this ensures that PE(p) is real. N is given in this case
+by
+N =
+1
+2 K(
+√
+2E)
+(F3)
+where, as in Appendix E, K is the complete elliptic inte-
+gral of the first kind.
+For motion above the separatrix we have
+√
+2E − 1 <
+|p| <
+√
+2E and PE(p) is zero outside this range. N is
+now given by
+N =
+√
+2E
+4 K(1/
+√
+2E)
+.
+(F4)
+To obtain the total thermal probability distribution
+PT (p) given in Eq. (F1) we need the density of states
+D(E) ≡ dn/dE, where n is the number of states be-
+low energy E. According to the Bohr-Sommerfeld rule
+n = S(E)/(2πℏ), where the action S(E) =
+�
+p dy is the
+area in phase space enclosed by the energy contour E.
+However, assuming that our Hamiltonian H is in units
+ℏω then the 2πℏ factor is absorbed into the definitions of
+p and y and we have D(E) = (d/dE)
+�
+p dy. Below the
+separatrix we have
+�
+p(y)dy = 4
+� arcsin
+√
+2E
+0
+�
+2E − sin2 y dy
+(F5)
+and putting 2E = sin2 y0 we find
+D<(E) = d
+dE
+�
+p(y)dy
+= 4
+� arcsin
+√
+2E
+0
+dy
+�
+sin2 y0 − sin2 y
+= 4K(
+√
+2E)
+(F6)
+where the integral is performed in a similar fashion to
+the one in Eq. (E3) and the subscript “<” indicates that
+this is the expression valid below the separatrix. Above
+the separatrix we find that the area enclosed in phase
+space between two oppositely rotating states of the same
+energy is
+�
+p(y)dy = 2
+� π/2
+−π/2
+�
+2E − sin2 y dy
+(F7)
+and thus
+D>(E) = d
+dE
+�
+p(y)dy
+= 2
+� π/2
+−π/2
+dy
+�
+2E − sin2 y
+=
+4
+√
+2E
+K
+�
+1
+√
+2E
+�
+.
+(F8)
+Due to the fact that above the separatrix 2E > sin2 y we
+no longer need to make the substitutions 2E = sin2 y0
+and sin y = sin y0 sin ζ, and the integral is straightfor-
+ward. The subscript “>” indicates that this expression
+holds above the separatrix.
+We now have all the necessary ingredients to perform
+the integral for PT (p) which we do numerically.
+The
+two contributions, one from below the separatrix and one
+from above, are added together to get the total. Inter-
+estingly, both density of states factors, Eqns. (F6) and
+(F8), diverge at the separatrix such that the two con-
+tributions individually display singular features but re-
+markably these cancel out when the two parts are added
+and result in the smooth gaussian curve plotted in Figure
+9.
+In order to compare the thermal distribution against
+the quenched (followed by integrable SG evolution) dis-
+tribution derived in Appendix E we need to choose a
+temperature T for the thermal distribution PT . We do
+this by matching the expectation value of the energy ⟨E⟩
+for both distributions. In the quenched case the initial
+state corresponds to an ensemble of pendula with dif-
+ferent starting angles y0 and zero kinetic energy. Each
+starting angle in the range −π/2 < y0 ≤ π/2 is equally
+probable in our J-quench. Therefore
+⟨E⟩quench = 1
+π
+� π/2
+−π/2
+1
+2 sin2 y0 dy0 = 1
+4 .
+(F9)
+To calculate ⟨E⟩ in the thermal case we compute
+⟨E⟩T = 1
+ζ
+� ∞
+0
+E e−E/T D(E) dE
+(F10)
+numerically for a large number of different values of
+T, performing the integrals below and above the sep-
+aratrix separately and adding the results.
+Here ζ =
+� ∞
+0
+e−E/T D(E) dE gives the normalization factor. We
+then fit a curve to the results and find the value of T
+
+25
+that best matches the result given in Eq. (F9). We find
+that T = 0.184 gives the best match. Putting back the
+units this result is
+kBT
+8J ℏc/ξh
+=
+kBT
+16JℏK/π = 0.184
+(F11)
+where c is the speed of sound and K is the Luttinger
+parameter and J is the tunnel coupling rate between the
+two wells. In this paper we take K = 25 and J = 30 Hz
+(see Table I) giving a temperature in SI units of 5.4 nK.
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+page_content='Caustics in the sine-Gordon model from quenches in coupled 1D Bose gases  Aman Agarwal,1, 2, 3, 4, 5, 6, ∗ Manas Kulkarni,3, † and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content=' O’Dell1, ‡  1Department of Physics and Astronomy, McMaster University,  1280 Main St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content=' Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Bengaluru – 560089,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' India  4Perimeter Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content=' N2L 2Y5  5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' University of Guelph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Guelph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content=' N1G 2W1  6Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' University of Greifswald,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 17489 Greifswald,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Germany  (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 2023)  Caustics are singularities that occur naturally in optical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' hydrodynamic and quantum waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  giving rise to high amplitude patterns that can be described using catastrophe theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this  paper we study caustics in a statistical field theory setting in the form of the sine-Gordon model  that describes a variety of physical systems including coupled 1D superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Specifically, we use  classical field simulations to study the dynamics of two ultracold 1D Bose gases (quasi-condensates)  that are suddenly coupled to each other and find that the resulting non-equilibrium dynamics are  dominated by caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thermal noise is included by sampling the initial states from a Boltzmann  distribution for phononic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We find that caustics pile up over time in both the number and  phase difference observables leading to a characteristic non-thermal ‘circus tent’ shaped probability  distribution at long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  INTRODUCTION  Wave focusing is ubiquitous in nature and leads to  localized regions of high amplitude called caustics that  dominate wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Everyday examples are provided  by rainbows and also the bright lines on the bottom of  water pools which are caused by the focusing of sunlight  by raindrops and surface water waves, respectively [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Caustics also occur in water waves themselves as ship  wakes [2] and more dramatically as tsunamis (focused by  the topography of the seabed [3–5]) and tidal bores (fo-  cused by v-shaped bays [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Astrophysical examples in-  clude gravitational lensing by matter and the twinkling of  starlight due to time-dependent fluctuations in the den-  sity of Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Natural focusing also leads  to the phenomenon of branched flow [7] and is speculated  to have given rise to the filamented nature of the large  scale structure of the universe [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In all these systems  caustics give rise to extreme amplitude fluctuations that  occur more frequently than those predicted by gaussian  statistics [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A remarkable property of caustics is that they com-  monly take on particular characteristic shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is  because caustics are singularities of the ray description,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' they are places where two or more rays coalesce lead-  ing to a diverging intensity in the short wavelength limit  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Such singularities are described by Thom’s catas-  trophe theory which rigorously shows that only certain  shapes of singularity are structurally stable against per-  turbations and hence occur under ‘natural’ or generic  ∗ aagarw03@uoguelph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ca  † manas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='kulkarni@icts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='in  ‡ dodell@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ca  conditions [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  These special shapes or catastro-  phes form a hierarchy organized by dimension where the  higher ones contain the lower ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each member of the  hierarchy represents a class of equivalent shapes that can  be smoothly transformed into each other, but each class  is distinct and cannot be smoothly transformed into any  of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In two dimensions the only structurally  stable shape is the cusp and we shall see it appear fre-  quently when we plot quantities such as number fluctu-  ations versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  It is worth noting in this context  that the humble point focus that we associate with lens-  ing is structurally unstable and unfolds into an extended  caustic in the presence of perturbations (aberrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Natural lenses are of course never perfect and so typi-  cally produce the shapes predicted by catastrophe theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The upshot of all this is that caustics represent a form of  universality in nonequilibrium wave dynamics: they fall  into equivalence classes each with their own shapes and  scaling properties analogous to, but a generalization of,  equilibrium phase transitions [13, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Caustics should equally be present in quantum waves  where, due to the probabilistic interpretation, they cor-  respond to regions of high probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Quantum  matter wave caustics have been seen in experiments with  cold neutrons [18, 19], electron microscopes [20], atom op-  tics [21–23], and most recently in atom lasers [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The-  oretical works on such matter wave caustics have also  considered their ‘fine structure’ [13] which features a lat-  tice of vortices [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Quantum fields are another area  where caustics are expected to form naturally during dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Early work centred on the electromagnetic field  [28, 29], including an interpretation of Hawking radiation  as a ‘quantum catastrophe’ [30], and more recently this  idea has been extended to quantum many-particle sys-  tems including bosonic Josephson junctions [26, 31, 32],  the XY model with long-range interactions (Hamiltonian  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='08410v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='quant-gas]  20 Jan 2023  2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Schematic of the setup we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The top fig-  ure shows two quasi one-dimensional gases that are prepared  independently and then suddenly coupled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We call  this process of sudden coupling a “J-quench”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' ρ1(z) and ρ2(z)  represent the density (red) in the first and second conden-  sates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Similarly, φ1(z) and φ2(z) represent the  phases (black) of the two condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Prior to the J-quench,  these fields in the two condensates are independent and con-  tain thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The bottom figure shows how a J-  quench could be implemented by suddenly reducing the tun-  neling barrier height in a double well potential from a higher  to a lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  mean field model) [33], quantum spin chains [27] and the  Bose-Hubbard model [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  One point to appreciate is  that the caustics in many-body systems can occur in the  wavefunction associated with an entire N-body configu-  ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Quantum many-particle caustics therefore live in  Fock space which can have a large number of dimensions  and hence lead to very complicated catastrophes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  However, catastrophes obey projection identities which  means that when projected down to lower dimensions one  obtains either the same catastrophe or one lower down  the hierarchy [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thus, low order correlation functions  obtained by integrating out most of the degrees of free-  dom will also generically contain caustics [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this paper we study caustics in the sine-Gordon (SG)  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The (classical) SG model obeys the nonlinear  wave equation  ∂2φ  ∂t2 − c2  0  ∂2φ  ∂z2 + ω2  0 sin φ = 0  (1)  where φ = φ(z, t) is a one dimensional field, and c0 and  ω0 represent a characteristic speed and frequency, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' If c0 is taken to be the speed of light then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (1) is  relativistically covariant, being a nonlinear version of the  Klein-Gordon equation and reducing to it when φ ≪ 1  such that sin φ ≈ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The SG model received attention  from the high energy physics community in the 1970s due  its soliton solutions [36–39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' but also describes the low en-  ergy physics of a considerable range of condensed matter  systems including crystal dislocations [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' domain walls  in magnetic [41] and binary superfluid [42] systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' the  Heisenberg spin chain with a field induced gap [43–45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  one-dimensional Bose gases in periodic potentials (that  can capture the Mott-insulator to superfluid transition  in one dimension) [46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' two-dimensional Bose gases  realizing the XY model [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' trapped ions [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' and two  tunnel-coupled one-dimensional Bose gases [50–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  fact that the SG model is both nonlinear and integrable  means that attention is often focused on its soliton so-  lutions, but part of our mission in this paper is to point  out that these same properties also imply that caustics  (which are associated with the existence of tori in phase  space [58]) are expected to occur generically, and we are  aware of only one previous study of caustics in this model  [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The particular physical realization we have in mind  for this paper is a system composed of two elongated  quasi-one dimensional Bose gases coupled by tunneling  along their length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' the field φ(z, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (1) gives the  relative phase between the two quantum gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Quasi  one-dimensional Bose gases have been created in a num-  ber of experiments over the last two decades using tightly  trapped ultracold atoms, and the remarkable tunability  of these systems allows the strongly interacting Tonks-  Girardeau regime [60, 61], the weakly interacting quasi-  condensate regime [62–65], and also the crossover be-  tween the two [66, 67], to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' It is important to  note that, in accordance with the Mermin-Wagner theo-  rem [68], one-dimensional Bose gases do not undergo true  Bose-Einstein condensation at low temperature, unlike  three dimensional gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Instead, they can form quasi-  condensates where density fluctuations are suppressed  but phase fluctuations remain [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this paper we  shall work in the weakly interacting regime and assume a  state of the system consisting of a quasi-condensate plus  small thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A system comprised of two coupled quasi-one dimen-  sional gases can be made by taking a single gas and  splitting it in two along its long axis by switching on an  elongated double well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is the experimen-  tal protocol typically adopted in a series of experiments  conducted by the Vienna group [63, 71–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The com-  bination of almost complete isolation from the environ-  ment, long relaxation times and spatially resolved mea-  surements of phase and number difference make these  experiments ideal for investigating many-particle quan-  tum dynamics, including fundamental questions such as  whether and how closed quantum systems reach equi-  librium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The gas can be split slowly so that it always  remains close to equilibrium leading to number squeezed  states [78, 79] or it can be split rapidly, leading to a so-  called quantum quench which launches the system into a  nonequilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this paper we shall consider the opposite quench  pi(2)  (2)d  P2(2)  p1(z)3  where two one-dimensional gases are suddenly connected  together (see schematic representation in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This  touches on rather fundamental considerations in quan-  tum mechanics since it describes the build-up of coher-  ence between two initially independent systems, and is  therefore related to the double-slit experiment for many-  particle systems [53, 80–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We shall refer to this as a  “J-quench” because J is often used to denote the cou-  pling strength between the two wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In a simple two-  mode description of a bosonic Josephson junction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  one that assumes a single mode in each well without the  quasi-continuum of low energy longitudinal modes that  are present in highly elongated traps, such a quench is  predicted to result in a periodic collapse and revival of  the atom number distribution between the two wells [84–  86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Essentially the same behavior, but π/2 out of phase,  occurs in the relative phase which is the conjugate vari-  able to number difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 26, 31, and 32 these  revivals are shown to be examples of quantum caustics  in a many-particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' One of our main aims here  is to investigate what happens to these caustics in the  presence of the dispersive longitudinal modes present in  the SG model, and is part of a wider program attempt-  ing to understand the role of caustics in quantum many  particle dynamics [17, 26, 27, 31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Due to the difficulty of solving the fully quantum SG  model we take a semiclassical-style approach based on  classical field configurations which are solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Each configuration is analogous to a single geo-  metric ray in optics and we include fluctuations by sum-  ming many configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The initial conditions for each  field configuration are randomly sampled from a Boltz-  mann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This approach is similar in spirit  to the truncated Wigner approximation (TWA) [87–92]  which includes quantum fluctuations around the classi-  cal field by summing many rays sampled from a quan-  tum probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The TWA has previously  been applied to one-dimensional Bose gases by Martin  and Ruostekoski [93, 94] who studied dark solitons, and  also to the connection problem of two zero temperature  one-dimensional Bose gases by Dalla Torre, Demler and  Polkovnikov [53], who proposed a universal scaling form  for the phase dynamics after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' More recently,  the TWA has been used by Horváth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' [95] to study  the surprisingly sudden relaxation of the phase seen in  the Vienna BEC splitting experiments [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this paper,  we include both the quantum fluctuations arising from  coupling two independent systems and thermal fluctua-  tions arising from thermal phonons in the longitudinal  modes and compare the time evolution of macroscopic  variables (the total number difference and phase differ-  ence) in the SG system against the simpler two mode  system [17, 26, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We find that following a quench  caustics dominate the dynamics of the macroscopic vari-  ables of both systems, even in the presence of thermal  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Due to the singular nature of caustics, and  combined with their structural stability, we therefore pro-  pose that strong nongaussian fluctuations are a generic  phenomenon following a quench in the SG model (and  indeed, in integrable or moderately chaotic many-body  systems in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The caustics we discuss in this paper also have implica-  tions for the question of relaxation towards equilibrium at  long times in many particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' While chaotic (non-  integrable) and open quantum systems should thermalize  (although a complete description is still the subject of ac-  tive research [96–103]), closed integrable models do not  reach a conventional Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We show here that in  the SG model there is a pile-up of caustics leading to a  singular shape for the long time probability distribution  for the macroscopic variables that resembles the shape of  a circus tent and is quite distinct from the thermal equi-  librium prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We find that an analytic approxima-  tion to the singular distribution based on an ergodic pen-  dulum (assuming a microcanonical or ‘equal-probability’  distribution) provides a good fit to the numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The plan for the rest of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We  start in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' II by deriving the SG hamiltonian from the  many-body description of two coupled 1D Bose gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' III we describe the natural length and time scales  and use them to write the SG hamiltonian and equa-  tions of motion in convenient dimensionless forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Sub-  sequently, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' IV we develop a method for finding the  initial conditions for the SG equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We  assume that prior to the quench the two Bose gases are  independent and at thermal equilibrium with a bath at  temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The initial conditions are obtained by  stochastically sampling the Fourier modes of a 1D quasi-  condensate obeying the Tomonaga-Luttinger liquid the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' With the initial conditions in hand, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' V we  give the main results of this paper which are the dy-  namics of the macroscopic number and phase difference  variables obtained by solving the equations of motion  numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' VI we consider the bigger picture  and examine the universal aspects of our results includ-  ing the influence of caustics on the coherence as well as  the long time dynamics and the establishment of (non-  thermal / non-Gaussian) equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We conclude in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' There are also six appendices where we give  the details of the calculations as well as bench marking  our numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  FROM TWO COUPLED CONDENSATES TO  THE SINE-GORDON PLUS MODEL  We begin by deriving the SG model as an effective low  energy description for two coupled one-dimensional Bose  gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For the sake of clarity, we list the main simplifica-  tions employed in this work:  the treatment of a quantum many body problem  by a semiclassical method (TWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  the neglect of a weak harmonic trap along the  long axis which would otherwise lead to a non-  uniform longitudinal density (this can be avoided  4  in box traps which, although rarer, can be realized  [76, 104])  the assumption of a constant value for the tunnel  coupling J along the entire length of the gases  the neglect of coupling to symmetric and higher  transverse modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Some more involved theoretical  models do include these effects [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  These simplifications are not expected to qualitatively  alter the main results of this work due to the structural  stability of caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In other words, caustics are known  to be robust to perturbations in both the Hamiltonian  and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A theoretical description of two ultracold quasi-one di-  mensional gases made up of bosonic atoms of mass m,  and held parallel to each other so that the atoms can  tunnel between them at rate J, can be obtained from the  following microscopic Hamiltonian [50, 51, 74]  ˆH =  �  j=1,2  � L/2  −L/2  dz  �  − ℏ2  2m  ˆψ†  j(z) ∂2  ∂z2 ˆψj(z) + U(z) ˆψ†  j(z) ˆψj(z) + g1D  2  ˆψ†  j(z) ˆψ†  j(z) ˆψj(z) ˆψj(z)  �  −  � L/2  −L/2  dz ℏJ  �  ˆψ†  1(z) ˆψ2(z) + ˆψ†  2(z) ˆψ1(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (2)  The indices j = 1, 2 label the two gases and each is as-  sumed to be tightly trapped in the x and y directions  so that those degrees of freedom are frozen into their  ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Only the longitudinal degree of freedom  z in each gas is taken to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In experiments there  will usually be a weak longitudinal trapping potential  U(z), although as mentioned above for simplicity we set  it to zero and hence consider a uniform system of length  L with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The quantum field  operator ˆψj(z) annihilates a particle at point z and to-  gether with its hermitian conjugate obeys bosonic com-  mutation relations [ ˆψj(z), ˆψ†  j′(z′)] = δjj′δ(z − z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  interaction constant g1D characterizes the effect of atom-  atom scattering within each gas on the longitudinal de-  gree of freedom and can be controlled both in magnitude  and sign either through Feshbach or confinement-induced  scattering resonances [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We note in passing that a  possible alternative physical realization of this problem  could be a spinor Bose gas in a single quasi-one dimen-  sional trap [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In fact, bosonic Josephson junctions  where the atoms are held in a single trap and two atomic  spin states are used for the two states have already been  realized experimentally [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A weakly interacting three-dimensional Bose gas at ul-  tracold temperatures will undergo Bose-Einstein conden-  sation and can be described to high accuracy by a clas-  sical field approximation (Gross-Pitaevskii theory [108]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In a quasi-one dimensional geometry quantum fluctua-  tions can still be small if the density is not too low, and  under these circumstances the gas can be treated as a  quasi-condensate where the quantum field operators are  replaced by classical fields [69, 109, 110]  ˆψj(z) → ψj(z) =  �  n1D + ρj(z) exp[iφj(z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (3)  Here n1D = N/L is the background density where N is  the number of atoms in each gas (for simplicity we as-  sume an equal number of atoms N in each gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' the struc-  tural stability of caustics means that they are stable to  small differences in n1D between the two gases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' ρj(z)  and φj(z) are the atom number density and phase fluc-  tuations at each point z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' These are canon-  ically conjugate variables and can even be quantized in  a semiclassical regime such that they obey the commu-  tation relations [ˆρj(z), ˆφj′ (z′)] ≈ δjj′δ(z − z′) in a coarse  grained sense [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, in the present paper ρj(z)  and φj(z) will be purely classical fields subject only to  thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We can further decompose the fields into their sym-  metric and antisymmetric components  ρs(z) = ρ1(z) + ρ2(z)  2  ,  ρa(z) = ρ1(z) − ρ2(z)  2  φs(z) = φ1(z) + φ2(z),  φa(z) = φ1(z) − φ2(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (4)  If the fluctuations are small ρa will be small whereas ρs  will be comparatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The particle-particle inter-  action energy will then typically cause the dynamics of  the symmetric modes to occur at higher energy than the  antisymmetric ones, and consequently we can ignore the  symmetric degrees of freedom as long as we restrict at-  tention to low energies [50, 55, 72, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The Hamiltonian  purely describing the antisymmetric variables is (see Ap-  pendix A for details)  HSG+ =  � L/2  −L/2  dz  �  g1D ρ2  a(z) + ℏ2n1D  4m  �∂φa  ∂z  �2  +  ℏ2  4mn1D  �∂ρa  ∂z  �2  − 2ℏJn1D cos φa(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (5)  We refer to this as the “sine-Gordon plus” (SG+) Hamil-  tonian because it includes an extra term (the third term)  in comparison to the standard SG Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This  term involves gradients of density fluctuations and results  5  in an energy cost which automatically suppresses den-  sity fluctuations at small length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' It is also worth  noting that including this term means that the density  and phase fluctuations [the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (5)] are  incorporated on an equal footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This is also in ac-  cordance with Gross-Pitaevskii theory which suppresses  density fluctuations with wavelengths below the healing  length [95]  ξh =  ℏ  √mg1Dn1D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (6)  However, when n1D is relatively large the third term is  naturally suppressed in comparison to the others and can  be dropped as long as the density gradients are small  leading to the SG Hamiltonian [55, 74]  HSG =  � L/2  −L/2  dz  �  g1D ρa(z)2 + ℏ2n1D  4m  �∂φa  ∂z  �2  − 2ℏJn1D cos φa(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (7)  The nonlinear piece in both Hamiltonians is the cosine  term which originates from tunneling between the two  wells and occurs in all Josephson junction type prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  It provides an effective potential well for phase  configurations φ(z, t) that play the role of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In fact,  as we shall see in Section V, it acts as an (imperfect) lens  that focuses rays excited by the quench to form caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For the sake of brevity, and when we deem no confusion  can arise, we will omit the ‘a’ subscript on antisymmet-  ric variables since we will not be dealing with symmetric  degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The fact that the two fields φ(z) and ρ(z) form a conju-  gate pair means that their equations of motion are given  by Hamilton’s equations  ˙φ = 1  ℏ  δH  δρ(z)  ˙ρ = −1  ℏ  δH  δφ(z)  (8)  where H is the Hamiltonian density defined via  H =  � L/2  −L/2  H dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (9)  Applying these equations to the SG+ Hamiltonian given  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (5) we find the following of equations of motion  dφ(z, t)  dt  = 2 g1D  ℏ ρ(z, t) + 2  ℏ  4mn1D  ∂2ρ(z, t)  ∂z2  dρ(z, t)  dt  = 2 ℏn1D  4m  ∂2φ(z, t)  ∂z2  − 2Jn1D sin[φ(z, t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (10)  These are the key equations we use to solve for the dy-  namics of the field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' They have the form of  Josephson’s equations [111] augmented by second order  spatial derivatives ∂2φ/∂z2 and ∂2ρ/∂z2 which account  for phase and density fluctuations along the longitudi-  nal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Combined with the sine term, they will  cause wavepackets to disperse along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the absence of  these terms we have exactly the equations of motion for  a pendulum where φ is the angular displacement from  equilibrium and ρ plays the role of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The dependence on z suggests an interpretation in terms  of a continuous chain of many pendula each coupled to its  neighbors by the spatial derivative terms and is reminis-  cent of the Fermi-Pasta-Ulam-Tsingou problem [50, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this paper the coupled equations of motion given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (10) will be solved numerically for a system of length  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' To perform the numerical computations we discretize  the system on a spatial grid with NL + 1 points which  makes the grid spacing a = L/NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The positions of the  grid points are given by z = ra where r is an integer  r = −NL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' , NL  2  (11)  and NL is chosen to be an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  There is in fact a physical limitation on the grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (10) is classical and valid only on length scales greater  that healing length ξh [51, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Therefore, any numerics  performed on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (10) are meaningful only when the lat-  tice grid size a is greater than ξh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In particular, NL  should be such that a > ξh which implies  N 2  L < mg1Dn1DL2  ℏ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (12)  We fulfil the condition given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (12) in our numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  NATURAL SCALES  Let us express the SG/SG+ Hamiltonians and equa-  tions of motion in terms of the natural scales for a one-  dimensional quantum fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For a length scale we chose  the healing length ξh given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The ratio of the  healing length to the mean interparticle spacing 1/n1D  motivates the definition of the Luttinger parameter  K =  �  n1D(ℏπ)2  4g1Dm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (13)  This dimensionless quantity measures how strongly in-  teracting the system is - when K ≫ 1 the healing length  is much greater than the interparticle spacing and the  system is in the weakly interacting (quasi-condensate)  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Another key physical quantity is the speed of  sound  c =  �g1Dn1D  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (14)  This can be used to define a characteristic energy, namely  that associated with phonons (quanta of sound)  E = ℏω = ℏc  ξh  (15)  6  where we have set the natural frequency ω to be the ratio  of the speed of sound to the healing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We therefore transform to the following dimensionless  variables  z −→ ˜z = z  ξh  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  t −→ ˜t = c  ξh  t  ρ −→ ˜ρ = ρ ξh  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  φ −→ ˜φ = φ  (16)  and defining ˜HSG = HSG/E and likewise for ˜HSG+ we  obtain the two Hamiltonians in dimensionless form  ˜HSG =  � L/2  −L/2  d˜z  �  Γ ˜ρ2 + ϵ  �  ∂ ˜φ  ∂˜z  �2  − 2J cos ˜φ  �  (17)  and  ˜HSG+ =  � L/2  −L/2  d˜z  �  Γ ˜ρ2 + ϵ  �  ∂ ˜φ  ∂˜z  �2  + Γ  4  �∂˜ρ  ∂˜z  �2  − 2J cos ˜φ  �  (18)  where the coefficients are given by  Γ =  π  2K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' ϵ = K  2π ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' J = K  2π  ξ2  h  ξ2s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (19)  In the last term we have introduced the spin healing  length  ξs =  �  ℏ  4mJ  (20)  which provides a measure for the distance over which  coherence between the two gases is restored due to the  tunnel coupling J [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' At finite temperatures another  useful length scale is the thermal phase coherence length  λT = 2ℏ2n1D  mkBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (21)  The dimensionless form of the equations of motion can  now be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For the SG model we find  d˜φ  d˜t = 2Γ˜ρ  d˜ρ  d˜t = 2ϵ∂2 ˜φ  ∂˜z2 − 2J sin ˜φ  (22)  and for the SG+ model we obtain  d˜φ  d˜t = 2Γ˜ρ − Γ  2  ∂2˜ρ  ∂˜z2  d˜ρ  d˜t = 2ϵ∂2 ˜φ  ∂˜z2 − 2J sin ˜φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (23)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  INITIAL CONDITIONS  The dynamics we seek to study in this paper start from  a J-quench where two independent one-dimensional gases  at thermal equilibrium are suddenly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In order  to obtain the initial density and phase fluctuations of  these gases we use the Tomonaga-Luttinger (TL) model  that provides the universal low energy effective theory for  one-dimensional systems (low energy limit of the Lieb-  Lininger theory, for example) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Tomonaga-Luttinger (TL) liquid  In our notation the TL Hamiltonian reads  HTL =  � L/2  −L/2  dz  �  g1Dρj(z)2 + ℏ2n1D  4m  �∂φj  ∂z  �2�  (24)  where j labels either of the two gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We henceforth,  omit this label for the sake of brevity with the under-  standing that in this section the density and phase fields  refer to just one of the two gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (24) has the same  mathematical structure as the SG model but without the  tunnelling term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  If we include density fluctuations we  find  HTL+ =  � L/2  −L/2  dz  �  g1Dρ(z)2 + ℏ2n1D  4m  �∂φ  ∂z  �2  +  ℏ2  4mn1D  �∂ρ  ∂z  �2 �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (25)  The TL model is quadratic and hence its thermal fluc-  tuations can be treated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' To this end it is useful  to work in Fourier space and we apply discrete Fourier  transforms defined on the numerical grid with NL points  as discussed at the end of Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The phase field φ  and its Fourier transform ϕ are related by  φr =  1  √NL + 1  NL/2  �  k=−NL/2  ϕk exp  �  i 2πkr  NL + 1  �  ϕk =  1  √NL + 1  NL/2  �  r=−NL/2  φr exp  �  −i 2πkr  NL + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (26)  The discrete data {φr} = {φ−NL/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' , φ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' , φNL/2}  and its transform are located symmetrically about r = 0  and k = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Since the value φr of the field  at each coordinate space grid point is a real number the  condition  ϕ−k = ϕ∗  k must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Similarly the density  fluctuation field ρ and its Fourier transform ϱ are related  by  ρr =  1  √NL + 1  NL/2  �  k=−NL/2  ϱk exp  �  i 2πkr  NL + 1  �  ϱk =  1  √NL + 1  NL/2  �  r=−NL/2  ρr exp  �  −i 2πkr  NL + 1  �  (27)  7  where again the reality of the field in coordinate space  requires that ϱ−k = ϱ∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Inserting these transformations  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (25) we obtain (see Appendix B for details)  HTL+ = a g1D  NL/2  �  k=−NL/2  |ϱk|2  + a ℏ n1D  NL/2  �  k=−NL/2  ℏπ2k2  mL2 |ϕk|2  + a  ℏ2  4mn1D  NL/2  �  k=−NL/2  4π2k2  L2  |ϱk|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (28)  Before proceeding with further analysis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (28), it is  worth noting that it can be recast in a standard Luttinger  liquid form  HTL+ = acℏ  2  NL/2  �  k=−NL/2  �K  π  4π2k2  L2  |ϕk|2 + π  K |ϱk|2  + K  π  4π2k2  N 2 |ϱk|2  �  (29)  where the strength of the terms depends either on K or  1/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Applying the transformations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (16), the  Fourier space variables can be written in dimensionless  form as  ϱk −→ ˜ϱk = ξhϱk  ,  ϕk −→ ˜ϕk = ϕk  (30)  and the TL+ Hamiltonian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (28) scaled by the  energy E = ℏc/ξh is given by  ˜HTL+ =  ˜L  NL  NL/2  �  k=−NL/2  �ϵ 4π2k2  ˜L2  | ˜ϕk|2 + Γ|˜ϱk|2  + Γ π2k2  ˜L2  |˜ϱk|2  �  (31)  where ˜L = L/ξh is the ratio of the system size to the  healing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Comparison with the spatial version of  HTL+ given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (25) shows where this factor comes  from: as the size is increased the range of the integration  increases linearly and this is accounted for by ˜L in the  Fourier transformed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Note that all parameters  and variables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (31) are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Thermal equilibrium  To find the initial conditions on the fields ρj(z) and  φj(z) we assume that each gas is at thermal equilibrium  such that the excitation (phonon) modes of the TL+  Hamiltonian are populated with a probability given by  the Boltzmann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The range of temperatures  we simulate is listed in Table I along with the values  of all the other key parameters, and is chosen so as to  correspond to realistic experimental conditions (the tem-  perature must be low enough that the quasi-condensate  description is valid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In the canonical ensemble of statistical mechanics the  probability that a system at thermal equilibrium has  the phase space configuration s = q1, p1, q2, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='qN, pN  is proportional to the Boltzmann weight exp[−βH(s)],  where β = 1/kBT and H = �  i p2  i /2m + V (qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The  Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (31) is quadratic and hence the Boltz-  mann weight becomes that of a series of independent har-  monic oscillators  e− ˜β ˜  HTL+ =  �  k  e−P 2  k /2σ2  ρ+ e−Q2  k/2σ2  φ+(k)  (32)  where ˜β = (ℏc/ξh)/kBT is the appropriately scaled tem-  perature parameter and we have introduced the real vari-  ables Qk and Pk which are related to the old variables  by  ˜ϕk = Qkeiαk,  ˜ϱk = Pkeiβk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (33)  The phases αk and βk allow for the fact that ˜ϕk and ˜ϱk  can be complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The variances in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (32) are  given by  σ2  ρ+(k) = NL  2˜β  1  Γ˜L(1 + π2k2/˜L2)  (34)  σ2  φ+(k) = NL  2˜β  ˜L  4π2k2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (35)  The partition function can now be written down as  Z =  �  k  � ∞  −∞  e− ˜β ˜  HTL+ dPkdQk  =  �  k  �  σρ+  √  2π  � �  σφ+(k)  √  2π  �  (36)  and hence the probability P of a particular configuration  (Q1, Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='., P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='.) is  P =  �  k  �  e−P 2  k /2σ2  ρ+  σρ+  √  2π  � �  e−Q2  k/2σ2  φ+(k)  σφ+(k)  √  2π  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (37)  This is seen to be the total probability distribution for  independent random variables Pk and Qk drawn from  normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thus, the absolute values of the  Fourier coefficients ˜ϱk and ˜ϕk are normally distributed  random variables with zero mean and variances given by  Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (34) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We sample these numerically from  normal distributions to generate the initial system con-  figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The phases αk and βk given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (33) do  not appear in the Boltzmann weight and are chosen ran-  domly from the range [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In fact, for both the phases  and the amplitudes we only need to choose the values for  8  terms with k ≥ 0 because the reality conditions imply  that we can put  Qk = Q−k ,  Pk = P−k,  αk = −α−k ,  βk = −β−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (38)  So far we have only considered the initial state of a sin-  gle gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By subtracting the results for two gases we can  obtain the initial values of the antisymmetric variables  ρa(z) and φa(z) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Actually, due to the  fact that the SG+ Hamiltonian with J = 0 and expressed  in terms of antisymmetric variables as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (5)  formally has the same structure as the TL+ Hamiltonian  given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (25), sampling initial data for two gases is  unnecessary and one can obtain ρa(z) and φa(z) directly  by sampling them as though they were from one gas de-  scribed by the TL+ Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, in doing so,  consideration needs to be given to the average value of  relative phase φa(z) because both the SG+ and TL+  Hamiltonians only contain the spatial derivative of the  phase but not the phase itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Its average value is there-  fore not determined by energy considerations and is left  to float freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is also apparent in the Fourier trans-  formed version of the TL Hamiltonian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (31)  where the k = 0 term involving ˜ϕ0 is absent due to the  vanishing of its coefficient which is proportional to k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  To take into account the random phase difference be-  tween the two gases one can chose ˜ϕ0 to be a random  number in the range [−π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' π) but multiplied by a factor  of √NL + 1 in order to respect the normalization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This gives values of the average value of φa(z) in  the desired range −π and +π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The random value of the initial phase difference is ac-  tually a key feature of the J-quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' It populates the  cosine potential landscape in the Hamiltonian with uni-  form probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As the trajectories roll back and forth  in this potential they form caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In effect, the co-  sine potential acts as an imperfect lens that focuses an  initially flat ‘wavefront’ over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Choice of parameters  There are three constraints which must be satisfied in  order to have a quasi-one dimensional condensate [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  To ensure minimal scattering into the transverse modes  we need the interaction to be sufficiently weak which im-  plies µ = g1Dn1D ≪ ℏω⊥ where µ is the chemical poten-  tial and ω⊥ is the transverse trapping frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' More-  over, the temperature needs to be low enough such that  transverse modes are not thermally excited leading to the  inequality kBT ≪ ℏω⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Finally, in order to have a quasi-  condensate which permits a semiclassical approach we  need weak interactions in comparison to the zero-point  kinetic energy associated with the density of the parti-  cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This implies n1Dg1D ≪ ℏ2n2  1D/m which means the  Symbol  Parameter  Value  ω⊥  trapping frequency  2π × 3 kHz  m  mass of Rb atom  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='41 × 10−25 kg  as  scattering length  98 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='52 Å  N  number of atoms  1200  L  system length  18 µm  n1D  average density  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='7 × 107m−1  g1D  2 ℏascatω⊥  2 × 10−38 Jm  K  Luttinger parameter  25  T  temperature  2 - 20 nK  J  J-quench  0 - 30 Hz  NL  number of grid points  50  c  speed of sound  3 × 10−3 m s−1  a  grid spacing  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='36 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  ξh  healing length  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='24 µm  λT  phase coherence length  38 − 380 µm  ξs  spin healing length  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5 µm  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Table containing important parameters and their val-  ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The parameters are chosen to be experimentally feasible  and correspond roughly to those reported in references [72–  77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Luttinger parameter should obey K ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' All the param-  eter values we use satisfy these three inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In quasi-one dimensional gases the interatomic inter-  action parameter g1D is related to the scattering length  as and transverse trapping frequency as g1D = 2ℏasω⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For 87Rb atoms we have as ≈ 98 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='52 Å[113] and we  will assume ω⊥ = 2 π×3 kHz [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The full list of pa-  rameters used in our simulations is given in Table I and  roughly corresponds to those used in the experiments by  the Vienna group [72–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For our numerical simulations we choose a grid size  that slightly exceeds the healing length because, as ex-  plained above, this cuts off unphysical density fluctua-  tions [51, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This condition is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (12) but can  be expressed succinctly in terms of Γ as N 2  L < ΓN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  magnitudes of ˜ρ and ˜φ also need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  phase difference can take the full range +π to −π, but  the number difference is limited by the condition that  the total number difference (integrated over the entire  system) cannot exceed the total number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In  fact, due to the random nature of sampled thermal fluc-  tuations, the integral of ˜ρ is always approximately zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  However, the validity of the SG/SG+ model requires that  local density fluctuations be small in comparison to the  background density n1D, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Translated  into the scaled variables this means that at any point  ˜ρ(˜z) ≪ n1Dξh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In practice we choose ˜ρ(˜z) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='6 so that  the fluctuations are an order of magnitude smaller than  the background density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  9  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Examples of Initial conditions  In Figure 2 we present typical spatial profiles of the  initial number difference field ˜ρ (upper row) and phase  difference field ˜φ (lower row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each profile provides the  initial conditions for a single classical field trajectory and  is obtained by summing up thermally activated phonons  (Fourier modes) using the Tomonaga-Luttinger model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The different columns show the effect of changing tem-  perature T or Luttinger parameter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As expected,  when T is increased the fluctuations in both ˜ρ and ˜φ  increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By contrast, if K is increased the maximum  magnitude and jaggedness of ˜ρ increases but the jagged-  ness of ˜φ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Referring to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (19) we can see that  this is because the coefficient multiplying the density fluc-  tuation term in the Hamiltonian is Γ = π/2K which de-  creases as K increases leading to increased variance of ϱk  modes according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The phase fluctuation term  shows the opposite behavior because its coefficient in the  Hamiltonian (which only appears as the spatial gradient  of ˜φ) is ϵ = K/2π which increases as K increases and  this reduces the variance of the ϕk modes according to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (35), thereby making the ˜φ profiles smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  NUMERICAL SIMULATIONS OF THE  DYNAMICS  In this section we explore the dynamics following a J-  quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Our approach is inspired by the TWA where  multiple classical field configurations are propagated in  time using the classical equations of motion, although in  our case the initial conditions are sampled from a ther-  mal distribution as described in Section IV rather than  a quantum distribution as in the standard TWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  J-quench dynamics have previously been explored for  the simpler case of a two-mode zero temperature bosonic  Josephson junction where it was found that caustics dom-  inate the number and phase difference probability distri-  butions [17, 26, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the two-mode case it is possible  to compute the exact quantum dynamics for some thou-  sands of particles and compare them against the TWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The results (see Figure 1 in [31]) show good qualitative  agreement and give us confidence that the TWA can cap-  ture the main features of the quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Fur-  thermore, the inevitable presence of decoherence due to  the environment will tend to reduce the quantum dy-  namics to their classical limit (this has been investigated  in the two-mode case for a J-quench in [32]) increasing  the relevance of semiclassical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the present  work we are interested in whether the phonons along the  long axis disrupt or sustain these caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We will start  by reproducing the caustics presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 31 for the  two-mode case and then add in the longitudinal modes  after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Numerical Methods  The initial conditions are generated via random sam-  pling from Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We then evolve the  equations of motion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 23 for the case of the full SG+  model) using a Runge-Kutta solver with a user-defined  time step [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The endpoints of our system are treated  by imposing periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In Appendix  C we demonstrate the numerical convergence of the solver  by varying the temporal and spatial steps by tracking the  time evolution of the total energy (hamiltonian) which  should be a constant of the motion and obtain the fidu-  cial time and space resolution for all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Special case: two-mode approximation  In the two-mode approximation only a single mode in  each well is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This description is rele-  vant to the SG/SG+ model in the limit where the entire  length of each quasicondensate is perfectly synchronized  so that the fields ˜ρ(˜z) and ˜φ(˜z) do not depend on ˜z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this case the spatial derivative terms vanish and the  equations of motion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (23) reduce to  d˜φ  d˜t = 2Γ˜ρ  ,  d˜ρ  d˜t = −2J sin ˜φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (39)  These are the standard Josephson equations of motion  and also correspond to those of a classical pendulum  [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Such synchronization can occur at very low tem-  peratures or when the coefficients ϵ and Γ are large  enough that they suppress spatial fluctuations in the ini-  tial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In Figure 3 we display the post-quench dynamics in  the two-mode approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The left hand and cen-  tral panels show the time dependence of 150 indepen-  dent solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (39) which give the trajectories for  the number difference and phase difference, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Note that in this paper we use the color blue for tra-  jectories calculated within the two mode approximation  and reserve red for the trajectories of the full many mode  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In accordance with our assumption that the two  wells start with an equal number of atoms, each solution  starts with ˜ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  And as discussed in Section IV B,  the initial value of ˜φ is randomly chosen from the range  [−π, π) because the two condensates are independent be-  fore the J-quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The most striking feature of Figure 3 is the series of  cusp-shaped caustics that form in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In or-  der to guide eye, we have have outlined the first cusp  caustic in the number difference variable using a black  curve (the calculation for this curve is given in Appendix  D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Like in optics, caustics are regions of high intensity  formed by the envelopes of families of rays (trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Each caustic is born at the centre of the distribution at  the tip of a cusp before spreading out in two arms that  move towards the edges of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The fact  10  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Examples of initial spatial profiles of the number difference ˜ρ (top row) and phase difference ˜φ (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each  profile is obtained by randomly sampling a thermal distribution using the method described in Section IV B, and each panel  includes ten different profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The parameter values common to all panels include the number of computational lattice points  NL = 50, grid spacing a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='36µm, and healing length ξh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='24 µm (the remaining parameters are listed in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  difference between the columns is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The left column has the Luttinger parameter K = 25, and temperature T = 2  nK giving a phase coherence length of λT = 380 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the middle column K = 25, but the temperature is increased to 20 nK,  giving λT = 38 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the right column, the value of K is artificially increased (without changing any other parameters)  to K = 250 and T = 2 nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Increases in temperature excite stronger fluctuations in the profiles as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Increases in  the Luttinger parameter have opposite effects on ˜ρ and ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The maximum value and jaggedness of ˜ρ is increased whereas the  jaggedness of ˜φ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' An explanation of this behavior is given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  that they are cusp shaped is in agreement with the pre-  diction of catastrophe theory that in two dimensions the  only structurally stable and hence generic singularities  are cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Each trajectory represents a single experimental run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The idea behind the TWA is that the number of tra-  jectories reaching a point ˜ρ at time ˜t is proportional to  the probability that a measurement of the true quantum  system would yield that value of ˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' An equivalent inter-  pretation holds for the ˜φ trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The caustics have  the highest probability density and hence give the values  most likely to be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Of course, if we only con-  sider the average values of ˜ρ or ˜φ we would get zero in  both cases due to the symmetry of the distributions and  hence miss the caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Many experimental runs must  be performed in order to obtain the probability distribu-  tion where these patterns live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The mechanism underlying caustics can be understood  from a phase space perspective, as shown in the right  hand panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each dot gives the number and  phase difference at a particular time for a different ini-  tial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The red dots are the initial values which  lie in a horizontal line because at ˜t = 0 all trajectories  have ˜ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As time evolves the dots rotate around the  origin: the green and blue dots show two successively  later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, the nonlinearity of the Josephson  equations means dots further from the origin rotate more  slowly and this leads to the formation of a spiral or whorl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  At places where the whorl has a vertical segment a range  of different solutions all have the same value of ˜φ and this  stationarity of the distribution with respect to changes in  the initial conditions is what generates a caustic, in this  case a ˜φ-caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Conversely, horizontal segments give  rise to ˜ρ-caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In the absence of nonlinearity the equations reduce to  those of a harmonic oscillator  d˜φ  d˜t = 2Γ˜ρ  ,  d˜ρ  d˜t = −2J ˜φ  (40)  giving rise to rigid rotation in phase space and the forma-  tion of perfect focal points in the number and phase dif-  ference variables, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, these  perfect revivals of the initial state are not stable: any  nonlinearity will cause the focal points to evolve into the  extended cusp caustics shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The frequency of the linearized motion is known in  Josephson junction terminology as the plasma frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  3  2  1  Initial  0  1  2  3  20  10  0  10  20  Grid points (r)3  2  1  Initial pr  0  1  2  3  20  10  0  10  20  Grid points (r)3  2  1  Initial  0  1  2  3  20  10  0  10  20  Grid points (r)3  2  Initial $r  1  0  1  2  3  20  10  0  10  20  Grid points (r)3  2  1  Initial $r  0  1  2  3  20  10  0  10  20  Grid points (r)3  2  1  0  Initi:  1  2  3  20  10  10  10  20  Grid points (r)11  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and phase space distribution (right) following  a J-quench from J = 0 to J = 30 Hz in the two mode approximation governed by the Josephson equations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The other parameter values are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each panel contains 150 trajectories: each trajectory starts with ˜ρ = 0 at time  ˜t = 0 but has an initial phase randomly sampled from [−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Both number and phase difference variables display a series of  cusp shaped caustics given by the envelopes of families of trajectories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' to guide the eye we have outlined the first cusp caustic  in the ˜ρ variable with a black curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the right panel three different time slices of the results are plotted in phase space (˜ρ  versus ˜φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each dot corresponds to a different initial condition (trajectory) and the colors indicate the time: ˜t=0 (red), ˜t=50  (green), ˜t=100 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' During time evolution the initial horizontal line winds into a whorl and the caustics in the ˜ρ and ˜φ plots  occur due to horizontal and vertical segments of a whorl, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and the phase space distribution (right) in  the linearized version of the two-mode approximation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (40)] following a J-quench from J = 0 to J = 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Like in Figure  3, there are 150 trajectories shown in each panel corresponding to different values of the initial value of ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, in this  linearized case we obtain a series of perfect focus points (revivals of the initial state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is because linearization gives rise to  rigid rotation in phase space without whorls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Unlike the extended cusp caustics seen in Figure 3 (which will be qualitatively  robust to details of the nonlinearity) perfect focus points are nongeneric because they are unstable to perturbations such as the  effects of nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' All parameter values and color labels are the same as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In our notation it reads  ωp =  √  4ΓJ  (41)  and the period of the motion is therefore given by 2π/ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For the case shown in Figure 4 we have Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='063 and  J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='037 giving a period ≈ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In fact, the tips of the  cusps in the nonlinear case also occur with this period  since they are formed from small amplitude trajectories  that only experience the quadratic bottom of the cosine  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  General case: many-mode SG+ model  Simulations of the full SG+ model are shown in Figure  5, which represents one of the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The trajectories in the left panel give the spatially av-  eraged number difference ⟨˜ρ(˜t)⟩z as a function of time  obtained by solving the equations of motion given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (23) for the many-mode system and then averaging over  its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The trajectories in the middle panel of Figure  5 give the equivalent spatial average of the phase differ-  ence ⟨˜φ(˜t)⟩z, and the right-hand panel is the phase space  picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Each trajectory is evolved from a single ran-  domly sampled field configuration (describing thermally  activated phonons) such as those shown in the top row  of Figure 2 and for the parameters given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We  observe that despite the inclusion of longitudinal modes  and the randomness of the initial conditions, the caustics  survive and are quite similar to those of the two-mode  approximation shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This suggests that  caustics are a generic feature of many particle dynamics  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content='5  2Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='3  1  0  1  2  32  1  2Q  0  1  2  0  25  50  75  5100 125 150 175 200  2+3  2  1  20  0  1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='3  0  25  50  75  100 125 150 175 200  2+2  1  2Q  0  1  2  3  1  0  1  2  w  iΦ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  10  25  50  75  100 125 150 175 2003  2  1  20  0  1  2  3  0  25  50  75  100125 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='175 20012  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Dynamics of the spatially averaged number difference ⟨˜ρ⟩z (left), phase difference ⟨˜φ⟩z (middle), and phase space  distribution (right) for the full many-mode SG+ model following a J-quench from J = 0 to J = 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each panel contains  150 trajectories which are solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The initial conditions are randomly sampled thermal phonons with the same  parameter values as those shown in the top row of Figure 2 and described in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In particular, the number of numerical  lattice points is NL = 50 separated by a grid spacing of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='36 µm, and the temperature is T = 2 nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The healing length is  ξh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='24 µm, the spin healing length is ξs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5 µm and the phase coherence length is λT = 380 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The different colors on  the phase space plot correspond to the same time slices as in the previous phase space plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  following quenches, at least for systems whose underlying  physics is based on coupled nonlinear oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each  oscillator starts with a random phase and a noisy momen-  tum but the quench acts so as to give all the oscillators  a momentum kick at the same time ˜t = 0 leading to an  initial partial synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As the system evolves in  time after the kick the different periods of nonlinear os-  cillators leads to cusp catastrophes in the distribution of  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' If we had instead calculated only the expec-  tation values of the number and phase differences then  this underlying structure would not have been visible be-  cause it lives in the probability distribution rather than  the mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A slice at fixed time through the probability distri-  bution for the spatially averaged phase variable ⟨˜φ⟩z is  shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is obtained by sorting the ⟨˜φ⟩z(˜t)  trajectories into bins each of which covers a small range  of ⟨˜φ⟩z and counting the number of trajectories in each  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The result is noisy due to the thermal fluctuations  but the caustics are clearly visible as strong peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' These  peaks display the characteristic ‘square root’ divergence  of fold caustics [1]  P(⟨˜φ⟩z) ∝  1  �  ˜φc − ⟨˜φ⟩z  (42)  where P(⟨˜φ⟩z) is the probability density and ˜φc is the  location of the caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The blue dashed lines in Figure 6  are fits of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (42) to the numerical data and we see that  the agreement is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Although the height of the singu-  larities predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (42) is infinite at the caustic, this  function is integrable so that a probability distribution  with caustics is still normalizable (of course, the peaks in  the numerical data are of finite height because the num-  ber of trajectories is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' A very similar pattern of  square root singularities at each caustic is obtained for a  time slice through the probability density for the number  difference variable so we shall not show it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The probability density (red curve) as a func-  tion of ⟨˜φ⟩z obtained from the density of trajectories at time  ˜t = 162 for the SG+ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This corresponds to a slice at  fixed time through the middle panel of Figure 5, although  calculated using 10000 trajectories to improve the statistics  and averaged over a short time window of ∆˜t = 1 to remove  rapid time fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The red curve has been drawn with  a bin width d˜φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='04 and is normalised such that the area  under the graph is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The caustics are clearly visible as di-  verging peaks and are well fitted (blue dashed curves) by the  inverse square root form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (42) that is expected  for fold catastrophes [1] (the satellite caustics also have this  shape but the fit is not shown to avoid obscuring the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A very similar profile is obtained for the probability density  in the ⟨˜ρ⟩z variable (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Effect of dispersion on the caustics  The double derivative terms in the SG+ equations of  motion given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 23 are responsible for transmitting  wave disturbances along the longitudinal axis and are not  present in the simpler two-mode case discussed in Section  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0  25  50  75  1001251501752003  2  1  0  1  2  3  0  25  50  75  100 125 150 175 200  2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0  1  2  3  ()z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='8  M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  3  1  0  1  2  3  (0)z13  V B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Initial thermal fluctuations in the SG+ model will  therefore disperse in z over time and it is interesting to  see what difference this makes to the caustics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' comparison  of Figures 3 and 5 suggests it makes little difference to  spatially averaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, this observation is  for only one choice of the parameters ϵ and Γ that govern  the size of the derivative terms and also for relatively  short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In particular, in Figure 5 the parameters  are ϵ ≈ 4 and Γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='06 which were chosen to match  experimental values [72–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In Figure 7 we compare the  long time dynamics of the two-mode approximation and  the SG+ model for the case where ϵ in the SG+ model  has been artificially increased by a factor of 10 (without  changing any other parameters), thereby increasing the  effect of spatial dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Apart from this change, the  initial conditions and J-quench are similar to those used  in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Note that we only use this increased value  of ϵ for the time propagation and not for the generation  of the thermal initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This avoids changing  the starting phase fluctuations from those used in Figure  5 which would otherwise be energetically suppressed and  would also lead to significantly different dynamics but is  not the comparison we would like to make here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' From  Figure 7 we see that the strong coupling of neighboring  ‘pendula’ does seem to largely wash out the caustics at  long times in comparison to the dispersionless two-mode  case, although some faint structure is still present which  underlines the structural stability of caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The long  time behavior will be further analyzed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Effect of J on the caustics  Another parameter that affects the dynamics is the  tunnel coupling strength J [or its dimensionless version  J which is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (19)] that becomes non-zero  after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The quench itself creates a strongly  nonequilibrium phase difference where all values of ˜φ are  equally probable independently of the value of J by virtue  of the fact that before the quench there is no phase co-  herence between the two quasicondensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, J  does control the post-quench dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' One way it does  this is via the frequency of the Josephson oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The cusps occur with a frequency given by the plasma  frequency in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (41) which goes as  √  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In Figure 8 we examine the effect of quenching to dif-  ferent J values, with the value of J increasing from left  to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We can see the expected increase in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The amplitude of the motion also increases with J be-  cause immediately after the quench each trajectory finds  itself at a random point on the cosine potential energy  surface whose depth between valley top and valley bot-  tom is 2J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The initial potential energy of a field config-  uration is therefore −2J ⟨cos ˜φ0⟩z, where ˜φ0 is the phase  field ˜φ(˜z, ˜t) at the initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This configuration evolves  under the full Hamiltonian and upon spatial averaging is  seen to execute oscillations about the potential minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The upper row in Figure 8 plots the spatially averaged  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Comparison of the long-time behavior of the phase  difference in the two-mode approximation (upper) and many-  mode SG+ model (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Both panels contain 150 different  runs and the initial conditions and J-quench are similar to  those of Figure 5 except that ϵ has been artificially multiplied  by 10 (without changing any other parameters) in the lower  panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This enhances the effect of the spatial derivative term  in φ in the SG+ model (this term does not appear in the two  mode model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We see that in the upper panel the caustics  are still visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By contrast, the stronger spatial interaction  causes dispersion and makes the caustics much less visible in  the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  number difference and according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (18) the maxi-  mum amplitude this can have is  ⟨˜ρ⟩max  z  =  �  2J (1 − ⟨cos ˜φ0⟩z)  Γ  (43)  where we have ignored the effects of spatial coupling (sec-  ond order derivative terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thus, ⟨˜ρ⟩max  z  also scales as  √  J, and this is in correspondence with Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The lower row of Figure 8 shows the behavior in phase  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In these figures we have also included the unaver-  aged data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' the ˜ρ and ˜φ values of each grid point at  the three selected times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This gives a sense of the size  of the statistical fluctuations due to the spatial degrees  3  2  1  0  1  2  3  600625650675700725 750775800  2+3  2  1  0  1  2  3  600 625 650 675 700 725 750 775 800  2+14  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Effect of quench strength J for J = 0 Hz, 3 Hz, and 30 Hz (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The top row shows the dynamics  of ⟨˜ρ⟩z with initial conditions sampled in the same way as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The bottom row plots the corresponding phase space  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Like in previous figures, the different colors give different time instants: ˜t=0 (red), ˜t=50 (green), ˜t=100 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The dots with intense colors are the spatially averaged values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We have also included the raw data (without spatial averaging)  as faint dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This gives an idea of the size of the statistical fluctuations due to the thermal initial conditions and is the same  for all values of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the left column there is no coupling between the two quasicondensates and hence no time evolution of  the spatially averaged data (the intense red, green, and blue dots sit on top of each other) although there can be evolution  of unaveraged data due to intrawell dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' without the J term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As we increase the magnitude of J time  evolution leads to whorls with a greater vertical extent because more energy can be extracted from the cosine potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (18) giving larger values of ⟨˜ρ⟩max  z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the left hand column J remains zero for  all time and the only dynamics that can occur is along  the long-axis of each quasicondensate individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  middle and right hand panels, which have J = 3 and  J = 30 Hz, respectively, have the same initial statistical  fluctuations as the left hand one because, as mentioned  above, the initial distribution is set by the pre-quench  thermal fluctuations in the two quasicondensates and is  independent of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, as time evolves the effects of  J described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (43) become apparent because larger  J allows a greater value of ⟨˜ρ⟩max  z  and this stretches the  distribution along the vertical direction in comparison to  a smaller value of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For a whorl to become apparent  ⟨˜ρ⟩max  z  should at least exceed the width of the statistical  fluctuations and becomes better and better defined as J  is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  UNIVERSALITY AND CAUSTICS  We have already discussed the relationship between  nonlinearity and caustics in the preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As  motivated earlier, and expounded in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 17, 26, 31, 33,  and 34, caustics also have implications for the universal  dynamics of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We explore a few of these  effects in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Long time distribution: the circus tent  The quench generates collective excitations that lead  to caustics as shown in Figures 3 and 5 for the two non-  linear models (two mode and SG+) discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  caustics are born at the center of the probability distri-  bution (in either the ˜ρ or the ˜φ variable) at intervals of  the plasma period and move out to the edges over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Figure 6 plots the probability distribution for the SG+  model as a function of ⟨˜φ⟩z at an intermediate time where  four pairs of fold caustics are discernible and shows how  they diminish in strength but are still present as they  move to the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The question then naturally arises  as to what happens at long times ˜t → ∞ when the dis-  tribution comprises of a large number of caustics and  whether it tends to a characteristic shape?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The answer  is yes, and is shown in Figure 9 which is made in the  same way as Figure 6 but this time by calculating the  density of ⟨˜ρ⟩z trajectories and averaging over a time  window extending between ˜t = 800 and ˜t = 980 in order  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  3  2  1  0  1  2  3  (0)z15  to remove rapid fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The probability distribu-  tion takes a shape reminiscent of a ‘circus tent’ or ‘big  top’ and can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The strongest  singularities present are the cusp tips born at the center  of the distribution which leads to this being the highest  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each cusp then splits into two fold arms (which  according to catastrophe theory are lower singularities)  that move outwards, reducing in height as they go, before  accumulating at the edges where there is a sharp drop to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The position of the outer edge is set by the maxi-  mum energy that can be extracted from the quench and  is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  An analytic expression for the circus tent distribution  is given by the integral  PCT(˜ρ) =  1  2πB  � 1  ˜ρ2/B2  U(m, ˜ρ)  K(m)  dm  (44)  where  U(m, ˜ρ) =  1  �  m(1 − m)(m − ˜ρ2/B2)(1 + ˜ρ2/B2 − m)  ,  (45)  K(m) is the complete elliptic integral of the first kind,  and B = 2  �  J /Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This expression is plotted in Figure  9 as the dashed line and is derived in Appendix E un-  der the assumption that at long times we can model the  system by an ensemble of independent pendulua where  each pendulum is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In other words, each pendu-  lum obeys a microcanonical distribution where there is  equal probability for it to be found anywhere on its en-  ergy shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The nature of the J-quench is such that it  leads to an ensemble with an equal probability for any  starting angle (this is different to an equal probability  for each energy due to the dependence of the density of  states on angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As can be seen from Figure 9, PCT(˜ρ)  gives a good fit to the numerical data generated by both  the SG+ and two-mode models considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In Figure 9 we also include the thermal probability  distribution  PT (˜ρ) = 1  Z  � ∞  0  PE(˜ρ) e−E/T D(E) dE  (46)  describing an ensemble of pendula at thermal equilibrium  at temperature T where PE(˜ρ) is the probability distri-  bution at fixed energy E, D(E) is the density of states  and Z is a normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The details of our cal-  culation of PT (˜ρ) are given in Appendix F, where, for  example, PE(˜ρ) is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The temperature  of this distribution is chosen such that the mean energy  of the thermal distribution ⟨E⟩T is equal to the mean  energy of the states excited by the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For a quench  to J = 30 Hz we show in Appendix F that the effective  temperature is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4 nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Clearly, the thermal distribution is very different to the  circus tent distribution: the thermal distribution takes  the form of a smooth gaussian with wings that extend  beyond ⟨˜ρ⟩max  z  because the thermal Boltzmann factor  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The long time probability distribution for the  number difference ˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The data points are from the different  nonlinear models considered in this paper averaged over the  spatial coordinate z and also over a time window ranging from  ˜t = 800 to ˜t = 980 to remove fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The pink dashed  line is the circus tent distribution PCT given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (44) and  derived in Appendix E under the assumption of ergodicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  the circus tent shape is due to the proliferation of caustics  at long times and gives a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The solid  black curve is the thermal distribution PT with a temperature  chosen so that the expectation value of the energy matches  that provided by the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  allows for excitations with any energy (albeit with ex-  ponentially small probability) including those involving  pendula undergoing rotation as well as libration, whereas  the J-quench only excites librational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The proba-  bility distribution for a thermal pendulum is in fact quite  delicate to compute because of the singularity in the den-  sity of states between libration and rotation but the com-  bined result is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' see Appendix F for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Structural stability of caustics  The defining characteristic of the singularities de-  scribed by catastrophe theory is structural stability  against perturbations and this ensures that they occur  generically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The same is not true of isolated singularities  as can be seen by comparing Figures 3 and 4 where it  is shown that point foci do not survive the introduction  of nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In two dimensions cusps are the unique  structurally stable catastrophe and from Figures 3 and 5  we see that cusp-shaped caustics are indeed stable against  random thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, thus far we have  imposed the symmetrical starting condition that the ini-  tial number difference between the two quasicondensates  is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' One may therefore wonder whether the caustics  we see are a consequence of this symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' To check that  this is not the case we show in Figure 10 the dynamics  for the case where the initial background density n1D in  the two quasicondensates differs by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We see that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  PcT  Thermal  two-mode  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='8  many-mode SG+  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  (p)z,t16  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Structural stability of caustics: here we investigate the effect of unbalanced densities on caustics by tracking the  same SG+ model dynamics as those shown in Figure 5 except for an initial density imbalance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='1 in the background of ˜ρ  at each point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We see that the cusp caustics in the plots of ⟨˜ρ⟩z and ⟨˜φ⟩z versus time are distorted but still maintain their  basic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is because the whorl in phase space is left intact despite having a displaced centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Caustics are resilient  against imperfections and perturbations and we expect them to be present under realistic experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  although the caustics in both ⟨˜ρ⟩z and ⟨˜φ⟩z are distorted  they maintain their basic cusp shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Furthermore, the  phase space whorls still occur and this guarantees the  existence of caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Coherence factor and relaxation towards  equilibrium  Cold atom experiments have the ability to measure cor-  relation functions in nonequilibrium many-body states  [74, 116–118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As a simple example let us consider the  coherence factor  C(˜t) =  �  ⟨cos ˜φ⟩z  �  (47)  which depends on the spatial average of the phase dif-  ference field ˜φ(˜z, ˜t) between points along the two qua-  sicondensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The outer brackets indicate an ensemble  average which means averaging over many trajectories  each sampled from the thermal distribution discussed in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the Vienna experiments, where one quasicon-  densate is suddenly split into two, the coherence starts  near unity and decays over time as the two quasiconden-  sates decohere [76, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the opposite case, where two  independent quasicondensates are suddenly coupled, one  expects the converse where the coherence starts at zero  and grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This situation has been previously modelled  by Horváth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' using both the TWA and a truncated  conformal space approach [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  They found that C(˜t)  initially grows and then undergoes damped oscillations  as it settles down towards a finite constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  coherence factor therefore provides a measure of how the  system reaches equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this context we note that  C(˜t) actually corresponds to an ensemble average of the  cosine term in the SG/SG+ Hamiltonian and thus gives  information on the exchange of energy between the dif-  ferent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In other words, since the total energy is a  constant of the motion, if the ‘potential’ part of the en-  ergy settles down to a constant this suggests the ‘kinetic’  parts of the energy are also constant, at least from an  ensemble averaged point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Our aim in this sec-  tion is to see if the dynamics of C(˜t) is connected to the  caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In Figure 11 we plot C(˜t) for two models: the full  SG+ model which is many-mode and nonlinear and a  linearized version which obeys the equations of motion  d˜φ  d˜t = 2Γ˜ρ − Γ  2  ∂2˜ρ  ∂˜z2  d˜ρ  d˜t = 2ϵ∂2 ˜φ  ∂˜z2 − 2J ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (48)  This differs from the linearized two-mode approximation  defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (40) because it describes an elongated  multi-mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' From Figure 11 we see that C(˜t) for  the SG+ model (dark blue curve) does indeed initially  grow, undergo damped oscillations and settle down to a  non-zero value (the fact that C(˜t) ̸= 0 at ˜t = 0 is due  to random fluctuations in the initial conditions: as we  include more trajectories we find that the initial value  gets smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Meanwhile, C(˜t) for the linear model (red  dashed curve) executes undamped oscillations and hence  does not settle down to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Both models agree  during the first oscillation but strongly differ after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  It is clear that nonlinearity is important for reaching  equilibrium at least as far as global quantities such as  C(˜t) are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We can understand this by inter-  preting the SG+ model as describing a chain of coupled  pendula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The nonlinearity of each pendulum means that  its period depends on the amplitude of its motion and  hence an ensemble of pendula whose motion is initiated  together by the quench, but all with different degrees of  excitation, will dephase from one another over time so  that collective oscillations are damped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By contrast,  linear oscillators have a period independent of their am-  plitudes of motion and hence remain in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Apart from the ensemble averages shown by the darker  curves in Figure 11, we have also included the individ-  ual trajectories for ⟨cos ˜φ⟩z as fainter curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The linear  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0  25  50  75  100 125 150 175 2003  2  1  0  1  2  3  0  25  50  75  100 125 150 175 200  2t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='0  3  2  1  0  1  2  3  (0)z17  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The two dark lines give the time evolution of the  coherence factor C(˜t) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (47) for a linear model  (dashed-dotted red) and the SG+ model (solid blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Both  models are multi-mode (many longitudinal modes along ˜z)  but the SG+ model is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Also included as faint lines  are the raw trajectories ⟨cos ˜φ⟩z from which C(˜t) is composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As everywhere in this paper, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='⟩z indicates a spatial aver-  age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This figure highlights that recurrences present in the  linear case are suppressed by nonlinearity in the SG+ sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The ensemble average over trajectories with different  periods causes C(˜t) to relax towards an equilibrium value in  the case of the SG+ model in line with previous experimental  observations [76, 77] and theory [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  model displays harmonic motion and hence perfect re-  vivals whereas the trajectories in the nonlinear model  give rise to half-cusp caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  These caustics overlap  in time such that averaging over them causes the coher-  ence to strongly relax after a single period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' It is not so  much that the caustics cause the relaxation, but rather  that both have a common origin in the nonlinearity of  the model and hence are generic features of dynamics in  complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  SUMMARY AND CONCLUSIONS  The sine-Gordon (SG) model is a nonlinear integrable  field theory that can be used to describe a wide range  of systems from high energy physics to condensed matter  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' A series of landmark experiments using two cou-  pled 1D atomic quasicondensates [63, 71–77] have real-  ized the SG model in a controllable quantum many body  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The key parameters can be varied in time  allowing the implementation of sudden quenches that ex-  cite many modes leading to nonequilibrium dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This is the setting we adopt for the current paper where  we use experimentally realistic parameters and compute  the dynamics of the number and phase difference fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  However, in contrast to the usual experimental protocol  where the tunnel coupling J is suddenly switched off,  we consider quenches where it is suddenly switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  While the former case is adapted to studying dephasing,  decay and thermalization between the two subsystems,  the many body dynamics is governed by the Tomonaga-  Luttinger Hamiltonian describing independent 1D quasi-  condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' If instead J is suddenly switched on then  the dynamics is that of the full SG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Our calculations employ a thermal version of the  semiclassical truncated Wigner approximation (TWA)  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' More specifically, we propagate a large num-  ber of classical field configurations over time with initial  conditions sampled from a distribution at thermal equi-  librium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The time evolved configurations (trajectories)  can be summed to obtain the probability distributions  for the observables and we find that these are dominated  by singular caustic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The natural mathematical  description of caustics is catastrophe theory that predicts  a hierarchy of structurally stable singularities with char-  acteristic shapes that depend on dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In two di-  mensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' number or phase difference versus time)  the structurally stable catastrophes are fold lines that  meet at cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is exactly what we find in both the  number and phase differences following a J-quench, see  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The probability distributions develop trains  of caustics that are born periodically as cusp points (lo-  cated at the center of the distribution if there is no tilt) at  each plasma period and evolve into pairs of fold lines that  gradually move out to the wings where they accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Fold catastrophes manifest as strong non-gaussian fluc-  tuations in the form of inverse square root divergences in  the intensity (probability density), as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  A special case is provided by the dynamics of a two  mode system as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Here the equa-  tions of motion are the Josephson equations given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The only fluctuations we include in this ex-  ample are the quantum fluctuations in the initial rela-  tive phase between the two condensates as mandated by  the uncertainty principle applied to systems in relative  number eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The two-mode case is relevant to  small systems where the higher modes are well above  the temperature scale and so any spatial fluctuations are  suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By contrast, the many-mode case shown in  all the other figures includes both quantum fluctuations  and thermal fluctuations in the longitudinal modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  thermal occupation of phonon modes in the 1D quasicon-  densates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Despite the presence of the many longitudinal  modes (typically 50 in our calculations, as set by the pa-  rameter NL) which give rise to highly random looking  phase and density profiles as seen in Figure 2, we find  that number and phase caustics survive for experimen-  tally realistic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Furthermore, the qualitative  features of the caustics are stable against variations in  quench strength and density imbalance, as seen in Fig-  ures 8 and 10, respectively, and also against the details  of the model (in this paper we use the SG+ model which  augments the SG model by including longitudinal den-  sity gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' All of these different examples confirm  the structural stability of caustics which is the reason why  they occur universally without the need for fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='50  (cos((z))z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='75  Linearmany-mode  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='00  Non-linear many-mode SG+  0  50 100 150 200 250 300 350 400  t18  The proliferation of caustics over time combined with  their migration to the edge of the probability distribution  has important consequences for the long time probability  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' It takes on the shape of a circus tent featur-  ing a strong central peak due to the cusp tips which are  the most singular part of a caustic, flatter intermediate  regions, and rapidly decaying edges where the caustics  pile up, see Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This shape is quite distinct from a  gaussian thermal distribution and can be derived assum-  ing an ergodic hypothesis in which individual pendula  have equal probability to be anywhere on their energy  shell (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The approach to this equilibrium  distribution can be tracked over time using the coherence  factor (Figure 11) which is a spatial and ensemble average  over the phase field and corresponds to the cosine term in  the Hamiltonian if the latter is ensemble averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  attainment of equilibrium relies on the nonlinearity of the  system to dephase itself when ensemble averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  caustics also rely on the nonlinearity without which they  would reduce to nongeneric perfect revivals (point foci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In this sense caustics are mutually exclusive to recur-  rences, at least in the statistical sense in which caustics  appear in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Caustics in the SG model could be observed experi-  mentally by measuring the probability density for either  the phase difference or the number difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For ex-  ample, the phase difference can be obtained by releasing  the two quasicondensates from their double well potential  and letting them overlap [80–82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This process must be  repeated many times and for as near identical initial con-  ditions and time evolution as possible in order to build  up a probability distribution, although due to the struc-  tural stability of caustics they will not be particularly  sensitive to differences in the experimental setup from  run to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' If the probability distribution is obtained for  a single time then we expect to see something like that  shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In order to observe the time evolu-  tion of a caustic, one must then repeat the whole process  for a range of different evolution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This is laborious  but technically possible, and since the first cusp caustic  appears at half the plasma period the experiment does  not need to run for long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The singular nature of caustics means that they dom-  inate wave fields and are well known in hydrodynam-  ics and optics through phenomena such as tsunamis and  gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The results of this paper show that  they also occur in the nonequilibrium dynamics of 1D su-  perfluids where a quench plays an analogous role to an  underwater earthquake by generating strong excitations  beyond the linear regime that are focused in this case by  the cosine term in the SG Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The universal  properties of catastrophes imply caustics likely also occur  in the post-quench dynamics of other condensed matter  systems too: systems with more degrees of freedom will  display higher catastrophes beyond folds and cusps such  as hyperbolic and elliptic umbilics [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' However, a spe-  cial feature of the SG model is that it is integrable and  so one may ask if that property plays a crucial role in  the existence of caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this context, we note that  in classical mechanics caustics are closely associated with  the existence of tori in phase space upon which trajec-  tories live [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Tori are broken up by chaos, and thus  caustics are not expected to survive for long in systems  which are deep in the chaotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Despite this, the  Kolmogorov-Arnold-Moser (KAM) theorem shows that  some tori survive in moderately chaotic systems [119],  which suggests caustics may also survive in cases where  the classical phase-space is mixed, which is the typical  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Indeed, they survive in the three site Bose-Hubbard  model [34] which is known to be chaotic [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The im-  portant problem of extending the KAM theorem to quan-  tum mechanics [121] is thus intertwined with the analysis  of caustics in quantum systems and provides an interest-  ing direction for extending the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank Ryan Plestid for contributions on ther-  mal field sampling in the early stages of this project,  Josh Hainge for suggesting the term ‘circus tent’, and  Igor Mazets for correspondence and advice about ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This work was supported by the Mitacs  Globalink research internship, by the Natural Sciences  and Engineering Research Council of Canada (NSERC),  and Research at the Perimeter Institute is supported  in part by the Government of Canada, through the  Department of Innovation, Science and Economic De-  velopment Canada, and by the Province of Ontario,  through the Ministry of Colleges and Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  would like to acknowledge support from the project  6004-1 of the Indo-French Centre for the Promotion of  Advanced Research (IFCPAR), Ramanujan Fellowship  (SB/S2/RJN-114/2016), SERB Early Career Research  Award (ECR/2018/002085) and SERB Matrics Grant  (MTR/2019/001101) from the Science and Engineering  Research Board (SERB), Department of Science and  Technology (DST), Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' acknowl-  edges support from the Infosys Foundation International  Exchange Program at ICTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='K acknowledges support  of the Department of Atomic Energy, Government of In-  dia, under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 19P1112R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Appendix A: Derivation of the sine-Gordon  Hamiltonian  In this appendix we derive the Hamiltonian HSG as  the effective low energy description of two cigar shaped  tunnel-coupled quasicondensates [50, 74] within a clas-  sical field description (Gross-Pitaevskii theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Along  the way we also obtain a slightly enhanced Hamiltonian  HSG+ that includes contributions from the gradient of  density fluctuations that are not included in the sine-  Gordon (SG) Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' These contributions are not  very important for our parameters but play an impor-  19  tant conceptual role by introducing an energetic price  for a rapidly varying density and hence effectively cut off  these fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Assuming tight radial trapping such that each quasi-  condensate is in its radial ground state, meaning that  only longitudinal excitations are taken into account, the  second quantized Hamiltonian for the total system be  written  H =  � ∞  −∞  dz  � �  j=1,2  �  − ℏ2  2m  ˆψ†  j(z)∂2 ˆψj(z)  ∂z2  +  U(z) ˆψ†  j(z) ˆψj(z) + g1D  2  ˆψ†  j(z) ˆψ†  j(z) ˆψj(z) ˆψj(z)  �  − ℏJ  �  ˆψ†  1(z) ˆψ2(z) + ˆψ†  2(z) ˆψ1(z)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A1)  The quantum field operator ˆψj(z) annihilates a particle  at the point z in the jth well, where z is the coordinate  along the longitudinal direction (long axis of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  m is the mass of the particles, U(z) is a possible external  potential (in this paper it will be set to zero), g1D con-  trols the interparticle interaction strength, and J is the  tunneling frequency between the two wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the classi-  cal field approximation we replace the field operators by  complex functions  ˆψj(z) → ψj(z) = eiφj(z)�  n1D + ρj(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A2)  Note that φj and ρj are the phase and density variables  for each well rather than their antisymmetric versions  which are used extensively in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Let us start by manipulating the kinetic energy term  −  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  � ∞  −∞  dz ℏ2  2m  ˆψ†  j(z)∂2 ˆψj(z)  ∂z2  (A3)  =  � ∞  −∞  dz  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  ℏ2  2m  � � ∂  ∂z e−iφj(z)�  n1D + ρj(z)  �  ×  � ∂  ∂z e+iφj(z)�  n1D + ρj(z)  � �  =  � ∞  −∞  dz  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  ℏ2  2m  �  − i∂φj  ∂z  ˆψ†  j +  e−iφj ∂ρj  ∂z  2√n1D + ρj  �  ×  �  i∂φj  ∂z  ˆψj +  eiφj ∂ρj  ∂z  2√n1D + ρj  �  =  � ∞  −∞  dz  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  ℏ2  2m  �  ˆψ†  j ˆψj  �∂φj  ∂z  �2  +  ( ∂ρj  ∂z )2  4(n1D + ρj)  + i  ∂ρj  ∂z  ∂φj  ∂z  2√n1D + ρj  [ ˆψje−iφj − ˆψ†  jeiφj]  �  =  � ∞  −∞  dz  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  ℏ2  2m  �  ˆψ†  j ˆψj  �∂φj  ∂z  �2  +  ( ∂ρj  ∂z )2  4(n1D + ρj)  �  ≈  � ∞  −∞  dz ℏ2  2m  �  n1D  2  ��∂φs  ∂z  �2  +  �∂φa  ∂z  �2�  +  1  2n1D  ��∂ρs  ∂z  �2  +  �∂ρa  ∂z  �2� �  (A4)  where  φa = φ1 − φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  φs = φ1 + φ2  (A5)  ρa = ρ1 − ρ2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  ρs = ρ1 + ρ2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A6)  and we assume that n1D ≫ ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Next we consider the  interactions  �  j=1,2  g1D  2 ψ†  jψ†  jψjψj =  �  j=1,2  g1D  2 [n1D + ρj(z)]2  =  �  j=1,2  �  g1Dn2  1D  2  + g1Dρ2  j  2  + g1Dn1Dρj  �  =g1Dn2  1D + g1D(ρ2  s + ρ2  a) + 2g1Dn1Dρs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A7)  Finally, we consider the tunneling term  −ℏJ  �  ψ†  1(z)ψ2(z) + ψ†  2(z)ψ1(z)  �  (A8)  = − ℏJ  �  (e−i(φ1−φ2) + e−i(φ2−φ1))√n1D + ρ1  √n1D + ρ2  �  = − 2ℏJ cos(φa)√n1D + ρ1  √n1D + ρ2  = − 2ℏJ cos(φa)  �  n2  1D + 2n1Dρs + ρ2s − ρ2a  ≈ − 2ℏJ cos(φa)(n1D + ρs) ≈ −2ℏn1DJ cos(φa) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A9)  20  At very low temperatures the symmetric and antisym-  metric components decouple and hence can be treated  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The lower energy terms are the antisymmet-  ric ones and we obtain the following Hamiltonian  HSG+ =  � ∞  −∞  dz  �  g1D ρa(z)2 + ℏ2n1D  4m  �∂φa  ∂z  �2  +  ℏ2  4mn1D  �∂ρa  ∂z  �2 �  −  � ∞  −∞  dz 2ℏJn1D cos [φa(z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A10)  When the higher wavelength ρ modes are suppressed this  reduces to the sine-Gordon model  HSG =  � ∞  −∞  dz  �  g1D ρa(z)2 + ℏ2n1D  4m  �∂φa  ∂z  �2  − 2ℏJ n1D cos [φa(z)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (A11)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (A11) is the finally obtained SG Hamiltonian HSG  which is the low energy description of two cigar shaped  tunnel-coupled quasicondensates [50, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Appendix B: Derivation of the Tomonaga-Luttinger  (TL) Hamiltonian in Fourier space  In this appendix we derive the Fourier space version  of the Tomonaga-Luttinger (TL) Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Starting  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (25), and applying the discrete Fourier decom-  positions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (26) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='HTL+(ra) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='g1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϱkei 2πkr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� ×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϱlei 2πlr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ℏ2n1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4ma2(NL + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϕkei 2πkr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� × ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϕlei 2πlr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ℏ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4mn1Da2(NL + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϱkei 2πkr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='� × ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ϱlei 2πlr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='= a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='r=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�g1Dϱkϱlei 2π(k+l)r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='− a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='r=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ℏ2n1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4ma2(NL + 1) ×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='klϕkϕlei 2π(k+l)r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='− a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='r=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='k=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='l=−NL/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='ℏ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4mn1Da2(NL + 1) ×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='klϱkϱlei 2π(k+l)r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='NL+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='(B1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='where we have split the z coordinate into NL + 1 grid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='points separated by distance a so that z = r a where r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='in an integer lying in the range specified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Using the fact that NLa = L, and applying the identity  �NL/2  r=−NL/2 ei 2π(k+l)r  NL+1  = (NL + 1)δk,−l we obtain  HTL+ ≈a  �  k  �  l  g1Dϱkϱlδk,−l  − a  �  k  �  l  �ℏ2n1Dπ2  mL2  �  klϕkϕlδk,−l  − a  �  k  �  l  �  ℏ2π2  mn1DL2  �  klϱkϱlδk,−l  (B2)  where in the second term we have also replaced a2(NL +  1)2 by L2 which holds when NL ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The limits of the  summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (B2) has been omitted for the sake of  21  brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We therefore find  HTL+ ≈  �  k  �  ag1Dϱkϱ−k+aℏ2n1Dπ2k2  mL2  ϕkϕ−k  + aℏ2π2k2  mn1DL2 ϱkϱ−k  �  =  �  k  �  ag1D|ϱk|2+aℏ2n1Dπ2k2  mL2  |ϕk|2  + aℏ2π2k2  mn1DL2 |ϱk|2  �  (B3)  where we used the property of real fields that  ϕ−k = ϕ⋆  k,  and  ϱ−k = ϱ⋆  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (B4)  Hence the Hamiltonian takes the form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (28)  of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Appendix C: Bench marking of the numerical  method  The results given in this paper rely on numerically  evolving the equations of motion over time for various  models [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' for the full SG+ model the equations of mo-  tion are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (22)], which we accomplish using  the Julia package DifferentialEquations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='jl [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This im-  plements a Runge-Kutta solver with a user-defined time  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As a measure of the accuracy of our numerical  method we use the deviation of the Hamiltonian from  its initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Since the Hamiltonian should be a con-  stant of motion this gives an indication of the size of the  numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In Figures 12 and 13 we plot the relative error in the  SG+ Hamiltonian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (18) for different time  and spatial resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' More precisely, Figure 12 shows  the effect of varying the time step d˜t, whereas Figure 13  shows the effect of varying the number of grid points NL  which sets the spatial step d˜z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In both cases we have  evolved the system for a total elapsed time of ˜t = 1000  which corresponds to the longest times we use in this  paper (for the calculation of the long-term distribution  shown in Figure 9), and also taken an ensemble average  over 100 different trajectories similar to those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Furthermore, we also performed a moving time average  of 30-time steps around ˜t = 1000 to average out the effect  of fast oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As expected, the relative error decreases as d˜t and d˜z  decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For all the calculations in this paper we chose  d˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2 and NL = 50 because this keeps the relative  error below 10 % and does not significantly slow down  the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The relative error in the SG+ Hamiltonian is  plotted here as a function of the time step d˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The definition  of the SG+ Hamiltonian is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 18 and should be  a constant of the motion were it not for numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The moving time average of relative error is evaluated after  propagating the equations of motion for a total elapsed time  of ˜t = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' All parameter values are the same as in Figure  5 including NL = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The relative error in the SG+ Hamiltonian is  plotted here as a function of the number of lattice points NL  on the numerical spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Like in Figure 12, the Hamil-  tonian is evaluated after evolving the equations of motion for  a total elapsed time of ˜t = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The moving time average  of the relative error fluctuates (at around 10 %) but does de-  crease as d˜z decreases (or NL increases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' All other parameter  values are the same as in Figure 5 with d˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  Appendix D: Caustic curve  In this appendix we use the exact solution for the mo-  tion of a pendulum to calculate the caustic curve plotted  as the solid black line in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The caustic is in fact  the envelope of a whole family of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' To begin,  we take the equations of motion for the SG model given in  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='5  (0) + 9SH  (1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='1  10-1  100  101  dt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='04  20  40  60  80  100  NL22  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (22) and drop the second order derivative term pro-  portional to ϵ which couples the different pendula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Next,  we make the change of variables  ˜t = At,  ˜ρ = Bp,  ˜φ = 2y  (D1)  where  A = 1  2  1  √J Γ  ,  B = 2  �  J  Γ  (D2)  so the equations of motion simplify to  dy  dt = p  (D3)  dp  dt = −1  2 sin 2y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (D4)  These equations are Hamilton’s equations obtained from  a standard pendulum hamiltonian of the form  H(y, p) = p2  2 + 1  2 sin2 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (D5)  The equations of motion given in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D3) and (D4)  have exact solutions in terms of the Jacobi elliptic func-  tions sn[u|m] and cn[u|m] [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' For the case relevant  to us where the pendulum starts at angle y0, with zero  initial angular momentum, they are  y(t, y0) = arcsin{sin y0 sn[t + K(sin y0)| sin y0]}(D6)  p(t, y0) = sin(y0) cn[t + K(sin y0)| sin y0]  (D7)  where K(m) =  � π/2  0  dθ/  �  1 − m2 sin2 θ is the complete  elliptic integral of the first kind [122] (we caution the  reader that some computer packages such as Mathematica  use the syntax K(m2) for this integral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Caustics occur when trajectories are focused, in other  words they are the places where the trajectory does not  change (to first order) when the initial conditions are  varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thus, caustics in the momentum variable p oc-  cur when dp/dy0 = 0 since the initial condition here is  specified by y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' By differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D7) an implicit  expression for the position of the caustics can be found  [123]  sn(u|m)dn(u|m)  �E(am(−t|m) |m)  cos(y0)  + t cos(y0)  �  − cos(y0)cn(u|m) = 0  (D8)  where u = t+K(sin y0), m = sin y0, E(u|m) is an elliptic  integral of the second kind, dn(u|m) is another Jacobi  elliptic function, and am(u|m) = arcsin[sin(φ)/m] is the  Jacobi amplitude [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Finding the roots y0 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D8)  numerically at each value of the time gives pairs of values  (y0, t) that can then be put back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D7) to yield  the black curve for the caustic shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The  match to the numerics is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Appendix E: Derivation of ergodic (“circus tent”)  probability distribution at long times  In this appendix we outline the derivation of an an-  alytic approximation to the probability distribution for  the number difference at long times, as shown in Figure  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This derivation is based upon a calculation given in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 124 and assumes that the average behaviour of a con-  tinuous chain of coupled pendula (the mechanical system  that underlies the sine-Gordon model) can be described  by a suitably ‘ergodized’ single pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  To keep the calculation general we use the pendulum  Hamiltonian in standard form as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' With  this hamiltonian we define a microcanonical probability  density in phase space:  dm(y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' y0) =  δ[H(y, p) − H(y0, p)]  � �  dy dp δ[H(y, p) − H(y0, p)]  (E1)  where y0 is the initial angle of the pendulum which fixes  its total energy to be E = (1/2) sin2 y0 if the the initial  angular momentum is zero (this is the appropriate ini-  tial condition for the tunneling quench considered in this  paper where the initial number difference is taken to be  zero), and the denominator ensures that dm is normalized  to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' A microcanonical distribution has equal prob-  ability to be anywhere on its energy shell (in this case  a closed curve in y, p phase space) and thus by adopt-  ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (E1) we are making an ergodic hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This  does not hold for a single pendulum starting at position  y0 since it will spend the most time at its turning points  y = ±y0, but when averaged over y0 and y (see below)  it gives a very good approximation at long times, as can  be seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The normalization integral can be evaluated exactly  by re-expressing the delta function using the relation  δ[g(x)] = �  i δ(x − xi)/|g′(xi)|, where xi are the roots  of g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the present case this gives  δ[(p2 + sin2 y − sin2 y0)/2] =δ[p − p1]  |p1|  + δ[p − p2]  |p2|  =2δ[p − p1]  |p1|  (E2)  where |p1| = |p2| =  �  sin2 y0 − sin2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In obtaining this  expression we have used the fact that for values of y  within the range accessed by the pendulum, there are  two values of p where the integral crosses the energy shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The integral over p is now trivial due to the delta func-  tion and the integral over y can be performed by putting  sin y = sin y0 sin ζ so that  23  2  � y0  −y0  dy  |p(y, y0)| = 2  � y0  −y0  dy  �  sin2 y0 − sin2 y  = 2  � π/2  −π/2  dζ  �  1 − sin2 y0 sin2 ζ  = 4  � π/2  0  dζ  �  1 − sin2 y0 sin2 ζ  = 4K(sin y0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (E3)  Therefore, the normalized microcanonical probability  density can be written as  dm(y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' y0) =  1  4K(sin y0)δ[(p2 + sin2 y − sin2 y0)/2]  =  1  2K(sin y0)δ(p2 + sin2 y − sin2 y0)  (E4)  where we have used the property of delta functions that  δ(αx) = (1/α)δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The initial condition for our dynamics is such that the  number difference is well defined but the phase differ-  ence is completely undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We must therefore average  the microcanonical probability density over all y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This  gives the phase space probability density relevant to J-  quenches as being  W(y, p) = 1  π  � π/2  −π/2  dy0 dm(y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' y0)  (E5)  where we employ the notation W to indicate that this is  a classical version of the Wigner function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The properties  of the delta function can once more be used to write  δ(p2 + sin2 y − sin2 y0) =  �  i  δ(y0 − y0i)θ(cos y − |p|)  2  �  p2 + sin2 y  �  cos2 y − p2  (E6)  where θ(x) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The integral  over y0 can now be evaluated exactly to give  W(y, p) = 2  4π  θ(cos y − |p|)  K(  �  p2 + sin2 y)  �  p2 + sin2 y  �  cos2 y − p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (E7)  The final step is to integrate out the y coordinate to  obtain the probability distribution PCT(p) for p alone  PCT(p) =  � π/2  −π/2  dy W(y, p) ,  (E8)  where “CT” stands for circus tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Although this integral  cannot be done analytically, it can be put in a form which  is convenient to evaluate numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Denoting m =  sin y0 =  �  p2 + sin2 y, one finds that  PCT(˜ρ) =  1  2πB  � 1  ˜ρ2/B2  dm  K(m)  �  m(1 − m)(m − ˜ρ2/B2)(1 + ˜ρ2/B2 − m)  (E9)  where we have also converted back from angular momen-  tum p to number difference ˜ρ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' This equa-  tion is given in the main text as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (44) and is plotted  in Figure 9 where it is compared against the long-time  spatially and temporally averaged numerical data for the  various nonlinear models considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  As  can be seen in Figure 9, PCT is characterized by a di-  verging (yet normalizable) peak at the center and then  relatively flat wings until it drops sharply to zero at the  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 124 it is shown that PCT(˜ρ) diverges log-  arithmically at the origin ˜ρ = 0 and also tends suddenly  to zero with logarithmic singularities at ˜ρ = ±B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Both  these non-thermal features can be attributed to the pres-  ence of caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Appendix F: Pendulum at thermal equilibrium  In Figure 9 the long time probability distribution for  the number difference is compared against the ergodic  prediction derived in Appendix E, and also against the  thermal equilibrium prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this Appendix we ex-  plain how to calculate the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In order to make  the calculation tractable we make the assumption that  the SG+ model can be approximated by a thermal en-  semble of independent pendula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We also adopt the same  notation as Appendix E and hence work with a pendulum  Hamiltonian in the standard form H = (1/2)(p2+sin2 y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  This is related to the two mode Hamiltonian H2M =  Γ˜ρ2 − 2J cos φ by H = H2M/8J + 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We proceed in two steps: we first calculate the prob-  ability distribution PE(p) for the momentum variable p  (that here plays the role of the number difference) for a  fixed energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Secondly, we assume our system is at  thermal equilibrium with a bath at temperature T such  that the relative probability of any energy is given by the  Boltzmann factor exp[−E/T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thus the thermal proba-  bility distribution is  PT (p) = 1  Z  � ∞  0  PE(p) e−E/T D(E) dE  (F1)  where Z is a normalizing factor (found numerically) and  24  D(E) is the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The probability distribution PE(p) at fixed E is pro-  portional to 1/ ˙p as this determines how long the pendu-  lum spends at each value of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' According to Hamilton’s  equation ˙p = −∂H/∂x = −(1/2) sin 2y, and using the  fact that sin y =  �  2E − p2, we find that this probability  distribution for a fixed value of E is  PE(p) =  N  (1/2) sin(2 arcsin  �  2E − p2)  ,  (F2)  where N is a normalization factor given by the period  of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Two cases must be distinguished: for  E < 1/2 the energy is less than the separatrix and the  pendulum undergoes vibrational motion (also known as  librational motion in some literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Conversely, when  E > 1/2 the energy is above the separatrix and the pen-  dulum undergoes rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For motion below the separatrix we have |p| < pmax =  √  2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We must therefore supplement the expression for  PE(p) with the condition that it is zero if |p| > pmax and  this ensures that PE(p) is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' N is given in this case  by  N =  1  2 K(  √  2E)  (F3)  where, as in Appendix E, K is the complete elliptic inte-  gral of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  For motion above the separatrix we have  √  2E − 1 <  |p| <  √  2E and PE(p) is zero outside this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' N is  now given by  N =  √  2E  4 K(1/  √  2E)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (F4)  To obtain the total thermal probability distribution  PT (p) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (F1) we need the density of states  D(E) ≡ dn/dE, where n is the number of states be-  low energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' According to the Bohr-Sommerfeld rule  n = S(E)/(2πℏ), where the action S(E) =  �  p dy is the  area in phase space enclosed by the energy contour E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  However, assuming that our Hamiltonian H is in units  ℏω then the 2πℏ factor is absorbed into the definitions of  p and y and we have D(E) = (d/dE)  �  p dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Below the  separatrix we have  �  p(y)dy = 4  � arcsin  √  2E  0  �  2E − sin2 y dy  (F5)  and putting 2E = sin2 y0 we find  D<(E) = d  dE  �  p(y)dy  = 4  � arcsin  √  2E  0  dy  �  sin2 y0 − sin2 y  = 4K(  √  2E)  (F6)  where the integral is performed in a similar fashion to  the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (E3) and the subscript “<” indicates that  this is the expression valid below the separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Above  the separatrix we find that the area enclosed in phase  space between two oppositely rotating states of the same  energy is  �  p(y)dy = 2  � π/2  −π/2  �  2E − sin2 y dy  (F7)  and thus  D>(E) = d  dE  �  p(y)dy  = 2  � π/2  −π/2  dy  �  2E − sin2 y  =  4  √  2E  K  �  1  √  2E  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (F8)  Due to the fact that above the separatrix 2E > sin2 y we  no longer need to make the substitutions 2E = sin2 y0  and sin y = sin y0 sin ζ, and the integral is straightfor-  ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The subscript “>” indicates that this expression  holds above the separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  We now have all the necessary ingredients to perform  the integral for PT (p) which we do numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  The  two contributions, one from below the separatrix and one  from above, are added together to get the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Inter-  estingly, both density of states factors, Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (F6) and  (F8), diverge at the separatrix such that the two con-  tributions individually display singular features but re-  markably these cancel out when the two parts are added  and result in the smooth gaussian curve plotted in Figure  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  In order to compare the thermal distribution against  the quenched (followed by integrable SG evolution) dis-  tribution derived in Appendix E we need to choose a  temperature T for the thermal distribution PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We do  this by matching the expectation value of the energy ⟨E⟩  for both distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In the quenched case the initial  state corresponds to an ensemble of pendula with dif-  ferent starting angles y0 and zero kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Each  starting angle in the range −π/2 < y0 ≤ π/2 is equally  probable in our J-quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Therefore  ⟨E⟩quench = 1  π  � π/2  −π/2  1  2 sin2 y0 dy0 = 1  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  (F9)  To calculate ⟨E⟩ in the thermal case we compute  ⟨E⟩T = 1  ζ  � ∞  0  E e−E/T D(E) dE  (F10)  numerically for a large number of different values of  T, performing the integrals below and above the sep-  aratrix separately and adding the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Here ζ =  � ∞  0  e−E/T D(E) dE gives the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We  then fit a curve to the results and find the value of T  25  that best matches the result given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' (F9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' We find  that T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='184 gives the best match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Putting back the  units this result is  kBT  8J ℏc/ξh  =  kBT  16JℏK/π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='184  (F11)  where c is the speed of sound and K is the Luttinger  parameter and J is the tunnel coupling rate between the  two wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' In this paper we take K = 25 and J = 30 Hz  (see Table I) giving a temperature in SI units of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='4 nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Nye, Natural Focusing and Fine Structure of Light:  Caustics and Wave Dislocations (Institute of Physics  Publishing: Bristol and Philadelphia, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [2] Lord Kelvin, Deep water ship-waves, Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 9, 733  (1905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Titov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Rabinovich, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Mofjeld, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  Thomson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' González, The global reach of the 26  December 2004 Sumatra tsunami, Science 309, 2045  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Berry, Focused tsunami waves, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Lon-  don A 463, 3055 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Degueldre, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Metzger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Geisel, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Fleis-  chmann, Random focusing of tsunami waves, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  12, 259 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Berry, Minimal analytical model for undular tidal  bore profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' quantum and hawking effect analogies, New  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 20, 053066 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Heller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Fleischmann, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Kramer, Branched  flow, Physics Today 74, 44 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Peebles, The Large-Scale Structure of the Uni-  verse (Princeton University Press, Princeton, NJ, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Arnold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Shandarin, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Zeldovich, The  large scale structure of the universe I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' General proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' One- and two-dimensional models, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' As-  trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 20, 111 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Gurbatov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Saichev, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Shandarin,  Large-scale structure of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' The Zeldovich ap-  proximation and the adhesion model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='-Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 55,  223 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Feldbrugge, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' van de Weygaert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Hidding, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Feldbrugge, Caustic skeleton & cosmic web, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Cos-  mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 5, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Berry, Regular and irregular semiclassical wave-  functions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' 10, 2083 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Berry, Singularities in waves and rays, in Physics  of Defects (Les Houches Session XXXV), edited by  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Balian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Kléman, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Poirier (North-Holland  Publishing, Amsterdam, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Thom, Structural Stability and Morphogenesis (Ben-  jamin, Reading, MA, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content='  [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Arnol’d, Critical points of smooth functions and  their normal forms, Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfChxb/content/2301.08410v1.pdf'}
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+Subgap states and quantum phase transitions in one-dimensional
+superconductor-ferromagnetic insulator heterostructures
+Javier Feijoo,1, 2 An´ıbal Iucci,1, 2 and Alejandro M. Lobos3, 4
+1Instituto de F´ısica La Plata - CONICET, Diag 113 y 64 (1900) La Plata, Argentina
+2Departamento de F´ısica, Universidad Nacional de La Plata, cc 67, 1900 La Plata, Argentina.
+3Instituto Interdisciplinario de Ciencias B´asicas (CONICET-UNCuyo)
+4Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, 5500 Mendoza, Argentina
+We
+theoretically
+study
+the
+spectral
+properties
+of
+a
+one
+dimensional
+semiconductor-
+superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recents
+experiments where such devices have been fabricated using epitaxial growing techniques. We model
+the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the si-
+multaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially
+inhomogeneous) Zeeman exchange field. The coexistence of these competing interactions generates
+a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum. By
+exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes
+equations into even and odd ABS with respect to the spatial inversion symmetry x → −x. We
+find the ABS spectrum of the device as a function of the different parameters of the model: the
+length L of the coexisting SU-FMI region, the induced Zeeman exchange field h0, and the induced
+superconducting coherence length ξ. In particular we analyze the evolution of the subgap spectrum
+as a function of the length L. Interestingly, we have found that depending on the ratio h0/∆, the
+emerging ABS can eventually cross below the Fermi energy at certain critical values Lc, and induce
+spin-and fermion parity-changing quantum phase transitions. We argue that this type of device
+constitute a promising highly-tunable platform to engineer subgap ABS.
+I.
+INTRODUCTION
+The interplay of superconductivity and magnetism at
+the microscopic scale has attracted a great deal of at-
+tention in recent years [1–4].
+For instance, the Yu-
+Shiba-Rusinov (YSR) states [5–7] arising from the ex-
+change interaction of an atomic magnetic moment in con-
+tact with a superconductor, have been proposed as fun-
+damental building blocks to engineer quantum devices
+with topologically non-trivial ground states. In partic-
+ular, the so-called “Shiba chains” (i.e., one-dimensional
+arrays of magnetic atoms deposited on top of a clean
+superconductor) are systems predicted to support Ma-
+jorana zero-modes at the ends of the chain [8–10], and
+could be used in topologically-protected quantum com-
+putation schemes. Low-temperature scanning-tunneling
+microscopy (STM) experiments have confirmed the pres-
+ence of intruiguing zero-energy end-modes [11–17].
+Other systems where the competition of superconduc-
+tivity and magnetism at the nanoscale generates ex-
+otic subgap states are superconductor (SU)- ferromag-
+net (FM) heterostructures, such as SU-FM-SU Josephson
+junctions and SU-FM proximity devices [18, 19]. Subgap
+states generated in these structures are usually referred
+to as Andreev bound states (ABS). More recently, a novel
+class of hybrid device, i.e., semiconductor (SE) nanowire
+systems combined with superconductors and ferromag-
+netic insulator (FMI) materials have been fabricated us-
+ing molecular-beam epitaxy techniques [20, 21]. These
+SE-SU-FMI hybrid structures allow to build nanostruc-
+tures with specific tailored properties which are impossi-
+ble to obtain with the isolated individual components.
+Despite the evident differences between the abovemen-
+x
+z
+SC Bulk
+FMI
+Semiconductor
+L
+x
+z
+L
+h0
+Magnetic profile
+FIG. 1. Schematic representation of the SC-FMI heterostruc-
+ture.
+tioned physical systems, from the theoretical perspective
+they can be described within the same unified theoretical
+model combining superconductivity and local exchange
+fields at the microscopic scale.
+The emerging subgap
+states (which can be referred to as either YSR states or
+ABS, depending on the context) appear symmetrically
+around the Fermi level EF , and localize spatially around
+the impurity or the FM region.
+Their energy-position
+within the gap depend on the value of the exchange
+field and on other experimental parameters.
+Interest-
+ingly, whenever one of these states crosses EF , a spin-
+and parity-changing quantum phase transition, usually
+arXiv:2301.03967v1  [cond-mat.supr-con]  10 Jan 2023
+
+2
+known as the “0 − π” phase transition, occurs [1, 22].
+In the case of atomic “Shiba impurities” or ultra-short
+SU-FM-SU junctions (i.e., junctions in which the length
+L of the FM region is much smaller than λF , the Fermi
+wavelength of the superconductor [23]), it is customary
+to consider the magnetic scatterer as a point-like classical
+spin S located at the point R0, interacting via a contact
+s-d exchange interaction HZ = J(r) S·s(r) with the host
+superconducting electrons [6]. Here J(r) = J0δ(r − R0)
+is the local exchange potential and s(r) is the spin den-
+sity vector of the electronic fluid. Subsequent theoretical
+works considered atomic-sized systems with finite- (al-
+beit short-ranged) exchange interactions with spherical
+symmetry [7, 24–26]. In that case, theory predicts the
+existence of multiple YSR states labelled by their orbital
+momentum ℓ, a prediction that has been recently ob-
+served in STM experiments [27–29].
+The behavior of subgap states and the associated 0−π
+quantum phase transitions has also been studied in the
+opposite limit L ≫ λF in the context of ballistic SU-
+FM-SU Josephson junctions with generic spin-dependent
+fields in the sandwiched region [30–32]. In this case the
+results differ from the well-known results of YSR states
+due to the finite extension of the magnetic profile. In
+particular, the subgap spectrum of long SU-FM-SU junc-
+tions with zero phase difference is known to be double
+degenerate [19, 31], showing the inherent complexity of
+these hybrid heterostructures. On the experimental side,
+the possibility to engineer and control the position of the
+subgap states by a modification of the fabrication para-
+maters (e.g., the length L or exchange field h0 via dif-
+ferent FM materials) opens interesting perspectives for
+potential electronic devices, where the precise knowledge
+of the subgap spectrum is crucial to control their trans-
+port properties.
+Motivated by the experimental developments men-
+tioned above, in this work we study the subgap states
+emerging in one-dimensional (1D) SE-SU-FMI het-
+erostructures where the SU and the FMI layers simul-
+taneously generate coexisting proximity-induced pairing
+and exchange interactions over a finite and arbitrary
+length L in the SE nanowire, as schematically shown in
+Fig. 1. This coexistence is a crucial aspect of this device,
+which makes it unique and different from the abovemen-
+tioned SU-FM-SU junctions, where such overlap occurs
+only at the SU-FM interface. Our main goal in this work
+is to study and understand the behavior of the subgap
+ABS in this device as a function of the experimentally rel-
+evant parameters of the model, i.e., the length L of the
+FMI region and the magnitude of the induced exchange
+field h0. As mentioned above, a device similar to that
+shown in Fig. 1 has been recently experimentally real-
+ized in SE nanowires with epitaxially-grown SU and FMI
+layers [20, 21]. While the main interest of that work was
+the fabrication of a device with non-trivial topological SU
+ground state hosting Majorana zero modes, here we will
+study the regime of parameters favoring a topologically-
+trivial ground state.
+As we will show below (see Sec.
+II), this case is already very complex and rich as a result
+of the antagonistic SU and FM interactions and, to the
+best of our knowledge, the detailed behavior of subgap
+states and the quantum phase diagram emerging in such
+a system have not been explicitly studied before.
+The article is organized as follows. In Section II, we
+introduce the model representing a 1D SE-SU-FMI hy-
+brid nanowire, discuss the solution to the Bogoliubov-
+de Gennes equations for the subgap states, and derive
+a generic equation for the subgap spectrum.
+In Sec-
+tion III, we analyze the results in two specific limits,
+where we recover well-known results: a) the semiclassical
+limit, where the superconducting coherence length ξ is
+much larger than the Fermi wavelength λF , and b) the
+atomic YSR limit, in which the exchange-field induced
+by the FMI region becomes a delta-function potential:
+i.e., infinitesimally narrow (L ≪ λF ), and infinitely deep
+(h0 ≫ EF ), in such a way that the product h0.L = J
+is kept constant.
+In both cases, well-known analytical
+solutions to the subgap spectrum can be recovered. In
+addition, we numerically solve the characteristic equation
+for the subgap states and provide a generic description
+of the subgap spectrum, not restricted to any of these
+limits.
+We find a rich behaviour of the subgap ABS,
+where the competing FM exchange and SU pairing inter-
+actions give rise to parity- and spin-changing quantum
+phase transitions. Finally, in Section IV, we present a
+summary and our conclusions.
+II.
+THEORETICAL MODEL
+We focus on the system schematically depicted in Fig.
+1, which represents a 1D SE-SU-FMI hybrid nanostruc-
+ture of total length Lw, similar to those fabricated in
+Refs. 20 and 21. We model this system with the Hamil-
+tonian H = Hw + H∆ + HZ, where
+Hw =
+�
+σ
+�
+Lw
+2
+− Lw
+2
+dx ψ†
+σ(x)
+�
+−ℏ2∂2
+x
+2m∗ − µ
+�
+ψ†
+σ(x),
+(1)
+H ∆ = ∆
+�
+Lw
+2
+− Lw
+2
+dx
+�
+ψ†
+↑(x)ψ†
+↓(x) + H.c.
+�
+,
+(2)
+HZ =
+�
+Lw
+2
+− Lw
+2
+dx h(x)
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+. (3)
+Here Hw is the Hamiltonian of a single-channel SE
+nanowire of length Lw, in which the fermionic operator
+ψσ(x) creates an electron at position x with spin projec-
+tion σ =↑, ↓ and effective mass m∗. The parameter µ is
+the chemical potential, which can be experimentally var-
+ied applying external gates beneath the nanostructure.
+The terms H∆ and HZ represent, respectively, the
+proximity-induced pairing interaction encoded by the pa-
+rameter ∆, and the Zeeman exchange interaction intro-
+duced by the FMI and described by a space-dependent
+exchange field h(x), which we assume oriented along the
+
+3
+z direction (see Fig. 1). Moreover, since these interac-
+tions are externally induced into the semiconductor, we
+make the additional assumption that ∆ is unaffected by
+the presence of h(x) (a renormalized value of ∆ does not
+change qualitatively our results). As mentioned before,
+these two terms can be effectively induced by the pres-
+ence of epitaxially-grown SU and FMI shells in contact
+with the SE nanowire [20, 21]. It has been experimen-
+tally confirmed [21] that the FMI shell (EuS in that case)
+consists of a single magnetic monodomain, and there-
+fore modelling this layer by the Hamiltonian HZ is a
+reasonable approximation. In addition, the epitaxially-
+generated interfaces are essentially disorder-free, a neces-
+sary condition to produce a proximity-induced hard-gap
+[33]. This feature allows to neglect the effects of disorder
+and considerably simplifies the theoretical description.
+The presence of both, a hard proximity-induced super-
+conductor gap and an effectively induced Zeeman field,
+in these nanowires have been reported in transport mea-
+surements in Refs.
+20 and 21.
+In addition, note that
+in the above model we have neglected the effect of the
+Rashba spin-orbit interaction. While this interaction is
+crucial for the emergence of a topologically non-trivial
+(i.e., D class) superconducting phase supporting Majo-
+rana zero-modes [34], here we will focus strictly on the
+topologically-trivial ground state. As we will show be-
+low, the competition of SU and FM interactions make
+this system already very complex and interesting in it-
+self.
+We note that since the total single-particle fermionic
+spin along z
+sz = 1
+2
+�
+Lw
+2
+− Lw
+2
+dx
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+,
+(4)
+is a conserved quantity which verifies [sz, H] = 0, we
+can label the electronic eigenstates of H with σ = {↑, ↓}.
+Therefore, we introduce the following Nambu spinors
+Ψ↑(x) =
+� ψ↑(x)
+ψ†
+↓(x)
+�
+,
+Ψ↓(x) =
+� ψ↓(x)
+ψ†
+↑(x)
+�
+,
+(5)
+related to each other via the charge-conjugation transfor-
+mation Ψ¯σ(x) = KτxΨσ(x), where τx is the 2 × 2 Pauli
+matrix, and K is the complex conjugation operator. In
+terms of these spinors the Hamiltonian writes
+H = 1
+2
+�
+σ
+�
+Lw
+2
+− Lw
+2
+dx Ψ†
+σ(x)HBdG,σ(x)Ψσ(x),
+(6)
+where the Bogoliubov-de Gennes (BdG) Hamiltonian is
+defined as
+HBdG,σ =
+�
+− ℏ2∂2
+x
+2m − µ + σh(x)
+σ∆
+σ∆
+ℏ2∂2
+x
+2m + µ + σh(x)
+�
+.
+(7)
+In this expression, the spin projection σ =↑ (↓) on
+the left-hand side corresponds to the + (−) sign in
+the definition of the BdG matrix.
+Using the above
+charge-conjugation transformation, we note that the
+BdG Hamiltonian Eq. (7) verifies the following symme-
+try transformation
+KτxHBdG,σ = −H∗
+BdG,¯σKτx,
+(8)
+and therefore, provided χσ(x) is a solution of the BdG
+eigenvalue equation
+HBdG,σ(x)χσ(x) = Eσχσ(x),
+(9)
+with eigenenergy Eσ, the transformed spinor χ¯σ(x) =
+Kτxχσ(x), is also a solution with eigenenergy E¯σ = −Eσ.
+In what follows, we assume for simplicity the thermo-
+dynamic limit Lw → ∞, and we focus on the features
+introduced by the magnitude and spatial dependence of
+h (x), which is crucial for the rest of this work. In addi-
+tion, we assume the following step-like spatial profile for
+the exchange field
+h(x) =
+�
+−h0
+if |x| < L
+2 ,
+0
+if |x| ⩾ L
+2 ,
+(10)
+which models a uniform FMI shell of length L in contact
+with the SE nanowire (see Fig. 1). This choice for h(x)
+allows to split the problem into regions with either |x| <
+L
+2 or |x| > L
+2 , with generic exponential solutions
+χσ(x) ∼
+�
+ασ
+βσ
+�
+eikx.
+(11)
+Linear combinations of Eq. (11), with appropriate coeffi-
+cients and with allowed values of k for each region, must
+be built so that continuity of the total wavefunction and
+its derivative at the interfaces is satisfied. With this re-
+quirement, the solution of Eq.(9) is finally obtained.
+Note that the BdG Hamiltonian (7) is even under space
+inversion x → −x, and therefore its eigenstates must be
+even or odd under this transformation of coordinates.
+This symmetry allows to reduce the number of unknowns
+of the problem (i.e., coefficients of the linear combinta-
+tion). Replacing the above ansatz Eq. (11) into the BdG
+eigenvalue Eq.
+(9), and looking for localized solutions
+with energy within the gap |Eσ| < ∆, we obtain the fol-
+lowing expressions for the eigenstates belonging to the
+even-symmetry subspace:
+
+4
+χe,σ
+�
+x > L
+2
+�
+= Ae
+1σ
+�
+1
+σe−iϕσ
+�
+e−κσx + Ae
+2σ
+�
+1
+σeiϕσ
+�
+e−κ∗
+σx,
+(12)
+χe,σ
+�
+−L
+2 ≤ x ≤ L
+2
+�
+= Be
+1σ
+�
+1
+σe−ησ
+�
+cos kσx + Be
+2σ
+�
+1
+σeησ
+�
+cos ¯kσx,
+(13)
+and the following expressions for the odd-symmetry eigenfunctions
+χo,σ
+�
+x > L
+2
+�
+= Ao
+1σ
+�
+1
+σe−iϕσ
+�
+e−κσx + Ao
+2σ
+�
+1
+σeiϕσ
+�
+e−κ∗
+σx,
+(14)
+χo,σ
+�
+−L
+2 ≤ x ≤ L
+2
+�
+= Bo
+1σ
+�
+1
+σe−ησ
+�
+sin kσx + Bo
+2σ
+�
+1
+σeησ
+�
+sin ¯kσx,
+(15)
+where the coefficients {Aν
+1σ, Aν
+2σ, Bν
+1σ, Bν
+2σ}, with ν =
+{e, o}, are unknowns to be fixed.
+In addition, in the
+above expressions we have introduced the parametriza-
+tion
+cos ϕσ = Eσ
+∆ ,
+(16)
+cosh ησ = Eσ + σh0
+∆
+,
+(17)
+where we fix the definition of ϕσ to the interval ϕσ ∈
+(0, π]. The phase variable ϕσ is associated to the An-
+dreev reflection taking place at the interface xb = L/2.
+Note that the parametrization in Eq. (17) makes sense
+whenever the right-hand side is positive. If this condi-
+tion is not satisfied, one can always use the symmetry
+Eq.(8) to send Eσ → −E¯σ and σ → ¯σ. In addition, note
+that whenever 1 ≤ (Eσ + σh0) /∆ the parameter ησ is
+purely real, while for 0 < (Eσ + σh0) /∆ < 1 it is purely
+imaginary. Finally, we have introduced the quantities
+κσ ≡ −ikF
+�
+1 + 2i
+kF ξ sin ϕσ,
+(18)
+kσ ≡ kF
+�
+1 +
+2
+kF ξ sinh ησ,
+(19)
+¯kσ ≡ kF
+�
+1 −
+2
+kF ξ sinh ησ,
+(20)
+and the definition of the coherence length of the
+(proximity-induced) 1D superconductor ξ = ℏvF /∆. No-
+tice also that the spatial dependence of the wavefunc-
+tions in the region x < −L/2 can be readily obtained by
+symmetry from the relations χe,σ (x) = χe,σ (−x), and
+χo,σ (x) = −χo,σ (−x).
+We can intuitively understand the form of the scatter-
+ing solutions in the regions x > L/2 and x < −L/2 in the
+limit kF ξ ≫ 1 (i.e., the semiclassical limit, see Sec.III A),
+where the momentum κσ in Eq. (18) can be expanded
+as κσ ≃ −ikF + sin ϕσ/kF ξ, and the eigenfunctions Eqs.
+(12) and (14) take the form
+χν,σ
+�
+x > L
+2
+�
+≈
+�
+Aν
+1σ
+�
+1
+σe−iϕσ
+�
+eikF x+
++Aν
+2σ
+�
+1
+σeiϕσ
+�
+e−ikF x
+�
+e− sin ϕσx
+ξ
+, (21)
+with ν = {e, o}. In this way, it becomes evident that the
+component proportional to Aν
+1σ corresponds to a right-
+moving particle ∼ eikF x while Aν
+2σ corresponds to a left-
+moving particle ∼ e−ikF x. In addition, the wavefunctions
+exponentially decay into the superconductor within a lo-
+calization length λloc = ξ/ sin ϕσ = ξ/
+�
+1 − (Eσ/∆)2.
+These results are in complete agreement with Ref. [32],
+where the spectrum of SU-FM-SU Josephson junctions
+has been recently studied as a function of the length L
+of the FM region. However, in our case, the presence of
+a finite pairing gap ∆ in the region −L/2 < x < L/2 (as
+opposed to the assumption ∆ = 0 in the FM region in
+that work), gives rise to important differences which we
+analyze below in Sec. III.
+A.
+Continuity conditions at the interface
+We now impose the continuity conditions on the wave-
+function and its derivative at the boundary xb = L/2:
+χν,σ
+�
+x−
+b
+�
+= χν,σ
+�
+x+
+b
+�
+(22)
+∂xχν,σ
+�
+x−
+b
+�
+= ∂xχν,σ
+�
+x+
+b
+�
+.
+(23)
+Note that the same equations are obtained by symme-
+try at the other boundary −xb. Inserting the solutions
+Eqs. (12)-(15), we can express the continuity equations
+in matrix form as
+
+5
+�
+1
+σe−iϕσ
+σe−iϕσ
+1
+� �
+aν
+1σ
+aν
+2σ
+�
+=
+�
+1
+σe−ησ
+σe−ησ
+1
+� �
+Fν
+� kσL
+2
+�
+0
+0
+Fν
+� ¯kσL
+2
+�
+� �
+bν
+1σ
+bν
+2σ
+�
+,
+(24)
+−
+�
+1
+σe−iϕσ
+σe−iϕσ
+1
+� �
+κσ
+0
+0
+κ∗
+σ
+� �
+aν
+1σ
+aν
+2σ
+�
+= −s(ν)
+�
+1
+σe−ησ
+σe−ησ
+1
+� �
+kσGν
+� kσL
+2
+�
+0
+0
+¯kσGν
+� ¯kσL
+2
+�
+� �
+bν
+1σ
+bν
+2σ
+�
+,
+(25)
+where we have conveniently redefined the unknown coef-
+ficients as
+Aν
+1σ → eκσL/2aν
+1σ
+Bν
+1σ → bν
+1σ
+(26)
+Aν
+2σ → σeκ∗
+σL/2e−iϕσaν
+2σ
+Bν
+2σ → σe−ησbν
+2σ,
+(27)
+in order to give these equations a more symmetric form.
+In addition, we have used the notation s(ν) = +1(−1) for
+ν = e(o), and Fe(x) = Go(x) ≡ cos(x), Ge(x) = Fo(x) ≡
+sin(x) for compactness.
+In each subspace (even or odd) we have four equa-
+tions and four unknowns.
+Eliminating the variables
+(bν
+1σ, bν
+2σ)T , and writing the equation for (aν
+1σ, aν
+2σ)T , we
+find from the nullification of the corresponding determi-
+nant the following equations:
+cosh ησ cos ϕσ − 1
+sinh ησ sin ϕσ
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+|κσ|2 −
+�
+Kσ + ¯Kσ
+�
+Re κσ + Kσ ¯Kσ
+� ¯Kσ − Kσ
+�
+Im κσ
+(even-symmetry subspace),
+|κσ|2 +
+�
+Qσ + ¯Qσ
+�
+Re κσ + Qσ ¯Qσ
+�
+Qσ − ¯Qσ
+�
+Im κσ
+(odd-symmetry subspace),
+(28)
+where we have defined the quantities
+Kσ = kσ tan
+�kσL
+2
+�
+,
+(29)
+¯Kσ = ¯kσ tan
+�¯kσL
+2
+�
+,
+(30)
+Qσ = kσ cot
+�kσL
+2
+�
+,
+(31)
+¯Qσ = ¯kσ cot
+�¯kσL
+2
+�
+.
+(32)
+From Eq. (28), the eigenvalue Eσ for each subspace is
+finally obtained. This equation summarizes our main the-
+oretical results. In the next Sec. III we analyze the nu-
+merical solution and different important limits.
+B.
+Spin-changing quantum phase transitions
+We now focus on the quantum phase transitions which
+occur whenever one of the subgap states crosses EF . To
+that end, let us analyze the spinors defined in Eq. (5),
+and consider the norm of the “up” spinor
+q↑ =
+� Lw/2
+−Lw/2
+dx
+�
+ψ†
+↑ (x) ψ↑ (x) + ψ↓ (x) ψ†
+↓ (x)
+�
+.
+Recalling the definition of the single-particle sz operator
+[see Eq. (4)], it is straightforward to associate these two
+quantities through the relation q↑ = 2sz − 1. Since sz
+is a conserved quantity, so is the norm q↑ of the “up”
+Nambu spinors. This connection allows to interpret q↑ as
+an effective “conserved charge”. Similar considerations
+allow to write the relation q↓ = −2sz − 1. Due to the
+particle-hole relation Eq.(8), the information about sz
+can be obtained with either q↑ or q↓. A more symmetric
+form involving both conserved charges is
+sz = q↑ − q↓
+4
+.
+(33)
+While redundant, this expression makes explicit that in
+the spin-symmetric case q↑ = q↓, the net spin sz must
+vanish (sz = 0).
+We now return to Hamiltonian Eq.
+(7), and let us
+separate the effect of the proximity-induced Zeeman field,
+by writing it as HBdG,σ = H0,σ + Vσ, where
+H0,σ =
+�
+− ℏ2∂2
+x
+2m − µ
+σ∆
+σ∆
+ℏ2∂2
+x
+2m + µ
+�
+,
+(34)
+Vσ =
+�
+σh(x)
+0
+0
+σh(x)
+�
+.
+(35)
+In this form, we can interpret the effect of the exchange
+field as a “perturbation” on an otherwise homogeneous
+
+6
+1D superconductor represented by H0,σ. Therefore, the
+full and the unperturbed single-particle Green’s functions
+in this problem are respectively defined as
+Gσ (z) = [z − H0,σ − Vσ]−1 ,
+(36)
+G0,σ (z) = [z − H0,σ]−1 ,
+(37)
+From here, the total number of effective “up” charges
+Q↑ induced in the ground state due to the potential Vσ,
+compared to the unperturbed homogeneous SU wire, can
+be computed as
+∆Q↑ = − 1
+π Im Tr
+� ∞
+−∞
+dϵ nF (ϵ) ∆G↑ (ϵ + iδ) .
+(38)
+where ∆Gσ (z) ≡ Gσ (z) − G0,σ (z). At T = 0, Eq. (38)
+can be easily computed from the well-known expression
+of the Friedel sum rule [35]
+∆Q↑ = 1
+π
+� 0
+−∞
+dϵ
+�∂η↑ (ϵ)
+∂ϵ
+− ∂η0,↑ (ϵ)
+∂ϵ
+�
+(39)
+= η↑ (0) − η0,↑ (0)
+π
+(40)
+where we have defined the phase shifts [32, 35]
+ησ (ϵ) = Im ln det Gσ (ϵ + iδ) ,
+(41)
+η0,σ (ϵ) = Im ln det G0,σ (ϵ + iδ) ,
+(42)
+and where we have used that the phase shifts vanish in
+the limit ϵ → ±∞.
+Since the system is non-interacting, the Green’s func-
+tion Eq. (36) can be written in terms of single-particle
+eigenstates |α, σ⟩, with α a generic label, as
+Gσ (z) =
+�
+α
+|α, σ⟩ ⟨α, σ|
+z − Eα,σ
+.
+(43)
+Therefore, after simple algebra, and using the above re-
+lations and the fact that in the absence of magnetic field
+sz = 0 [see Eq. (33)], the total Sz of the ground state is
+Sz = ∆Q↑
+2
+= 1
+2
+��
+α
+Θ (−Eα,↑) −
+�
+α′
+Θ
+�
+−E0
+α′,↑
+�
+�
+,
+(44)
+where Θ(ϵ) is the unit-step function. The above expres-
+sion allows to interpret the total Sz of the ground state
+as a function of the “up” Nambu spinors with energy be-
+low EF = 0, as compared to the (unperturbed) situation
+h0 = 0. Since the effective charges are quantized in inte-
+ger numbers, the total spin Sz can only change in discrete
+“jumps” of 1/2 whenever a subgap state with projection
+up crosses below EF (note that we have defined dimen-
+sionless spin operators). This interpretation makes sense
+since the ground state becomes spin-polarized when the
+exchange field h0 becomes large enough [i.e., the Zeeman
+energy of up-spin electron is decreased, see Eqs. (3) and
+(10)]. While the result of Eq. (44) has been obtained re-
+cently by the authors of Ref. [32], we note that here we
+have rederived it in a different physical situation which
+allows a more generic regime of parameters.
+III.
+RESULTS
+We start this section by analyzing different limits of
+the general result given in Eq. (28). In particular, in
+Sec. III A we focus on the semiclassical limit, and in Sec.
+III B we study the atomic limit, where we recover the
+YSR results. In both cases, Eq. (28) reduces to well-
+known analytical results. Finally in Sec. III C we show
+results corresponding to intermediate regimes, obtained
+by solving numerically Eq. (28).
+A.
+Semiclassical limit
+Generally speaking, the semiclassical limit is verified
+when EF is the largest scale of the problem [36]. In par-
+ticular, the condition EF ≫ ∆ (which is very well satis-
+fied in most experimental systems) can be expressed as
+kF ξ ≫ 1, recalling that after linearization of the normal
+quasiparticle dispersion, i.e., ϵk,σ ≃ ±ℏvF k, where the
++(−) sign corresponds to right-(left-)movers, the Fermi
+energy can be approximated as EF ≃ ℏkF vF . In this
+case, Eqs. (18)-(20) reduce to
+rσ ≡ κσ
+kF
+≃ −i + sin ϕσ
+kF ξ ,
+(45)
+ζσ ≡ kσ
+kF
+≃ 1 + sinh ησ
+kF ξ
+,
+(46)
+¯ζσ ≡
+¯kσ
+kF
+≃ 1 − sinh ησ
+kF ξ
+,
+(47)
+to leading order in O(kF ξ)−1, and Eq. (28) becomes
+cosh ησ cos ϕσ − 1
+sinh ησ sin ϕσ
+≃ s(ν)
+1 + tan
+�
+kF Lζσ
+2
+�
+tan
+�
+kF L¯ζσ
+2
+�
+tan
+�
+kF Lζσ
+2
+�
+− tan
+�
+kF L¯ζσ
+2
+� ,
+= s(ν) cot
+�L sinh ησ
+ξ
+�
+,
+(48)
+where
+we
+have
+used
+the
+trigonometric
+identity
+tan (x + y) = (tan (x) + tan y)/(1 + tan x tan y). In gen-
+eral this transcendental equation cannot be solved ana-
+lytically. However, in the regime of parameters EF ≫
+h0 ≫ ∆, where the exchange field h0 is much larger
+than ∆, we can write cosh ησ ≈ sinh ησ ≈
+�� h0
+∆
+�� ≫ 1
+[see Eqs.
+(16) and (17) ], and Eq.
+(48) reduces to
+cot ϕσ = s(ν) cot (Lh0/ℏvF ). Equivalently we can write
+this result as
+arccos
+�Eσ
+∆
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+LEσ
+ℏvF
++ σ Lh0
+ℏvF
++ 2nπ,
+(even)
+LEσ
+ℏvF
++ σ Lh0
+ℏvF
++ (2n + 1) π.
+(odd)
+(49)
+This result can be interpreted as a semiclassical Bohr-
+Sommerfeld quantization condition for particles which
+
+7
+perform a complete a closed loop in the region −L/2 <
+x < L/2 [36]. In particular, it exactly coincides with the-
+oretical results obtained for SU-FM-SU Josephson junc-
+tions with a normal (i.e., ∆ = 0) FM region [30–32], the
+only difference being that within our theoretical treat-
+ment, we can distinguish the symmetry of the solutions.
+The similarity of these results can be rationalized noting
+that considering a normal sandwiched region in an SU-
+FM-SU junction corresponds to taking the limit h0 ≫ ∆
+in our Eq. (48) while keeping the ratio Eσ/∆ finite (since
+Eσ corresponds to a subgap state, it is always bounded
+by ∆), thus resulting in Eq. (49). This shows that our
+Eq. (28) is a generic relation describing different situa-
+tions regardless of the magnitude of the ratio h0/∆.
+B.
+YSR-impurity limit
+We now consider the atomic YSR (or simply Shiba)
+limit, in which the exchange profile becomes point-like,
+L → 0, while h0 → ∞, in such a way that the product
+Lh0 = J = const. Under these assumptions the magnetic
+barrier becomes a delta function and the Hamiltonian in
+Eq. (3) can be written as
+HZ ≈ −J
+� ∞
+−∞
+dx δ(x)
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+.
+(50)
+In this case, it is easy to see that the odd-symmetry solu-
+tions decouple from the above Hamiltonian (50), as they
+vanish at x = 0, and only even solutions can couple to
+the delta-potential.
+As in the previous section, note that the limit h0 → ∞
+implies cosh ησ ≈ sinh ησ ≈
+�� h0
+∆
+�� ≫ 1. However, the limit
+h0 → ∞ is not compatible with the semiclassical ap-
+proach, as it violates the requirement h0 ≪ EF . There-
+fore we cannot use here our previous Eq. (49). Instead,
+we must first take the limit ησ ≫ 1 together with the
+limit L → 0, which applied to Eqs. (19) and (20) yield
+kσ → kF
+�
+2h0
+ℏvF kF
+,
+(51)
+¯kσ → ikF
+�
+2h0
+ℏvF kF
+.
+(52)
+In addition Eqs. (29)-(32) become
+Kσ → kF h0L
+ℏvF
+= kF ρ0J,
+(53)
+¯Kσ → −kF h0L
+ℏvF
+= −kF ρ0J,
+(54)
+where the expressions for the density of states per spin
+of 1D quasiparticles at the Fermi energy ρ0 = 1/ℏvF ,
+and the exchange coupling J = h0L, have been used.
+Replacing these expressions into Eq. (28) for the even-
+symmetry solutions, we obtain
+σ
+Ee
+σ
+�
+∆2 − (Ee)2
+σ
+= 1 − (ρ0J)2
+(2Jρ0)
+.
+(55)
+From this expression, we can easily solve for Ee
+σ
+Ee
+σ
+∆ = σ 1 − (ρ0J)2
+1 + (ρ0J)2 ,
+(56)
+which is the well-known expression for the energy of YSR-
+impurity subgap level [1]. This result indicates that any
+finite value of J produces a YSR in-gap state. This type
+of subgap YSR states has been observed in several STM
+experiments on atomic magnetic adsorbates on supercon-
+ducing substrates [27, 37–41].
+For completeness, and in order to illustrate the general
+scope of Eq. (28), here we also show the result for the
+YSR odd states for a small (but finite) L. Using similar
+approximations, we obtain the expression
+Eo
+σ
+∆ = σ
+1
+�
+1 +
+�ρ0Jk2
+F L2
+6
+�2 ,
+(57)
+where it becomes evident that in addition to a finite value
+of J, a finite value of kF L is needed to observe an odd-
+symmetry subgap YSR state.
+C.
+Subgap ABS spectrum in generic cases
+As stated in Section II, Eq. (28) implicitly defines the
+energy of the subgap states as a function of the param-
+eters h0/∆ , kF ξ, and kF L. These parameters can be
+directly or indirectly controlled in experiments, i.e., the
+parameter h0 can be controlled by modifying the FMI
+material, the length L of the FMI region can be modified
+varying the length Lw of the semiconductor via vapor-
+liquid-solid (VLS) method and subsequent evaporation of
+the FMI material [20], and the parameter kF in the semi-
+conductor can be varied by changing the SE material or
+by introducing external gates to modify the chemical po-
+tential µ. Therefore, due to this high degree of tunability,
+hybrid heterostructures might offer a unique platform to
+produce and control engineered subgap states. Probably
+the easiest way to experimentally control the subgap elec-
+tronic structure is by producing different devices with the
+same FMI material and different lengths L. Therefore, in
+this section we show the numerical solutions of Eq. (28)
+with fixed parameters h0/∆ and kF ξ (which control the
+“operation regime” of the device), and calculate both the
+energy dependence of the even- and odd-symmetry ABS,
+and the total spin Sz of the device as a function of L (i.e.,
+dimensionless variable kF L).
+Generally speaking, the overall evolution of the ABS
+spectrum from L = 0 to L → ∞ is quite complex and de-
+serves a detailed explanation. As shown in Fig. 2, as the
+
+8
+parameter kF L increases, more and more subgap states
+emerge from the gap edges. This behavior is reminiscent
+of a quantum particle in a square-well potential, tipically
+taught in introductory quantum mechanics courses [42],
+where increasing the width L of the well increases the
+number of allowed bound states. In our case, the emer-
+gence of new ABS as kF L increases can be intuitively
+understood in terms of a competition between supercon-
+ductivity and magnetic field: the magnetic field tends
+to break Cooper-pairs and to locally disrupt supercon-
+ductivity in the magnetic region by introducing subgap
+states that become macroscopic in number for large L,
+eventually populating the whole gap.
+We note that for any finite L, even- and odd-symmetry
+states are generically non-degenerate (except at isolated
+points). However, as it is clear from Figs. 2 and 3, their
+energy difference (evidenced as oscillations of the blue
+and red lines around the semiclassical value) decreases
+very rapidly and the solutions become degenerate in the
+limit L → ∞. This transition from non-degenerate YSR
+states in the limit L → 0, to double degenerate ABS
+states for L → ∞ has been discussed in previous works
+on ballistic SU-FM-SU junctions [19, 30–32], and in the
+case of extended Shiba impurities in 1D nanowires [43].
+It is also clearly visible in Fig. 2, and more dramatically
+in Fig. 3 below. In our 1D geometry, this degeneracy
+in the limit L → ∞ can be intuitively understood by
+linearizing the spectrum around the Fermi energy, and
+expressing the original fermionic operators in terms of
+right- and left-moving fields slowly varying in the scale
+of k−1
+F
+[44], i.e., ψσ (x) ≈ eikF xψR,σ (x) + e−ikF xψL,σ (x).
+The slowly-varying fields ψR,σ(x) and ψL,σ(x) are two
+independent chiral fermionic fields obeying the usual an-
+ticommutation relations, in terms of which the original
+Hamiltonian becomes [43]
+Hw ≈
+�
+σ
+� ∞
+−∞
+dx
+�
+−iℏvF ψ†
+R,σ(x)∂xψR,σ(x)
++ iℏvF ψ†
+L,σ(x)∂xψL,σ(x)
+�
+(58)
+H ∆ ≈ ∆
+� ∞
+−∞
+dx
+�
+ψ†
+R,↑(x)ψ†
+L,↓(x) + ψ†
+L,↑(x)ψ†
+R,↓(x) + H.c.
+�
+,
+(59)
+HZ ≈ −
+� ∞
+−∞
+dx h0
+�
+ψ†
+R,↑(x)ψR,↑(x) − ψ†
+R,↓(x)ψR,↓(x)
++ ψ†
+L,↑(x)ψL,↑(x) − ψ†
+L,↓(x)ψL,↓(x)
+�
+,
+(60)
+where oscillating terms proportional to e±2ikF x have been
+neglected as they cancel out in the limit L → ∞ due to
+destructive interference. Defining the new chiral Nambu
+spinors
+Ψ1,σ(x) =
+� ψR,σ(x)
+ψ†
+L,¯σ(x)
+�
+,
+Ψ2,σ(x) =
+� ψL,σ(x)
+ψ†
+R,¯σ(x)
+�
+,
+(61)
+the Hamiltonian of the system can be expressed in terms
+of two decoupled chiral sectors
+H = 1
+2
+�
+σ=↑,↓
+�
+j=1,2
+� ∞
+−∞
+dx Ψ†
+j,σ(x)Hj,σ(x)Ψj,σ(x), (62)
+with the definitions of the chiral BdG Hamiltonians
+Hj,σ =
+�
+(−1)jivF ∂x − σh0
+σ∆
+σ∆
+(−1)j+1ivF ∂x − σh0
+�
+.
+(63)
+The Nambu spinors Eq. (61) define two independent chi-
+ral subspaces related by the inversion symmetry of the
+original Hamiltonian, i.e., under the space inversion op-
+eration x ↔ −x, the fermionic operators transform as
+ψL,σ(x) ↔ ψR,σ(x), and consequently we conclude that
+Ψ1,σ(x) ↔ Ψ2,σ(x), which must then be degenerate. In
+addition, the particle-hole symmetry Eq. (8) in this rep-
+resentation produces Ψ1,σ(x) → Ψ2,¯σ(x), and therefore
+H1,σ → −H2,¯σ, implying that the solutions verify the
+particle-hole symmetry property E1,σ = −E2,¯σ. More-
+over, notice that assuming periodic boundary conditions,
+the problem can be solved with the solutions ψR,σ(x) ∼
+eikx and ψL,σ(x) ∼ e−ikx, and the dispersion relation
+becomes E1,σ(k) = E2,σ(k) = ±
+�
+(ℏvF k)2 + ∆2 − σh0.
+From here, a renormalized quasiparticle gap 2∆ren =
+2 |∆ − h0| is obtained, consistent with our previous re-
+sult.
+In terms of the chiral Nambu spinors, the most general
+solution is the linear combination
+Ψσ(x) = AeikF xΨ1,σ(x) + Be−ikF xΨ2,σ(x).
+(64)
+This is exactly the same form that can be obtained by
+combining the degenerate even and odd solutions in Eqs.
+(13) and (15) in the semiclassical limit where kF ξ ≫ 1.
+From the analysis of the linearized Hamiltonian, we
+conclude that the degeneracy in the limit L → ∞
+arises from the absence of chirality-breaking terms, i.e.,
+terms ∼ Ψ†
+1,σ(x)Ψ2,σ(x) arising from, e.g., single par-
+ticle backscattering terms ψ†
+R,σ(x)ψL,σ(x) or Cooper-
+pairing channels ψ†
+R(L),↑(x)ψ†
+R(L),↓(x) carrying momen-
+tum ∓2kF .
+For this to occur, the magnetic FMI re-
+gion must be uniform and its length L must be much
+larger than k−1
+F
+in order to produce the required cancel-
+lation of the rapidly oscillating exponentials ∼ e±2ikF x.
+In other words, the product kF L must be kF L ≫ 1,
+consistent with our numerical results in Figs. 2 and 3.
+Only for small values of kF L, where this destructive in-
+terference is incomplete, residual couplings of the type
+∼ Ψ†
+1,σ(x)Ψ2,σ(x) remain, and the degeneracy is lifted.
+Finally, we stress that the degeneracy in the limit L → ∞
+is a robust property to the presence of interactions, as
+shown in previous works [43].
+On the other hand, in the limit L → 0 and for any
+finite value of the Zeeman field h0, both (even and odd)
+solutions converge to Eσ/∆ → ±1, indicating that the
+FMI region is no longer relevant (i.e., it physically drops
+
+9
+−1
+1
+0
+E/∆
+−1
+1
+0
+0
+10
+20
+30
+40
+50
+0
+1
+2
+3
+4
+5
+6
+kF L
+Sz
+0
+2
+4
+6
+8
+10
+12
+14
+kF L
+FIG. 2. Energy of the Andreev bound states (upper panel) and total spin Sz(lower panel) as a function of kF L, for kF ξ = 7.8
+and h0/∆ = 3.0 (left panel) and kF ξ = 3.4 and h0/∆ = 2.1 (right panel). Blue and red colors correspond to even and odd states
+respectively. Lines starting from the top gap edge at positive energy E/∆ = 1 (bottom gap edge at negative energy E/∆ = −1)
+correspond to up (down) spin projections of the states. For smaller values of kF L (right panel), plateaus corresponding to
+regions of integer and half-integer spin are more separated and might become easier to observe in experiments.
+from the description). However, the behavior near L = 0
+is quite different for each case: while the even-symmetry
+solution tends to E/∆ → 1 as [see Eq. (56)]
+Ee
+σ
+∆ ≈ σ
+�
+1 − 2
+�h0L
+ℏvF
+�2
+. . .
+�
+,
+(65)
+from Eq. (57) we conclude that the odd solution behaves
+as
+Eo
+σ
+∆ ≈ σ
+�
+1 − 1
+2
+�h0k2
+F L3
+6ℏvF
+�2
+. . .
+�
+,
+(66)
+therefore approaching the gap edge much faster as L → 0.
+Besides the general features of the spectrum discussed
+up to this point, its evolution as L increases is strongly
+affected by the values of the parameters kF ξ and h0/∆.
+In what follows, we analyze their effects on Figs. 2 and
+Fig. 3 respectively.
+1.
+Effect of varying the parameter kF ξ
+This parameter can be considered as a “knob” which
+tunes the device from the semiclassical behavior (kF ξ
+large, see left panel in Fig. 2) into a “quantum” regime
+(kF ξ small, see right panel) where the spectrum is dom-
+inated by quantum oscillations. The hybrid heterostruc-
+ture under study is promising in this sense since, due to
+the combination of materials (in particular, semiconduc-
+tors with a much smaller kF as compared to metals), it
+is in principle possible that kF ξ can be experimentally
+controlled. In addition, kF could be further modified by
+introducing external gating leads (through the modifica-
+tion of the chemical potential µ). To illustrate the dra-
+matic changes in the spectrum as kF ξ varies, in Fig 2 we
+show the numerically obtained subgap spectra as a func-
+tion of kF L for kF ξ = 7.8 and h0/∆ = 3.0 (left panel),
+for and kF ξ = 3.4 and h0/∆ = 2.1 (right panel). Solid
+blue (red) lines correspond to even(odd)-symmetry solu-
+tions. Moreover, since we always assume h0 > 0, solu-
+tions emerging from the top edge E/∆ = 1 (bottom edge
+E/∆ = −1) correspond to spin up (spin down) solutions.
+In addition, note the reflection symmetry of the solutions
+around the horizontal E = 0 axis, a consequence of the
+particle-hole symmetry of the BdG Hamiltonian, Eq. (8).
+Upon decreasing kF ξ, the subgap spectrum becomes
+much more intricate due to the enhanced even-odd
+energy-splitting, which results in an amplified oscillatory
+behavior of the ABS (we have reduced the range of kF L in
+the right panel for clarity in the figure). Unfortunately,
+in the regime kF ξ ∼ 1 no analytic expressions for the
+subgap ABS are possible, but qualitative considerations
+
+10
+can be provided. In fact, the amplified oscillations can
+be traced back to the larger energy dependence of the
+momenta Eq. (18)-(20) as kF ξ decreases. Then, whereas
+for large kF ξ all these quantities converge to a static (i.e.,
+energy-independent) value ∼ kF , the limit of small kF ξ
+produces a larger effect on the space-dependence of the
+wave functions through the exponential factors in Eqs.
+(12)-(15). This in turn produces larger interference ef-
+fects, and an enhanced lifting of the even-odd degener-
+acy.
+This phenomenological behavior enables interesting
+possibilities, such as the chance to observe half-integer
+spin (and fermion parity-switching) quantum phase tran-
+sitions in the ground state. To illustrate this effect, we
+show the ground-state Sz transitions in the bottom pan-
+els of Fig.
+2 in each case.
+While for larger kF ξ, the
+half-integer Sz steps are very narrow due to the almost-
+degenerate even-odd solutions (i.e., the even and odd so-
+lutions cross zero energy almost at the same value of
+kF L), for smaller kF ξ the Sz transitions occur in well-
+defined half-integer steps. This behavior is well explained
+by the enhanced lifting of the even-odd degeneracy, which
+allows to observe one ABS crossing zero energy at a time.
+2.
+Effect of varying the parameter h0/∆
+In Fig. 3 we show the evolution of the subgap spectrum
+as a function of kF L, for different values of the Zeeman
+field h0/∆ = 0.8, 1.54 and 2.2, and for a fixed relatively
+large value kF ξ = 8.2, allowing to interpret these results
+in terms of the semiclassical approximation. Here we can
+clearly distinguish three qualitatively different regimes:
+a) the “weak field” regime h0 < ∆ (top panel) where
+the ABS do not cross E = 0, b) the “intermediate field”
+regime ∆ < h0 < 2∆ (middle panel) where the ABS can
+evenually cross zero energy, and quantum phase transi-
+tions can be induced, and finally c) the “strong field”
+(2∆ < h0) regime (bottom panel), where the ABS can
+be found anywhere in the region −1 < Eσ/∆ < 1. In all
+cases, the value of h0 determines the asymptotic limit to
+which the ABS approach for large L (see dashed black
+lines in Fig. 3). Below we briefly discuss the main fea-
+tures of the spectrum in each regime.
+a.
+Weak-field regime 0 < h0 < ∆:
+This regime
+is characterized by a Zeeman field which is not strong
+enough to destroy the superconducting gap.
+In this
+case none of the ABS is able to cross E = 0 and in
+the limit L → ∞ they asymptotically approach the
+value Eσ/∆ → σ (1 − h0/∆) (see horizontal dashed black
+lines), and therefore a renormalized gap remains (see top
+panel in Fig. 3). More quantitatively, in the semiclassi-
+cal limit [Eq. (48)] they obey the asymptotic expression
+−1
+1
+0
+h0
+E/∆
+−1
+1
+0
+h0
+E/∆
+0
+20
+40
+60
+80
+100
+−1
+1
+0
+kF L
+E/∆
+FIG. 3. Energy of the Andreev bound states as a function of
+kF L for the three different values of h0 (h0/∆ = 0.8, 1.54, 2.2
+for the lower, middle and upper panels) and kF ξ = 8.2. Blue
+and red colors correspond to even and odd states respectively.
+Lines starting from negative (positive) energies correspond to
+down (up) spin projections of the state. Note that the value
+of h0/∆ sets the asymptotic limit for the Andreev states and
+is crucial to determine the overall subgap spectrum.
+valid for kF L → ∞
+Eν
+σ
+∆ ≃ σ
+�
+�1 − h0
+∆ + π2
+2
+� ξ
+L
+�2 �
+1 − s(ν)ξ
+L
+�
+2∆
+h0
+− 1
+�2�
+� ,
+(67)
+with s(ν) = 1(−1) for ν = e(o).
+From here, we can
+clearly see that whereas the even-odd averaged quanti-
+ties (i.e., the semiclassical values) approach the asymp-
+totic limit as L−2, the energy difference between even
+and odd solutions (i.e., the amplitude of the oscillation
+around the semiclassical limit) decreases as L−3, and the
+solutions become degenerate in the limit L → ∞. On the
+other hand, the quasiparticle gap in the limit L → ∞ is
+renormalized to 2∆ren = 2 |∆ − h0|. Note that this gap
+renormalization is quite specific to this setup, and is not
+present, for instance, in the case of Ref. [32], where the
+magnetic region is normal and not superconducting, and
+in addition the system corresponds to a “short” SU-FM-
+SU junction with L < ξ, and therefore only few subgap
+states are allowed.
+Another feature of the weak-field regime is that the
+ABS require a minimal length Lmin to emerge in the sub-
+
+11
+gap region. This can be easily understood in terms of Eq.
+49, where a minimal magnetic phase, represented by the
+product Lh0/ℏvF , must be accumulated in order to pro-
+duce an observable in-gap ABS. Finally, concerning the
+spin quantum number of the ground state, since none of
+the ABS cross EF , no quantum phase transitions are ex-
+pected according to the results of Sec. II B and the value
+of the ground state spin remains a spin-singlet Sz = 0.
+b. Intermediate field regime ∆ < h0 < 2∆: In this
+case the Zeeman field h0 is sufficiently strong to force the
+ABS to cross zero energy, eventually inducing quantum
+phase transitions (see middle panel in Fig. 3). The n-th
+critical value Lc,n can be obtained imposing the condition
+Eσ = 0 on the semiclassical approximation in Eq. (48),
+Lν
+c,n = ξ
+arctan
+�
+−s (ν) ∓
+�� h0
+∆
+�2 − 1
+�
++ nπ
+�� h0
+∆
+�2 − 1
+,
+(68)
+with s(ν) = 1(−1) for ν = e(o).
+In this regime, the ABS follow the same asymp-
+totic behavior as in Eq.
+(67), approaching Eσ/∆ →
+σ (1 − h0/∆), although the overall subgap spectrum is
+completely different due to the closing of the gap, and
+due to the overlap of the E↑ and E↓ spectrum as L
+increases beyond the first critical Lc,0. In fact, in the
+regime L > Lc,0 the quasiparticle gap becomes com-
+pletely populated (and washed away) by subgap states.
+Moreover, we predict an accumulation of levels in the re-
+gion −∆ + h0 < E < ∆ − h0, which can eventually form
+a peak structure in the total density of states.
+c. Strong field regime 2∆ < h0: Finally, in this regime
+(see bottom panel in Fig. 3), the asymptotic dashed lines
+fall within the continuum and it is no longer possible to
+obtain an analytic expression for the ABS behavior in
+the limit L → ∞. As a result, the subgap ABS can be
+found anywhere in the subgap region −1 < Eσ/∆ < 1.
+In addition, we note that the minimal length required to
+observe in-gap ABS has reduced to Lmin ≈ 0.
+IV.
+SUMMARY AND CONCLUSIONS
+In this work we have analyzed the subgap electronic
+structure in the one dimensional SE-SU-FMI heterostruc-
+ture schematically depicted in Fig. 1, a novel physical
+system recently fabricated using molecular beam epitaxy
+techniques (MBE). The main motivation to study this
+type of hybrid systems is that, via a careful combina-
+tion of different materials, the emergent characteristics
+can be completely different from those of the individ-
+ual components, providing a way to build devices with
+tailored properties and specific functionalities. In partic-
+ular, much of the experimental effort has focused on the
+realization of topological superconducting phases host-
+ing Majorana zero modes, with possible applications in
+topological quantum computing [20, 21]. A distinguish-
+ing feature of these heterostructures is the coexistence
+of antagonistic superconductor and ferromagnetic insu-
+lating layers over a finite and arbitrary length L in a
+semiconductor wire, a combination that confers unique
+spectral properties which cannot be found in elemental
+materials in nature.
+In particular, we have modelled the hybrid struc-
+ture
+assuming
+non-interacting
+fermions
+in
+a
+one-
+dimensional single-channel nanowire under the effect of
+two proximity-induced interactions: a SU pairing and
+a space-dependent Zeeman exchange coupling [see Eqs.
+(1)-(3)].
+We have solved the associated Bogoliubov-de
+Gennes equations and, by imposing standard continuity
+conditions on the wave functions, we have obtained an
+equation [Eq. (28)] defining the subgap ABS spectrum
+of the device.
+This single equation encodes our main
+theoretical results. We stress that our approach is equiv-
+alent to other works using the scattering-matrix formal-
+ism. We have analytically solved Eq. (28) in two paradig-
+matic limits: the semiclassical limit (Sec. III A) and the
+Yu-Shiba-Rusinov limit, typical of atomic magnetic mo-
+ments interacting with a superconductor (Sec. III B). In
+both cases, we have been able to recover well-known ana-
+lytical results, providing important sanity checks for our
+theoretical results. As a consequence of the symmetries
+of the Hamiltonian (i.e., inversion x → −x and sz spin
+symmetries), it was possible to classify the solutions into
+even- and odd-symmetry, and with sz labels σ =↑, ↓. In
+particular, we note that the even-odd classification, aris-
+ing in the present case due to the inversion symmetry of
+the Hamiltonian, is nothing but the 1D analog of the clas-
+sification in angular momentum eigenstates ℓ occurring
+in 3D spherically-symmetric Hamiltonians [7, 25, 26].
+We have studied the subgap spectrum of ABS as a
+function of different parameters, namely: the length of
+the magnetic region (through the dimensionless parame-
+ter kF L), the strength of the Zeeman exchange induced
+by the FMI (parameter h0/∆), and the superconducting
+coherence length (parameter kF ξ). We stress that each
+one of these parameters could in principle (directly or in-
+directly) be controlled in experiments. However, due to
+its potential relevance for on-going experimental efforts,
+we have in particular focused our study on the evolution
+of the subgap spectrum as a function of the length L (i.e.,
+as it is probably the easiest parameter to vary in experi-
+ments), for fixed parameters kF ξ and h0/∆. The parame-
+ter L can be controlled by, e.g., changing the experimen-
+tal growing conditions of the semiconductor nanowires
+using the VLS growth method. In Figs. 2 and 3 we have
+analyzed the evolution of the subgap spectrum in terms
+of the parameter kF L for different values of h0/∆ and
+kF ξ. Roughly speaking, while kF ξ controls the “semi-
+classical vs quantum” operation regime of the device,
+and the magnitude of the even-odd energy separation,
+the parameter h0/∆ essentially controls the energy sep-
+aration of the E↑ and E↓ solutions, eventually enabling
+many interesting physical phenomena such as the possi-
+
+12
+bility to observe multiple ABS crossing zero-energy, the
+existence of multiple spin- and parity-changing quantum
+phase transitions in the device, quasiparticle gap renor-
+malization ∆ → ∆ren = |∆ − h0| in the limit of large
+kF L, etc.. An important conclusion here is that in order
+to experimentally observe a quantum phase transition,
+the condition h0 > ∆ must be fulfilled.
+Interpreting L as a “tunable” parameter has another
+theoretical advantage, as it enables to address the in-
+teresting fundamental question of how to connect two
+paradigmatic limits in SU-FM hybrid devices: the atomic
+limit (kF L → 0), where the physics is that of the well-
+known non-degenerate YSR states, and the ballistic limit
+(kF L ≫ 1) where the spectrum of the subgap ABS be-
+comes double degenerate. Until very recently, these lim-
+its were treated as disconnected from each other. In Ref.
+[32] this issue was addressed in the particular case of SU-
+FM-SU junctions in the limit L < ξ. Here we have revis-
+ited this intriguing question for a different setup where
+such constraint does not exist, and have studied the evo-
+lution of the subgap spectrum as a function of L. The
+abovementioned symmetry classification into even and
+odd solutions is critically important to allow the inter-
+pretation of the degeneracy in the limit kF L → ∞ as an
+“even-odd degeneracy”. At the same time, it enables to
+explain the degeneracy lifting in the limit L → 0, where
+only even states prevail in the subgap region of ener-
+gies.
+Using an approximate model of one-dimensional
+fermions with linearized dispersion, we have provided a
+simple picture where the even-odd degeneracy naturally
+emerges as a consequence of destructive interferences of
+terms e±i2kF x arising from single-particle backscattering
+mechanisms.
+The continuous evolution of the subgap spectrum as a
+function of kF L allows a better understanding of previ-
+ous experimental STM results on atomic magnetic adsor-
+bates on superconducting substrates, where the subgap
+YSR states are usually interpreted in terms of a point-like
+magnetic moment [27, 37–41]. While the delta-function
+limit is obviously a mathematical idealization, in terms
+of our model the observed YSR states can be rationalized
+assuming a finite value of kF L and a (more physically ap-
+pealing) finite value of the atomic local field h0. This is
+precisely the case if we note that for magnetic impurities
+(e.g. Fe, Co or Mn atoms) deposited on top of bulk metal-
+lic S surfaces (e.g., Pb or Al), the spatial extension of the
+short-ranged Zeeman field can be estimated as the size of
+the d-shell orbitals L ∼ 1 ˚A, while the Fermi wavevector
+of bulk superconductors (e.g., Pb) is kF ∼ 1−2×1010m−1
+(see Ref. [45]). This type of adsorbate/substrate combi-
+nation yields a parameter kF L ∼ 1, which is within the
+regime where we recover observable subgap states (see
+Figs. 2 and 3). On the other hand, in 1D semiconduc-
+tor heterostructures as those of Refs. 20 and 21, kF is
+usually much smaller than in metallic superconductors.
+Measurements of the number of carriers from the Hall
+conductance RH in 2D InGaAl quantum wells [46] yield
+the estimated value kF ∼ 2.2 × 107m−1, three orders of
+magnitude smaller as compared to bulk Pb. This much
+smaller value of kF allows for much larger, experimentally
+accessible values of L, while keeping values of h0 also
+within experimental reach. All together, this combina-
+tion makes these hybrid materials a much more versatile
+platform to control the spectrum of YSR/ABS subgap
+states.
+To characterize the quantum phase transitions occur-
+ring in the device, we have computed the value of the to-
+tal Sz using a spin version of the Friedel sum rule [see Eq.
+(44) and also Ref. [32]. We stress that these transitions
+are a generalization of the well-known “0-π” transition
+occurring in atomic Shiba impurities [22, 47] or quantum
+dots coupled to superconductors [48–50]. From this per-
+spective, the difference with respect to atomic systems
+is that instead of a single transition, actually multiple
+transitions can occur due to the finite extension L of
+the “impurity” and the many ABS states with different
+symmetry which can eventually cross below EF . Inter-
+estingly, we stress that the ocurrence of these quantum
+phase transitions can be tuned varying the length L.
+We now briefly address the effect of the Rashba spin-
+orbit interaction, which has been neglected in our work.
+As mentioned previously, this interaction was neglected
+to simplify the theoretical description of this (already
+quite complex and rich) problem. This interaction can
+drive the system into the topological superconductor
+class D [51, 52], hosting Majorana zero modes at the ends
+(see e.g., Ref. 34 for a related setup), and in that case
+we expect qualitative changes with respect to the results
+presented here. Consequently our results apply to exper-
+imental SE-SU-FMI systems where the spin-orbit energy
+term ESOC = α2
+Rm∗/2, with αR the Rashba parameter,
+is negligible compared to ∆ and h0.
+Finally, we consider the effect of disorder in this setup.
+This might be a relevant effect as a random disorder po-
+tential will eventually break the inversion symmetry of
+the model and might lift the predicted even-odd degen-
+eracy in the limit kF L ≫ 1. However, we believe the
+energy-lifting effect might be weak in epitaxially-grown
+samples, where disorder is a relatively small effect.
+ACKNOWLEDGMENTS
+This work was partially supported by CONICET un-
+der grant PIP 0792, UNLP under grant PID X497, and
+Agencia I+D+i under PICT 2017-2081, Argentina. AML
+is grateful to Liliana Arrachea for pointing out crucial
+bibliographic references.
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diff --git a/3NE2T4oBgHgl3EQfjQer/content/tmp_files/load_file.txt b/3NE2T4oBgHgl3EQfjQer/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf,len=974
+page_content='Subgap states and quantum phase transitions in one-dimensional  superconductor-ferromagnetic insulator heterostructures  Javier Feijoo,1, 2 An´ıbal Iucci,1, 2 and Alejandro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lobos3, 4  1Instituto de F´ısica La Plata - CONICET, Diag 113 y 64 (1900) La Plata, Argentina  2Departamento de F´ısica, Universidad Nacional de La Plata, cc 67, 1900 La Plata, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  3Instituto Interdisciplinario de Ciencias B´asicas (CONICET-UNCuyo)  4Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, 5500 Mendoza, Argentina  We  theoretically  study  the  spectral  properties  of  a  one  dimensional  semiconductor-  superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recents  experiments where such devices have been fabricated using epitaxial growing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We model  the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the si-  multaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially  inhomogeneous) Zeeman exchange field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The coexistence of these competing interactions generates  a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' By  exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes  equations into even and odd ABS with respect to the spatial inversion symmetry x → −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We  find the ABS spectrum of the device as a function of the different parameters of the model: the  length L of the coexisting SU-FMI region, the induced Zeeman exchange field h0, and the induced  superconducting coherence length ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In particular we analyze the evolution of the subgap spectrum  as a function of the length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Interestingly, we have found that depending on the ratio h0/∆, the  emerging ABS can eventually cross below the Fermi energy at certain critical values Lc, and induce  spin-and fermion parity-changing quantum phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We argue that this type of device  constitute a promising highly-tunable platform to engineer subgap ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  INTRODUCTION  The interplay of superconductivity and magnetism at  the microscopic scale has attracted a great deal of at-  tention in recent years [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  For instance, the Yu-  Shiba-Rusinov (YSR) states [5–7] arising from the ex-  change interaction of an atomic magnetic moment in con-  tact with a superconductor, have been proposed as fun-  damental building blocks to engineer quantum devices  with topologically non-trivial ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In partic-  ular, the so-called “Shiba chains” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', one-dimensional  arrays of magnetic atoms deposited on top of a clean  superconductor) are systems predicted to support Ma-  jorana zero-modes at the ends of the chain [8–10], and  could be used in topologically-protected quantum com-  putation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Low-temperature scanning-tunneling  microscopy (STM) experiments have confirmed the pres-  ence of intruiguing zero-energy end-modes [11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Other systems where the competition of superconduc-  tivity and magnetism at the nanoscale generates ex-  otic subgap states are superconductor (SU)- ferromag-  net (FM) heterostructures, such as SU-FM-SU Josephson  junctions and SU-FM proximity devices [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Subgap  states generated in these structures are usually referred  to as Andreev bound states (ABS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' More recently, a novel  class of hybrid device, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', semiconductor (SE) nanowire  systems combined with superconductors and ferromag-  netic insulator (FMI) materials have been fabricated us-  ing molecular-beam epitaxy techniques [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' These  SE-SU-FMI hybrid structures allow to build nanostruc-  tures with specific tailored properties which are impossi-  ble to obtain with the isolated individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Despite the evident differences between the abovemen-  x  z  SC Bulk  FMI  Semiconductor  L  x  z  L  h0  Magnetic profile  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Schematic representation of the SC-FMI heterostruc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  tioned physical systems, from the theoretical perspective  they can be described within the same unified theoretical  model combining superconductivity and local exchange  fields at the microscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The emerging subgap  states (which can be referred to as either YSR states or  ABS, depending on the context) appear symmetrically  around the Fermi level EF , and localize spatially around  the impurity or the FM region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Their energy-position  within the gap depend on the value of the exchange  field and on other experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Interest-  ingly, whenever one of these states crosses EF , a spin-  and parity-changing quantum phase transition, usually  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='03967v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='supr-con]  10 Jan 2023  2  known as the “0 − π” phase transition, occurs [1, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In the case of atomic “Shiba impurities” or ultra-short  SU-FM-SU junctions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', junctions in which the length  L of the FM region is much smaller than λF , the Fermi  wavelength of the superconductor [23]), it is customary  to consider the magnetic scatterer as a point-like classical  spin S located at the point R0, interacting via a contact  s-d exchange interaction HZ = J(r) S·s(r) with the host  superconducting electrons [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Here J(r) = J0δ(r − R0)  is the local exchange potential and s(r) is the spin den-  sity vector of the electronic fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Subsequent theoretical  works considered atomic-sized systems with finite- (al-  beit short-ranged) exchange interactions with spherical  symmetry [7, 24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In that case, theory predicts the  existence of multiple YSR states labelled by their orbital  momentum ℓ, a prediction that has been recently ob-  served in STM experiments [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The behavior of subgap states and the associated 0−π  quantum phase transitions has also been studied in the  opposite limit L ≫ λF in the context of ballistic SU-  FM-SU Josephson junctions with generic spin-dependent  fields in the sandwiched region [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In this case the  results differ from the well-known results of YSR states  due to the finite extension of the magnetic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  particular, the subgap spectrum of long SU-FM-SU junc-  tions with zero phase difference is known to be double  degenerate [19, 31], showing the inherent complexity of  these hybrid heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' On the experimental side,  the possibility to engineer and control the position of the  subgap states by a modification of the fabrication para-  maters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the length L or exchange field h0 via dif-  ferent FM materials) opens interesting perspectives for  potential electronic devices, where the precise knowledge  of the subgap spectrum is crucial to control their trans-  port properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Motivated by the experimental developments men-  tioned above, in this work we study the subgap states  emerging in one-dimensional (1D) SE-SU-FMI het-  erostructures where the SU and the FMI layers simul-  taneously generate coexisting proximity-induced pairing  and exchange interactions over a finite and arbitrary  length L in the SE nanowire, as schematically shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This coexistence is a crucial aspect of this device,  which makes it unique and different from the abovemen-  tioned SU-FM-SU junctions, where such overlap occurs  only at the SU-FM interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Our main goal in this work  is to study and understand the behavior of the subgap  ABS in this device as a function of the experimentally rel-  evant parameters of the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the length L of the  FMI region and the magnitude of the induced exchange  field h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As mentioned above, a device similar to that  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1 has been recently experimentally real-  ized in SE nanowires with epitaxially-grown SU and FMI  layers [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' While the main interest of that work was  the fabrication of a device with non-trivial topological SU  ground state hosting Majorana zero modes, here we will  study the regime of parameters favoring a topologically-  trivial ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  As we will show below (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  II), this case is already very complex and rich as a result  of the antagonistic SU and FM interactions and, to the  best of our knowledge, the detailed behavior of subgap  states and the quantum phase diagram emerging in such  a system have not been explicitly studied before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In Section II, we  introduce the model representing a 1D SE-SU-FMI hy-  brid nanowire, discuss the solution to the Bogoliubov-  de Gennes equations for the subgap states, and derive  a generic equation for the subgap spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In Sec-  tion III, we analyze the results in two specific limits,  where we recover well-known results: a) the semiclassical  limit, where the superconducting coherence length ξ is  much larger than the Fermi wavelength λF , and b) the  atomic YSR limit, in which the exchange-field induced  by the FMI region becomes a delta-function potential:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', infinitesimally narrow (L ≪ λF ), and infinitely deep  (h0 ≫ EF ), in such a way that the product h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='L = J  is kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In both cases, well-known analytical  solutions to the subgap spectrum can be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  addition, we numerically solve the characteristic equation  for the subgap states and provide a generic description  of the subgap spectrum, not restricted to any of these  limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We find a rich behaviour of the subgap ABS,  where the competing FM exchange and SU pairing inter-  actions give rise to parity- and spin-changing quantum  phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Finally, in Section IV, we present a  summary and our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  THEORETICAL MODEL  We focus on the system schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  1, which represents a 1D SE-SU-FMI hybrid nanostruc-  ture of total length Lw, similar to those fabricated in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We model this system with the Hamil-  tonian H = Hw + H∆ + HZ, where  Hw =  �  σ  �  Lw  2  − Lw  2  dx ψ†  σ(x)  �  −ℏ2∂2  x  2m∗ − µ  �  ψ†  σ(x),  (1)  H ∆ = ∆  �  Lw  2  − Lw  2  dx  �  ψ†  ↑(x)ψ†  ↓(x) + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  �  ,  (2)  HZ =  �  Lw  2  − Lw  2  dx h(x)  �  ψ†  ↑(x)ψ↑(x) − ψ†  ↓(x)ψ↓(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (3)  Here Hw is the Hamiltonian of a single-channel SE  nanowire of length Lw, in which the fermionic operator  ψσ(x) creates an electron at position x with spin projec-  tion σ =↑, ↓ and effective mass m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The parameter µ is  the chemical potential, which can be experimentally var-  ied applying external gates beneath the nanostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The terms H∆ and HZ represent, respectively, the  proximity-induced pairing interaction encoded by the pa-  rameter ∆, and the Zeeman exchange interaction intro-  duced by the FMI and described by a space-dependent  exchange field h(x), which we assume oriented along the  3  z direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Moreover, since these interac-  tions are externally induced into the semiconductor, we  make the additional assumption that ∆ is unaffected by  the presence of h(x) (a renormalized value of ∆ does not  change qualitatively our results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As mentioned before,  these two terms can be effectively induced by the pres-  ence of epitaxially-grown SU and FMI shells in contact  with the SE nanowire [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' It has been experimen-  tally confirmed [21] that the FMI shell (EuS in that case)  consists of a single magnetic monodomain, and there-  fore modelling this layer by the Hamiltonian HZ is a  reasonable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In addition, the epitaxially-  generated interfaces are essentially disorder-free, a neces-  sary condition to produce a proximity-induced hard-gap  [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This feature allows to neglect the effects of disorder  and considerably simplifies the theoretical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The presence of both, a hard proximity-induced super-  conductor gap and an effectively induced Zeeman field,  in these nanowires have been reported in transport mea-  surements in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In addition, note that  in the above model we have neglected the effect of the  Rashba spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' While this interaction is  crucial for the emergence of a topologically non-trivial  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', D class) superconducting phase supporting Majo-  rana zero-modes [34], here we will focus strictly on the  topologically-trivial ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As we will show be-  low, the competition of SU and FM interactions make  this system already very complex and interesting in it-  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We note that since the total single-particle fermionic  spin along z  sz = 1  2  �  Lw  2  − Lw  2  dx  �  ψ†  ↑(x)ψ↑(x) − ψ†  ↓(x)ψ↓(x)  �  ,  (4)  is a conserved quantity which verifies [sz, H] = 0, we  can label the electronic eigenstates of H with σ = {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Therefore, we introduce the following Nambu spinors  Ψ↑(x) =  � ψ↑(x)  ψ†  ↓(x)  �  ,  Ψ↓(x) =  � ψ↓(x)  ψ†  ↑(x)  �  ,  (5)  related to each other via the charge-conjugation transfor-  mation Ψ¯σ(x) = KτxΨσ(x), where τx is the 2 × 2 Pauli  matrix, and K is the complex conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  terms of these spinors the Hamiltonian writes  H = 1  2  �  σ  �  Lw  2  − Lw  2  dx Ψ†  σ(x)HBdG,σ(x)Ψσ(x),  (6)  where the Bogoliubov-de Gennes (BdG) Hamiltonian is  defined as  HBdG,σ =  �  − ℏ2∂2  x  2m − µ + σh(x)  σ∆  σ∆  ℏ2∂2  x  2m + µ + σh(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (7)  In this expression, the spin projection σ =↑ (↓) on  the left-hand side corresponds to the + (−) sign in  the definition of the BdG matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Using the above  charge-conjugation transformation, we note that the  BdG Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (7) verifies the following symme-  try transformation  KτxHBdG,σ = −H∗  BdG,¯σKτx,  (8)  and therefore, provided χσ(x) is a solution of the BdG  eigenvalue equation  HBdG,σ(x)χσ(x) = Eσχσ(x),  (9)  with eigenenergy Eσ, the transformed spinor χ¯σ(x) =  Kτxχσ(x), is also a solution with eigenenergy E¯σ = −Eσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In what follows, we assume for simplicity the thermo-  dynamic limit Lw → ∞, and we focus on the features  introduced by the magnitude and spatial dependence of  h (x), which is crucial for the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In addi-  tion, we assume the following step-like spatial profile for  the exchange field  h(x) =  �  −h0  if |x| < L  2 ,  0  if |x| ⩾ L  2 ,  (10)  which models a uniform FMI shell of length L in contact  with the SE nanowire (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This choice for h(x)  allows to split the problem into regions with either |x| <  L  2 or |x| > L  2 , with generic exponential solutions  χσ(x) ∼  �  ασ  βσ  �  eikx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (11)  Linear combinations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (11), with appropriate coeffi-  cients and with allowed values of k for each region, must  be built so that continuity of the total wavefunction and  its derivative at the interfaces is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' With this re-  quirement, the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (9) is finally obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Note that the BdG Hamiltonian (7) is even under space  inversion x → −x, and therefore its eigenstates must be  even or odd under this transformation of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  This symmetry allows to reduce the number of unknowns  of the problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', coefficients of the linear combinta-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Replacing the above ansatz Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (11) into the BdG  eigenvalue Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' and looking for localized solutions  with energy within the gap |Eσ| < ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' we obtain the fol-  lowing expressions for the eigenstates belonging to the  even-symmetry subspace:  4  χe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='σ  �  x > L  2  �  = Ae  1σ  �  1  σe−iϕσ  �  e−κσx + Ae  2σ  �  1  σeiϕσ  �  e−κ∗  σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (12)  χe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='σ  �  −L  2 ≤ x ≤ L  2  �  = Be  1σ  �  1  σe−ησ  �  cos kσx + Be  2σ  �  1  σeησ  �  cos ¯kσx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (13)  and the following expressions for the odd-symmetry eigenfunctions  χo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='σ  �  x > L  2  �  = Ao  1σ  �  1  σe−iϕσ  �  e−κσx + Ao  2σ  �  1  σeiϕσ  �  e−κ∗  σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (14)  χo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='σ  �  −L  2 ≤ x ≤ L  2  �  = Bo  1σ  �  1  σe−ησ  �  sin kσx + Bo  2σ  �  1  σeησ  �  sin ¯kσx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (15)  where the coefficients {Aν  1σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Aν  2σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Bν  1σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Bν  2σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' with ν =  {e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' o},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' are unknowns to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In addition, in the  above expressions we have introduced the parametriza-  tion  cos ϕσ = Eσ  ∆ ,  (16)  cosh ησ = Eσ + σh0  ∆  ,  (17)  where we fix the definition of ϕσ to the interval ϕσ ∈  (0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The phase variable ϕσ is associated to the An-  dreev reflection taking place at the interface xb = L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Note that the parametrization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (17) makes sense  whenever the right-hand side is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' If this condi-  tion is not satisfied, one can always use the symmetry  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (8) to send Eσ → −E¯σ and σ → ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In addition, note  that whenever 1 ≤ (Eσ + σh0) /∆ the parameter ησ is  purely real, while for 0 < (Eσ + σh0) /∆ < 1 it is purely  imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Finally, we have introduced the quantities  κσ ≡ −ikF  �  1 + 2i  kF ξ sin ϕσ,  (18)  kσ ≡ kF  �  1 +  2  kF ξ sinh ησ,  (19)  ¯kσ ≡ kF  �  1 −  2  kF ξ sinh ησ,  (20)  and the definition of the coherence length of the  (proximity-induced) 1D superconductor ξ = ℏvF /∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' No-  tice also that the spatial dependence of the wavefunc-  tions in the region x < −L/2 can be readily obtained by  symmetry from the relations χe,σ (x) = χe,σ (−x), and  χo,σ (x) = −χo,σ (−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We can intuitively understand the form of the scatter-  ing solutions in the regions x > L/2 and x < −L/2 in the  limit kF ξ ≫ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the semiclassical limit, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='III A),  where the momentum κσ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (18) can be expanded  as κσ ≃ −ikF + sin ϕσ/kF ξ, and the eigenfunctions Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (12) and (14) take the form  χν,σ  �  x > L  2  �  ≈  �  Aν  1σ  �  1  σe−iϕσ  �  eikF x+  +Aν  2σ  �  1  σeiϕσ  �  e−ikF x  �  e− sin ϕσx  ξ  , (21)  with ν = {e, o}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In this way, it becomes evident that the  component proportional to Aν  1σ corresponds to a right-  moving particle ∼ eikF x while Aν  2σ corresponds to a left-  moving particle ∼ e−ikF x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In addition, the wavefunctions  exponentially decay into the superconductor within a lo-  calization length λloc = ξ/ sin ϕσ = ξ/  �  1 − (Eσ/∆)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  These results are in complete agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' [32],  where the spectrum of SU-FM-SU Josephson junctions  has been recently studied as a function of the length L  of the FM region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, in our case, the presence of  a finite pairing gap ∆ in the region −L/2 < x < L/2 (as  opposed to the assumption ∆ = 0 in the FM region in  that work), gives rise to important differences which we  analyze below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Continuity conditions at the interface  We now impose the continuity conditions on the wave-  function and its derivative at the boundary xb = L/2:  χν,σ  �  x−  b  �  = χν,σ  �  x+  b  �  (22)  ∂xχν,σ  �  x−  b  �  = ∂xχν,σ  �  x+  b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (23)  Note that the same equations are obtained by symme-  try at the other boundary −xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Inserting the solutions  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (12)-(15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' we can express the continuity equations  in matrix form as  5  �  1  σe−iϕσ  σe−iϕσ  1  � �  aν  1σ  aν  2σ  �  =  �  1  σe−ησ  σe−ησ  1  � �  Fν  � kσL  2  �  0  0  Fν  � ¯kσL  2  �  � �  bν  1σ  bν  2σ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (24)  −  �  1  σe−iϕσ  σe−iϕσ  1  � �  κσ  0  0  κ∗  σ  � �  aν  1σ  aν  2σ  �  = −s(ν)  �  1  σe−ησ  σe−ησ  1  � �  kσGν  � kσL  2  �  0  0  ¯kσGν  � ¯kσL  2  �  � �  bν  1σ  bν  2σ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (25)  where we have conveniently redefined the unknown coef-  ficients as  Aν  1σ → eκσL/2aν  1σ  Bν  1σ → bν  1σ  (26)  Aν  2σ → σeκ∗  σL/2e−iϕσaν  2σ  Bν  2σ → σe−ησbν  2σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (27)  in order to give these equations a more symmetric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In addition, we have used the notation s(ν) = +1(−1) for  ν = e(o), and Fe(x) = Go(x) ≡ cos(x), Ge(x) = Fo(x) ≡  sin(x) for compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In each subspace (even or odd) we have four equa-  tions and four unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Eliminating the variables  (bν  1σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' bν  2σ)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' and writing the equation for (aν  1σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' aν  2σ)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' we  find from the nullification of the corresponding determi-  nant the following equations:  cosh ησ cos ϕσ − 1  sinh ησ sin ϕσ  =  �  �  �  �  �  �  �  �  �  �  �  �  �  |κσ|2 −  �  Kσ + ¯Kσ  �  Re κσ + Kσ ¯Kσ  � ¯Kσ − Kσ  �  Im κσ  (even-symmetry subspace),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  |κσ|2 +  �  Qσ + ¯Qσ  �  Re κσ + Qσ ¯Qσ  �  Qσ − ¯Qσ  �  Im κσ  (odd-symmetry subspace),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (28)  where we have defined the quantities  Kσ = kσ tan  �kσL  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (29)  ¯Kσ = ¯kσ tan  �¯kσL  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (30)  Qσ = kσ cot  �kσL  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (31)  ¯Qσ = ¯kσ cot  �¯kσL  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (32)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28), the eigenvalue Eσ for each subspace is  finally obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This equation summarizes our main the-  oretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In the next Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III we analyze the nu-  merical solution and different important limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Spin-changing quantum phase transitions  We now focus on the quantum phase transitions which  occur whenever one of the subgap states crosses EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' To  that end, let us analyze the spinors defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (5),  and consider the norm of the “up” spinor  q↑ =  � Lw/2  −Lw/2  dx  �  ψ†  ↑ (x) ψ↑ (x) + ψ↓ (x) ψ†  ↓ (x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Recalling the definition of the single-particle sz operator  [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (4)], it is straightforward to associate these two  quantities through the relation q↑ = 2sz − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Since sz  is a conserved quantity, so is the norm q↑ of the “up”  Nambu spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This connection allows to interpret q↑ as  an effective “conserved charge”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Similar considerations  allow to write the relation q↓ = −2sz − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Due to the  particle-hole relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (8), the information about sz  can be obtained with either q↑ or q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' A more symmetric  form involving both conserved charges is  sz = q↑ − q↓  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (33)  While redundant, this expression makes explicit that in  the spin-symmetric case q↑ = q↓, the net spin sz must  vanish (sz = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We now return to Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (7), and let us  separate the effect of the proximity-induced Zeeman field,  by writing it as HBdG,σ = H0,σ + Vσ, where  H0,σ =  �  − ℏ2∂2  x  2m − µ  σ∆  σ∆  ℏ2∂2  x  2m + µ  �  ,  (34)  Vσ =  �  σh(x)  0  0  σh(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (35)  In this form, we can interpret the effect of the exchange  field as a “perturbation” on an otherwise homogeneous  6  1D superconductor represented by H0,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Therefore, the  full and the unperturbed single-particle Green’s functions  in this problem are respectively defined as  Gσ (z) = [z − H0,σ − Vσ]−1 ,  (36)  G0,σ (z) = [z − H0,σ]−1 ,  (37)  From here, the total number of effective “up” charges  Q↑ induced in the ground state due to the potential Vσ,  compared to the unperturbed homogeneous SU wire, can  be computed as  ∆Q↑ = − 1  π Im Tr  � ∞  −∞  dϵ nF (ϵ) ∆G↑ (ϵ + iδ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (38)  where ∆Gσ (z) ≡ Gσ (z) − G0,σ (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' At T = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (38)  can be easily computed from the well-known expression  of the Friedel sum rule [35]  ∆Q↑ = 1  π  � 0  −∞  dϵ  �∂η↑ (ϵ)  ∂ϵ  − ∂η0,↑ (ϵ)  ∂ϵ  �  (39)  = η↑ (0) − η0,↑ (0)  π  (40)  where we have defined the phase shifts [32, 35]  ησ (ϵ) = Im ln det Gσ (ϵ + iδ) ,  (41)  η0,σ (ϵ) = Im ln det G0,σ (ϵ + iδ) ,  (42)  and where we have used that the phase shifts vanish in  the limit ϵ → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Since the system is non-interacting, the Green’s func-  tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (36) can be written in terms of single-particle  eigenstates |α, σ⟩, with α a generic label, as  Gσ (z) =  �  α  |α, σ⟩ ⟨α, σ|  z − Eα,σ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (43)  Therefore, after simple algebra, and using the above re-  lations and the fact that in the absence of magnetic field  sz = 0 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (33)], the total Sz of the ground state is  Sz = ∆Q↑  2  = 1  2  ��  α  Θ (−Eα,↑) −  �  α′  Θ  �  −E0  α′,↑  �  �  ,  (44)  where Θ(ϵ) is the unit-step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The above expres-  sion allows to interpret the total Sz of the ground state  as a function of the “up” Nambu spinors with energy be-  low EF = 0, as compared to the (unperturbed) situation  h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Since the effective charges are quantized in inte-  ger numbers, the total spin Sz can only change in discrete  “jumps” of 1/2 whenever a subgap state with projection  up crosses below EF (note that we have defined dimen-  sionless spin operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This interpretation makes sense  since the ground state becomes spin-polarized when the  exchange field h0 becomes large enough [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the Zeeman  energy of up-spin electron is decreased, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (3) and  (10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' While the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (44) has been obtained re-  cently by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' [32], we note that here we  have rederived it in a different physical situation which  allows a more generic regime of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  RESULTS  We start this section by analyzing different limits of  the general result given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In particular, in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III A we focus on the semiclassical limit, and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  III B we study the atomic limit, where we recover the  YSR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In both cases, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) reduces to well-  known analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Finally in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III C we show  results corresponding to intermediate regimes, obtained  by solving numerically Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Semiclassical limit  Generally speaking, the semiclassical limit is verified  when EF is the largest scale of the problem [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In par-  ticular, the condition EF ≫ ∆ (which is very well satis-  fied in most experimental systems) can be expressed as  kF ξ ≫ 1, recalling that after linearization of the normal  quasiparticle dispersion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', ϵk,σ ≃ ±ℏvF k, where the  +(−) sign corresponds to right-(left-)movers, the Fermi  energy can be approximated as EF ≃ ℏkF vF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In this  case, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (18)-(20) reduce to  rσ ≡ κσ  kF  ≃ −i + sin ϕσ  kF ξ ,  (45)  ζσ ≡ kσ  kF  ≃ 1 + sinh ησ  kF ξ  ,  (46)  ¯ζσ ≡  ¯kσ  kF  ≃ 1 − sinh ησ  kF ξ  ,  (47)  to leading order in O(kF ξ)−1, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) becomes  cosh ησ cos ϕσ − 1  sinh ησ sin ϕσ  ≃ s(ν)  1 + tan  �  kF Lζσ  2  �  tan  �  kF L¯ζσ  2  �  tan  �  kF Lζσ  2  �  − tan  �  kF L¯ζσ  2  � ,  = s(ν) cot  �L sinh ησ  ξ  �  ,  (48)  where  we  have  used  the  trigonometric  identity  tan (x + y) = (tan (x) + tan y)/(1 + tan x tan y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In gen-  eral this transcendental equation cannot be solved ana-  lytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, in the regime of parameters EF ≫  h0 ≫ ∆, where the exchange field h0 is much larger  than ∆, we can write cosh ησ ≈ sinh ησ ≈  �� h0  ∆  �� ≫ 1  [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (16) and (17) ], and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (48) reduces to  cot ϕσ = s(ν) cot (Lh0/ℏvF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Equivalently we can write  this result as  arccos  �Eσ  ∆  �  =  �  �  �  �  �  �  �  �  �  LEσ  ℏvF  + σ Lh0  ℏvF  + 2nπ,  (even)  LEσ  ℏvF  + σ Lh0  ℏvF  + (2n + 1) π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (odd)  (49)  This result can be interpreted as a semiclassical Bohr-  Sommerfeld quantization condition for particles which  7  perform a complete a closed loop in the region −L/2 <  x < L/2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In particular, it exactly coincides with the-  oretical results obtained for SU-FM-SU Josephson junc-  tions with a normal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', ∆ = 0) FM region [30–32], the  only difference being that within our theoretical treat-  ment, we can distinguish the symmetry of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The similarity of these results can be rationalized noting  that considering a normal sandwiched region in an SU-  FM-SU junction corresponds to taking the limit h0 ≫ ∆  in our Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (48) while keeping the ratio Eσ/∆ finite (since  Eσ corresponds to a subgap state, it is always bounded  by ∆), thus resulting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This shows that our  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) is a generic relation describing different situa-  tions regardless of the magnitude of the ratio h0/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  YSR-impurity limit  We now consider the atomic YSR (or simply Shiba)  limit, in which the exchange profile becomes point-like,  L → 0, while h0 → ∞, in such a way that the product  Lh0 = J = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Under these assumptions the magnetic  barrier becomes a delta function and the Hamiltonian in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (3) can be written as  HZ ≈ −J  � ∞  −∞  dx δ(x)  �  ψ†  ↑(x)ψ↑(x) − ψ†  ↓(x)ψ↓(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (50)  In this case, it is easy to see that the odd-symmetry solu-  tions decouple from the above Hamiltonian (50), as they  vanish at x = 0, and only even solutions can couple to  the delta-potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  As in the previous section, note that the limit h0 → ∞  implies cosh ησ ≈ sinh ησ ≈  �� h0  ∆  �� ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, the limit  h0 → ∞ is not compatible with the semiclassical ap-  proach, as it violates the requirement h0 ≪ EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' There-  fore we cannot use here our previous Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Instead,  we must first take the limit ησ ≫ 1 together with the  limit L → 0, which applied to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (19) and (20) yield  kσ → kF  �  2h0  ℏvF kF  ,  (51)  ¯kσ → ikF  �  2h0  ℏvF kF  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (52)  In addition Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (29)-(32) become  Kσ → kF h0L  ℏvF  = kF ρ0J,  (53)  ¯Kσ → −kF h0L  ℏvF  = −kF ρ0J,  (54)  where the expressions for the density of states per spin  of 1D quasiparticles at the Fermi energy ρ0 = 1/ℏvF ,  and the exchange coupling J = h0L, have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Replacing these expressions into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) for the even-  symmetry solutions, we obtain  σ  Ee  σ  �  ∆2 − (Ee)2  σ  = 1 − (ρ0J)2  (2Jρ0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (55)  From this expression, we can easily solve for Ee  σ  Ee  σ  ∆ = σ 1 − (ρ0J)2  1 + (ρ0J)2 ,  (56)  which is the well-known expression for the energy of YSR-  impurity subgap level [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This result indicates that any  finite value of J produces a YSR in-gap state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This type  of subgap YSR states has been observed in several STM  experiments on atomic magnetic adsorbates on supercon-  ducing substrates [27, 37–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  For completeness, and in order to illustrate the general  scope of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28), here we also show the result for the  YSR odd states for a small (but finite) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Using similar  approximations, we obtain the expression  Eo  σ  ∆ = σ  1  �  1 +  �ρ0Jk2  F L2  6  �2 ,  (57)  where it becomes evident that in addition to a finite value  of J, a finite value of kF L is needed to observe an odd-  symmetry subgap YSR state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Subgap ABS spectrum in generic cases  As stated in Section II, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) implicitly defines the  energy of the subgap states as a function of the param-  eters h0/∆ , kF ξ, and kF L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' These parameters can be  directly or indirectly controlled in experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the  parameter h0 can be controlled by modifying the FMI  material, the length L of the FMI region can be modified  varying the length Lw of the semiconductor via vapor-  liquid-solid (VLS) method and subsequent evaporation of  the FMI material [20], and the parameter kF in the semi-  conductor can be varied by changing the SE material or  by introducing external gates to modify the chemical po-  tential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Therefore, due to this high degree of tunability,  hybrid heterostructures might offer a unique platform to  produce and control engineered subgap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Probably  the easiest way to experimentally control the subgap elec-  tronic structure is by producing different devices with the  same FMI material and different lengths L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Therefore, in  this section we show the numerical solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28)  with fixed parameters h0/∆ and kF ξ (which control the  “operation regime” of the device), and calculate both the  energy dependence of the even- and odd-symmetry ABS,  and the total spin Sz of the device as a function of L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=',  dimensionless variable kF L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Generally speaking, the overall evolution of the ABS  spectrum from L = 0 to L → ∞ is quite complex and de-  serves a detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2, as the  8  parameter kF L increases, more and more subgap states  emerge from the gap edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This behavior is reminiscent  of a quantum particle in a square-well potential, tipically  taught in introductory quantum mechanics courses [42],  where increasing the width L of the well increases the  number of allowed bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In our case, the emer-  gence of new ABS as kF L increases can be intuitively  understood in terms of a competition between supercon-  ductivity and magnetic field: the magnetic field tends  to break Cooper-pairs and to locally disrupt supercon-  ductivity in the magnetic region by introducing subgap  states that become macroscopic in number for large L,  eventually populating the whole gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We note that for any finite L, even- and odd-symmetry  states are generically non-degenerate (except at isolated  points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, as it is clear from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2 and 3, their  energy difference (evidenced as oscillations of the blue  and red lines around the semiclassical value) decreases  very rapidly and the solutions become degenerate in the  limit L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This transition from non-degenerate YSR  states in the limit L → 0, to double degenerate ABS  states for L → ∞ has been discussed in previous works  on ballistic SU-FM-SU junctions [19, 30–32], and in the  case of extended Shiba impurities in 1D nanowires [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  It is also clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2, and more dramatically  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In our 1D geometry, this degeneracy  in the limit L → ∞ can be intuitively understood by  linearizing the spectrum around the Fermi energy, and  expressing the original fermionic operators in terms of  right- and left-moving fields slowly varying in the scale  of k−1  F  [44], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', ψσ (x) ≈ eikF xψR,σ (x) + e−ikF xψL,σ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The slowly-varying fields ψR,σ(x) and ψL,σ(x) are two  independent chiral fermionic fields obeying the usual an-  ticommutation relations, in terms of which the original  Hamiltonian becomes [43]  Hw ≈  �  σ  � ∞  −∞  dx  �  −iℏvF ψ†  R,σ(x)∂xψR,σ(x)  + iℏvF ψ†  L,σ(x)∂xψL,σ(x)  �  (58)  H ∆ ≈ ∆  � ∞  −∞  dx  �  ψ†  R,↑(x)ψ†  L,↓(x) + ψ†  L,↑(x)ψ†  R,↓(x) + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  �  ,  (59)  HZ ≈ −  � ∞  −∞  dx h0  �  ψ†  R,↑(x)ψR,↑(x) − ψ†  R,↓(x)ψR,↓(x)  + ψ†  L,↑(x)ψL,↑(x) − ψ†  L,↓(x)ψL,↓(x)  �  ,  (60)  where oscillating terms proportional to e±2ikF x have been  neglected as they cancel out in the limit L → ∞ due to  destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Defining the new chiral Nambu  spinors  Ψ1,σ(x) =  � ψR,σ(x)  ψ†  L,¯σ(x)  �  ,  Ψ2,σ(x) =  � ψL,σ(x)  ψ†  R,¯σ(x)  �  ,  (61)  the Hamiltonian of the system can be expressed in terms  of two decoupled chiral sectors  H = 1  2  �  σ=↑,↓  �  j=1,2  � ∞  −∞  dx Ψ†  j,σ(x)Hj,σ(x)Ψj,σ(x), (62)  with the definitions of the chiral BdG Hamiltonians  Hj,σ =  �  (−1)jivF ∂x − σh0  σ∆  σ∆  (−1)j+1ivF ∂x − σh0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (63)  The Nambu spinors Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (61) define two independent chi-  ral subspaces related by the inversion symmetry of the  original Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', under the space inversion op-  eration x ↔ −x, the fermionic operators transform as  ψL,σ(x) ↔ ψR,σ(x), and consequently we conclude that  Ψ1,σ(x) ↔ Ψ2,σ(x), which must then be degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  addition, the particle-hole symmetry Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (8) in this rep-  resentation produces Ψ1,σ(x) → Ψ2,¯σ(x), and therefore  H1,σ → −H2,¯σ, implying that the solutions verify the  particle-hole symmetry property E1,σ = −E2,¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' More-  over, notice that assuming periodic boundary conditions,  the problem can be solved with the solutions ψR,σ(x) ∼  eikx and ψL,σ(x) ∼ e−ikx, and the dispersion relation  becomes E1,σ(k) = E2,σ(k) = ±  �  (ℏvF k)2 + ∆2 − σh0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  From here, a renormalized quasiparticle gap 2∆ren =  2 |∆ − h0| is obtained, consistent with our previous re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In terms of the chiral Nambu spinors, the most general  solution is the linear combination  Ψσ(x) = AeikF xΨ1,σ(x) + Be−ikF xΨ2,σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (64)  This is exactly the same form that can be obtained by  combining the degenerate even and odd solutions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (13) and (15) in the semiclassical limit where kF ξ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  From the analysis of the linearized Hamiltonian, we  conclude that the degeneracy in the limit L → ∞  arises from the absence of chirality-breaking terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=',  terms ∼ Ψ†  1,σ(x)Ψ2,σ(x) arising from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', single par-  ticle backscattering terms ψ†  R,σ(x)ψL,σ(x) or Cooper-  pairing channels ψ†  R(L),↑(x)ψ†  R(L),↓(x) carrying momen-  tum ∓2kF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  For this to occur, the magnetic FMI re-  gion must be uniform and its length L must be much  larger than k−1  F  in order to produce the required cancel-  lation of the rapidly oscillating exponentials ∼ e±2ikF x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In other words, the product kF L must be kF L ≫ 1,  consistent with our numerical results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Only for small values of kF L, where this destructive in-  terference is incomplete, residual couplings of the type  ∼ Ψ†  1,σ(x)Ψ2,σ(x) remain, and the degeneracy is lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Finally, we stress that the degeneracy in the limit L → ∞  is a robust property to the presence of interactions, as  shown in previous works [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  On the other hand, in the limit L → 0 and for any  finite value of the Zeeman field h0, both (even and odd)  solutions converge to Eσ/∆ → ±1, indicating that the  FMI region is no longer relevant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', it physically drops  9  −1  1  0  E/∆  −1  1  0  0  10  20  30  40  50  0  1  2  3  4  5  6  kF L  Sz  0  2  4  6  8  10  12  14  kF L  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Energy of the Andreev bound states (upper panel) and total spin Sz(lower panel) as a function of kF L, for kF ξ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='8  and h0/∆ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='0 (left panel) and kF ξ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='4 and h0/∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='1 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Blue and red colors correspond to even and odd states  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lines starting from the top gap edge at positive energy E/∆ = 1 (bottom gap edge at negative energy E/∆ = −1)  correspond to up (down) spin projections of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' For smaller values of kF L (right panel), plateaus corresponding to  regions of integer and half-integer spin are more separated and might become easier to observe in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  from the description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, the behavior near L = 0  is quite different for each case: while the even-symmetry  solution tends to E/∆ → 1 as [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (56)]  Ee  σ  ∆ ≈ σ  �  1 − 2  �h0L  ℏvF  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  �  ,  (65)  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (57) we conclude that the odd solution behaves  as  Eo  σ  ∆ ≈ σ  �  1 − 1  2  �h0k2  F L3  6ℏvF  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  �  ,  (66)  therefore approaching the gap edge much faster as L → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Besides the general features of the spectrum discussed  up to this point, its evolution as L increases is strongly  affected by the values of the parameters kF ξ and h0/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In what follows, we analyze their effects on Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2 and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Effect of varying the parameter kF ξ  This parameter can be considered as a “knob” which  tunes the device from the semiclassical behavior (kF ξ  large, see left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2) into a “quantum” regime  (kF ξ small, see right panel) where the spectrum is dom-  inated by quantum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The hybrid heterostruc-  ture under study is promising in this sense since, due to  the combination of materials (in particular, semiconduc-  tors with a much smaller kF as compared to metals), it  is in principle possible that kF ξ can be experimentally  controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In addition, kF could be further modified by  introducing external gating leads (through the modifica-  tion of the chemical potential µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' To illustrate the dra-  matic changes in the spectrum as kF ξ varies, in Fig 2 we  show the numerically obtained subgap spectra as a func-  tion of kF L for kF ξ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='8 and h0/∆ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='0 (left panel),  for and kF ξ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='4 and h0/∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='1 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Solid  blue (red) lines correspond to even(odd)-symmetry solu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Moreover, since we always assume h0 > 0, solu-  tions emerging from the top edge E/∆ = 1 (bottom edge  E/∆ = −1) correspond to spin up (spin down) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In addition, note the reflection symmetry of the solutions  around the horizontal E = 0 axis, a consequence of the  particle-hole symmetry of the BdG Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Upon decreasing kF ξ, the subgap spectrum becomes  much more intricate due to the enhanced even-odd  energy-splitting, which results in an amplified oscillatory  behavior of the ABS (we have reduced the range of kF L in  the right panel for clarity in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Unfortunately,  in the regime kF ξ ∼ 1 no analytic expressions for the  subgap ABS are possible, but qualitative considerations  10  can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In fact, the amplified oscillations can  be traced back to the larger energy dependence of the  momenta Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (18)-(20) as kF ξ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Then, whereas  for large kF ξ all these quantities converge to a static (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=',  energy-independent) value ∼ kF , the limit of small kF ξ  produces a larger effect on the space-dependence of the  wave functions through the exponential factors in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (12)-(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This in turn produces larger interference ef-  fects, and an enhanced lifting of the even-odd degener-  acy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  This phenomenological behavior enables interesting  possibilities, such as the chance to observe half-integer  spin (and fermion parity-switching) quantum phase tran-  sitions in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' To illustrate this effect, we  show the ground-state Sz transitions in the bottom pan-  els of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  2 in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  While for larger kF ξ, the  half-integer Sz steps are very narrow due to the almost-  degenerate even-odd solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the even and odd so-  lutions cross zero energy almost at the same value of  kF L), for smaller kF ξ the Sz transitions occur in well-  defined half-integer steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This behavior is well explained  by the enhanced lifting of the even-odd degeneracy, which  allows to observe one ABS crossing zero energy at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Effect of varying the parameter h0/∆  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3 we show the evolution of the subgap spectrum  as a function of kF L, for different values of the Zeeman  field h0/∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='54 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='2, and for a fixed relatively  large value kF ξ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='2, allowing to interpret these results  in terms of the semiclassical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Here we can  clearly distinguish three qualitatively different regimes:  a) the “weak field” regime h0 < ∆ (top panel) where  the ABS do not cross E = 0, b) the “intermediate field”  regime ∆ < h0 < 2∆ (middle panel) where the ABS can  evenually cross zero energy, and quantum phase transi-  tions can be induced, and finally c) the “strong field”  (2∆ < h0) regime (bottom panel), where the ABS can  be found anywhere in the region −1 < Eσ/∆ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In all  cases, the value of h0 determines the asymptotic limit to  which the ABS approach for large L (see dashed black  lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Below we briefly discuss the main fea-  tures of the spectrum in each regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Weak-field regime 0 < h0 < ∆:  This regime  is characterized by a Zeeman field which is not strong  enough to destroy the superconducting gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In this  case none of the ABS is able to cross E = 0 and in  the limit L → ∞ they asymptotically approach the  value Eσ/∆ → σ (1 − h0/∆) (see horizontal dashed black  lines), and therefore a renormalized gap remains (see top  panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' More quantitatively, in the semiclassi-  cal limit [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (48)] they obey the asymptotic expression  −1  1  0  h0  E/∆  −1  1  0  h0  E/∆  0  20  40  60  80  100  −1  1  0  kF L  E/∆  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Energy of the Andreev bound states as a function of  kF L for the three different values of h0 (h0/∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='54, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='2  for the lower, middle and upper panels) and kF ξ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Blue  and red colors correspond to even and odd states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Lines starting from negative (positive) energies correspond to  down (up) spin projections of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Note that the value  of h0/∆ sets the asymptotic limit for the Andreev states and  is crucial to determine the overall subgap spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  valid for kF L → ∞  Eν  σ  ∆ ≃ σ  �  �1 − h0  ∆ + π2  2  � ξ  L  �2 �  1 − s(ν)ξ  L  �  2∆  h0  − 1  �2�  � ,  (67)  with s(ν) = 1(−1) for ν = e(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  From here, we can  clearly see that whereas the even-odd averaged quanti-  ties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the semiclassical values) approach the asymp-  totic limit as L−2, the energy difference between even  and odd solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', the amplitude of the oscillation  around the semiclassical limit) decreases as L−3, and the  solutions become degenerate in the limit L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' On the  other hand, the quasiparticle gap in the limit L → ∞ is  renormalized to 2∆ren = 2 |∆ − h0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Note that this gap  renormalization is quite specific to this setup, and is not  present, for instance, in the case of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' [32], where the  magnetic region is normal and not superconducting, and  in addition the system corresponds to a “short” SU-FM-  SU junction with L < ξ, and therefore only few subgap  states are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Another feature of the weak-field regime is that the  ABS require a minimal length Lmin to emerge in the sub-  11  gap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This can be easily understood in terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  49, where a minimal magnetic phase, represented by the  product Lh0/ℏvF , must be accumulated in order to pro-  duce an observable in-gap ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Finally, concerning the  spin quantum number of the ground state, since none of  the ABS cross EF , no quantum phase transitions are ex-  pected according to the results of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' II B and the value  of the ground state spin remains a spin-singlet Sz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Intermediate field regime ∆ < h0 < 2∆: In this  case the Zeeman field h0 is sufficiently strong to force the  ABS to cross zero energy, eventually inducing quantum  phase transitions (see middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The n-th  critical value Lc,n can be obtained imposing the condition  Eσ = 0 on the semiclassical approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (48),  Lν  c,n = ξ  arctan  �  −s (ν) ∓  �� h0  ∆  �2 − 1  �  + nπ  �� h0  ∆  �2 − 1  ,  (68)  with s(ν) = 1(−1) for ν = e(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In this regime, the ABS follow the same asymp-  totic behavior as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (67), approaching Eσ/∆ →  σ (1 − h0/∆), although the overall subgap spectrum is  completely different due to the closing of the gap, and  due to the overlap of the E↑ and E↓ spectrum as L  increases beyond the first critical Lc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In fact, in the  regime L > Lc,0 the quasiparticle gap becomes com-  pletely populated (and washed away) by subgap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Moreover, we predict an accumulation of levels in the re-  gion −∆ + h0 < E < ∆ − h0, which can eventually form  a peak structure in the total density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Strong field regime 2∆ < h0: Finally, in this regime  (see bottom panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3), the asymptotic dashed lines  fall within the continuum and it is no longer possible to  obtain an analytic expression for the ABS behavior in  the limit L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As a result, the subgap ABS can be  found anywhere in the subgap region −1 < Eσ/∆ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In addition, we note that the minimal length required to  observe in-gap ABS has reduced to Lmin ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  SUMMARY AND CONCLUSIONS  In this work we have analyzed the subgap electronic  structure in the one dimensional SE-SU-FMI heterostruc-  ture schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 1, a novel physical  system recently fabricated using molecular beam epitaxy  techniques (MBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The main motivation to study this  type of hybrid systems is that, via a careful combina-  tion of different materials, the emergent characteristics  can be completely different from those of the individ-  ual components, providing a way to build devices with  tailored properties and specific functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In partic-  ular, much of the experimental effort has focused on the  realization of topological superconducting phases host-  ing Majorana zero modes, with possible applications in  topological quantum computing [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' A distinguish-  ing feature of these heterostructures is the coexistence  of antagonistic superconductor and ferromagnetic insu-  lating layers over a finite and arbitrary length L in a  semiconductor wire, a combination that confers unique  spectral properties which cannot be found in elemental  materials in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  In particular, we have modelled the hybrid struc-  ture  assuming  non-interacting  fermions  in  a  one-  dimensional single-channel nanowire under the effect of  two proximity-induced interactions: a SU pairing and  a space-dependent Zeeman exchange coupling [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (1)-(3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We have solved the associated Bogoliubov-de  Gennes equations and, by imposing standard continuity  conditions on the wave functions, we have obtained an  equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28)] defining the subgap ABS spectrum  of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  This single equation encodes our main  theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We stress that our approach is equiv-  alent to other works using the scattering-matrix formal-  ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We have analytically solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' (28) in two paradig-  matic limits: the semiclassical limit (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III A) and the  Yu-Shiba-Rusinov limit, typical of atomic magnetic mo-  ments interacting with a superconductor (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  both cases, we have been able to recover well-known ana-  lytical results, providing important sanity checks for our  theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' As a consequence of the symmetries  of the Hamiltonian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', inversion x → −x and sz spin  symmetries), it was possible to classify the solutions into  even- and odd-symmetry, and with sz labels σ =↑, ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In  particular, we note that the even-odd classification, aris-  ing in the present case due to the inversion symmetry of  the Hamiltonian, is nothing but the 1D analog of the clas-  sification in angular momentum eigenstates ℓ occurring  in 3D spherically-symmetric Hamiltonians [7, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We have studied the subgap spectrum of ABS as a  function of different parameters, namely: the length of  the magnetic region (through the dimensionless parame-  ter kF L), the strength of the Zeeman exchange induced  by the FMI (parameter h0/∆), and the superconducting  coherence length (parameter kF ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We stress that each  one of these parameters could in principle (directly or in-  directly) be controlled in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, due to  its potential relevance for on-going experimental efforts,  we have in particular focused our study on the evolution  of the subgap spectrum as a function of the length L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=',  as it is probably the easiest parameter to vary in experi-  ments), for fixed parameters kF ξ and h0/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The parame-  ter L can be controlled by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', changing the experimen-  tal growing conditions of the semiconductor nanowires  using the VLS growth method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2 and 3 we have  analyzed the evolution of the subgap spectrum in terms  of the parameter kF L for different values of h0/∆ and  kF ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Roughly speaking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' while kF ξ controls the “semi-  classical vs quantum” operation regime of the device,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  and the magnitude of the even-odd energy separation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  the parameter h0/∆ essentially controls the energy sep-  aration of the E↑ and E↓ solutions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' eventually enabling  many interesting physical phenomena such as the possi-  12  bility to observe multiple ABS crossing zero-energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' the  existence of multiple spin- and parity-changing quantum  phase transitions in the device,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' quasiparticle gap renor-  malization ∆ → ∆ren = |∆ − h0| in the limit of large  kF L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='. An important conclusion here is that in order  to experimentally observe a quantum phase transition,  the condition h0 > ∆ must be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Interpreting L as a “tunable” parameter has another  theoretical advantage, as it enables to address the in-  teresting fundamental question of how to connect two  paradigmatic limits in SU-FM hybrid devices: the atomic  limit (kF L → 0), where the physics is that of the well-  known non-degenerate YSR states, and the ballistic limit  (kF L ≫ 1) where the spectrum of the subgap ABS be-  comes double degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Until very recently, these lim-  its were treated as disconnected from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [32] this issue was addressed in the particular case of SU-  FM-SU junctions in the limit L < ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Here we have revis-  ited this intriguing question for a different setup where  such constraint does not exist, and have studied the evo-  lution of the subgap spectrum as a function of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' The  abovementioned symmetry classification into even and  odd solutions is critically important to allow the inter-  pretation of the degeneracy in the limit kF L → ∞ as an  “even-odd degeneracy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' At the same time, it enables to  explain the degeneracy lifting in the limit L → 0, where  only even states prevail in the subgap region of ener-  gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Using an approximate model of one-dimensional  fermions with linearized dispersion, we have provided a  simple picture where the even-odd degeneracy naturally  emerges as a consequence of destructive interferences of  terms e±i2kF x arising from single-particle backscattering  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  The continuous evolution of the subgap spectrum as a  function of kF L allows a better understanding of previ-  ous experimental STM results on atomic magnetic adsor-  bates on superconducting substrates, where the subgap  YSR states are usually interpreted in terms of a point-like  magnetic moment [27, 37–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' While the delta-function  limit is obviously a mathematical idealization, in terms  of our model the observed YSR states can be rationalized  assuming a finite value of kF L and a (more physically ap-  pealing) finite value of the atomic local field h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This is  precisely the case if we note that for magnetic impurities  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Fe, Co or Mn atoms) deposited on top of bulk metal-  lic S surfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', Pb or Al), the spatial extension of the  short-ranged Zeeman field can be estimated as the size of  the d-shell orbitals L ∼ 1 ˚A, while the Fermi wavevector  of bulk superconductors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', Pb) is kF ∼ 1−2×1010m−1  (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This type of adsorbate/substrate combi-  nation yields a parameter kF L ∼ 1, which is within the  regime where we recover observable subgap states (see  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' On the other hand, in 1D semiconduc-  tor heterostructures as those of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 20 and 21, kF is  usually much smaller than in metallic superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Measurements of the number of carriers from the Hall  conductance RH in 2D InGaAl quantum wells [46] yield  the estimated value kF ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='2 × 107m−1, three orders of  magnitude smaller as compared to bulk Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This much  smaller value of kF allows for much larger, experimentally  accessible values of L, while keeping values of h0 also  within experimental reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' All together, this combina-  tion makes these hybrid materials a much more versatile  platform to control the spectrum of YSR/ABS subgap  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  To characterize the quantum phase transitions occur-  ring in the device, we have computed the value of the to-  tal Sz using a spin version of the Friedel sum rule [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  (44) and also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' We stress that these transitions  are a generalization of the well-known “0-π” transition  occurring in atomic Shiba impurities [22, 47] or quantum  dots coupled to superconductors [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' From this per-  spective, the difference with respect to atomic systems  is that instead of a single transition, actually multiple  transitions can occur due to the finite extension L of  the “impurity” and the many ABS states with different  symmetry which can eventually cross below EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Inter-  estingly, we stress that the ocurrence of these quantum  phase transitions can be tuned varying the length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  We now briefly address the effect of the Rashba spin-  orbit interaction, which has been neglected in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  As mentioned previously, this interaction was neglected  to simplify the theoretical description of this (already  quite complex and rich) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' This interaction can  drive the system into the topological superconductor  class D [51, 52], hosting Majorana zero modes at the ends  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 34 for a related setup), and in that case  we expect qualitative changes with respect to the results  presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Consequently our results apply to exper-  imental SE-SU-FMI systems where the spin-orbit energy  term ESOC = α2  Rm∗/2, with αR the Rashba parameter,  is negligible compared to ∆ and h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Finally, we consider the effect of disorder in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  This might be a relevant effect as a random disorder po-  tential will eventually break the inversion symmetry of  the model and might lift the predicted even-odd degen-  eracy in the limit kF L ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' However, we believe the  energy-lifting effect might be weak in epitaxially-grown  samples, where disorder is a relatively small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was partially supported by CONICET un-  der grant PIP 0792, UNLP under grant PID X497, and  Agencia I+D+i under PICT 2017-2081, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' AML  is grateful to Liliana Arrachea for pointing out crucial  bibliographic references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  13  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Balatsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Vekhter, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Zhu, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 78, 373 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Heinrich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Pascual, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Franke, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 93, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Pawlak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Klinovaja, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Loss, and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Meyer, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 107, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Choi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lorente, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Wiebe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' von Bergmann,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Otte, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Heinrich, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 91,  041001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Yu, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Sin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 21, 75 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Shiba, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 40, 435 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Rusinov, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Pisma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 9, 146  (1968), [JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 9, 85 (1969)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Nadj-Perge, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Drozdov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Bernevig, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Yazdani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' B 88, 020407(R) (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Klinovaja, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Stano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Yazdani, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Loss, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 111, 186805 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Chen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Drozdov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Yazdani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Bernevig, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' MacDonald, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Jeon,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Seo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' MacDonald, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Bernevig, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Yaz-  dani, Science 346, 602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Pientka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Peng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' von Oppen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Heinrich, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Franke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Feldman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Wang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Yaz-  dani, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Kawai,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Glatzel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Loss, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Meyer, npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Heinrich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Peng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' von Oppen, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Franke, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Palot´as, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Szunyogh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Thorwart, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' J¨ack, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Xie, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' 3, 541  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Bergeret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Volkov, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Vaitiek˙enas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Mart´ı-S´anchez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Koch,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Cui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Khan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Krogstrup, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Heinrich, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+page_content=' Tokatly, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE2T4oBgHgl3EQfjQer/content/2301.03967v1.pdf'}
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+On the gate-error robustness of variational quantum algorithms
+Daniil Rabinovich,1 Ernesto Campos,1 Soumik Adhikary,1 Ekaterina
+Pankovets,1, 2 Dmitry Vinichenko,1, 3 and Jacob Biamonte4
+1Skolkovo Institute of Science and Technology, Moscow, Russian Federation
+2Moscow Institute of Physics and Technology, Moscow, Russian Federation
+3Moscow Engineering Physics Institute, Moscow, Russian Federation
+4Beijing Institute of Mathematical Sciences and Applications, Beijing, China
+Variational algorithms are designed to work within the limitations of contemporary devices and
+suffer from performance limiting errors.
+Here we identify an experimentally relevant model for
+gate errors, natural to variational quantum algorithms. We study how a quantum state prepared
+variationally decoheres under this noise model, which manifests as a perturbation to the energy
+approximation in the variational paradigm. A perturbative analysis of an optimized circuit allows
+us to determine the noise threshold for which the acceptance criteria imposed by the stability
+lemma remains satisfied. We benchmark the results against the variational quantum approximate
+optimization algorithm for 3-SAT instances and unstructured search with up to 10 qubits and 30
+layers. Finally, we observe that errors in certain gates have a significantly smaller impact on the
+quality of the prepared state. Motivated by this, we show that it is possible to reduce the execution
+time of the algorithm with minimal to no impact on the performance.
+I.
+INTRODUCTION
+Noisy Intermediate Scale Quantum (NISQ) quantum
+computing [1] suffers from limited coherence times and
+opeartion precision [2–5]. In practice we are severely lim-
+ited by the number of qubits and circuit depths that
+one may implement with reasonable fidelity.
+This has
+piratical implications in that it limits contemporary ex-
+perimental demonstrations. A host of theoretical results
+are now emerging, leading to improved understanding
+of the use of random circuit sampling as the basis of a
+scalable experimental violation of the extended Church-
+Turing thesis [6] and on the complexity analysis of NISQ
+[7]. The variational model of quantum computation is
+designed to work within these practical limitations [8–
+10]. More generally, the variational model is known to
+be computationally universal, yet these results are highly
+idealized and do not account for noise [11].
+Reminiscent of machine learning, a variational algo-
+rithm makes use of a short parameterized quantum cir-
+cuit, known as ansatz, in which parameters are itera-
+tively tuned to minimize a cost function in a quantum-to-
+classical feedback loop [12]. The cost function is typically
+given in the form of the expectation of a so called prob-
+lem Hamiltonian; where the ground state of the problem
+Hamiltonian encodes the solution of a given problem in-
+stance. Thus, by the way of cost function (energy) min-
+imization, a variational algorithm attempts to approx-
+imate the ground state of a given Hamiltonian.
+This
+strategy, however, does not provide us with a guarantee
+in regards to the quality of the approximate solution,
+where the latter is typically quantified as the overlap
+between the state prepared by the ansatz and the true
+ground state. Nevertheless, the overlap can be bounded.
+It has been shown using the stability lemma that the
+bounds can be directly related to the energy, thus allow-
+ing us to determine the energy threshold (upper bound)
+required to guarantee a fixed minimum overlap. We call
+this the acceptance threshold; a state with energy below
+this threshold is said to be accepted by the algorithm
+[11].
+Variational algorithms by their design alleviate the ef-
+fects of certain systematic limitations of NISQ devices.
+Nevertheless, variational algorithms are not immune to
+stochastic noise. While there exist some evidence that
+variational algorithms can in fact benefit from certain
+level of stochastic noise [13], in general, it is detrimental
+to the performance; stochastic noise leads to decoherence
+thus typically reducing solution quality.
+In this paper we study the extent to which errors, in the
+form of parameter alterations, affects the performance of
+variational algorithms.
+We analytically show that the
+shift in energy varies quadratically with the strength of
+noise (for small amounts of noise). We demonstrate this
+numerically for variational quantum approximate opti-
+misation in two common problems—3-SAT [14] and un-
+structured search [15, 16]. Furthermore we also found the
+performance to be more resilient to alterations in certain
+parameters. With that in mind we propose avenues to
+potentially improve performance and reduce the execu-
+tion time of variational quantum algorithms.
+II.
+PRELIMINARIES
+A.
+Variational Quantum Approximate
+Optimization
+The quantum approximate optimization algorithm
+(QAOA) [17], originally designed to approximately solve
+combinatorial optimization problems [14, 17–28], consists
+of ansatze circuits expressive enough to (in theory) emu-
+late any quantum cirucuit [19, 20].
+Consider a pseudo-Boolean function C : {0, 1}×n → R,
+the objective of the algorithm is to approximate a bit
+string that minimizes C. To accomplish this, C is first
+arXiv:2301.00048v1  [quant-ph]  30 Dec 2022
+
+2
+encoded as a problem Hamiltonian H, diagonal in the
+computational basis. The ground state H encodes the
+solution to the problem; in other words QAOA searches
+for a solution |g⟩ such that ⟨g|H|g⟩ = min H.
+The algorithm begins with an ansatz state |ψp(γ, β)⟩—
+prepared by a circuit of depth p — parameterized as:
+|ψp(γ, β)⟩ =
+p
+�
+k=1
+e−iβkHxe−iγkH |+⟩⊗n ,
+(1)
+with real parameters γk ∈ [0, 2π), βk ∈ [0, π).
+Here
+Hx = �n
+j=1 Xj is the standard one-body mixer Hamil-
+tonian with Pauli matrix Xj applied to the j-th qubit.
+The cost function is given by the expectation of the prob-
+lem Hamiltonian with respect to the ansatz state. The
+algorith minimizes this cost function to output:
+E∗ = minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩
+(2)
+γ∗, β∗ ∈ arg minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩
+(3)
+Here, |ψp(γ∗, β∗)⟩ is the approximate ground state of
+H and hence the approximate solution to C. Indeed, the
+quality of the approximation, quantified as the overlap
+between the true solution and the approximate solution,
+is not known a priori from (2).
+Nevertheless one can
+establish bounds on this quantity using the so called sta-
+bility lemma.
+B.
+Stability lemma
+The stability lemma states that if |g⟩ is the true ground
+state of H with energy Eg and ∆ is the spectral gap
+(the difference between the ground state energy and the
+energy of the first excited state) the following relation
+holds [11, 29]:
+1 − E∗ − Eg
+∆
+≤ |⟨ψp(γ∗, β∗)|g⟩|2 ≤ 1 − E∗ − Eg
+Em − Eg
+(4)
+where Em is the maximum eigenvalue of H.
+Thus to
+guarantee a non-trivial overlap one must ensure that
+E∗ ≤ Eg + ∆. We call the latter the acceptance con-
+dition.
+III.
+VARIATIONAL QUANTUM ALGORITHMS
+IN THE PRESENCE OF REALISTIC GATE
+ERRORS
+Implementation of unitary operations depends signif-
+icantly on the considered hardware. However, typically
+the implementation makes use of electromagnetic pulses,
+such as in superconducting quantum computers [30, 31],
+neutral atom based quantum computers [32, 33], and
+trapped ion based quantum computers [34, 35].
+Such
+pulses can change the population of the energy levels
+that constitute a qubit or introduce phases to the quan-
+tum amplitudes, thus controlling the state of the qubits.
+Consequently, the main contribution to gate errors comes
+from variation in pulse shaping, meaning that amplitude
+and timing of electromagnetic pulse can stochasticaly
+vary.
+In certain experimental setups, such as ground
+state ion qubits, where entangling operations are per-
+formed using the radial phonon modes [36], the variabil-
+ity in pulse shaping is the main source of gate errors.
+Angles of rotation in a typical gate operation depend
+on time averaged intensity I(t) of the electromagnetic
+pulse; θ ∝
+�
+I(t)dt. Thus, variations in the pulse shap-
+ing lead to stochastic deviations of the angles of rota-
+tions from the desired values. In other words, if a cir-
+cuit is composed of the parameterised gates {Uk(θk)}k;
+θ ∈ [0, 2π) and one tries to prepare a state |ψ(θ)⟩ =
+�
+k Uk(θk) |ψ0⟩, a different state
+|ψ(θ + δθ)⟩ =
+�
+k
+U(θk + δθk) |ψ0⟩ ,
+(5)
+is prepared instead due to the presence of errors. No-
+tice here that the perturbation δθ to the parameters is
+stochastic and is sampled with a certain probability den-
+sity p(δθ). This implies that the prepared state can be
+described by an ensemble {|ψ(θ + δθ)⟩ , p(δθ)}, which we
+can equivalently view as a density matrix
+ρ(θ) =
+�
+|ψ(θ + δθ)⟩⟨ψ(θ + δθ)|p(δθ)d(δθ).
+(6)
+Eq. (6) represents a noise model native to the vari-
+ational paradigm of quantum computing. For the rest
+of this paper we systematically study the effect of this
+noise model on the performance of QAOA for instances
+of 3-SAT and the unstructured search problem (see ap-
+pendix A for more details on the considered problems).
+In particular we study the energy perturbation around
+E∗ in different scenarios subsequently recovering the
+strength of noise under which the acceptance condition
+continues to be satisfied.
+IV.
+RESULTS
+A.
+Perturbative analysis in presence of gate errors
+Consider a problem Hamiltonian H and a variational
+ansatz |ψ(θ)⟩ = U1(θ1) . . . Uq(θq) |ψ0⟩ used to mini-
+mize H. Here the gates Uk(θk) have the form:
+Uk(θk) = eiAkθk, A2
+k = 1,
+(7)
+A typical example of such an ansatz is the checkerboard
+ansatz, with Mølmer-Sørensen (MS) gates as the entan-
+gling two qubit gates. Nevertheless, any quantum circuit
+can admit a decomposition in terms of operations that
+satisfy (7); this adds generality to this assumption.
+
+3
+In the presence of gate errors the prepared quantum
+state decoheres as |ψ(θ)⟩ → ρ(θ) as per (6). To obtain
+the analytic form of ρ(θ) we first note that
+Uk(θk + δθk) = Uk(θk)Uk(δθk)
+= cos δθkUk(θk) + sin δθkUk
+�
+θk + π
+2
+�
+.
+(8)
+This follows directly from (7). Therefore we get:
+|ψ(θ + δθ)⟩⟨ψ(θ + δθ)| =
+1
+�
+k1,...,kq,m1,...,mq=0
+(cos2 δθ1 tank1+m1 δθ1) . . . (cos2 δθq tankq+mq δθq)|ψk1...kq⟩⟨ψm1...mq|, (9)
+where
+|ψk1...kq⟩ = U1(θ1 + k1
+π
+2 ) . . . Uq(θq + kq
+π
+2 ) |ψ0⟩ .
+(10)
+Here we make three realistic assumptions—(a) pertur-
+bations to all the angles are independent, (b) average
+perturbation ⟨δθk⟩ = 0 and (c) the distribution p(δθk)
+vanishes quickly outside the range (−σk, σk); that is, the
+error is localized on the scale σk ≪ 1. Note that if as-
+sumption (b) does not hold, one can always shift the
+parameters as θ → θ + ⟨δθ⟩.
+Substituting (9) in (6) we arrive at the expression:
+ρ(θ) = |ψ(θ)⟩⟨ψ(θ)| + δρ,
+(11)
+where
+δρ ≈ −
+q
+�
+k=1
+ak|ψ(θ)⟩⟨ψ(θ)|+
+q
+�
+k=1
+ak|ψk⟩⟨ψk|+o(σ2
+k). (12)
+Here |ψk⟩ = |ψ00...1...00⟩ with 1 placed in the k-th posi-
+tion, and
+ak ≡ ⟨sin2 δθk⟩ =
+�
+sin2 δθkp(δθk)d(δθk) ∼ σ2
+k.
+(13)
+Notice that the derivation above does not require θ to
+be a minimum of the noiseless cost function.
+Let us
+now assume that θ∗ is a vector of parameters such that
+|ψ(θ∗)⟩ approximates the ground state of H. The noise
+induced energy perturbation around the optimal energy
+E∗ is given as:
+δE = Tr(ρ(θ∗)H) − ⟨ψ(θ∗)| H |ψ(θ∗)⟩
+≤ (Em − E∗)
+�
+k
+ak.
+(14)
+For the simplest case where each parameter is sampled
+from the same distribution (σk = σ) we can roughly es-
+timate:
+δE ≤ qσ2(Em − E∗).
+(15)
+Thus, requesting an energy threshold E ≤ Eg + ∆, we
+conclude that for σ <∼
+�
+∆ − (E∗ − Eg)
+q(Em − E∗)
+the acceptance
+condition is still satisfied.
+While our perturbative analysis holds for all varia-
+tional algorithms, we substantiate our findings numer-
+ically using QAOA. In particular we solve instances of
+3-SAT and unstructured search problems to study the
+behaviour of energy perturbation around E∗ caused by
+the presence of gate errors.
+1.
+Constant perturbation
+We begin with a simplified version of the noise model
+proposed in (6). We ran QAOA for 100 uniformly gen-
+erated 3-SAT instances of 6,8, and 10 variables with 26,
+34 and 42 clauses respectively.
+All the instances were
+selected to have a unique satisfying assignment. The in-
+stances were minimized by QAOA sequences of 15, 25
+and 30 layers respectively in order to obtain expected
+values well below the energy gap. In order to numeri-
+cally verify the behaviour of the energy perturbation, we
+vary all optimal parameters by a constant angle δ. Fig-
+ure 1 illustrates the shift in the energy for the minimized
+instances, which can be seen to have a quadratic depen-
+dence of the perturbed energy δE with respect to the
+shift δ. This is natural to expect since the parameters
+deviate from the local minimum, where linear contribu-
+tion must have vanished (a rigorous expression showing
+the quadratic behavior is derived in appendix B).
+Similar to the case of 3-SAT, for the problem of un-
+structured search we perturb optimal parameters of the
+circuit by an angle δ and plot corresponding energy in
+Fig. 2. Again, as expected, for small values of δ the en-
+ergy perturbation is quadratic which comes from the fact
+that the deviation happens around the minimum.
+
+4
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+δ
+−0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+δE
+6.0 qubits
+71.8δ2
+8.0 qubits
+160.5δ2
+10.0 qubits
+250.3δ2
+FIG. 1. Energy shift obtained by perturbing the ansatz state
+as |ψp(γ∗ + δ, β∗ + δ)⟩. The curves illustrate averages over
+100 uniformly generated 3-SAT instances of 6, 8 and 10 qubits
+with clause to variable ratio of 4.2 and unique satisfying as-
+signment. The error bars depict standard error. Polynomial
+fits of data indicates δ ∈ [0, 0.02] follow quadratic curves.
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+δ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+δE
+6 qubits
+208.9δ2
+8 qubits
+1228.4δ2
+10 qubits
+4664.2δ2
+FIG. 2.
+Energy shift for the problem of unstructured
+search
+obtained
+by
+perturbing
+of
+the
+ansatz
+state
+as
+|ψp(γ∗ + δ, β∗ + δ)⟩.
+Polynomial fits for data points of 6,
+8 and 10 qubits follow quadratic curves in the ranges δ ∈
+[0, 0.02], [0, 0.01], [0, 0.008] respectively.
+2.
+Stochastic perturbation
+We now consider the complete noise model in (6) and
+verify our analytical prediction as shown in (15).
+For
+each 3-SAT instance, we randomly sample perturbations
+δ to each of the gates from a uniform distribution on the
+interval (−σ, σ) and average the obtained energy. Then
+we average energies over instances of the same number
+of qubits as depicted in Fig. 3. It is seen that for small
+values of σ the behaviour is quadratic as per (15). It is
+seen, that the value σ ∼ 0.075 could never violate the
+acceptance criteria, as corresponding energy error never
+exceeds the gap ∆ ≥ 1. For smaller number of qubits
+and gates the threshold value of σ increases.
+For unstructured search, we average the energy over
+δ sampled for each gate from the uniform distribution
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+σ
+0
+1
+2
+3
+4
+5
+δE
+6.0 qubits
+61.0σ2
+8.0 qubits
+118.5σ2
+10.0 qubits
+172.8σ2
+FIG. 3. Average energy shift of 100 uniformly generated 3-
+SAT instances of 6, 8 and 10 qubits with clause to variable
+ratio of 4.2 and unique satisfying assignment. The shifts are
+obtained by the perturbation of γ∗, β∗ by δ uniformly sam-
+pled from the range (−σ, σ). Error bars depict standard error.
+Polynomial fits of data indicates σ ∈ [0, 0.1] follow quadratic
+curves.
+(−σ, σ). We again recover quadratic behaviour in σ, as
+depicted in Fig. 4.
+It is seen that the same threshold
+σ ∼ 0.075 now increases energy by no more then 0.6,
+which guaranties 40% overlap with the target state.
+0.00
+0.04
+0.08
+0.12
+0.16
+0.20
+0.24
+0.28
+σ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+δE
+6 qubits
+21.4σ2
+8 qubits
+60.3σ2
+10 qubits
+133.2σ2
+FIG. 4.
+Average energy for the problem of unstructured
+search obtained by the perturbation of γ∗, β∗ by δ uni-
+formly sampled from the range (−σ, σ).
+Error bars de-
+pict standard error.
+Polynomial fits of data points of 6,
+8 and 10 qubits follow quadratic curves in the ranges σ ∈
+[0, 0.1], [0, 0.07], [0, 0.05], respectively.
+B.
+Perturbation to individual parameters
+Here we consider a modified version of (6), where pa-
+rameters are perturbed one at a time while the rest are
+kept intact. Effect of this model on the energy is illus-
+trated in Figures 5 and 6. The results are numerical and
+are yet to be explained analytically. We observe that per-
+turbations to certain angles have a significantly smaller
+
+5
+tbh
+γ
+β
+n = 6
+p = 8
+1
+2
+3
+4
+5
+6
+7
+8
+k
+0.0105
+0.0110
+0.0115
+0.0120
+0.0125
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+1
+2
+3
+4
+5
+6
+7
+8
+k
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+n = 8
+p = 15
+2
+4
+6
+8
+10
+12
+14
+k
+0.0190
+0.0195
+0.0200
+0.0205
+0.0210
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+2
+4
+6
+8
+10
+12
+14
+k
+0.05
+0.10
+0.15
+0.20
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+n = 10
+p = 25
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.0850
+0.0855
+0.0860
+0.0865
+0.0870
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.10
+0.15
+0.20
+0.25
+0.30
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+FIG. 5. Energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ from the unstructured search problem, where βk (right column) or γk (left
+column), from the k-th layer, are perturbed.
+effect on the energy.
+Thus we can infer that reducing
+the value of such angles would not have a significant ef-
+fect on performance but will reduce the execution time of
+the algorithm, that is texec = �p
+k=1 βk + γk. Conversely,
+we could limit the execution time as texec ≤ tmax and
+increase the number of layers, since
+min ⟨ψp| H |ψp⟩ ≥ min ⟨ψp+1| H |ψp+1⟩
+(16)
+for the same tmax.
+Reducing the execution time is important to quantum
+algorithms, since variational parameters are proportional
+to the time required to execute a gate experimentally.
+NISQ era devices suffers from limited coherence, thus
+reducing execution times can lead to more efficient hard-
+ware utilization [37, 38]. We test these ideas in the setting
+of unstructured search, as depicted in Fig. 7. Here we
+show the optimized QAOA energies for 6 qubits at mul-
+tiple depths with execution time limited to tmax. The
+highlighted green and orange rectangles depict the two
+groups of optimal angles that minimize the energy at
+each depth, as presented in [15]. Green rectangles also
+indicate the depth and texec at which an ansatz will not
+be able to decrease its energy by either increasing depth
+or tmax. Following the observations of Fig. 5, by slightly
+reducing tmax the optimizer will reduce the parameters
+to which the energy is less sensitive. This results in a
+
+6
+γ
+β
+n = 6
+p = 15
+2
+4
+6
+8
+10
+12
+14
+k
+0.08
+0.10
+0.12
+0.14
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+2
+4
+6
+8
+10
+12
+14
+k
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+n = 8
+p = 25
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.08
+0.10
+0.12
+0.14
+0.16
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.10
+0.15
+0.20
+0.25
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+n = 10
+p = 30
+0
+4
+8
+12
+16
+20
+24
+28
+k
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+0
+4
+8
+12
+16
+20
+24
+28
+k
+0.10
+0.15
+0.20
+0.25
+0.30
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+FIG. 6.
+Average energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ of 100 uniformly generated 3-SAT instances where βk (right
+column) or γk (left column), from the k-th layer, are perturbed. The instances are of 6, 8 and 10 qubits with clause to variable
+ratio of 4.2 and unique satisfying assignment.
+slight energy increase as illustrated in Fig. 7 where to
+the left of the green rectangles we can observe darkening
+gradients.
+By contrast, orange rectangles highlight longer execu-
+tion times corresponding to different sets of angles that
+also minimize the energy for a given number of layers.
+Therefore if the optimization routine finds the a solu-
+tion corresponding to the orange rectangle, setting tmax
+to be slightly less than the texec of the orange rectangle
+will lead the optimizer to find angles corresponding to
+the green rectangle. This will amount to a considerable
+reduction in execution time.
+Alternatively increasing the number of layers while
+keeping tmax will reduce the energy. In general, for an
+arbitrary problem Hamiltonian we can not be sure if our
+optimization has returned the ideal set of angles (green
+ones in our example). For this reason, the best strategy
+would be to reduce tmax or increase depth while fixing
+tmax until performance stagnates.
+V.
+DISCUSSION
+In this study we considered a realistic noise model—
+one where the variational gate parameters are stochasti-
+cally perturbed—and demonstrated its effect on the per-
+
+7
+4.1
+6.1
+8.1
+10.1
+12.1
+14.1
+16.1
+18.1
+20.1
+22.1
+24.1
+26.1
+28.1
+30.1
+32.1
+34.1
+36.1
+38.1
+40.0
+42.0
+max execution time tmax
+8
+7
+6
+5
+4
+3
+2
+1
+depth p
+energy
+0.2
+0.4
+0.6
+0.8
+FIG. 7. Expected value for multiple combinations of depth for maximum execution times. Green and orange rectangles depict
+the two branches of angles that minimize expectation value for a given depth.
+formance of variational algorithms.
+Using a perturba-
+tive analysis we showed that the change in energy δE
+(from optimised energy E∗), caused due to the pres-
+ence of the considered gate errors, behaves quadratically
+with respect to the angle perturbations for small values
+of the perturbations. This allows us to establish upper
+bounds on the amount of perturbation such that the ac-
+ceptance condition continues to be satisfied. This guar-
+antees a fixed overlap between the target sate and the
+state prepared by the noisy variational circuit. We con-
+firm our analytical findings numerically in QAOA for two
+common problems—3-SAT and unstructured search, us-
+ing different modifications of the considered noise model.
+Moreover we observed form our numerical results that
+the algorithmic performance is more resilient to pertur-
+bations of certain variational parameters. Motivated by
+this observation we demonstrated that performance of
+QAOA with a total execution time texec = �
+k γk + βk
+is stable if retrained with a maximum execution time
+tmax = texec ± ϵ for ϵ ≪ texec. We also show that in
+some cases (a) reduction in tmax can lead to dramatic
+reductions in texec, and (b) increasing depth while fixing
+texec can lead to an energy reduction.
+While our study is primarily focused on energy pertur-
+bations around the noiseless optimum θ∗, in practice one
+has to train in the presence of noise. This would change
+optimal angles θ∗ → θ∗ + δθ∗, where shift δθ∗ increases
+with increase of the strength of the noise. Nevertheless,
+using perturbation theory around the noiseless optimum
+one can estimate δθ∗ = O(σ2), and the corresponding
+change in the energy is Tr(ρ(θ∗+δθ∗)H)−Tr(ρ(θ∗)H) =
+O(σ4). Therefore, working in the regime of weak noise
+one can safely use noiseless optimum θ∗. See appendix
+C for detailed calculation.
+VI.
+ACKNOWLEDGEMENT
+* D.R., E.C., S.A., E.P., D.V. acknowledge support
+from the research project, Leading Research Center on
+Quantum Computing (agreement No. 014/20).
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+amonte. Ion-native variational ansatz for quantum ap-
+proximate optimization. Phys. Rev. A, 106:032418, Sep
+2022.
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+long-range ising model with a trapped-ion quantum sim-
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+arXiv:2211.00350, 2022.
+
+9
+Appendix A: 3-SAT and unstructured search
+problems
+1.
+3-SAT
+Boolean satifyability, or SAT, is the problem of deter-
+mining weather a boolean formula written in conjunctive
+normal form (CNF) is satisfiable. It is possible to map
+any SAT instance via Karp reduction into 3-SAT, which
+are restricted to 3 literals per clause.
+In order to ap-
+proximate solutions to SAT we embed the instance into
+a Hamiltonian as
+HSAT =
+�
+j
+P(j),
+(A1)
+where j indexes clauses of an instance, and P(j) is the
+tensor product of projectors that penalizes bit string as-
+signments that do not satisfy the j-th clause.
+2.
+Unstructured search
+Consider an unstructured database S indexed by j ∈
+{0, 1}×n. Let f : {0, 1}×n → {0, 1} be a Boolean function
+(a.k.a. black box) such that:
+f(j) =
+�
+1
+iff j = t
+0
+otherwise.
+(A2)
+The task is to find t ∈ {0, 1}×n. The corresponding prob-
+lem Hamiltonian for QAOA is
+Ht = 1 − |t⟩⟨t|,
+(A3)
+thus the expected value is given by
+⟨H⟩ = 1 − |⟨t|ψp(γ, β)⟩|2.
+(A4)
+QAOA performance for unstructured search is not sen-
+sitive to the particular target state |t⟩ in the computa-
+tional basis. For any target state |t⟩ representing a binary
+string, there is a U = U † composed of X and 1 opera-
+tors such that U |0⟩⊗n = |t⟩. The overlap of an arbitrary
+state prepared by a QAOA sequence with |t⟩ is then:
+⟨t|ψp(γ, β)⟩ = ⟨t|
+p
+�
+k=1
+e−iβkHxe−iγk|t⟩⟨t| |+⟩⊗n
+= ⟨0|⊗n U
+p
+�
+k=1
+e−iβkHxe−iγkU(|0⟩⟨0|)⊗nU |+⟩⊗n
+= ⟨0|⊗n U
+p
+�
+k=1
+e−iβkHxUe−iγk(|0⟩⟨0|)⊗nU |+⟩⊗n
+= ⟨0|⊗n
+p
+�
+k=1
+e−iβkHxe−iγk(|0⟩⟨0|)⊗n |+⟩⊗n ,
+which is independent on t.
+Appendix B: Energy variation in presence of
+constant perturbations to gate parameters
+Using (9) one can calculate perturbation to the energy
+caused by a shift of the optimal angles by a constant δθ
+as
+δE = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ − ⟨ψ(θ∗)| H |ψ(θ∗)⟩
+= −
+q
+�
+k=1
+δθ2
+kE∗ +
+q
+�
+m̸=k
+δθkδθm(⟨ψ(θ∗)| H |ψkm⟩ + h.c.)
++
+q
+�
+m,k
+δθkδθk ⟨ψm| H |ψk⟩ + o(δθkδθm)
+= 1
+2(δθ)T Hδθ + o(δθkδθm),
+(B1)
+where |ψmk⟩ = |ψ0...1...1...0⟩ with 1 placed only at m-th
+and k-th positions. H is the Hessian of the energy at
+noiseless optimum, Hij =
+∂2
+∂θi∂θj
+⟨ψ(θ)| H |ψ(θ)⟩ |θ=θ∗.
+Here we use the fact that at the optimal position linear
+contribution to the cost function necessarily vanishes. It
+is seen now that for the constant perturbation δθk = δ
+the energy changes as δE ∝ δ2.
+Appendix C: Optimal parameters variation in the
+presence of noise
+Let us use expressions (11) and (12) to estimate change
+in the energy if one accounts for shift of optimal param-
+eters θ∗ → θ∗ + δθ∗:
+Tr(ρ(θ∗ + δθ∗)H) = (1 −
+q
+�
+k=1
+ak) ⟨ψ(θ∗ + δθ∗)| H |ψ(θ∗ + δθ∗)⟩ +
+q
+�
+k=1
+ak ⟨ψk(θ∗ + δθ∗)| H |ψk(θ∗ + δθ∗)⟩ + o(σ2
+k)
+(C1)
+We introduce gradients of the noisy terms Bk
+=
+∂
+∂θ ⟨ψk(θ)| H |ψk(θ)⟩ |θ=θ∗. Notice that gradients of the
+
+10
+noiseless function ⟨ψ(θ)| H |ψ(θ)⟩ vanish at optimum.
+Then,
+Tr(ρ(θ∗ + δθ∗)H) ≈ (1 −
+q
+�
+k=1
+ak)E∗ + 1
+2(δθ∗)T Hδθ∗
++
+q
+�
+k=1
+ak[⟨ψk(θ∗)| H |ψk(θ∗)⟩ + (δθ∗)T Bk].
+(C2)
+Minimizing it with respect to δθ∗ one gets δθ∗ =
+�q
+k=1 akH−1Bk. Thus, if we account for the change of
+optimal parameters in the presence of noise, the energy
+shifts by
+Tr(ρ(θ∗ + δθ∗)H) − Tr(ρ(θ∗)H) ≈
+(δθ∗)T Hδθ∗ +
+q
+�
+k=1
+ak(δθ∗)T Bk = O(σ4).
+(C3)
+
diff --git a/69AyT4oBgHgl3EQfQfbA/content/tmp_files/load_file.txt b/69AyT4oBgHgl3EQfQfbA/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf,len=673
+page_content='On the gate-error robustness of variational quantum algorithms  Daniil Rabinovich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1 Ernesto Campos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1 Soumik Adhikary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1 Ekaterina  Pankovets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 2 Dmitry Vinichenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 3 and Jacob Biamonte4  1Skolkovo Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Russian Federation  2Moscow Institute of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Russian Federation  3Moscow Engineering Physics Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Russian Federation  4Beijing Institute of Mathematical Sciences and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' China  Variational algorithms are designed to work within the limitations of contemporary devices and  suffer from performance limiting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Here we identify an experimentally relevant model for  gate errors, natural to variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We study how a quantum state prepared  variationally decoheres under this noise model, which manifests as a perturbation to the energy  approximation in the variational paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' A perturbative analysis of an optimized circuit allows  us to determine the noise threshold for which the acceptance criteria imposed by the stability  lemma remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We benchmark the results against the variational quantum approximate  optimization algorithm for 3-SAT instances and unstructured search with up to 10 qubits and 30  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Finally, we observe that errors in certain gates have a significantly smaller impact on the  quality of the prepared state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Motivated by this, we show that it is possible to reduce the execution  time of the algorithm with minimal to no impact on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  INTRODUCTION  Noisy Intermediate Scale Quantum (NISQ) quantum  computing [1] suffers from limited coherence times and  opeartion precision [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' In practice we are severely lim-  ited by the number of qubits and circuit depths that  one may implement with reasonable fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  This has  piratical implications in that it limits contemporary ex-  perimental demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' A host of theoretical results  are now emerging, leading to improved understanding  of the use of random circuit sampling as the basis of a  scalable experimental violation of the extended Church-  Turing thesis [6] and on the complexity analysis of NISQ  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The variational model of quantum computation is  designed to work within these practical limitations [8–  10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' More generally, the variational model is known to  be computationally universal, yet these results are highly  idealized and do not account for noise [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Reminiscent of machine learning, a variational algo-  rithm makes use of a short parameterized quantum cir-  cuit, known as ansatz, in which parameters are itera-  tively tuned to minimize a cost function in a quantum-to-  classical feedback loop [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The cost function is typically  given in the form of the expectation of a so called prob-  lem Hamiltonian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' where the ground state of the problem  Hamiltonian encodes the solution of a given problem in-  stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Thus, by the way of cost function (energy) min-  imization, a variational algorithm attempts to approx-  imate the ground state of a given Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  This  strategy, however, does not provide us with a guarantee  in regards to the quality of the approximate solution,  where the latter is typically quantified as the overlap  between the state prepared by the ansatz and the true  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Nevertheless, the overlap can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  It has been shown using the stability lemma that the  bounds can be directly related to the energy, thus allow-  ing us to determine the energy threshold (upper bound)  required to guarantee a fixed minimum overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We call  this the acceptance threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' a state with energy below  this threshold is said to be accepted by the algorithm  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Variational algorithms by their design alleviate the ef-  fects of certain systematic limitations of NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Nevertheless, variational algorithms are not immune to  stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' While there exist some evidence that  variational algorithms can in fact benefit from certain  level of stochastic noise [13], in general, it is detrimental  to the performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' stochastic noise leads to decoherence  thus typically reducing solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  In this paper we study the extent to which errors, in the  form of parameter alterations, affects the performance of  variational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  We analytically show that the  shift in energy varies quadratically with the strength of  noise (for small amounts of noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We demonstrate this  numerically for variational quantum approximate opti-  misation in two common problems—3-SAT [14] and un-  structured search [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Furthermore we also found the  performance to be more resilient to alterations in certain  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' With that in mind we propose avenues to  potentially improve performance and reduce the execu-  tion time of variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  PRELIMINARIES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Variational Quantum Approximate  Optimization  The quantum approximate optimization algorithm  (QAOA) [17], originally designed to approximately solve  combinatorial optimization problems [14, 17–28], consists  of ansatze circuits expressive enough to (in theory) emu-  late any quantum cirucuit [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Consider a pseudo-Boolean function C : {0, 1}×n → R,  the objective of the algorithm is to approximate a bit  string that minimizes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' To accomplish this, C is first  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00048v1  [quant-ph]  30 Dec 2022  2  encoded as a problem Hamiltonian H, diagonal in the  computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The ground state H encodes the  solution to the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' in other words QAOA searches  for a solution |g⟩ such that ⟨g|H|g⟩ = min H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  The algorithm begins with an ansatz state |ψp(γ, β)⟩—  prepared by a circuit of depth p — parameterized as:  |ψp(γ, β)⟩ =  p  �  k=1  e−iβkHxe−iγkH |+⟩⊗n ,  (1)  with real parameters γk ∈ [0, 2π), βk ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Here  Hx = �n  j=1 Xj is the standard one-body mixer Hamil-  tonian with Pauli matrix Xj applied to the j-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  The cost function is given by the expectation of the prob-  lem Hamiltonian with respect to the ansatz state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The  algorith minimizes this cost function to output:  E∗ = minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩  (2)  γ∗, β∗ ∈ arg minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩  (3)  Here, |ψp(γ∗, β∗)⟩ is the approximate ground state of  H and hence the approximate solution to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Indeed, the  quality of the approximation, quantified as the overlap  between the true solution and the approximate solution,  is not known a priori from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Nevertheless one can  establish bounds on this quantity using the so called sta-  bility lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Stability lemma  The stability lemma states that if |g⟩ is the true ground  state of H with energy Eg and ∆ is the spectral gap  (the difference between the ground state energy and the  energy of the first excited state) the following relation  holds [11, 29]:  1 − E∗ − Eg  ∆  ≤ |⟨ψp(γ∗, β∗)|g⟩|2 ≤ 1 − E∗ − Eg  Em − Eg  (4)  where Em is the maximum eigenvalue of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Thus to  guarantee a non-trivial overlap one must ensure that  E∗ ≤ Eg + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We call the latter the acceptance con-  dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  VARIATIONAL QUANTUM ALGORITHMS  IN THE PRESENCE OF REALISTIC GATE  ERRORS  Implementation of unitary operations depends signif-  icantly on the considered hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' However, typically  the implementation makes use of electromagnetic pulses,  such as in superconducting quantum computers [30, 31],  neutral atom based quantum computers [32, 33], and  trapped ion based quantum computers [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Such  pulses can change the population of the energy levels  that constitute a qubit or introduce phases to the quan-  tum amplitudes, thus controlling the state of the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Consequently, the main contribution to gate errors comes  from variation in pulse shaping, meaning that amplitude  and timing of electromagnetic pulse can stochasticaly  vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  In certain experimental setups, such as ground  state ion qubits, where entangling operations are per-  formed using the radial phonon modes [36], the variabil-  ity in pulse shaping is the main source of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Angles of rotation in a typical gate operation depend  on time averaged intensity I(t) of the electromagnetic  pulse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' θ ∝  �  I(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Thus, variations in the pulse shap-  ing lead to stochastic deviations of the angles of rota-  tions from the desired values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' In other words, if a cir-  cuit is composed of the parameterised gates {Uk(θk)}k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  θ ∈ [0, 2π) and one tries to prepare a state |ψ(θ)⟩ =  �  k Uk(θk) |ψ0⟩, a different state  |ψ(θ + δθ)⟩ =  �  k  U(θk + δθk) |ψ0⟩ ,  (5)  is prepared instead due to the presence of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' No-  tice here that the perturbation δθ to the parameters is  stochastic and is sampled with a certain probability den-  sity p(δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This implies that the prepared state can be  described by an ensemble {|ψ(θ + δθ)⟩ , p(δθ)}, which we  can equivalently view as a density matrix  ρ(θ) =  �  |ψ(θ + δθ)⟩⟨ψ(θ + δθ)|p(δθ)d(δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (6)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' (6) represents a noise model native to the vari-  ational paradigm of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' For the rest  of this paper we systematically study the effect of this  noise model on the performance of QAOA for instances  of 3-SAT and the unstructured search problem (see ap-  pendix A for more details on the considered problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  In particular we study the energy perturbation around  E∗ in different scenarios subsequently recovering the  strength of noise under which the acceptance condition  continues to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Perturbative analysis in presence of gate errors  Consider a problem Hamiltonian H and a variational  ansatz |ψ(θ)⟩ = U1(θ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Uq(θq) |ψ0⟩ used to mini-  mize H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Here the gates Uk(θk) have the form:  Uk(θk) = eiAkθk, A2  k = 1,  (7)  A typical example of such an ansatz is the checkerboard  ansatz, with Mølmer-Sørensen (MS) gates as the entan-  gling two qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Nevertheless, any quantum circuit  can admit a decomposition in terms of operations that  satisfy (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' this adds generality to this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  3  In the presence of gate errors the prepared quantum  state decoheres as |ψ(θ)⟩ → ρ(θ) as per (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' To obtain  the analytic form of ρ(θ) we first note that  Uk(θk + δθk) = Uk(θk)Uk(δθk)  = cos δθkUk(θk) + sin δθkUk  �  θk + π  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (8)  This follows directly from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Therefore we get:  |ψ(θ + δθ)⟩⟨ψ(θ + δθ)| =  1  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=',kq,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=',mq=0  (cos2 δθ1 tank1+m1 δθ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' (cos2 δθq tankq+mq δθq)|ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='kq⟩⟨ψm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='mq|, (9)  where  |ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='kq⟩ = U1(θ1 + k1  π  2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Uq(θq + kq  π  2 ) |ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (10)  Here we make three realistic assumptions—(a) pertur-  bations to all the angles are independent, (b) average  perturbation ⟨δθk⟩ = 0 and (c) the distribution p(δθk)  vanishes quickly outside the range (−σk, σk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' that is, the  error is localized on the scale σk ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Note that if as-  sumption (b) does not hold, one can always shift the  parameters as θ → θ + ⟨δθ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Substituting (9) in (6) we arrive at the expression:  ρ(θ) = |ψ(θ)⟩⟨ψ(θ)| + δρ,  (11)  where  δρ ≈ −  q  �  k=1  ak|ψ(θ)⟩⟨ψ(θ)|+  q  �  k=1  ak|ψk⟩⟨ψk|+o(σ2  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' (12)  Here |ψk⟩ = |ψ00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00⟩ with 1 placed in the k-th posi-  tion, and  ak ≡ ⟨sin2 δθk⟩ =  �  sin2 δθkp(δθk)d(δθk) ∼ σ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (13)  Notice that the derivation above does not require θ to  be a minimum of the noiseless cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Let us  now assume that θ∗ is a vector of parameters such that  |ψ(θ∗)⟩ approximates the ground state of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The noise  induced energy perturbation around the optimal energy  E∗ is given as:  δE = Tr(ρ(θ∗)H) − ⟨ψ(θ∗)| H |ψ(θ∗)⟩  ≤ (Em − E∗)  �  k  ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (14)  For the simplest case where each parameter is sampled  from the same distribution (σk = σ) we can roughly es-  timate:  δE ≤ qσ2(Em − E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (15)  Thus, requesting an energy threshold E ≤ Eg + ∆, we  conclude that for σ <∼  �  ∆ − (E∗ − Eg)  q(Em − E∗)  the acceptance  condition is still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  While our perturbative analysis holds for all varia-  tional algorithms, we substantiate our findings numer-  ically using QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' In particular we solve instances of  3-SAT and unstructured search problems to study the  behaviour of energy perturbation around E∗ caused by  the presence of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Constant perturbation  We begin with a simplified version of the noise model  proposed in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We ran QAOA for 100 uniformly gen-  erated 3-SAT instances of 6,8, and 10 variables with 26,  34 and 42 clauses respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  All the instances were  selected to have a unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The in-  stances were minimized by QAOA sequences of 15, 25  and 30 layers respectively in order to obtain expected  values well below the energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' In order to numeri-  cally verify the behaviour of the energy perturbation, we  vary all optimal parameters by a constant angle δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Fig-  ure 1 illustrates the shift in the energy for the minimized  instances, which can be seen to have a quadratic depen-  dence of the perturbed energy δE with respect to the  shift δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This is natural to expect since the parameters  deviate from the local minimum, where linear contribu-  tion must have vanished (a rigorous expression showing  the quadratic behavior is derived in appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Similar to the case of 3-SAT, for the problem of un-  structured search we perturb optimal parameters of the  circuit by an angle δ and plot corresponding energy in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Again, as expected, for small values of δ the en-  ergy perturbation is quadratic which comes from the fact  that the deviation happens around the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
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+page_content='0200  δ  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  δE  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='8δ2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='5δ2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='3δ2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Energy shift obtained by perturbing the ansatz state  as |ψp(γ∗ + δ, β∗ + δ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The curves illustrate averages over  100 uniformly generated 3-SAT instances of 6, 8 and 10 qubits  with clause to variable ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2 and unique satisfying as-  signment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The error bars depict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Polynomial  fits of data indicates δ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02] follow quadratic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δE  6 qubits  208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='9δ2  8 qubits  1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='4δ2  10 qubits  4664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2δ2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Energy shift for the problem of unstructured  search  obtained  by  perturbing  of  the  ansatz  state  as  |ψp(γ∗ + δ, β∗ + δ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Polynomial fits for data points of 6,  8 and 10 qubits follow quadratic curves in the ranges δ ∈  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='01], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='008] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Stochastic perturbation  We now consider the complete noise model in (6) and  verify our analytical prediction as shown in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  For  each 3-SAT instance, we randomly sample perturbations  δ to each of the gates from a uniform distribution on the  interval (−σ, σ) and average the obtained energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Then  we average energies over instances of the same number  of qubits as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' It is seen that for small  values of σ the behaviour is quadratic as per (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' It is  seen, that the value σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='075 could never violate the  acceptance criteria, as corresponding energy error never  exceeds the gap ∆ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' For smaller number of qubits  and gates the threshold value of σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  For unstructured search, we average the energy over  δ sampled for each gate from the uniform distribution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='40  σ  0  1  2  3  4  5  δE  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0σ2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='5σ2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0 qubits  172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='8σ2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Average energy shift of 100 uniformly generated 3-  SAT instances of 6, 8 and 10 qubits with clause to variable  ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2 and unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The shifts are  obtained by the perturbation of γ∗, β∗ by δ uniformly sam-  pled from the range (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Error bars depict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Polynomial fits of data indicates σ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1] follow quadratic  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We again recover quadratic behaviour in σ, as  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  It is seen that the same threshold  σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='075 now increases energy by no more then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='6,  which guaranties 40% overlap with the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='28  σ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δE  6 qubits  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='4σ2  8 qubits  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='3σ2  10 qubits  133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2σ2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Average energy for the problem of unstructured  search obtained by the perturbation of γ∗, β∗ by δ uni-  formly sampled from the range (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Error bars de-  pict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Polynomial fits of data points of 6,  8 and 10 qubits follow quadratic curves in the ranges σ ∈  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='07], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Perturbation to individual parameters  Here we consider a modified version of (6), where pa-  rameters are perturbed one at a time while the rest are  kept intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Effect of this model on the energy is illus-  trated in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The results are numerical and  are yet to be explained analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We observe that per-  turbations to certain angles have a significantly smaller  5  tbh  γ  β  n = 6  p = 8  1  2  3  4  5  6  7  8  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0110  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0125  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  1  2  3  4  5  6  7  8  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  n = 8  p = 15  2  4  6  8  10  12  14  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0190  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
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+page_content='0205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0210  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  2  4  6  8  10  12  14  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
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+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  n = 10  p = 25  0  3  6  9  12  15  18  21  24  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0855  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0860  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0865  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0870  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  0  3  6  9  12  15  18  21  24  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='30  ⟨H⟩  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ from the unstructured search problem, where βk (right column) or γk (left  column), from the k-th layer, are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  effect on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Thus we can infer that reducing  the value of such angles would not have a significant ef-  fect on performance but will reduce the execution time of  the algorithm, that is texec = �p  k=1 βk + γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Conversely,  we could limit the execution time as texec ≤ tmax and  increase the number of layers, since  min ⟨ψp| H |ψp⟩ ≥ min ⟨ψp+1| H |ψp+1⟩  (16)  for the same tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Reducing the execution time is important to quantum  algorithms, since variational parameters are proportional  to the time required to execute a gate experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  NISQ era devices suffers from limited coherence, thus  reducing execution times can lead to more efficient hard-  ware utilization [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We test these ideas in the setting  of unstructured search, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Here we  show the optimized QAOA energies for 6 qubits at mul-  tiple depths with execution time limited to tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The  highlighted green and orange rectangles depict the two  groups of optimal angles that minimize the energy at  each depth, as presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Green rectangles also  indicate the depth and texec at which an ansatz will not  be able to decrease its energy by either increasing depth  or tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Following the observations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 5, by slightly  reducing tmax the optimizer will reduce the parameters  to which the energy is less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This results in a  6  γ  β  n = 6  p = 15  2  4  6  8  10  12  14  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='14  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  2  4  6  8  10  12  14  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='200  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  n = 8  p = 25  0  3  6  9  12  15  18  21  24  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='16  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  0  3  6  9  12  15  18  21  24  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='25  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  n = 10  p = 30  0  4  8  12  16  20  24  28  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  0  4  8  12  16  20  24  28  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='30  ⟨H⟩  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='02  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='05  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='08  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Average energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ of 100 uniformly generated 3-SAT instances where βk (right  column) or γk (left column), from the k-th layer, are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The instances are of 6, 8 and 10 qubits with clause to variable  ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2 and unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  slight energy increase as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 7 where to  the left of the green rectangles we can observe darkening  gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  By contrast, orange rectangles highlight longer execu-  tion times corresponding to different sets of angles that  also minimize the energy for a given number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Therefore if the optimization routine finds the a solu-  tion corresponding to the orange rectangle, setting tmax  to be slightly less than the texec of the orange rectangle  will lead the optimizer to find angles corresponding to  the green rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This will amount to a considerable  reduction in execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Alternatively increasing the number of layers while  keeping tmax will reduce the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' In general, for an  arbitrary problem Hamiltonian we can not be sure if our  optimization has returned the ideal set of angles (green  ones in our example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' For this reason, the best strategy  would be to reduce tmax or increase depth while fixing  tmax until performance stagnates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  DISCUSSION  In this study we considered a realistic noise model—  one where the variational gate parameters are stochasti-  cally perturbed—and demonstrated its effect on the per-  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0  max execution time tmax  8  7  6  5  4  3  2  1  depth p  energy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Expected value for multiple combinations of depth for maximum execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Green and orange rectangles depict  the two branches of angles that minimize expectation value for a given depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  formance of variational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Using a perturba-  tive analysis we showed that the change in energy δE  (from optimised energy E∗), caused due to the pres-  ence of the considered gate errors, behaves quadratically  with respect to the angle perturbations for small values  of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This allows us to establish upper  bounds on the amount of perturbation such that the ac-  ceptance condition continues to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This guar-  antees a fixed overlap between the target sate and the  state prepared by the noisy variational circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We con-  firm our analytical findings numerically in QAOA for two  common problems—3-SAT and unstructured search, us-  ing different modifications of the considered noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Moreover we observed form our numerical results that  the algorithmic performance is more resilient to pertur-  bations of certain variational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Motivated by  this observation we demonstrated that performance of  QAOA with a total execution time texec = �  k γk + βk  is stable if retrained with a maximum execution time  tmax = texec ± ϵ for ϵ ≪ texec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' We also show that in  some cases (a) reduction in tmax can lead to dramatic  reductions in texec, and (b) increasing depth while fixing  texec can lead to an energy reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  While our study is primarily focused on energy pertur-  bations around the noiseless optimum θ∗, in practice one  has to train in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' This would change  optimal angles θ∗ → θ∗ + δθ∗, where shift δθ∗ increases  with increase of the strength of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Nevertheless,  using perturbation theory around the noiseless optimum  one can estimate δθ∗ = O(σ2), and the corresponding  change in the energy is Tr(ρ(θ∗+δθ∗)H)−Tr(ρ(θ∗)H) =  O(σ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Therefore, working in the regime of weak noise  one can safely use noiseless optimum θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' See appendix  C for detailed calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' acknowledge support  from the research project, Leading Research Center on  Quantum Computing (agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' 014/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [1] John Preskill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Quantum computing in the nisq era and  beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Quantum, 2:79, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [2] Johannes Weidenfeller, Lucia C Valor, Julien Gacon,  Caroline Tornow, Luciano Bello, Stefan Woerner, and  Daniel J Egger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Scaling of the quantum approximate  optimization algorithm on superconducting qubit based  hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='03459, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [3] Alexander K Ratcliffe, Richard L Taylor, Joseph J Hope,  and Andr´e RR Carvalho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Scaling trapped ion quantum  computers using fast gates and microtraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Physical Re-  view Letters, 120(22):220501, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [4] Swathi S Hegde, Jingfu Zhang, and Dieter Suter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Toward  the speed limit of high-fidelity two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Physical  Review Letters, 128(23):230502, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [5] Adam R Mills, Charles R Guinn, Michael J Gullans, An-  thony J Sigillito, Mayer M Feldman, Erik Nielsen, and  Jason R Petta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Two-qubit silicon quantum processor  with operation fidelity exceeding 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
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+page_content=' Becker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Kaplan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
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+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Gong,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Monroe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Observation of a many-body dynami-  cal phase transition with a 53-qubit quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Nature, 551(7682):601–604, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [36] Laird Nicholas Egan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Scaling Quantum Computers with  Long Chains of Trapped Ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' PhD thesis, University of  Maryland, College Park, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [37] Zhi-Cheng Yang,  Armin Rahmani,  Alireza Shabani,  Hartmut Neven, and Claudio Chamon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Optimizing vari-  ational quantum algorithms using pontryagin’s minimum  principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Physical Review X, 7(2):021027, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  [38] Mohannad  Ibrahim,  Hamed  Mohammadbagherpoor,  Cynthia Rios, Nicholas T Bronn, and Gregory T Byrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Pulse-level optimization of parameterized quantum cir-  cuits for variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' arXiv preprint  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='00350, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  9  Appendix A: 3-SAT and unstructured search  problems  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  3-SAT  Boolean satifyability, or SAT, is the problem of deter-  mining weather a boolean formula written in conjunctive  normal form (CNF) is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' It is possible to map  any SAT instance via Karp reduction into 3-SAT, which  are restricted to 3 literals per clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  In order to ap-  proximate solutions to SAT we embed the instance into  a Hamiltonian as  HSAT =  �  j  P(j),  (A1)  where j indexes clauses of an instance, and P(j) is the  tensor product of projectors that penalizes bit string as-  signments that do not satisfy the j-th clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Unstructured search  Consider an unstructured database S indexed by j ∈  {0, 1}×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Let f : {0, 1}×n → {0, 1} be a Boolean function  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' black box) such that:  f(j) =  �  1  iff j = t  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (A2)  The task is to find t ∈ {0, 1}×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The corresponding prob-  lem Hamiltonian for QAOA is  Ht = 1 − |t⟩⟨t|,  (A3)  thus the expected value is given by  ⟨H⟩ = 1 − |⟨t|ψp(γ, β)⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (A4)  QAOA performance for unstructured search is not sen-  sitive to the particular target state |t⟩ in the computa-  tional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' For any target state |t⟩ representing a binary  string, there is a U = U † composed of X and 1 opera-  tors such that U |0⟩⊗n = |t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' The overlap of an arbitrary  state prepared by a QAOA sequence with |t⟩ is then:  ⟨t|ψp(γ, β)⟩ = ⟨t|  p  �  k=1  e−iβkHxe−iγk|t⟩⟨t| |+⟩⊗n  = ⟨0|⊗n U  p  �  k=1  e−iβkHxe−iγkU(|0⟩⟨0|)⊗nU |+⟩⊗n  = ⟨0|⊗n U  p  �  k=1  e−iβkHxUe−iγk(|0⟩⟨0|)⊗nU |+⟩⊗n  = ⟨0|⊗n  p  �  k=1  e−iβkHxe−iγk(|0⟩⟨0|)⊗n |+⟩⊗n ,  which is independent on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Appendix B: Energy variation in presence of  constant perturbations to gate parameters  Using (9) one can calculate perturbation to the energy  caused by a shift of the optimal angles by a constant δθ  as  δE = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ − ⟨ψ(θ∗)| H |ψ(θ∗)⟩  = −  q  �  k=1  δθ2  kE∗ +  q  �  m̸=k  δθkδθm(⟨ψ(θ∗)| H |ψkm⟩ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=')  +  q  �  m,k  δθkδθk ⟨ψm| H |ψk⟩ + o(δθkδθm)  = 1  2(δθ)T Hδθ + o(δθkδθm),  (B1)  where |ψmk⟩ = |ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='0⟩ with 1 placed only at m-th  and k-th positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' H is the Hessian of the energy at  noiseless optimum, Hij =  ∂2  ∂θi∂θj  ⟨ψ(θ)| H |ψ(θ)⟩ |θ=θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Here we use the fact that at the optimal position linear  contribution to the cost function necessarily vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' It  is seen now that for the constant perturbation δθk = δ  the energy changes as δE ∝ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Appendix C: Optimal parameters variation in the  presence of noise  Let us use expressions (11) and (12) to estimate change  in the energy if one accounts for shift of optimal param-  eters θ∗ → θ∗ + δθ∗:  Tr(ρ(θ∗ + δθ∗)H) = (1 −  q  �  k=1  ak) ⟨ψ(θ∗ + δθ∗)| H |ψ(θ∗ + δθ∗)⟩ +  q  �  k=1  ak ⟨ψk(θ∗ + δθ∗)| H |ψk(θ∗ + δθ∗)⟩ + o(σ2  k)  (C1)  We introduce gradients of the noisy terms Bk  =  ∂  ∂θ ⟨ψk(θ)| H |ψk(θ)⟩ |θ=θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Notice that gradients of the  10  noiseless function ⟨ψ(θ)| H |ψ(θ)⟩ vanish at optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  Then,  Tr(ρ(θ∗ + δθ∗)H) ≈ (1 −  q  �  k=1  ak)E∗ + 1  2(δθ∗)T Hδθ∗  +  q  �  k=1  ak[⟨ψk(θ∗)| H |ψk(θ∗)⟩ + (δθ∗)T Bk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (C2)  Minimizing it with respect to δθ∗ one gets δθ∗ =  �q  k=1 akH−1Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content=' Thus, if we account for the change of  optimal parameters in the presence of noise, the energy  shifts by  Tr(ρ(θ∗ + δθ∗)H) − Tr(ρ(θ∗)H) ≈  (δθ∗)T Hδθ∗ +  q  �  k=1  ak(δθ∗)T Bk = O(σ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
+page_content='  (C3)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}
diff --git a/6NFJT4oBgHgl3EQflSzb/content/tmp_files/2301.11583v1.pdf.txt b/6NFJT4oBgHgl3EQflSzb/content/tmp_files/2301.11583v1.pdf.txt
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+Tunable Strong Magnon-Magnon Coupling in Two-
+Dimensional Array of Diamond Shaped Ferromagnetic 
+Nanodots 
+Sudip Majumder1, Samiran Choudhury1, Saswati Barman2, Yoshichika Otani3, 4, 
+Anjan Barman1,* 
+1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for 
+Basic Sciences, Block JD, Sector III, Salt Lake, 700106, Kolkata, India 
+2Institute for Engineering and Management, Sector V, Salt Lake, 700091, Kolkata, India 
+3 CEMS-RIKEN, 2-1 Hirosawa, Saitama, 3510198, Wako, Japan 
+4Institute for Solid State Physics, University of Tokyo, 515 Kashiwanoha, Chiba, 277 8581, 
+Kashiwa, Japan 
+*Email: abarman@bose.res.in 
+ 
+ 
+Abstract 
+Hybrid magnonics involving coupling between magnons and different quantum particles have 
+been extensively studied during past few years for varied interests including quantum 
+electrodynamics. In such systems, magnons in magnetic materials with high spin density are 
+utilized where the “coupling strength” is collectively enhanced by the square root of the number 
+of spins to overcome the weaker coupling between individual spins and the microwave field. 
+However, achievement of strong magnon-magnon coupling in a confined nanomagnets would 
+be essential for on-chip integration of such hybrid systems. Here, through intensive study of  
+interaction between different magnon modes in a Ni80Fe20 (Py) nanodot array, we demonstrate 
+that the intermodal coupling can approach the strong coupling regime with coupling strength 
+up to 0.82 GHz and cooperativity of 2.51. Micromagnetic simulations reveal that the 
+intermodal coupling is mediated by the exchange field inside each nanodot. The coupling 
+strength could be continuously tuned by varying the bias field (Hext) strength and orientation 
+(), opening routes for external control over hybrid magnonic systems. These findings could 
+greatly enrich the rapidly evolving field of quantum magnonics. 
+ 
+1. Introduction 
+Hybrid quantum systems [1] have recently attracted great attention due to their fundamental 
+importance and potential applications. It provides a new paradigm for the coherent transfer of 
+
+quantum states from one platform to another to execute quantum information processing [2,3]. 
+This significantly facilitates the research on the fundamental physics of coupling between 
+different platforms which may lead to varied applications of quantum technologies, such as: 
+quantum computing [4,5], quantum communications [6,7], and quantum sensing [8]. The 
+introduction of magnons in hybrid systems was initiated from the exploration of spin ensembles 
+coupled to microwave photons [8-10]. The higher densities of spin in magnetic materials and 
+their collective dynamics as magnons, provide ultra-strong coupling with cooperativity up to 
+103-104 [11,12]. During the last decade, extensive research has been done on magnon-magnon 
+coupling [13-19]. However, on-chip integration of hybrid systems requires downscaling the 
+dimensions of the systems to the nanometer range. The microwave cavity usually has the 
+dimension of millimeters. The coupling strength (g) is proportional to the square root of the 
+number of spins present in the magnetic material [20,21]. To increase the coupling strength the 
+number of spins in the magnetic material is usually required to be large enough (N  1013), 
+thereby restricting the size of the microwave cavity and magnet and the ensuing device 
+miniaturization towards CMOS integration. 
+To overcome this geometrical limitation of a microwave cavity, it becomes imperative to 
+search for different systems to act as nanometric resonators. In this context, the recent 
+development of interlayer magnon coupling or exchange-driven magnon-magnon coupling in 
+the magnetic systems has opened a new avenue for quantum magnonics [22-24]. In the last 
+decade, extensive studies have been done using both confined and propagating magnons in the 
+field of magnonics, which emerged as an exciting field of research. To this end single 
+nanomagnets have been studied extensively due to their geometrically confined rich volume 
+and localized magnetic modes [25-29] in nanometer dimension and their tunability with 
+different external parameters. Therefore, such systems possess great potential in quantum 
+magnonics with the possibility of developing magnon-based on-chip quantum information 
+processing systems in the GHz and THz frequency range with high energy efficiency. Recently 
+magnon-magnon coupling has been observed experimentally in ferromagnetic nanowire 
+array[15] and in single nanomagnet using micromagnetic simulation[30]. Furthermore, 
+moderate to strong magnon-magnon coupling have also been observed in Ni80Fe20 (Py) 
+nanocross array mediated by dynamic dipolar interaction [31] and anisotropic dipolar 
+interaction[32]. These studies have opened a new approach for executing and controlling this 
+phenomenon in a large variety of systems by tailoring the geometric and material parameters 
+of these artificially patterned systems and the external bias field. This leads the quest for 
+
+optimal solutions for applications in magnon-based quantum information technology. 
+ 
+Here, we have explored magon-magnon coupling in diamond-shaped Py nanodot array with 
+the aid of a broadband ferromagnetic resonance (FMR) spectrometer[33,34] and 
+micromagnetic simulations. Remarkably, we observe an avoided crossing (anticrossing) of 
+magnon modes [1] characteristic of the formation of hybrid system. Anticrossing gap of up to 
+0.82 GHz and the ensuing cooperativity value as high as 2.51 are observed. Micromagnetic 
+simulations reveal that the coupling between two magnon modes is mediated by the exchange 
+field within each nanodot. Furthermore, the coupling strength is found to be highly dependent 
+on the orientation and strength of the bias magnetic field, leading towards the possibility of 
+externally controlled hybrid magnonic devices. 
+ 
+ 
+2. Experimental Details 
+The 20-nm-thick diamond shaped Py nanodots, arranged in an array of dimensions 25 μm × 
+200 μm, were prepared on self-oxidized Si [100] substrate by using electron beam evaporation 
+(EBE), electron beam lithography (EBL), and Ar+ ion milling tools. A coplanar waveguide 
+(CPW) made of Au, having 150 nm thickness, 30 μm wide central conducting (signal) line and 
+50 Ω characteristic impedance (Fig. 1(a)) was deposited on top of each array for broadband 
+FMR measurements. The CPW is separated from the nanodot array by a 60-nm-thick insulating 
+Al2O3 layer. The fabrication details are described in section S1 of the Supplementary Materials. 
+Fig. 1(b) exhibits the scanning electron microscope (SEM) image of the diamond nanodot array 
+arranged in a square lattice having width and height of the nanodots as 325 nm (dx) and 350 
+nm (dy) and lattice constant of 400 nm. The nanomagnet’s lateral dimensions and pitch are 
+shown in the SEM image of Fig. 1(b). The SEM image shows that the fabricated structures 
+suffer from slight edge deformations and rounded corners. All these deformations have been 
+incorporated in the micromagnetic simulations as described later. The applied bias magnetic 
+
+field orientation is shown in the inset of Fig. 1(b). The spin-wave (SW) spectra from the 
+samples were measured using a broadband FMR spectrometer, consisting of a high-frequency 
+Vector Network Analyzer (VNA, Agilent PNA-L, model no.: N5230C, frequency range: 10 
+MHz to 50 GHz) and a homemade high-frequency probe station equipped with nonmagnetic 
+ground-signal-ground (GSG)-type picoprobe (GGB Industries, model no.: 40A-GSG-150-
+EDP) and a coaxial cable. One end of the CPW is shorted and the back-reflected signal is 
+collected and fed back to the VNA by the same GSG probe and the coaxial cable. From the 
+frequency dependent real part of the S-parameter in the reflection geometry (Re (S11)), different 
+SW frequencies are identified, which results in the characteristic SW spectrum of the sample. 
+Additional details of the experimental setup are given in section S2 of the Supplementary 
+Materials. 
+ 
+ 
+ 
+FIG. 1. (a) Schematic of the experimental geometry. The directions of the bias magnetic field 
+(Hext) and rf magnetic field (hrf) are shown in the schematic. (b) SEM image of diamond-shaped 
+Ni80Fe20 (Py) nanodots arranged in a square lattice having lattice constant a = 400 nm and nanodot 
+width dx = 325 nm, height dy = 350 nm. The inset again shows the orientation of Hext with respect 
+to hrf. (c) Real parts of the forward scattering parameter (S11) representing the FMR spectra at Hext 
+= 400 Oe applied at an azimuthal angle  = 0°. The observed spin-wave (SW) modes are marked 
+by down arrows. (d) Bias field (Hext) dependent SW absorption spectra of Py nanodots is shown 
+at  = 0°. The surface plots correspond to the experimental results, while the symbols represent 
+the simulated data. The color map for the surface plots and the schematic of Hext are given at the 
+bottom right corner of the figure. 
+ 
+3
+6
+9
+0.0
+0.5
+1.0
+ 
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ 
+Frequency (GHz)
+Hext (kOe)
+Frequency (GHz)
+Re S11 (Normalized) 
+M1
+M2
+M3
+Hext= 400 Oe
+(a)
+(b)
+(c)
+(d)
+500 nm
+x
+y
+Hext
+
+dx
+a
+dy
+Re S11
+Normalised
+1
+0
+(b)
+
+G
+s
+G 
+3. Results and Discussion 
+3.1. Experimental Result 
+3.1.1. Field Dependence of SW 
+       The SW absorption spectra (Re (S11)) are acquired from FMR measurements for a broad 
+range of bias magnetic field. Fig. 1(c) shows representative raw spectra at Hext = 400 Oe.  At 
+first, the magnetization of the samples are saturated along the +x direction by applying Hext = 
+1800 Oe, followed by gradual reduction of the field from 1600 Oe to 0 Oe at steps of 20 Oe in 
+a single trace. The surface plot in Fig. 1(d) displays the bias-field-dependent of SW absorption 
+spectra with their maximum power normalized to 1.0. These surface plots are generated from 
+the individual Re (S11) spectra acquired at a given applied magnetic field. Here, the bright 
+regions represent the experimental data while the symbols represent the micromagnetic 
+simulation results. The normalized surface plots help to identify three separate branches of SW, 
+among which the lowest frequency branch M1 shows maximum intensity in the entire field 
+regime. As we decrease the bias field M1 shows a dip (minimum) in f-Hext at Hext ≈ 300 Oe, 
+which indicates a mode softening due to transition in magnetization state of the nanomagnet 
+array. Other two SW modes M2 and M3 do not show any such transition and monotonically 
+decrease with the reduction in the bias field. 
+ 
+Fig. 2 shows the magnetic field dependences of the frequencies at different bias field angles. 
+The variation of magnetic field orientation creates some remarkable changes. First, the dip in 
+M1 occurring at ~300 Oe gradually disappears. Fig. 2(a) shows the f-Hext plot at  = 5, where 
+the dip shows an upward shift. At  = 15, the dip completely disappears and the M1 shows a 
+monotonic variation of frequency with the field, as shown in Fig. 2(b). Secondly, the relative 
+intensity of M2 and M3 shows a clear variation with the bias field orientation. For 5 ≤   ≤ 
+15, M2 gradually losses its intensity at the expense of gradual increment of intensity of M3, 
+which starts to dominate over M2 at  = 15. With further increment of angle, M2 further loses 
+its intensity and at  = 23 it completely disappears. Fig. 2(c) shows the f-Hext plot at  = 23 
+where a clear anticrossing between the branches representing modes M1 and M3 is observed 
+at Hext = 1060 Oe. The vertical dotted line represents the anticrossing field (Hac) in the f-Hext 
+plot. The value of Hac gradually shifts towards the lower field regime as we keep increasing . 
+
+Fig. 2(d) shows the magnetic field dispersion of SW frequencies at  = 30 where an 
+anticrossing is observed at Hext = 920 Oe in between the SW modes M1 and M3. Here, the mid 
+frequency SW mode M2* reappears, though the intensity of this mode is low. With further 
+increment of , this mode becomes more prominent and two different anticrossings are now 
+observed instead of one. One of those appears in between M1 and M2* and another one in 
+between M2* and M3. At  = 45, both of the anticrossings are observed at Hext = 475 Oe as 
+shown in Fig. 2(e). With further increment of , the first anticrossing shifts towards lower bias 
+magnetic field values, whereas the second one appears in higher bias field values. Fig. 2(f) 
+shows the magnetic field dispersion of SW frequencies at  = 60 where the first anticrossing 
+in between M1 and M2* appear at Hext = 410 Oe and second one at Hext = 600 Oe. 
+ 
+3.1.2. Angular Dependence of SW 
+ 
+The variation of SW modes and their mutual interactions show high dependence on the in-
+plane magnetic field orientation. For this reason, -dependence of SW spectra were acquired 
+at a constant bias field magnitude Hext in the range 0º ≤  ≤ 360º. In Fig. 3(a-d), we have 
+ 
+ 
+ 
+FIG. 2. Bias field (Hext) dependent SW absorption plots of Py diamond shaped nanodot array are shown 
+for the bias field orientation () of (a) 5°, (b) 15°, (c) 23°, (d) 30°, (e) 45° and (f) 60°. The surface plots 
+correspond to the experimental results, while the symbols represent the simulated data. The color map 
+for the surface plots and the schematic of the external applied field (Hext) are given at the bottom right 
+corner of the figure. 
+ 
+0
+400
+800
+1200
+3
+6
+9
+12
+ M1
+ M2*
+ M3
+ 
+0
+500
+1000
+1500
+ M1
+ M3
+ 
+ 
+ 
+Frequency (GHz)
+Hext (kOe)
+0
+400
+800
+1200
+ M1
+ M2
+ M3
+ 
+ = 15 
+ = 30 
+0
+400
+800
+1200
+ M1
+ M2*
+ M3
+ 
+ = 45 
+ = 23 
+0
+400
+800
+1200
+ M1
+ M2*
+ M3
+ 
+ = 60 
+0
+400
+800
+1200
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ 
+ = 5 
+x
+y
+Hext
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Re S11
+Normalized
+1
+0
+
+presented the -dependence at Hext = 200, 400, 600 and 800 Oe. To show the anticrossing points 
+we have magnified the relevant regions of the -dependent SW spectra. In the Supplementary 
+Information figure S4, we have shown the full range of -dependence. At a lower field value 
+like Hext = 200 Oe, only M1 shows angular dispersion as shown in Fig. 3(a). With an increment 
+in Hext, two more modes start to show angular dispersion. Here, mode M1 shows a sharp 
+variation of frequency with a minimum at  = 0, corresponding to the minimum observed in 
+Fig. 1(d). As we increase the field this sharp modulation gradually transforms into a continuous 
+angular variation. Fig. 3(b) shows the angular dispersion at Hext = 400 Oe. For   between 50 
+and 55, an anticrossing gap appears in between M1 and M2* which is shown by a white dotted 
+line. At a higher field of Hext = 600 Oe instead of one, two different anticrossings are observed. 
+The first one appears in between M1 and M3 at  = 40 while the 2nd one appears in between 
+M2* and M3 at  = 60. With an increment of magnetic field (e.g., 800 Oe) the first anticrossing 
+shifts towards lower angle (e.g. 35), while the second one gradually disappears as shown in 
+Fig. 3(d). Due to four fold symmetry[35] of diamond shaped nanodot array these anticrossing 
+also appear in other three quadrants of angular variation spectra of SW, which is shown in 
+section S4 of supplementary section.  
+ 
+
+   
+ 
+3.1.3. Anticrossing Strength 
+Fig. 4(a) shows the power spectrum measured at Hext = 1060 Oe, which is the anticrossing field 
+(Hac) for  = 23 configuration. The blue line represents the FMR spectra whereas the red line 
+represent the fitted spectra using an antisymmetric lorentzian function. Other FMR spectra for 
+varying anticrossing fields are presented in section S5 of Supplementary Information. The 
+magnon–magnon coupling strength g is defined as half of the peak-to-peak frequency spacing 
+at the anticrossing field, which is shown in Fig. 4(a). In order to estimate the strength of 
+interaction between these two modes, we have extracted the value of g13 and the corresponding 
+dissipation rates 1, 3 as shown in Fig. 4(a). Here, 1 and 3 are defined as half-width at half-
+maximum of the FMR peak of SW mode M1 and M3, respectively.  
+ 
+ 
+ 
+ 
+FIG. 3. Variation of SW frequency as a function of the azimuthal angle () varying from 0° to 360° for 
+bias field value fixed at (a) Hext = 200 Oe, (b) 400 Oe, (c) 600 Oe and (d) 800 Oe. The surface plots 
+correspond to the experimental results, while the symbols represent the simulated data. The colour map 
+for the surface plots and the schematic of Hext are shown on the right side of the figure.  
+ 
+ 
+ 
+6
+9
+ M2*
+ M3
+ M1
+ M2
+ 
+ 
+ 0
+-60
+60
+3
+6
+9
+ M1
+ M2
+ M3
+ M1
+ M2
+ M3
+ M2*
+ 
+0
+-60
+60
+Frequency (GHz)
+x
+y
+Hext
+
+6
+9
+ M2*
+ M3
+ M1
+ M2
+ 
+0
+-60
+60
+Azimuthal Angle,  (Degree)
+3
+6
+9
+ M2
+*
+ M3
+ M1
+ M2
+ 
+0
+-60
+60
+(a)
+(b)
+(c)
+(d)
+Re S11
+Normalized
+1
+0
+200 Oe
+400 Oe
+600 Oe
+800 Oe
+
+ 
+ 
+ 
+ 
+At  = 23 the extracted value of g13 is 0.592 GHz, while the values of 1 and 3 are found to 
+be 0.60 GHz and 0.711 GHz, respectively. Since g13  1 and 3, therefore the interaction 
+between M1 and M3 can be considered as weak coupling. In the opposite case, i.e. when g13 > 
+1 and 3 it will be considered as strong coupling between two SW branches. We have also 
+calculated magnon–magnon cooperativity (C), which is defined as C = g2/() (, = 1, 2, 
+3) and obtained C13 = 0.821 for the coupling between M1 and M3. The extracted value of g, 
+k, k, and the estimated value of C for anticrossing points corresponds to different bias field 
+ 
+ 
+g13 
+(GHz) 
+g12 
+(GHz) 
+g23 
+(GHz) 
+1(GHz) 
+2(GHz) 
+3(GHz) 
+C13 
+C12 
+C23 
+23o 
+0.592 
+- 
+- 
+0.60 
+- 
+0.711 
+0.821 
+ 
+ 
+30o 
+0.82 
+- 
+- 
+0.423 
+- 
+0.660 
+2.515 
+ 
+ 
+45o 
+- 
+0.745 
+0.255 
+0.426 
+0.645 
+0.645 
+- 
+2.019 
+0.113 
+60o 
+- 
+0.915 
+0.205 
+1.35 
+0.69 
+0.707 
+- 
+0.878 
+0.675 
+ 
+Table 1 The extracted values of coupling strength (g), FWHM (2k) and calculated cooperativity factor 
+(C) for different orientation of bias field at the anticrossing points. Values of g and k are extracted 
+from the FMR spectra). 
+ 
+ 
+FIG. 4. Real part of S11 parameter as a function of frequency to highlight the anticrossing field are 
+shown for  = (a) 23°. The frequency gap in the anticrossing mode reveals the coupling strength g. (b) 
+Variation of cooperativity factor with the orientation of bias field. It shows that coupling strength is 
+stronger at  = 30 and 45. The schematic of Hext are shown on the right side of the figure.  
+ 
+ 
+ 
+ 
+ 
+8
+10
+0.0
+0.3
+0.6
+0.9
+ 
+ 
+ = 23 
+1062 Oe
+Frequency (GHz)
+Re S11  (Normalized)
+x
+y
+Hext
+
+2k1
+2k3
+2g
+(a)
+20
+40
+60
+0
+1
+2
+3
+ C13
+ C23
+ C12
+ 
+ 
+ (Degree)
+Cooperativity 
+(b)
+
+angles are listed in Table 1. At  = 30  obtained value for g13, 1, 3 and C13 are estimated 
+0.82, 0.423, 0.66, and 2.515, respectively and here this magnon-magnon coupling falls in the 
+strong coupling regime. From Table 1, we can see that first anticrossing at  = 45 also shows 
+strong magnon-magnon coupling with C = 2.019, while the second one shows weak interaction. 
+At  = 60  both the interactions are in the weak coupling regime. Fig. 4(b) shows the -
+dependence of the C where it shows the tunability of coupling strength with the in-plane 
+magnetic field orientation. It also exhibits that the interaction between different SW branches 
+show strong coupling in-between 30 to 45 orientation.  
+ 
+3.2.  Micromagnetic Simulation 
+3.2.1. Static Magnetic Configuration 
+ 
+In Fig. 1(d) at  = 0, a sharp minimum is observed which gradually vanishes for higher values 
+of . The answer to this lies in the nanodot structure and its rich and flexible spin configurations 
+which we have simulated using OOMMF software[36]. Details of the micromagnetic 
+simulations are given in section S3 of the Supplementary Materials. The simulations reproduce 
+important features of the experimental SW spectra with nearly identical frequencies and 
+number of modes besides their relative intensity variations. The simulated static spin textures 
+within the nanomagnet array for different bias field magnitudes Hext at  = 0 and 45 are shown 
+in Fig. 5. At  = 0, the nanodot structure shows drastic variation in spin configurations with 
+Hext. It shows the formation of an S-state at the lower field regime (Hext = 100 Oe) as shown in 
+Fig. 5. At larger bias fields (e.g., Hext = 800 Oe), the spins are nearly aligned along the bias-
+field direction (x-axis) and switch to a leaf-state (Fig. 5). This transformation from S- to leaf-
+state occurs for 250 Oe ≤ Hext ≤ 350 Oe, where the SW frequency shows a minimum as a 
+function of Hext. At   = 45 , this transformation is not observed. Here, for the entire field 
+range, the static magnetic configuration shows a leaf state.  
+
+ 
+ 
+3.2.2. SW mode Characterization 
+ 
+To interpret the nature of the SW modes, we have further simulated the spatial profiles of power 
+and phase of each SW mode by using a home-built MATLAB based code Dotmag[37]. 
+OOMMF simulation provides magnetization (M (r, t)) information of each rectangular prism-
+like cell at different simulation times. By performing discrete Fourier transformation with 
+respect to time in each of these cells and subsequently extracting the power and phase of the 
+dynamic magnetization for a desired frequency gives rise to the spatial distribution of the power 
+phase profile for that particular mode. In Fig. 6, we have shown the power distribution profile 
+of SW mode at  = 45 orientation for five different fields, Hext = 200 Oe (Hext << Hac), 400 
+Oe (Hext < Hac), 475 Oe (Hac), 600 Oe (Hext  Hac) and 1000 Oe (Hext >> Hac), while the phase 
+profile for each case is shown in the inset. The power profile at Hext = 1000 Oe indicates that 
+at high bias field only existing mode is M3, which is boosted by all the available energy. With 
+a gradual decrement of bias field, two additional modes M1 and M2 appear and the power of 
+  
+ 
+ 
+FIG. 5. Simulated static magnetic configurations for Py nanodot array at four different bias magnetic-
+field magnitude (Hext) at  = 0 and  = 45. We have shown here a single nanodot from the center of 
+the array for clarity in spin configurations. The nanodot structure shows a drastic variation in spin 
+configurations with bias magnetic-field strength. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+100 Oe
+250 Oe
+800 Oe
+350 Oe
+
+x
+y
+Hext
+
+0 
+45 
+-Y
++Y
+0.0
+0.3
+0.6
+0.9
+1.2
+M1
+M2
+M3
+M4
+M5
+M'
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Frequency (GHz)
+H1
+H2
+0.0
+0.3
+0.6
+0.9
+1.2
+ M*
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+H3
+0.0
+0.3
+0.6
+0.9
+1.2
+ M '
+  M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Applied Field Hext (kOe)
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ 
+ 
+O1
+O2
+O3
+H2
+Fig 2
+Hext
+x
+y
+1
+0
+Re (S11)
+Normalised
+
+M3 is gradually transferred to these two modes. At the anticrossing field, Hext = 475 Oe, M2 
+appears as the most intense mode although M1 and M3 have significant power at this field. At 
+lower fields, this power is gradually transferred to M1, and at 200 Oe, barring M1 other modes 
+  
+ 
+ 
+FIG. 6. Simulated spatial distribution of power and phase (in the inset) profiles corresponding to 
+different SW modes at five different bias field values for  = 45 for the Py nanodot array. The 
+applied field direction is shown at the bottom left of the figure. Symbols with different colors 
+represent different SW modes. The color map is shown at the upper right side of the figure. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+200 Oe
+400 Oe
+475 Oe
+600 Oe
+1000 Oe
+M1
+M2*
+M3
+20
+0
+Power 
+(dB)
+Phase
+(rad)
++
+-
+x
+y Hext
+
+ = 45 
+
+disappear. Similar to this energy exchange, the phase profiles also exhibit interchange of mode 
+behavior. At high bias fields (e.g., 1000 Oe), M3 shows quantized nature in BV-like geometry 
+with a quantization number n = 3. With a decrease in the field, this mode gradually transforms 
+into higher-order quantized mode and M2* is transformed into a quantized mode with n = 3. 
+At Hext = 475 Oe the quantization number of M1, and M3 are n = 5, and 7, respectively, while 
+for M2*, n = 3, which is identical to the quantization number of M3 at Hext = 1000 Oe. This 
+transformation of mode quantization number is also seen in-between M1 and M2* as we further 
+reduce the bias field and finally at Hext = 200 Oe, M1 shows a quantized behavior with n = 3. 
+This transformation of power as well as mode property from one branch of SW mode to another 
+at the anticrossing region indicates a strong interaction between these modes. For other 
+orientation like  = 23, 30 and 60, similar kind of behavior are observed, which are shown 
+in section S6 of the Supplementary Materials. 
+ 
+3.2.3. Distribution of Exchange field 
+To understand the origin of the magnon-magnon coupling and its modulation with bias 
+magnetic field, we have simulated the spatial distribution of the dipole-exchange field 
+(Exchange field distribution of each dot, which is modulated by dipolar interaction of nanodot 
+array) lines at the equilibrium for different bias field orientations. Fig. 7 shows the exchange 
+field map of nanodots array at eight different fields for  = 45 orientation. Due to inter-dot 
+dipolar interactions, a dynamic variation of exchange field line with the bias field amplitude 
+(for better viewing purpose, we just present a single nanodot) is observed. The Supplementary 
+Movie A1 shows the dynamics of this exchange field in more detail. At lower bias fields (Hext 
+<< Hac), due to dominating effect of demagnetizing field, spins take a configuration such that 
+at equilibrium condition the exchange field lines create three different regions within a single 
+dot. The field lines of center and edge regions are configured in opposite direction as denoted 
+with yellow and green arrows in Fig. 7(a). As we increase the bias field, the region around the 
+edge of the dot start to vanish and the center region gradually expands. At a very high bias field 
+(Hext   Hac), e.g., Hext = 1000 Oe, only the central region with unidirectional field lines are 
+observed inside a dot. This transformation from three mutually opposite (antiparallel) field-line 
+configuration to uniform (parallel) configuration occurs for 450 Oe ≤ Hext ≤ 500 Oe, which is 
+exactly the anticrossing field region for  = 45 orientation. This change in exchange field 
+profile can be observed much more clearly if we take a linescan along the bias field direction 
+(white dotted line in Fig. 7(a)) as shown in Fig. 7(b). In the inset, we have magnified the end 
+
+part of the linescan. Here, it is clearly visible that below the anticrossing field (Hext = Hac = 475 
+Oe) the linescan has two different local maxima  
+ 
+which transform into one maximum as we increase Hext. The exchange field profile for other 
+values of   are shown in section S7 of the Supplementary Materials, where similar 
+transformation is observed in the anticrossing field region. Our observation of correlation 
+between these two phenomena indicates that the anticrossing gap appears only when such a 
+variation of exchange field occurs due to the bias field strength as well as its orientation. The 
+internal field distribution in presence and absence of the exchange field leads to similar 
+conclusion, which we have described in section S8 of the Supplementary Materials. 
+ 
+4. Conclusion 
+In summary, the interaction between magnons confined in a sole magnonic cavity has been 
+realized in the strong coupling regime. We have investigated a bias field strength and angle-
+dependent magnetization dynamics in diamond-shaped Py nanodot arrays using the broadband 
+ferromagnetic resonance technique. Our study has demonstrated that the coupling between two 
+magnon modes is mediated by the exchange coupling inside individual nanodot. Furthermore, 
+the coupling strength is found to be highly dependent on the orientation and strength of 
+the bias magnetic field, leading towards the possibility of externally controlled hybrid 
+ 
+ 
+ 
+ 
+FIG. 7. Exchange field distributions for (a) single nanodot for eight different bias field values at  
+= 45 . Yellow and green arrows represent the direction of exchange field at the center and edge 
+position of the nanodot. We have shown here a single nanodot from the center of the array for clarity 
+in spin configurations. The color bars are shown at the right side of the figure. (b) Linescan of the 
+simulated exchange field for nanodot array along the field direction. In the inset magnified portion 
+of simulated exchange field is shown. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+360
+420
+480
+540
+0
+250
+500
+  1200 Oe
+  550 Oe
+  450 Oe
+  200 Oe
+ 
+ 
+Exchange Field (Oe)
+Distance (nm)
+x
+y
+Hext
+
+(a)
+(b)
+200 Oe
+300 Oe
+450 Oe
+500 Oe
+550 Oe
+700 Oe
+800 Oe
+1000 Oe
+-8
+-6
+-4
+-2
+  1200 Oe
+  550 Oe
+  450 Oe
+  200 Oe
+ 
+ 
+Power(dB)
+0.0
+0.3
+0.6
+0.9
+1.2
+M1
+M2
+M3
+M4
+M5
+M'
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Frequency (GHz)
+H1
+H2
+0.0
+0.3
+0.6
+0.9
+1.2
+ M*
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+H3
+0.0
+0.3
+0.6
+0.9
+1.2
+ M '
+  M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Applied Field Hext (kOe)
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ 
+ 
+O1
+O2
+O3
+H2
+Fig 2
+Hext
+x
+y
+1
+0
+Re (S11)
+Normalised
+
+800 Oe7000e600 0e200 0e500 0emagnonic devices. The experimental results have been well reproduced by micromagnetic 
+simulation. The power and phase profiles of the resonant modes have been numerically 
+calculated to gain insight into the spatial nature of the dynamics. The transformation of power 
+as well as mode property from one branch of SW to another, apparently support the strong 
+interaction in-between these modes. Numerical study shows that the anticrossing gap appears 
+when the symmetry of exchange configuration inside each nanodot is broken due to the applied 
+bias magnetic field. We have also observed mode softening phenomena when the static 
+magnetic configuration switches from the S-state to the leaf state and with the variation of bias 
+field angle it gradually disappears.  Our findings offer a new approach toward tunable magnon-
+magnon coupling in ferromagnetic nanostructures for applications in quantum transduction 
+using magnons. 
+ 
+ 
+ 
+5. Acknowledgements 
+ 
+AB gratefully acknowledges the financial support from S. N. Bose National Centre for 
+Basic Sciences, India (Grant No. SNB/AB/18-19/211). SB acknowledges Science and 
+Engineering Research Board (SERB), India for funding (Grant no. CRG/2018/002080). SM 
+and SC acknowledge S. N. Bose National Centre for Basic Sciences for senior research 
+fellowship 
+ 
+ 
+References 
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf,len=613
+page_content='Tunable Strong Magnon-Magnon Coupling in Two- Dimensional Array of Diamond Shaped Ferromagnetic Nanodots Sudip Majumder1, Samiran Choudhury1, Saswati Barman2, Yoshichika Otani3, 4, Anjan Barman1,* 1Department of Condensed Matter Physics and Material Sciences, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, 700106, Kolkata, India 2Institute for Engineering and Management, Sector V, Salt Lake, 700091, Kolkata, India 3 CEMS-RIKEN, 2-1 Hirosawa, Saitama, 3510198, Wako, Japan 4Institute for Solid State Physics, University of Tokyo, 515 Kashiwanoha, Chiba, 277 8581, Kashiwa, Japan *Email: abarman@bose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='in  Abstract Hybrid magnonics involving coupling between magnons and different quantum particles have been extensively studied during past few years for varied interests including quantum electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In such systems, magnons in magnetic materials with high spin density are utilized where the “coupling strength” is collectively enhanced by the square root of the number of spins to overcome the weaker coupling between individual spins and the microwave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' However, achievement of strong magnon-magnon coupling in a confined nanomagnets would be essential for on-chip integration of such hybrid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, through intensive study of interaction between different magnon modes in a Ni80Fe20 (Py) nanodot array, we demonstrate that the intermodal coupling can approach the strong coupling regime with coupling strength up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='82 GHz and cooperativity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Micromagnetic simulations reveal that the intermodal coupling is mediated by the exchange field inside each nanodot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The coupling strength could be continuously tuned by varying the bias field (Hext) strength and orientation (\uf066), opening routes for external control over hybrid magnonic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' These findings could greatly enrich the rapidly evolving field of quantum magnonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Introduction Hybrid quantum systems [1] have recently attracted great attention due to their fundamental importance and potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' It provides a new paradigm for the coherent transfer of  quantum states from one platform to another to execute quantum information processing [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This significantly facilitates the research on the fundamental physics of coupling between different platforms which may lead to varied applications of quantum technologies, such as: quantum computing [4,5], quantum communications [6,7], and quantum sensing [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The introduction of magnons in hybrid systems was initiated from the exploration of spin ensembles coupled to microwave photons [8-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The higher densities of spin in magnetic materials and their collective dynamics as magnons, provide ultra-strong coupling with cooperativity up to 103-104 [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' During the last decade, extensive research has been done on magnon-magnon coupling [13-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' However, on-chip integration of hybrid systems requires downscaling the dimensions of the systems to the nanometer range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The microwave cavity usually has the dimension of millimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The coupling strength (g) is proportional to the square root of the number of spins present in the magnetic material [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' To increase the coupling strength the number of spins in the magnetic material is usually required to be large enough (N \uf03e 1013), thereby restricting the size of the microwave cavity and magnet and the ensuing device miniaturization towards CMOS integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' To overcome this geometrical limitation of a microwave cavity, it becomes imperative to search for different systems to act as nanometric resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  In this context, the recent development of interlayer magnon coupling or exchange-driven magnon-magnon coupling in the magnetic systems has opened a new avenue for quantum magnonics [22-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In the last decade, extensive studies have been done using both confined and propagating magnons in the field of magnonics, which emerged as an exciting field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' To this end single nanomagnets have been studied extensively due to their geometrically confined rich volume and localized magnetic modes [25-29] in nanometer dimension and their tunability with different external parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Therefore, such systems possess great potential in quantum magnonics with the possibility of developing magnon-based on-chip quantum information processing systems in the GHz and THz frequency range with high energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Recently magnon-magnon coupling has been observed experimentally in ferromagnetic nanowire array[15] and in single nanomagnet using micromagnetic simulation[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Furthermore, moderate to strong magnon-magnon coupling have also been observed in Ni80Fe20 (Py) nanocross array mediated by dynamic dipolar interaction [31] and anisotropic dipolar interaction[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' These studies have opened a new approach for executing and controlling this phenomenon in a large variety of systems by tailoring the geometric and material parameters of these artificially patterned systems and the external bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This leads the quest for  optimal solutions for applications in magnon-based quantum information technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  Here, we have explored magon-magnon coupling in diamond-shaped Py nanodot array with the aid of a broadband ferromagnetic resonance (FMR) spectrometer[33,34] and micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Remarkably, we observe an avoided crossing (anticrossing) of magnon modes [1] characteristic of the formation of hybrid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Anticrossing gap of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='82 GHz and the ensuing cooperativity value as high as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='51 are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Micromagnetic simulations reveal that the coupling between two magnon modes is mediated by the exchange field within each nanodot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Furthermore, the coupling strength is found to be highly dependent on the orientation and strength of the bias magnetic field, leading towards the possibility of externally controlled hybrid magnonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Experimental Details The 20-nm-thick diamond shaped Py nanodots, arranged in an array of dimensions 25 μm × 200 μm, were prepared on self-oxidized Si [100] substrate by using electron beam evaporation (EBE), electron beam lithography (EBL), and Ar+ ion milling tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' A coplanar waveguide (CPW) made of Au, having 150 nm thickness, 30 μm wide central conducting (signal) line and 50 Ω characteristic impedance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(a)) was deposited on top of each array for broadband FMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The CPW is separated from the nanodot array by a 60-nm-thick insulating Al2O3 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The fabrication details are described in section S1 of the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(b) exhibits the scanning electron microscope (SEM) image of the diamond nanodot array arranged in a square lattice having width and height of the nanodots as 325 nm (dx) and 350 nm (dy) and lattice constant of 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The nanomagnet’s lateral dimensions and pitch are shown in the SEM image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The SEM image shows that the fabricated structures suffer from slight edge deformations and rounded corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' All these deformations have been incorporated in the micromagnetic simulations as described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The applied bias magnetic  field orientation is shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The spin-wave (SW) spectra from the samples were measured using a broadband FMR spectrometer, consisting of a high-frequency Vector Network Analyzer (VNA, Agilent PNA-L, model no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' : N5230C, frequency range: 10 MHz to 50 GHz) and a homemade high-frequency probe station equipped with nonmagnetic ground-signal-ground (GSG)-type picoprobe (GGB Industries, model no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' : 40A-GSG-150- EDP) and a coaxial cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' One end of the CPW is shorted and the back-reflected signal is collected and fed back to the VNA by the same GSG probe and the coaxial cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' From the frequency dependent real part of the S-parameter in the reflection geometry (Re (S11)), different SW frequencies are identified, which results in the characteristic SW spectrum of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Additional details of the experimental setup are given in section S2 of the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (a) Schematic of the experimental geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The directions of the bias magnetic field (Hext) and rf magnetic field (hrf) are shown in the schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (b) SEM image of diamond-shaped Ni80Fe20 (Py) nanodots arranged in a square lattice having lattice constant a = 400 nm and nanodot width dx = 325 nm, height dy = 350 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The inset again shows the orientation of Hext with respect to hrf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (c) Real parts of the forward scattering parameter (S11) representing the FMR spectra at Hext = 400 Oe applied at an azimuthal angle \uf066 = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The observed spin-wave (SW) modes are marked by down arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (d) Bias field (Hext) dependent SW absorption spectra of Py nanodots is shown at \uf066 = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The surface plots correspond to the experimental results, while the symbols represent the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The color map for the surface plots and the schematic of Hext are given at the bottom right corner of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3  6  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2  3  6  9  12  M1  M2  M3  Frequency (GHz)  Hext (kOe)  Frequency (GHz)  Re S11 (Normalized)  M1  M2  M3  Hext= 400 Oe  (a)  (b)  (c)  (d)  500 nm  x  y  Hext  \uf066  dx  a  dy  Re S11  Normalised  1  0  (b)  G s G 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Experimental Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Field Dependence of SW The SW absorption spectra (Re (S11)) are acquired from FMR measurements for a broad range of bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(c) shows representative raw spectra at Hext = 400 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At first, the magnetization of the samples are saturated along the +x direction by applying Hext = 1800 Oe, followed by gradual reduction of the field from 1600 Oe to 0 Oe at steps of 20 Oe in a single trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The surface plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(d) displays the bias-field-dependent of SW absorption spectra with their maximum power normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' These surface plots are generated from the individual Re (S11) spectra acquired at a given applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, the bright regions represent the experimental data while the symbols represent the micromagnetic simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The normalized surface plots help to identify three separate branches of SW, among which the lowest frequency branch M1 shows maximum intensity in the entire field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' As we decrease the bias field M1 shows a dip (minimum) in f-Hext at Hext ≈ 300 Oe, which indicates a mode softening due to transition in magnetization state of the nanomagnet array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Other two SW modes M2 and M3 do not show any such transition and monotonically decrease with the reduction in the bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2 shows the magnetic field dependences of the frequencies at different bias field angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The variation of magnetic field orientation creates some remarkable changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' First, the dip in M1 occurring at ~300 Oe gradually disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(a) shows the f-Hext plot at \uf066 = 5\uf0b0, where the dip shows an upward shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At \uf066 = 15\uf0b0, the dip completely disappears and the M1 shows a monotonic variation of frequency with the field, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Secondly, the relative intensity of M2 and M3 shows a clear variation with the bias field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' For 5\uf0b0 ≤ \uf066  ≤ 15\uf0b0, M2 gradually losses its intensity at the expense of gradual increment of intensity of M3, which starts to dominate over M2 at \uf066 = 15\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With further increment of angle, M2 further loses its intensity and at \uf066 = 23\uf0b0 it completely disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(c) shows the f-Hext plot at \uf066 = 23\uf0b0 where a clear anticrossing between the branches representing modes M1 and M3 is observed at Hext = 1060 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The vertical dotted line represents the anticrossing field (Hac) in the f-Hext plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The value of Hac gradually shifts towards the lower field regime as we keep increasing \uf066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(d) shows the magnetic field dispersion of SW frequencies at \uf066 = 30\uf0b0 where an anticrossing is observed at Hext = 920 Oe in between the SW modes M1 and M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, the mid frequency SW mode M2* reappears, though the intensity of this mode is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With further increment of \uf066, this mode becomes more prominent and two different anticrossings are now observed instead of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' One of those appears in between M1 and M2* and another one in between M2* and M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At \uf066 = 45\uf0b0, both of the anticrossings are observed at Hext = 475 Oe as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With further increment of \uf066, the first anticrossing shifts towards lower bias magnetic field values, whereas the second one appears in higher bias field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2(f) shows the magnetic field dispersion of SW frequencies at \uf066 = 60\uf0b0 where the first anticrossing in between M1 and M2* appear at Hext = 410 Oe and second one at Hext = 600 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Angular Dependence of SW  The variation of SW modes and their mutual interactions show high dependence on the in- plane magnetic field orientation\uf066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' For this reason, \uf066-dependence of SW spectra were acquired at a constant bias field magnitude Hext in the range 0º ≤ \uf066 ≤ 360º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 3(a-d), we have  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Bias field (Hext) dependent SW absorption plots of Py diamond shaped nanodot array are shown for the bias field orientation (\uf066) of (a) 5°, (b) 15°, (c) 23°, (d) 30°, (e) 45° and (f) 60°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The surface plots correspond to the experimental results, while the symbols represent the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The color map for the surface plots and the schematic of the external applied field (Hext) are given at the bottom right corner of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  0  400  800  1200  3  6  9  12  M1  M2 M3  0  500  1000  1500  M1  M3  Frequency (GHz)  Hext (kOe)  0  400  800  1200  M1  M2  M3  \uf066 = 15  \uf066 = 30  0  400  800  1200  M1  M2 M3  \uf066 = 45  \uf066 = 23  0  400  800  1200  M1  M2 M3  \uf066 = 60  0  400  800  1200  3  6  9  12  M1  M2  M3  \uf066 = 5  x  y  Hext  \uf066  (a)  (b)  (c)  (d)  (e)  (f)  Re S11  Normalized  1  0  presented the \uf066-dependence at Hext = 200, 400, 600 and 800 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' To show the anticrossing points we have magnified the relevant regions of the \uf066-dependent SW spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In the Supplementary Information figure S4, we have shown the full range of \uf066-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At a lower field value like Hext = 200 Oe, only M1 shows angular dispersion as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With an increment in Hext, two more modes start to show angular dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, mode M1 shows a sharp variation of frequency with a minimum at \uf066 = 0\uf0b0, corresponding to the minimum observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' As we increase the field this sharp modulation gradually transforms into a continuous angular variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 3(b) shows the angular dispersion at Hext = 400 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' For \uf066  between 50\uf0b0 and 55\uf0b0, an anticrossing gap appears in between M1 and M2* which is shown by a white dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At a higher field of Hext = 600 Oe instead of one, two different anticrossings are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The first one appears in between M1 and M3 at \uf066 = 40\uf0b0 while the 2nd one appears in between M2* and M3 at \uf066 = 60\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With an increment of magnetic field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=', 800 Oe) the first anticrossing shifts towards lower angle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 35\uf0b0), while the second one gradually disappears as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Due to four fold symmetry[35] of diamond shaped nanodot array these anticrossing also appear in other three quadrants of angular variation spectra of SW, which is shown in section S4 of supplementary section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Anticrossing Strength Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 4(a) shows the power spectrum measured at Hext = 1060 Oe, which is the anticrossing field (Hac) for \uf066 = 23\uf0b0 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The blue line represents the FMR spectra whereas the red line represent the fitted spectra using an antisymmetric lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Other FMR spectra for varying anticrossing fields are presented in section S5 of Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The magnon–magnon coupling strength g is defined as half of the peak-to-peak frequency spacing at the anticrossing field, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In order to estimate the strength of interaction between these two modes, we have extracted the value of g13 and the corresponding dissipation rates \uf06b1, \uf06b3 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, \uf06b1 and \uf06b3 are defined as half-width at half- maximum of the FMR peak of SW mode M1 and M3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Variation of SW frequency as a function of the azimuthal angle (\uf066) varying from 0° to 360° for bias field value fixed at (a) Hext = 200 Oe, (b) 400 Oe, (c) 600 Oe and (d) 800 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The surface plots correspond to the experimental results, while the symbols represent the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The colour map for the surface plots and the schematic of Hext are shown on the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  6  9  M2 M3  M1  M2  0 60  60  3  6  9  M1  M2  M3  M1  M2  M3  M2*  0 60  60  Frequency (GHz)  x  y  Hext  \uf066  6  9  M2 M3  M1  M2  0 60  60  Azimuthal Angle, \uf066 (Degree)  3  6  9  M2 M3  M1  M2  0 60  60  (a)  (b)  (c)  (d)  Re S11  Normalized  1  0  200 Oe  400 Oe  600 Oe  800 Oe  At \uf066 = 23\uf0b0 the extracted value of g13 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='592 GHz, while the values of \uf06b1 and \uf06b3 are found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='60 GHz and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='711 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Since g13 \uf03c \uf06b1 and \uf06b3, therefore the interaction between M1 and M3 can be considered as weak coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In the opposite case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' when g13 > \uf06b1 and \uf06b3 it will be considered as strong coupling between two SW branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' We have also calculated magnon–magnon cooperativity (C), which is defined as C\uf061\uf062 = g2/(\uf06b\uf061\uf06b\uf062) (\uf061,\uf062 = 1, 2, 3) and obtained C13 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='821 for the coupling between M1 and M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The extracted value of g\uf061\uf062, k\uf061, k\uf062, and the estimated value of C\uf061\uf062 for anticrossing points corresponds to different bias field  g13  (GHz)  g12  (GHz)  g23  (GHz)  \uf06b1(GHz)  \uf06b2(GHz)  \uf06b3(GHz)  C13  C12  C23  23o  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='711  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='821  30o  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='660  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='515  45o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='745  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='255  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='426  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='645  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='645 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='113  60o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='915  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='205  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='878  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='675  Table 1 The extracted values of coupling strength (g), FWHM (2k) and calculated cooperativity factor (C) for different orientation of bias field at the anticrossing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Values of g and k are extracted from the FMR spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Real part of S11 parameter as a function of frequency to highlight the anticrossing field are shown for \uf066 = (a) 23°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The frequency gap in the anticrossing mode reveals the coupling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (b) Variation of cooperativity factor with the orientation of bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' It shows that coupling strength is stronger at \uf066 = 30\uf0b0 and 45\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The schematic of Hext are shown on the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  \uf066 = 23  1062 Oe  Frequency (GHz)  Re S11  (Normalized)  x  y  Hext  \uf066  2k1  2k3  2g  (a)  20  40  60  0  1  2  3  C13  C23  C12  \uf066 (Degree)  Cooperativity  (b)  angles are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At \uf066 = 30\uf0b0  obtained value for g13, \uf06b1, \uf06b3 and C13 are estimated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='82, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='423, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='66, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='515, respectively and here this magnon-magnon coupling falls in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' From Table 1, we can see that first anticrossing at \uf066 = 45\uf0b0 also shows strong magnon-magnon coupling with C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='019, while the second one shows weak interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At \uf066 = 60\uf0b0  both the interactions are in the weak coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 4(b) shows the \uf066- dependence of the C where it shows the tunability of coupling strength with the in-plane magnetic field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' It also exhibits that the interaction between different SW branches show strong coupling in-between 30\uf0b0 to 45\uf0b0 orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Micromagnetic Simulation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Static Magnetic Configuration  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 1(d) at \uf066 = 0\uf0b0, a sharp minimum is observed which gradually vanishes for higher values of \uf066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The answer to this lies in the nanodot structure and its rich and flexible spin configurations which we have simulated using OOMMF software[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Details of the micromagnetic simulations are given in section S3 of the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The simulations reproduce important features of the experimental SW spectra with nearly identical frequencies and number of modes besides their relative intensity variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The simulated static spin textures within the nanomagnet array for different bias field magnitudes Hext at \uf066 = 0\uf0b0 and 45\uf0b0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At \uf066 = 0\uf0b0, the nanodot structure shows drastic variation in spin configurations with Hext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' It shows the formation of an S-state at the lower field regime (Hext = 100 Oe) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At larger bias fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=', Hext = 800 Oe), the spins are nearly aligned along the bias- field direction (x-axis) and switch to a leaf-state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This transformation from S- to leaf- state occurs for 250 Oe ≤ Hext ≤ 350 Oe, where the SW frequency shows a minimum as a function of Hext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At  \uf066 = 45\uf0b0 , this transformation is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, for the entire field range, the static magnetic configuration shows a leaf state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' SW mode Characterization  To interpret the nature of the SW modes, we have further simulated the spatial profiles of power and phase of each SW mode by using a home-built MATLAB based code Dotmag[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' OOMMF simulation provides magnetization (M (r, t)) information of each rectangular prism- like cell at different simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' By performing discrete Fourier transformation with respect to time in each of these cells and subsequently extracting the power and phase of the dynamic magnetization for a desired frequency gives rise to the spatial distribution of the power phase profile for that particular mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 6, we have shown the power distribution profile of SW mode at \uf066 = 45\uf0b0 orientation for five different fields, Hext = 200 Oe (Hext << Hac), 400 Oe (Hext < Hac), 475 Oe (Hac), 600 Oe (Hext \uf03e Hac) and 1000 Oe (Hext >> Hac), while the phase profile for each case is shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The power profile at Hext = 1000 Oe indicates that at high bias field only existing mode is M3, which is boosted by all the available energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With a gradual decrement of bias field, two additional modes M1 and M2 appear and the power of  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Simulated static magnetic configurations for Py nanodot array at four different bias magnetic- field magnitude (Hext) at \uf066 = 0\uf0b0 and \uf066 = 45\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' We have shown here a single nanodot from the center of the array for clarity in spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The nanodot structure shows a drastic variation in spin configurations with bias magnetic-field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  100 Oe  250 Oe  800 Oe  350 Oe  \uf066  x  y  Hext  \uf066  0  45 Y  +Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content="2  M1  M2  M3  M4  M5  M'  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2  M1  M2  M3  M4  M5  M6  M7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2  3  6  9  12  M1  M2  M3  M4  M5  M6  Frequency (GHz)  H1  H2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2  M M1  M2  M3  M4  M5  M6  M7  H3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content="2  M '  M1  M2  M3  M4  M5  M6  Applied Field Hext (kOe)  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2  3  6  9  12  M1  M2  M3  M4  O1  O2  O3  H2  Fig 2  Hext  x  y  1  0  Re (S11)  Normalised  M3 is gradually transferred to these two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At the anticrossing field, Hext = 475 Oe, M2 appears as the most intense mode although M1 and M3 have significant power at this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At lower fields, this power is gradually transferred to M1, and at 200 Oe, barring M1 other modes  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Simulated spatial distribution of power and phase (in the inset) profiles corresponding to different SW modes at five different bias field values for \uf066 = 45\uf0b0 for the Py nanodot array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The applied field direction is shown at the bottom left of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Symbols with different colors represent different SW modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The color map is shown at the upper right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  200 Oe  400 Oe  475 Oe  600 Oe  1000 Oe  M1  M2 M3  20  0  Power  (dB)  Phase  (rad)  +\uf070 \uf070  x  y Hext  \uf066  \uf066 = 45  disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Similar to this energy exchange, the phase profiles also exhibit interchange of mode behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At high bias fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=', 1000 Oe), M3 shows quantized nature in BV-like geometry with a quantization number n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' With a decrease in the field, this mode gradually transforms into higher-order quantized mode and M2* is transformed into a quantized mode with n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At Hext = 475 Oe the quantization number of M1, and M3 are n = 5, and 7, respectively, while for M2*, n = 3, which is identical to the quantization number of M3 at Hext = 1000 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This transformation of mode quantization number is also seen in-between M1 and M2* as we further reduce the bias field and finally at Hext = 200 Oe, M1 shows a quantized behavior with n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This transformation of power as well as mode property from one branch of SW mode to another at the anticrossing region indicates a strong interaction between these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' For other orientation like \uf066 = 23\uf0b0, 30\uf0b0 and 60\uf0b0, similar kind of behavior are observed, which are shown in section S6 of the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Distribution of Exchange field To understand the origin of the magnon-magnon coupling and its modulation with bias magnetic field, we have simulated the spatial distribution of the dipole-exchange field (Exchange field distribution of each dot, which is modulated by dipolar interaction of nanodot array) lines at the equilibrium for different bias field orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 7 shows the exchange field map of nanodots array at eight different fields for \uf066 = 45\uf0b0 orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Due to inter-dot dipolar interactions, a dynamic variation of exchange field line with the bias field amplitude (for better viewing purpose, we just present a single nanodot) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The Supplementary Movie A1 shows the dynamics of this exchange field in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At lower bias fields (Hext << Hac), due to dominating effect of demagnetizing field, spins take a configuration such that at equilibrium condition the exchange field lines create three different regions within a single dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The field lines of center and edge regions are configured in opposite direction as denoted with yellow and green arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' As we increase the bias field, the region around the edge of the dot start to vanish and the center region gradually expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' At a very high bias field (Hext \uf03e\uf03e  Hac), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=', Hext = 1000 Oe, only the central region with unidirectional field lines are observed inside a dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  This transformation from three mutually opposite (antiparallel) field-line configuration to uniform (parallel) configuration occurs for 450 Oe ≤ Hext ≤ 500 Oe, which is exactly the anticrossing field region for \uf066 = 45\uf0b0 orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' This change in exchange field profile can be observed much more clearly if we take a linescan along the bias field direction (white dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 7(a)) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In the inset, we have magnified the end  part of the linescan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Here, it is clearly visible that below the anticrossing field (Hext = Hac = 475 Oe) the linescan has two different local maxima  which transform into one maximum as we increase Hext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The exchange field profile for other values of \uf066  are shown in section S7 of the Supplementary Materials, where similar transformation is observed in the anticrossing field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Our observation of correlation between these two phenomena indicates that the anticrossing gap appears only when such a variation of exchange field occurs due to the bias field strength as well as its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The internal field distribution in presence and absence of the exchange field leads to similar conclusion, which we have described in section S8 of the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Conclusion In summary, the interaction between magnons confined in a sole magnonic cavity has been realized in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' We have investigated a bias field strength and angle- dependent magnetization dynamics in diamond-shaped Py nanodot arrays using the broadband ferromagnetic resonance technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Our study has demonstrated that the coupling between two magnon modes is mediated by the exchange coupling inside individual nanodot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Furthermore, the coupling strength is found to be highly dependent on the orientation and strength of the bias magnetic field, leading towards the possibility of externally controlled hybrid  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Exchange field distributions for (a) single nanodot for eight different bias field values at \uf066 = 45\uf0b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Yellow and green arrows represent the direction of exchange field at the center and edge position of the nanodot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' We have shown here a single nanodot from the center of the array for clarity in spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The color bars are shown at the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' (b) Linescan of the simulated exchange field for nanodot array along the field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' In the inset magnified portion of simulated exchange field is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  360  420  480  540  0  250  500  1200 Oe  550 Oe  450 Oe  200 Oe  Exchange Field (Oe)  Distance (nm)  x  y  Hext  \uf066  (a)  (b)  200 Oe  300 Oe  450 Oe  500 Oe  550 Oe  700 Oe  800 Oe  1000 Oe 8 6 4 2  1200 Oe  550 Oe  450 Oe  200 Oe  Power(dB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+page_content='2  3  6  9  12  M1  M2  M3  M4  O1  O2  O3  H2  Fig 2  Hext  x  y  1  0  Re (S11)  Normalised  800 Oe7000e600 0e200 0e500 0emagnonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The experimental results have been well reproduced by micromagnetic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The power and phase profiles of the resonant modes have been numerically calculated to gain insight into the spatial nature of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' The transformation of power as well as mode property from one branch of SW to another, apparently support the strong interaction in-between these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Numerical study shows that the anticrossing gap appears when the symmetry of exchange configuration inside each nanodot is broken due to the applied bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' We have also observed mode softening phenomena when the static magnetic configuration switches from the S-state to the leaf state and with the variation of bias field angle it gradually disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Our findings offer a new approach toward tunable magnon- magnon coupling in ferromagnetic nanostructures for applications in quantum transduction using magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Acknowledgements  AB gratefully acknowledges the financial support from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Bose National Centre for Basic Sciences, India (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' SNB/AB/18-19/211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' SB acknowledges Science and Engineering Research Board (SERB), India for funding (Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' CRG/2018/002080).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' SM and SC acknowledge S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Bose National Centre for Basic Sciences for senior research fellowship  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+page_content=' Choudhury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Majumder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Otani, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Active control of mode crossover and mode hopping of spin waves in a ferromagnetic antidot lattice, Physical Review Applied 10, 064044 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+page_content=' Majumder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Choudhury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Otani, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Reconfigurable spin-wave dynamics in two-dimensional quasiperiodic magnonic crystals, Physica E: Low-dimensional Systems Nanostructures 134, 114901 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+page_content=' Mahato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Choudhury, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Mandal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Otani, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Tunable configurational anisotropy in collective magnetization dynamics of Ni80Fe20 nanodot arrays with varying dot shapes, Journal of Applied Physics 117, 213909 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+page_content=' Porter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Lau, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' McMichael, Interagency report NISTIR 6376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' National institute of standards and technology, Gaithersburg, NIST J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' 114, 57 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Kumar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Dmytriiev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Ponraj, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
+page_content=' Barman, Numerical calculation of spin wave dispersions in magnetic nanostructures, Journal of Physics D: Applied Physics 45, 015001 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFJT4oBgHgl3EQflSzb/content/2301.11583v1.pdf'}
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+arXiv:2301.02630v1  [math.AG]  15 Aug 2022
+HEIGHT PAIRING AND NEARBY CYCLES
+A. Beilinson
+To Yuri Ivanovich Manin with deepest gratitude
+Abstract. We prove that, as was conjectured by Spencer Bloch, the Hodge period
+of some limit Hodge structures equals the height pairing of algebraic cycles on the
+resolution of singularities of the singular fiber.
+§1. Introduction: the theorem and the idea of the proof
+1.1. The Hodge period.
+Suppose we have a Q-Hodge structure E with weights
+in [−2, 0] equiped with isomorphisms ι0 : grW
+0 E = Q(0), ι−2 : grW
+−2E = Q(1).
+One defines the Hodge period ⟨E⟩ = ⟨E, ι0, ι−2⟩ ∈ R as follows.
+Consider the
+R-Hodge structure E ⊗ R. Since the weight filtration on any R-Hodge structure
+with two consequitive weights (canonically) splits one has E ⊗ R = G ⊕ grW
+−1E ⊗ R
+where G is an extension of R(0) by R(1). Our ⟨E⟩ is the class of this extension in
+Ext1(R(0), R(1)) = R.
+Remark. One computes ⟨E⟩ explicitly as follows. Let ER be E ⊗ R viewed a plain
+R-vector space, EC be its complexification. Let 1F 0 ∈ F 0 ⊂ EC be any lifting of
+ι−1
+0 (1). Then ⟨E⟩ is the image of 1F 0 in (ER + W−1EC)/(ER + (F 0 ∩ W−1EC))
+∼
+←
+W−2EC/W−2ER = C/2πiR
+∼
+← R.
+1.2. A geometric example. Let Y be a smooth proper equidimensional algebraic
+variety over C. We denote by Hi(Y ) the homology of Hi(Y (C), Q) seen as an object
+of the category of Q-Hodge structures; ditto for relative homology, etc. Let Zm(Y )
+be the group of algebraic m-cycles on Y with Q-coefficients, Zm(Y )0 := Ker(cl :
+Zm(Y ) → H2m(Y )(−m)) be the subgroup of cycles homologically equivalent to
+zero. For a closed subset P ⊂ Y let Zm(P) ⊂ Zm(Y ) be the subgroup of cycles
+supported on P, Zm(P)0 := Zm(P) ∩ Zm(Y )0. For an m-cycle A on Y we denote
+by |A| its support (which is a closed subset of Y ).
+Suppose m + m′ = dim Y − 1 and we have A ∈ Zm(Y )0, B ∈ Zm′(Y )0 such that
+|A| ∩ |B| = ∅. Set E|A|,|B| := H2m+1(Y ∖ |B|, |A|)(−m). Notice that E|B|,|A| =
+E∗
+|A|,|B|(1) by the Poicar´e duality.
+Lemma. E|A|,|B| has weights in [−2, 0]. One has grW
+−2E|A|,|B| = Zm′(|B|)∗
+0(1),
+grW
+−1E|A|,|B| = H2m+1(Y )(−m), grW
+0 E|A|,|B| = Zm(|A|)0.
+Proof. Notice that H2m(|A|)(−m) = Zm(|A|) and H>2m(|A|) = 0. By the Poincar´e
+duality Hi(Y, Y ∖|B|)(− dim Y ) = H2 dim Y −i(|B|)∗, hence H2m+2(Y, Y ∖|B|)(−m) =
+1991 Mathematics Subject Classification. Primary 14C25; Secondary 14D07.
+Key words and phrases. height pairing, nearby cycles, Hodge periods.
+Typeset by AMS-TEX
+1
+
+2
+A. BEILINSON
+(H2m′(|B|)(−m′))∗(1) and H<2m+2(Y, Y ∖ |B|) = 0. Now use the long exact ho-
+mology sequences for (Y ∖ |B|, |A|) and (Y, Y ∖ |B|).
+□
+Denote by EA,B the Hodge structure obtained from H|A|,|B| by pullback by A
+and pushforward by B:
+(1.2.1)
+Zn(|B|)∗
+0(1)
+֒→
+E|A|,|B|
+։
+Zm(|A|)0
+B ↓
+↑ A
+Q(1)
+֒→
+EA,B
+։
+Q(0)
+Our EA,B is as in 1.1, so we have ⟨EA,B⟩ ∈ R.
+1.3. The height pairing (cf. [B], [Bl1]). Let k be a subfield of C and suppose that
+Y comes from a variety Yk over k, Y = Yk ⊗ C. Let Zm(Yk) ⊂ Zm(Y ) be the
+group of algebraic cycles with Q-coefficients on Yk, Zm(Yk)0 := Zm(Yk) ∩ Zm(Y )0,
+and let CHm(Yk)0 ⊂ CHm(Yk) be their quotients modulo the rational equivalence
+relation. One checks (see §2) that if A, B as above are cycles on Yk then the class
+of ⟨EA,B⟩ in R/Q log |k×| depends only on linear equivalence classes of A and B,
+and so one has a bilinear height pairing
+(1.3.1)
+⟨ , ⟩Yk : CHm(Yk)0 ⊗ CHm′(Yk)0 → R/Q log |k×|.
+Namely ⟨a, b⟩Yk = ⟨EA,B⟩ where A, B are any cycles on Yk of classes a, b such that
+|A| ∩ |B| = ∅.
+Remark. If k = Q and we assume some motivic rationality conjectures (see (2.2.1),
+(2.2.3) of [B]) then ⟨EA,B⟩ can be corrected (by adding a finite sum of corrections
+log(p)⟨EA,B⟩p where p is a prime, ⟨EA,B⟩p is defined using the Gal(Qp)-action on
+EA,B ⊗ Qℓ) so that the resulting real number depends only on rational equivalence
+classes of A and B. In this manner (1.3.1) lifts naturally to an R-valued pairing.
+1.4. Finding elements of Chow groups that are homologically equivalent to zero
+is an art. Spencer Bloch described one situation where they naturally arise, and
+conjectured that the height pairing of his cycles can be computed in s different way,
+namely, as Hodge periods of some nearby cycles. We start with preliminaries.
+Let X be a smooth variety over C of pure dimension n ≥ 2, S be a smooth curve,
+0 ∈ S be a closed point, and f : X → S be a proper map which is smooth otside
+a finite subset {xα} of the fiber X0 = f −1(0). Let Zα be the projectivized tangent
+cone to X0 at xα; this is a hypersurface in the projectivization Pα := P(TxαX) of
+the tangent space; denote by dα its degree. We assume the next condition:
+(∗) All hypersurfaces Zα are smooth.
+Let π : Y → X0 be the blowup of X0 at {xα}. Condition (∗) implies that Y is
+a smooth variety, and Zα are pairwise disjoint divisors on Y . Set Z := ⊔Zα and
+K := Ker(Hn−2(Z) → Hn−2(Y )) = Im(Hn−1(Y, Z) → Hn−2(Z)). If n = 2 then let
+K0 ⊂ K be the subgroup of those elements A = ΣAα that deg Aα = 0 for every
+α. One has a natural map Hn−1(X0) → Hn−1(Y, Z) defined as the composition
+Hn−1(X0) → Hn−1(X0, {xα})
+∼
+← Hn−1(Y, Z).
+Lemma. (i) The map Hn−1(X0) → Hn−1(Y, Z) is an isomorphism if n > 2. If
+n = 2 it is injective and its image equals the preimage of K0 in Hn−1(Y, Z).
+(ii) Hn−1(Y, Z) has weights 1 − n and 2 − n, and grW
+2−nHn−1(Y, Z) = K. The map
+
+HEIGHT PAIRING AND NEARBY CYCLES
+3
+Hn−1(Y ) → Hn−1(Y, Z) has image W1−nHn−1(Y, Z). If n is even then Hn−1(Y )
+∼
+→
+W1−nHn−1(Y, Z).
+Proof. (i) Replace Hn−1(Y, Z) by Hn−1(X0, {xα}) and use the long exact homology
+sequence. (ii) The first assertions follow from the exact homology sequence and
+purity of weights on H·(Y ), H·(Z). The last one comes because Hn−1(Z) = 0 if n
+is even (since Zα are hypersurfaces).
+□
+1.5. Consider a variation of Q-Hodge structures V on S ∖ {0} with fibers Vs =
+Hn−1(Xs). One has a nondegenerate intersection pairing ( , ) : V ⊗ V → Q(n − 1).
+Choose a parameter t at 0 ∈ S and consider the limiting (a.k.a. nearby cycles)
+Hodge structure ψtV. Let ψun
+t V be its direct summand where the monodromy acts
+unipotently. Since ψun
+t
+commutes with duality, ( , ) yields self-duality pairing on
+it that we denote again by ( , ). One has the log of monodromy morphism N =
+NV : ψun
+t V(1) → ψun
+t V and the specialization morphism sp : ψun
+t V → Hn−1(X0).
+Let (ψun
+t V)N := Coker(NV) be the monodromy coinvariants. The next assertion
+follows from the local invariant cycles theorem, see 3.5 for a detailed proof:
+Proposition. sp factors through the isomorphism (ψun
+t V)N
+∼
+→ Hn−1(X0).
+Corollary. ψun
+t V has weights in [−n, 2 − n]. One has grW
+2−nψun
+t V = K if n > 2
+and grW
+2−nψun
+t V = K0 if n = 2. By self-duality, grW
+−nψun
+t V = (grW
+2−nψun
+t V)∗(n − 1).
+If n is even then grW
+1−nψun
+t V = Hn−1(Y ).
+Proof. Since ψun
+t V is self-dual and N is nilpotent, the claim follows from the propo-
+sition and the lemma in 1.4.
+□
+1.6. Bloch cycles. We are in the setting of 1.4; suppose n is even, n = 2m + 2. Let
+A = ΣAα be an m-cycle on Z. We say that A is a Bloch cycle if it is homologically
+equivalent to zero on Y , i.e., cl(A) lies in K(−m) ⊂ Hn−2(Z)(−m). If m = 0 then
+we demand, in addition, that cl(A) ∈ K0 ⊂ K.
+Lemma. If A is a Bloch cycle then each cl(Aα) ∈ Hn−2(Zα)(−m) is primitive.
+Proof. The composition Hn−2(Z)(−m) → Hn−2(Y )(−m) → Hn−4(Zα)(−m + 1),
+where the second arrow is the pullback by Zα ֒→ Y , sends any class c = Σcα to
+cα ∩ c1(O(−1)) (for O(−1) is the normal bundle to Zα in Y ). This composition
+kills cl(Aα) since the first arrow does.
+□
+If A, B are two Bloch cycles then we denote by Eψ
+A,B = Eψ
+A,B,t the Hodge struc-
+ture obtained from ψun
+t V(−m) by pullback by cl(A) and pushforward by cl(B)∗:
+(1.6.1)
+K∗(m + 1)
+→
+ψun
+t V(−m)
+→
+K(−m)
+cl(B)∗ ↓
+↑ cl(A)
+Q(1)
+֒→
+Eψ
+A,B
+։
+Q(0)
+Our Eψ
+A,B is as in 1.1 so we have ⟨Eψ
+A,B⟩ ∈ R.
+1.7. Examples. Consider the case when we have single singular point x0 ∈ X0 of
+f and the singularity at x0 is quadratic. Then the monodromy action on ψtV is
+unipotent, the only possible Bloch cycle is the difference A of the rulings of the
+quadric Z0, and it is actually a Bloch cycle if and only if the monodromy action on
+ψtV is nontrivial or, equivalently, the Hodge structure on Hn−1(X0) is not pure.
+
+4
+A. BEILINSON
+Lemma. (i) If m = 0 then the curve X0 can have either 1 or 2 irreducible compo-
+nents, and A is a Bloch cycle if and only if X0 is irreducible.
+(ii) If X/S is a family of quadratic hypersurfaces in Pn then A is not a Bloch cycle.
+(iii) If X/S is a family of hypersurfaces of degree d on a given smooth projective
+variety P then A is a Bloch cycle if d is large enough.
+Proof. (i) is clear. (ii) follows since the global monodromy for quadratic hypersur-
+faces is ±1, and so it can’t contain non-trivial unipotent local monodromy.
+(iii) Consider the corresponding map r : S → B := {hypersurfaces of degree
+d on P}. Since X is smooth r is transversal to the locus D ⊂ B of degenerate
+hypersurfaces. Replacing S by a germ of another transversal to D that intersects
+D near r(0) would not change the topology of X over a small disc around 0. So we
+can assume that S is a Zariski open subset of the base of a Lefschetz pencil on P.
+Then, since local monodromies of a Lefschetz pencil are all conjugate, triviality of
+one local monodromy amounts to triviality of the global monodromy. Thus A is a
+Bloch cycle if and only if the global monodromy on V is not trivial. Let us check
+that this happens for large enough d.
+If R ⊂ P is the axis of our pencil then H·(X) = H·(P) ⊕ H·−2(R)(−1), and so
+hn−1,0(P) = hn−1,0(X) which equals hn−1,0(Xs) if the global monodromy is trivial.
+Thus the monodromy is not trivial when hn−1,0(Xs) > hn−1,0(P). To finish the
+argument it remains to notice that hn−1,0(Xs) ≥ dim(H0(P, Ωn
+P (d))/H0(P, Ωn
+P )),
+and so it tends to ∞ when d → ∞.
+□
+1.8. Statement of the theorem. Now suppose we have a subfield k ⊂ C and our
+datum is defined over k, i.e., there is Xk/Sk, a closed point 0 of Sk, a parameter t
+on Sk at 0, and Bloch cycles A, B on Zk such that X/S, etc., come by base change
+k → C. Let a ∈ CHm(Yk)0, b ∈ CHm(Yk)0 be the classes of A and B. The next
+result was conjectured by Spencer Bloch:
+Theorem. One has ⟨a, b⟩Yk = ⟨Eψ
+A,B⟩ mod Q log |k×|.
+In case n = 1 the theorem was proven in [BlJS].
+Remark. Suppose we are in the situation of Remark in 1.3. If ⟨Eψ
+A,B⟩ is corrected
+in the same way as was discussed there, then the theorem lifts to an equality of real
+numbers. The proof does not change; we will not discuss it below.
+1.9. Reformulation of the theorem that discards Hodge periods; the idea of the
+proof. Let A′, B′ be cycles on Yk of classes a, b such that |A′| ∩ |B′| = ∅ (no-
+tice that they are, most probably, not supported on Zk). We want to show that
+⟨EA′,B′⟩ = ⟨Eψ
+A,B⟩ (see 1.2, 1.6). Let us compare the Hodge structures E = EA′,B′
+and Eψ = Eψ
+A,B themselves. Their weights lie in [−2, 0], and one has a canonical
+identification grW
+· E = grW
+· Eψ. Indeed, grW
+0 E(ψ) = Q(0), grW
+−2E(ψ) = Q(1) by the
+constructions, and grW
+−1E = H2m+1(Y )(−m) = grW
+−1E(ψ) by the lemma in 1.2, and
+the one in 1.4 combined with the corollary in 1.5. This identification lifts (uniquely)
+to W−1E = W−1Eψ and E/W−2E = Eψ/W−2Eψ. Indeed, the classes of extensions
+0 → H2m+1(Y )(−m) → E(ψ)/W−2E(ψ) → Q(0) → 0 both equal Deligne cohomol-
+ogy class clD(A) (a.k.a. Griffiths’ Abel-Jacobi periods) of A; by duality, the classes
+of (the duals to) extensions 0 → Q(1) → W−1E(ψ) → H2m+1(Y )(−m) → 0 both
+equal to clD(B) (see loc.cit.).
+
+HEIGHT PAIRING AND NEARBY CYCLES
+5
+Now suppose we have a Q-Hodge structure H of weight −1 and two classes
+a ∈ Ext1(Q(0), H), b ∈ Ext1(H, Q(1)).
+Consider the set EH
+a,b = EH(H)a,b of
+all Hodge structures E with weights in [−2, 0] and equipped with identifications
+grW
+0 E = Q(0), grW
+−1E = H, grW
+−2E = Q(1) such that the extensions E/W−2E and
+W−1E have classes a and b. The group Ext1(Q(0), Q(1)) = C× ⊗ Q acts on EH
+a,b by
+the Baer sum action, and EH
+a,b is a C× ⊗ Q-torsor. Notice that for q ∈ C× one has
+⟨q · E⟩ = log |q| + ⟨E⟩. Applying this format to H = H2m+1(Y )(−m), a = clD(A),
+b = clD(B) and EA′,B′, Eψ
+A,B ∈ EH
+a,b we get EA′,B′ − Eψ
+A,B ∈ C× ⊗ Q. Now the
+theorem in 1.8 follows immediately from the next result (notice that the Hodge
+periods and the height pairing play no role here):
+Theorem. One has EA′,B′ − Eψ
+A,B ∈ k× ⊗ Q ⊂ C× ⊗ Q.
+The theorem would be an immediate corollary of the motivic formalism if all
+the above constructions could be spelled in motivic world: Indeed, we would have
+then a motivic version EM of EH which is an Ext1
+M(Q(0), Q(1)) = k× ⊗ Q-torsor
+equipped with the Hodge realization embedding EM ֒→ EH; our EA′,B′, Eψ
+A,B
+would come from elements of EM, and so their difference lies in k× ⊗ Q. The only
+problem is that in the present day formalism of motives, due to Voevodsky, Ayoub,
+and Cisinski-D´eglise, the t-structure is not available, so we do not have the motivic
+version of separate homology groups like Hi(Y ). The actual proof is an exercise in
+spelling out the constructions in a way that makes the t-structure redundant.
+I am very grateful to Spencer Bloch for explaining me his conjecture and stimu-
+lating discussions (pity Spencer refused to coauthor the article), to Volodya Drinfeld
+for valuable comments and discussions, and to Luc Illusie for calling my attention
+to the construction of [I] which helped to clearify and simplify the argument.
+§2. The height pairing and the construction of EM
+a,b ∈ EM
+a,b ⊂ EH
+a,b
+This section is a variation on the theme of [Bl2] and [G].
+2.1. Let C be a stable dg category. It yields two other dg categories C(1) and C(2)
+constructed as follows:
+An object of C(1) is a closed morphism α : M → N of degree 0 in C. One has
+Hom((M, N, α), (M ′, N ′, α′))i = Hom(M, M ′)i ×Hom(N, N ′)i ×Hom(M, N ′)i+1 ⊂
+Hom(Cone(α), Cone(α′))i, and the differential is defined so that the latter embed-
+ding is a morphism of complexes; the composition of morphisms is defined in a sim-
+ilar way. There are three dg functors C(1) → C which send (M, N, α) to M, N, and
+Cone(α) respectively. We can view C(1) as the category of distinguished triangles,
+and the rotation yields an autoequivalence ρ : C(1) → C(1) which sends α : M → N
+to ρ(α) : N → Cone(α); the inverse autoequivalence is ρ−1(α) : Cone(α)[−1] → M.
+An object of C(2) is a datum (P, M, Q, α, β, κ) where P, M, Q are objects of C,
+α ∈ Hom(P, M)1, β ∈ Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1 is such
+that d(κ) = βα; we sometimes abbreviate it to (α, β, κ). One can assign to such a
+datum an object E = E(α, β, κ) ∈ C which equals P ⊕ M ⊕ Q with α, β, and −κ
+added as the components to the differential.1 There is a filtration Q ⊂ Cone(β :
+M[−1], Q) ⊂ E, and morphisms in C(2) are the same as morphisms between the
+1Thus E = Cone((α, κ) : P [−1] → Cone(β : M[−1] → Q)) = Cone((κ, β) : Cone(α : P [−2] →
+M[−1]) → Q).
+
+6
+A. BEILINSON
+corresponding objects E that preserve this filtration.
+We have two dg functors
+C(2) → C(1) which send to (α, β, κ) to α : P[−1] → M and β : M → Q[1], and
+six dg functors C(2) → C which send (α, β, κ) to P, M, Q, Cone(α : P[−1] → M),
+Cone(β : M[−1] → Q), and E(α, β, κ) respectively.
+The dg category C(3) carries a natural involution σ which sends (P, M, Q, α, β, κ)
+to the object (Q[−1], E(α, β, κ), P[1], ασ, βσ, 0) where ασ and βσ are the evident
+embedding and projection.
+Remark. One can view an object (α, β, κ) ∈ C(2) as an object of C equipped with
+a 3-step filtration in two different ways. Namely, this could be E(α, β, κ) equipped
+with an evident filtration with successive quotients Q, M, and P. Or this could be
+M equipped with a filtration whose successive quotients are P[−1], E(α, β, κ), and
+Q[1]. The involution σ exchanges the two perspectives.
+2.2. For C as above we denote by C× the ∞-groupoid of its homotopy equivalences,
+by C×τ the corresponding 1-truncaded plain groupoid, and by HC the homotopy
+category of C. For S, T ∈ C set Exti(S, T ) := HiHom(S, T ) = HomHC(S, T [i]).
+Denote by Ext(S, T ) the plain Picard groupoid of extensions that corresponds to
+the two-term complex τ [0,1]Hom(S, T ).
+For M, N ∈ C let C(1)×
+M,N be the ∞-groupoid of collections (α′ : M ′ → N ′, ιM, ιN)
+where (α′ : M ′ → N ′) ∈ C(1) and ιM : M → M ′, ιN : N → N ′ are homotopy equiv-
+alences. It is equivalent to the Picard ∞-groupoid that corresponds to the complex
+τ ≤0Hom(M, N). The 1-truncated plain Picard groupoid C(1)×τ
+M,N
+corresponds to the
+two-term complex τ [−1,0]Hom(M, N).
+Similarly, for three objects P, M, Q ∈ C we have the ∞-groupoid C(2)×
+P,M,Q whose
+objects are data (P ′, M ′, Q′, α′, β′, κ′, ιP , ιM, ιQ) where (P ′, M ′, Q′, α′, β′, κ′) ∈ C(2)
+and ιP : P → P ′, ιM : M → M ′, ιQ : Q → Q′ are homotopy equivalences.
+The 1-truncated plain groupoid C(2)×τ
+P,M,Q contains a normal subgroup Ext0(P, Q) =
+HomHC(P, Q).
+Let by E = E(M) = E(P, M, Q) be the quotient groupoid.
+It
+is equivalent to the groupoid of triples (α, β, κ) where α ∈ Hom(P, M)1, β ∈
+Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1/d(Hom(P, Q)0) is such that
+d(κ) = βα; a morphism (α, β, κ) → (α′, β′, κ′) in E is a pair (φ, ψ) where φ ∈
+Hom(P, M)0/d(Hom(P, M)−1), ψ ∈ Hom(M, Q)0/d(Hom(M, Q)−1) are such that
+α′ − α = d(φ), β′ − β = d(ψ), κ′ − κ = βφ + ψα + ψd(φ).
+The projection C(2)
+P,M,Q → C(1)
+P [−1],M × C(1)
+M,Q[1] yields a map of plain groupoids
+E(P, M, Q) → C(1)×τ
+P [−1],M × C(1)×τ
+M,Q[1] = Ext(P, M) × Ext(M, Q), (α, β, κ) �→ (α, β).
+The group Ext1(P, Q) acts on E by translations of κ, and non-empty fibers Eα,β
+are Ext1(P, Q)-torsors.
+Remark. E(P, M, Q) is naturally functorial with respect to P and Q: every pair
+of closed morphisms µ : P1 → P and ν : Q → Q1 yields a map E(P, M, Q) →
+E(P1, M, Q1), (α, β, κ) �→ (αµ, νβ, νκµ); is compatible with the Ext1(P, Q)-action
+via the map (µ∗, ν∗) : Ext1(P, Q) → Ext1(P1, Q1).
+Suppose Ext2(P, Q) = 0.
+Then Eα,β are non-empty, and the addition maps
+Eα1,β × Eα2,β → Eα1+α2,β, Eα,β1 × Eα,β2 → Eα,β1+β2 define on E the structure of
+an Ext1(P, Q)-biextension of (Ext(P, M), Ext(M, Q)).
+2.3. In our first example C is the dg category whose homotopy category is the
+bounded derived category DH of the category H of Q-Hodge structures, and
+
+HEIGHT PAIRING AND NEARBY CYCLES
+7
+P = Q(0), Q = Q(1). We denote the corresponding E by EH = EH(M). Then
+Ext̸=1
+DH(P, Q) = 0 and Ext1
+DH(P, Q) = C× ⊗ Q, so EH is a C× ⊗ Q-biextension of
+(Ext(Q(0), M), Ext(M, Q(1))).
+Let Ext1
+0(Q(0), M) ⊂ Ext1(Q(0), M), Ext1
+0(M, Q(1)) ⊂ Ext1(M, Q(1)) be the
+subgroups of those elements a, b that the maps H0a : Q(0) → H1M, H−1b :
+H−1M → Q(1) vanish. Let Ext0(Q(0), M) ⊂ Ext(Q(0), M), etc., be the Picard
+groupoids of such extensions.
+Lemma. Suppose that Hom(Q(0), H0M) = Hom(H0M, Q(1)) = 0.
+(i) The restriction of EH to (Ext0(Q(0), M), Ext0(M, Q(1))) descends to the C×⊗Q-
+biextension of (Ext1
+0(Q(0), M), Ext1
+0(M, Q(1))).
+(ii) EH is naturally functorial with respect to M: if ϕ : M → M ′ is a morphism,
+and we have a′ ∈ Ext1
+0(Q(0), M ′), b′ ∈ Ext1
+0(M ′, Q(1)) with ϕ∗(a) = a′, ϕ∗(b′) = b
+then there is a canonical identification EH(M)a,b = EH(M ′)a′,b′.
+(iii) The isomorphisms Ext1
+0(Q(0), M)
+∼
+→ Ext1(Q(0), H0M), Ext1
+0(M, Q(1))
+∼
+→ Ext1
+(H0M, Q(1)) which assign to an extension its zero cohomology, lifts naturally to an
+isomorphism of biextensions H0 : EH(M)
+∼
+→ EH(H0M). One has EH0 = H0E.
+Proof. Let us prove (i); the rest is clear. We need to check that for every closed α ∈
+Hom1
+0(Q(0), M), β ∈ Hom1
+0(M, Q(1)) the action of Aut(α)×Aut(β) = Hom(Q(0), M)
+×Hom(M, Q(1)) on EH
+α,β is trivial.
+Since H has homological dimension 1 our M is isomorphic to the direct sum of its
+homologies and so Aut(α) = Ext1(Q(0), H−1M), Aut(β) = Ext1(H1(M), Q(1)) by
+the condition on M. The action of (e, h) ∈ Ext1(Q(0), H−1M)×Ext1(H1(M), Q(1))
+on EH
+α,β is the translation by H−1(β)e + hH0(α) which is 0 since α, β ∈ Ext1
+0.
+□
+2.4. Lemma. Suppose that H0M is pure of weight −1 (which implies the condition
+of the lemma in 2.3). Then the function EH(M) → R, (α, β, κ) �→ ⟨E(α, β, κ)⟩ :=
+⟨H0E(α, β, κ)⟩, see 1.1, is a natural trivialization of the R-biextension log |EH(M)|.
+Proof. Everything said in 2.3 works for the category HR of R-Hodge structures. The
+extension of scalars functor H → HR, ? �→? ⊗ R, yields a morphism of our biex-
+tensions EH(M) → EHR(M ⊗ R). The map Ext1(Q(0), Q(1)) → Ext1(R(0), R(1))
+equals log | | after the standard identifications of the Ext groups with, respectively,
+C× ⊗ Q and R.
+Since Ext1(R(0), H0M ⊗ R) = Ext1(H0M ⊗ R, R(1)) = 0 by
+the condition on M, one has EHR(M ⊗ R) = EHR(H0M ⊗ R) = R.
+The map
+EH(M) → EHR(M ⊗ R) = R is ⟨ ⟩ of 1.1.
+□
+2.5. Let k ⊂ C be a subfield. Denote by DM(k) the dg category of geometric
+Voevodsky Q-motives over k. We have the Hodge realization dg functor DM(k) →
+DH, M �→ M H.
+Consider the story of 2.2 for C = DM(k) with P = Q(0),
+Q = Q(1). As before one has Ext̸=1
+DM(k)(Q(0), Q(1)) = 0, and there is a canonical
+identification Ext1(Q(0), Q(1)) = k× ⊗ Q such that the Hodge realization map
+between the Ext1’s is the embedding k× ⊗ Q ֒→ C× ⊗ Q. So for any M ∈ DM(k)
+we get a k× ⊗ Q-biextension of (Ext1(Q(0), M), Ext1(M, Q(1))) together with the
+Hodge realization morphism EM(M) → EH(M) := EH(M H) of the biextensions.
+Remark. Since the homomorphism k× ⊗ Q ֒→ C× ⊗ Q is injective, the maps of
+torsors EM(M)α,β → EH(M)α,β := EH(M)αH,βH are injective too.
+We define Ext1
+0(Q(0), M) ⊂ Ext1
+0(Q(0), M) and Ext1
+0(M, Q(1)) ⊂ Ext1(M, Q(1))
+as preimages of the Ext1
+0 subgroups of the Hodge setting by the Hodge realiza-
+
+8
+A. BEILINSON
+tion maps.
+Assume that H0M H is pure of weight −1.
+Then (i) and (ii) of
+the lemma in 2.3 remain true in the DM(k) setting (with C× replaced by k×):
+this follows from loc.cit. by Remark above. Thus we have a k× ⊗ Q-biextension
+EM(M) of (Ext1
+0(Q(0), M), Ext1
+0(M, Q(1))) together with a map of biextensions
+EM(M) → EH(M), so the lemma in 2.4 provides a natural trivialization of the
+R-biextension log |EM(M)|. The image of EM
+a,b in R/Q log |k×| depends only on
+a, b ∈ Ext1
+0(M, Q(1)) × Ext1
+0(Q(0), M), and we denote it by ⟨a, b⟩M. It is clearly
+biadditive with respect to a, b.2 We have defined a canonical height pairing
+(2.5.1)
+⟨ ⟩M : Ext1
+0(Q(0), M) × Ext1
+0(M, Q(1)) → R/Q log |k×|.
+2.6. We return to the situation of 1.3 and set M := M(Yk)(−m)[−1 − 2m] where
+M(Yk) is the motive of Yk. One has Ext1(Q(0), M) = CHm(Yk), Ext1(M, Q(1)) =
+CHm′(Yk) by the Poincar´e duality, and Ext1
+0 are the subgroups CH(Yk)0 of cycles
+homologically equivalent to zero. Therefore we get a k× ⊗ Q-biextension EM of
+(CHm(Yk)0, CHm′(Yk)0), the map of biextensions EM → EH, the trivialization of
+log |EM|, and the height pairing ⟨ , ⟩M : CHm(Yk)0 × CHm′(Yk)0 → R/Q log |k×|.
+By (iii) of the lemma in 2.3 one has H0 : EH(M)
+∼
+→ EH(H2m+1(Y )(−m)). For
+a ∈ CHm(Yk)0, b ∈ CHm′(Yk)0 pick, as in 1.3, cycles A, B that represent them
+such that |A| ∩ |B| = ∅.3 Let us construct (a, b, κA,B) ∈ EM
+a,b such that the Hodge
+realization EH
+A,B of EM
+A,B := E(a, b, κA,B) (see 2.1) has zero cohomology H0EH
+A,B
+equal to the Hodge structure EA,B from 1.3. This would imply that for our M the
+height pairing (2.5.1) equals (1.3.1).
+The composition of the maps M(|A|)
+α→ M(Yk)
+β→ M(Yk, Yk ∖ |B|) is naturally
+homotopic to 0: indeed, M(Yk, Yk ∖ |B|) := Cone(M(Yk ∖ |B|) → M(Yk)), and the
+homotopy κ|A|,|B| is M(|A|) → M(Yk ∖|B|) ⊂ Cone. Thus we have (α, β, κ|A|,|B|) ∈
+DM(2) (see 2.1). Notice that E(α, β, κ|A|,|B|) = M(Yk ∖ |B|, |A|).
+One has Ext−2m(Q(m), M(|A|)) = Zm(|A|) := the group of m-cycles supported
+on |A| (recall that dim |A| = m), and Ext2m+2(M(Yk, Yk ∖ |B|), Q(m + 1)) =
+Zm′(|B|) by the Poincar´e duality.
+Therefore we have (αA, Bβ, Bκ|A|,|B|A) =
+(Q(m)[2m+1], M(Yk), Q(m)[2m+2], αA, Bβ, Bκ|A|,|B|A) ∈ DM(2). The promised
+(a, b, κA,B) ∈ EM
+a,b is (αA, Bβ, Bκ|A|,|B|A)(−m)[−1 − 2m]. The fact that H0EH
+A,B
+equals the Hodge structure EA,B from 1.3 follows from the construction.
+§3. The unipotent nearby cycles in the Hodge setting
+3.1. A nearby cycles reminder. In this section we play with algebraic varieties over
+C. For an algebraic variety X we denote by H(X) the abelian category of perverse
+Hodge Q-sheaves of M. Saito on X, by DH(X) its bounded derived category. It sat-
+isfies the usual Grothendieck six functors formalism. Below ∗ is the Verdier duality.
+Every object of H(X), hence of DH(X), carries a canonical weight filtration.
+For F ∈ DH(X) let Γ(X, F), Γc(X, F) ∈ DH be the complex of chains, resp.
+chains with compact support, with coefficients in F equipped with the natural
+Hodge structure, H·
+(c)(X, F) := H·Γ(c)(X, F) ∈ H; set Γ(c)(X) := Γ(c)(X, Q(0)X),
+H·
+(c)(X) := H·
+(c)(X, Q(0)), and denote by ( , ) the Poincar´e duality pairing. Simi-
+larly for a closed subvariety A ⊂ X we set ΓA(X) := ΓA(X, Q(0)) ∈ DH, etc.
+2Indeed, a morphism from a biextension by a trivial group to a trivialized biextension amounts
+to a biadditive pairing.
+3Recall that |A|, |B| ⊂ Yk are supports of the cycles.
+
+HEIGHT PAIRING AND NEARBY CYCLES
+9
+Let g : X → A1 be a function on X; set X0 := g−1(0), and let v : X ∖ X0 ֒→ X,
+iX0 : X0 ֒→ X be the open and closed embeddings. One has the unipotent nearby
+cycles functor ψun
+g
+: DH(X ∖ X0) → DH(X0) that carries a natural logarithm
+of monodromy morphism N = Ng = NF : ψun
+g (F)(1) → ψun
+g (F) where F ∈
+D(X ∖ X0). It has ´etale local origin with respect to X0. For sheaves on X there
+is a natural morphism of functors ι : i∗
+X0 → ψun
+g v∗.
+There are basic canonical
+identifications:
+(i) Compatibility with Verdier duality: One has ψun
+g (F∗) = (ψun
+g F)∗(1)[2].
+(ii) Compatibility with proper direct images: Suppose h : X → T is a proper map
+and t is a function on T such that g = th; then one has ψun
+t h∗F = h∗ψun
+g F.
+(iii) One has Cone(NF) = i∗
+X0v∗F(1)[1].
+(iv) For every n > 0 one has ψun
+gnF
+∼
+→ ψun
+g F.
+These identifications are mutually compatible; (i) and (ii) are compatible with
+the action of N, and (iv) identifies Ngn with nNg. Finally, one has
+(v) ψun[−1] is t-exact for the perverse t-structure.
+Examples. Suppose that X is smooth of dimension n and F = Q(0)X∖X0. Then
+F∗ = F(n)[2n] hence ψun
+g (F)∗ = (ψun
+g F)(n − 1)[2n − 2].
+(a) If g is smooth then ιQ(0)X : Q(0)X0
+∼
+→ ψun
+g F, NF = 0.
+(b) Suppose g is semi-stable and X0 has two irreducible components Y and Y ′. By
+(a) one has natural morphisms jY ′∖Y !QY ′∖Y → ψun
+g F → jY ∖Y ′∗QY ∖Y ′ compatible
+with the N-action (we take it that on the left and right object N acts trivially).
+They form an exact triangle; its Verdier dual is the same triangle with Y and Y ′
+interchanged.
+3.2. We are in the setting of 1.4 and follow the notation there.
+Let j : U := X0 ∖ {xα} ֒→ X0 ←֓ {xα} : ⊔ixα be the complementary open
+and closed embeddings. Let I be the intersection cohomology sheaf j!∗Q(0)U =
+τ ≤n−2j∗Q(0)U 4 on X0; set I+ := π∗Q(0)Y . One has natural self-duality isomor-
+phisms I∗ = I(n − 1)[2n − 2], I+∗ = I+(n − 1)[2n − 2] (recall that Y is smooth of
+dimension n − 1 and π is proper).
+The decomposition theorem for π is easy and explicit:
+Proposition. There is a natural orthogonal direct sum decomposition
+(3.2.1)
+I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)
+compatible with the self-dualities.
+Proof. One has a natural orthogonal direct sum decomposition
+(3.2.2)
+Γ(Zα) = Hn−2
+prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)
+defined as follows. Consider the embedding Zα ֒→ Pα. The pullback and Gysin
+maps Γ(Pα) → Γ(Zα) → Γ(Pα)(1)[2] are mutually dual for the Poincar´e duality
+pairings, and their composition in either direction equals to the multiplication by
+c1(O(dα)).5 Thus the composition of τ ≤2n−4Γ(Pα) → Γ(Zα) → τ ≥0(Γ(Pα)(1)[2]) is
+an isomorphism. This yields a direct sum decomposition Γ(Zα) =?⊕τ ≤2n−4Γ(Pα).
+4Below τ is the usual truncation, pτ is the perverse one.
+5Since O(dα) is the normal bundle to Zα in Pα.
+
+10
+A. BEILINSON
+Since multiplication by c1(O(dα)) preserves the direct sum decomposition, the only
+nonzero cohomology of ? is Hn−2
+prim(Zα) ⊂ Hn−2(Zα), q.e.d.
+Consider the embeddings of smooth divisors iZα : Zα ֒→ Y . One has i!
+ZαQ(0)Y =
+Q(−1)[−2]Zα, i∗
+ZαQ(0)Y = Q(0)Zα, and the composition of the adjunction maps
+iZα∗i!
+ZαQ(0)Y → Q(0)Y → iZα∗i∗
+ZαQ(0)Y equals the multiplication by c1(O(−1))
+map Q(−1)[−2]Zα → Q(0)Zα.6 Apply π∗; then i!
+xαI+ = Γ(Zα)(−1)[−2], i∗
+xαI+ =
+Γ(Zα) by base change, and the composition of the adjunctions ixα∗i!
+xαI+ → I+ →
+ixα∗i∗
+xαI+ is multiplication by c1(O(−1)) map ixα∗Γ(Zα)(−1)[−2] → ixα∗Γ(Zα).
+Composing the maps τ ≤2n−6Γ(Pα) ֒→ Γ(Zα) and Γ(Zα) ։ τ [2,2n−4]Γ(Pα) that
+come from decomposition (3.2.2) from the left and from the right with the latter ad-
+junctions, we get the maps ixα∗(τ ≤2n−6Γ(Pα))(−1)[−2] → I+ → ixα∗τ [2,2n−4]Γ(Pα).
+Their composition is an isomorphism, which yields a decomposition I+ = I? ⊕
+ixα∗τ [2,2n−4]Γ(Pα). Since the adjunctions are mutually dual, the decomposition is
+orthogonal.
+By (3.2.2) one has i!
+xαI? = Hn−2
+prim(Zα)(−1)[−n] ⊕ Q(n − 1)[2 − 2n], i∗
+xαI? =
+Hn−2
+prim(Zα)[2 − n] ⊕ Q(0).
+Thus I?[n − 1] is a perverse sheaf which equals Q(0)[n − 1]U on U and has no
+subquotients supported on {xα}, and so I? = I. We are done.
+□
+Remarks. (i) The adjunction map Q(0)X0 → π∗Q(0)Y = I+ takes value in I ⊂ I+
+since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = 0.
+(ii) Set B := ⊕ixα∗Hn−2
+prim(Zα)[1 − n]. By the formula for i∗
+xαI at the end of the
+previous paragraph, one has an exact triangle Q(0)X0 → I → B[1].
+3.3. As in 1.5, t is a local coordinate at 0 ∈ S; shrinking S we can assume that
+t is defined and invertible on S ∖ {0}, so X0 = (tf)−1(0). Consider the functor
+ψun
+tf : DH(X ∖ X0) → DH(X0) (see 3.1). Set R := ψun
+tf Q(0)X∖X0. By 3.1(i) one
+has a canonical self-duality identification R∗ = R(n − 1)[2n − 2] and the mutually
+dual maps Q(0)X0
+ι→ R
+ι∗
+→ Q(0)∗
+X0(1 − n)[2 − 2n] which are isomorphisms over U.
+The next result is due to Illusie [Il]; we will need it in 4.5. The reader can skip
+it at the moment and jump directly to section 3.4.
+Proposition. For every critical point xα one has canonical isomorphisms
+(3.3.1)
+i!
+xαR = Γc(Pα ∖ Zα),
+i∗
+xαR = Γ(Pα ∖ Zα)
+interchanged by the duality. The N-action on i!
+xαR, i∗
+xαR is trivial.
+Proof. (a) The claim is local at xα, so for the proof we remove from X the rest
+of critical points, and still call it X by the abuse of notation. Let S♭ → S be the
+covering of degree dα obtained by adding t♭ = t1/dα to the sheaf of functions; its
+Galois group is µdα. Set X♭ := X×SS♭ and let f ♭ : X♭ → S♭ be the projection. Our
+X♭ is a hypersurface {(x, t♭) : (tf)(x) − t♭dα = 0} in X × A1; its only singular point
+is (xα, 0). The projectivized tangent cone Qα of X♭ at (xα, 0) is a hypersurface in
+P +
+α := P(T(xα,0)X × A1). The Galois group µdα acts on X♭ hence on Qα.
+(b) Let us check that Qα is a µdα-covering of Pα completely ramified along Zα
+and ´etale over its complement, and Qα is smooth. To see this, consider the leading
+term [tf]dα(x) (of the Taylor expansion) of tf at xα; then the leading term of
+6Since O(−1) is the normal bundle to Zα in Y .
+
+HEIGHT PAIRING AND NEARBY CYCLES
+11
+(tf)(x) − t♭dα at (xα, 0) is [tf]dα(x) − t♭dα. The zeros of [tf]dα is Zα ⊂ Pα, of
+[tf]dα(x) − t♭dα is Qα ⊂ P +
+α , and so the projection Qα → Pα (x, t♭) �→ x, is as
+claimed. The smoothness of Qα follows from that of Zα.
+(c) Let π+ : X+ → X♭ be the blowup of X♭ at (xα, 0). By (b) X+ is smooth
+and the map f + := f ♭π+ : X+ → S♭ has semistable reduction at 0 ∈ S♭. The
+fiber X+
+0 has two irreducible components: one equals Y and the other Qα, and
+their intersection equals Zα. The action of µdα on X♭ yields one on X+. The
+µdα-action on X+
+0
+fixes Y and acts on Qα as described in (b).
+The projection
+π+
+0 : X+
+0 → X♭
+0 = X0 contracts Qα to xα.
+Set R+ := ψun
+tf +Q(0)X+∖X+
+0 , R♭ := ψun
+tf ♭Q(0)X♭∖X♭
+0. These are sheaves on X+
+0
+and X♭
+0 = X0 respectively that are naturally µdα-equivariant.
+By 3.1(ii) (with
+h = π+) one has a natural identification π+
+0∗R+ = R♭ compatible with the µdα-
+actions. Since the projection p : X♭ → X is a µdα-torsor over X ∖ X0 one has
+Q(0)X∖X0 = (p∗Q(0)X♭∖X♭
+0)µdα , and so, by 3.1(ii) with h = p, one has R = R♭µdα .
+Therefore R = (π+
+0∗R+)µdα .
+(d) By 3.1(iv) with g = t♭f +, n = dα, one has ψun
+tf + = ψun
+t♭f +. Our t♭f + is semi-
+stable, so we have the exact triangle jY ∖Zα!QY ∖Zα → R+ → jQα∖Zα∗QQα∖Zα
+as in Example (b) in 3.1. Applying π+
+0∗ we get an exact triangle j!QU → R♭ →
+ixα∗Γ(Qα∖Zα). Passing to µdα-invariants we get, by (b), an exact triangle j!QU →
+R → ixα∗Γ(Pα ∖ Zα); here we use the identification Γ(Qα ∖ Zα)µdα
+∼
+→ Γ(Pα ∖ Zα)
+defined as the composition Γ(Qα ∖ Zα)µdα ⊂ Γ(Qα ∖ Zα)
+tr
+→ Γ(Pα ∖ Zα). Thus
+we get the isomorphism i∗
+xαR
+∼
+→ Γ(Pα ∖ Zα) in (3.3.1). The second isomorphism
+there comes in the dual manner from the dual exact triangle jQα∖Zα!QQα∖Zα →
+R+ → jY ∖Zα∗QY ∖Zα. Since π+
+0∗ commutes with duality, the two isomorphisms are
+mutually dual, and we are done.
+□
+Let αR be the composition B
+∂→ Q(0)X0
+ι→ R where ∂ is the boundary map of the
+triangle from Remark (ii) in 3.2, so I = Cone(∂). Let us compute the map i!
+xα(αR).
+Consider the standard triangle Hn−2
+prim(Zα)[1−n]
+δ→ Γc(Pα ∖Zα)
+tr
+→ Q(1−n)[2−2n]
+that comes from (3.2.2).
+Lemma. −i!
+xα(αR) equals the composition δR of the maps Hn−2
+prim(Zα)[1 − n]
+δ→
+Γc(Pα ∖ Zα)
+(3.3.1)
+=
+i!
+xαR.
+Proof. Consider the exact triangle
+(3.3.2)
+jQα∖Zα!Q(0)Qα∖Zα ⊕ jY ∖Zα!Q(0)Y ∖Zα → Q(0)X+
+0 → Q(0)Zα.
+Let (δQ, δY ) : Q(0)Zα[−1] → jQα∖Zα!Q(0)Qα∖Zα ⊕ jY ∖Zα!Q(0)Y ∖Zα be the bound-
+ary map. Its composition with the map to Q(0)X+
+0 , and hence with the further
+composition with Q(0)X+
+0
+ι→ R+, is 0.
+Therefore the sum of the compositions
+Q(0)Zα[−1]
+δQ
+−→ jQα∖Zα!
+ι→ R+ and Q(0)Zα[−1]
+δY
+−→ jY ∖Zα!
+ι→ R+ is 0.
+Ap-
+ply i!
+xαπ+
+∗ and consider the restriction of our compositions to Hn−2
+prim(Zα)[1 − n] ⊂
+Γ(Zα)[−1]. For the first one it is δR, for the second one it is i!
+xα(αR), and we are
+done.
+□
+
+12
+A. BEILINSON
+3.4. Set P := R[n − 1] = ψun
+tf Q(0)X∖X0[n − 1]; this is a perverse sheaf on X0; one
+has a canonical self-duality identification P∗ = P(n − 1). Consider the perverse
+sheaves PN := Ker(N : P → P(−1)), PN := Coker(N : P(1) → P).
+Lemma. (i) Q(0)X0[n − 1] is a perverse sheaf of weights n − 1 and n − 2 with
+grW
+n−1 = I[n − 1], grW
+n−2 = ⊕α ixα∗Hn−2
+prim(Zα).
+(ii) One has PN = Q(0)X0[n − 1], PN = (Q(0)X0[n − 1])∗(1 − n).
+(iii) P has weights in [n − 2, n]. One has Wn−1P = Q(0)X0[n − 1], P/Wn−2P =
+(Q(0)X0[n − 1])∗(1 − n), grW
+n−2P = ⊕α ixα∗Hn−2
+prim(Zα), grW
+n−1P = I[n − 1], grW
+n P =
+(grW
+n−2P)∗(1 − n).
+Proof. (i) The exact triangle from Remark (ii) in 3.2 amounts to an exact triangle
+⊕ixα∗Hn−2
+prim(Zα) → Q(0)X0[n − 1] → I[n − 1], and we are done since its left and
+right terms are pure perverse sheaves of weights n − 2 and n − 1 respectively.
+(ii) For any sheaf A on X one has a canonical exact triangle i∗
+X0A → i∗
+X0v∗v∗A →
+i!
+X0A[1]: Indeed, the map v!v∗A → v∗v∗A factors as composition v!v∗A → A →
+v∗v∗A, and so one has an exact triangle Cone(v!v∗A → A) → Cone(v!v∗A →
+v∗v∗A) → Cone(A → v∗v∗A) which is supported on X0.
+The promised exact
+triangle is its restriction to X0.
+Now take for A the perverse sheaf Q(0)X[n]. The first term of the triangle is
+Q(0)X0[n] which is perverse sheaf shifted by 1, its third term is (Q(0)X0[n−1])∗(−n)
+which is a perverse sheaf. Therefore they equal, respectively, pH−1 and pH0 of
+i∗
+X0v∗v∗Q(0)X[2n], i.e., of Cone(N : P → P(−1)) by 3.1(iii), and we are done.
+(iii) Since N is nilpotent, the weights of P are bounded from below by the
+minimum of weights of PN, which is n − 2 by (ii) and (i). By self-duality of P they
+are bounded then from above by n, and we have the first assertion. It implies that
+Wn−2P ⊂ PN. The rest follows directly from (i), (ii), and self-duality of P.
+□
+3.5. Proof of the proposition in 1.5. We use the notation in loc.cit. Injectivity of
+sp : (ψun
+t H)N → Hn−1(X0) follows from the local invariant cycles theorem. Let us
+check the surjectivity. By 3.1(ii) applied to h = f (recall that f is proper) and 3.1(v)
+applied to ψun
+t , one has ψun
+t H = H0(X0, P)(n−1). By 3.4 we have exact sequence of
+perverse sheaves 0 → ⊕α ixα∗Hn−2
+prim(Zα)(n−1) → P(n−1) → (Q(0)X0[n−1])∗ → 0.
+Its left term has finite support, and so has no cohomology in degrees ̸= 0. Therefore
+the map H0(X0, P)(n − 1) → H0(X0, (Q(0)X0[n − 1])∗) = Hn−1(X0) is surjective.
+This map equals sp, and we are done.
+□
+§4. The motivic setting and the construction of EψM
+a,b
+∈ EM
+a,b
+4.1. We are in the setting of 1.8 so k ⊂ C is a subfield and we play with varieties
+over k. Changing slightly the notation of 1.3 and 1.8, for a variety Z = Zk we set
+ZC := Z ⊗k C. The notation of §3 is preserved except that we equip from now on
+all Hodge sheaves and Hodge structures met previously with extra upper index H.
+We play with motives (a.k.a. motivic sheaves) over varieties, see [A1] and [CD].
+For a variety Z the category of constructible Q-motives over Z is denoted by
+DM(Z).
+We use Grothendieck’s six functors formalism for DM as developed
+in [CD]. Recall that DM(Spec k) = DM(k) is the category of Voevodsky’s geo-
+metric Q-motives over k.
+For a variety Z one has M(Z) = πZ!π!
+ZQ(0) where
+πZ : Z → Spec k is the structure map. For a motivic sheaf F on Z set Γ(Z, F) :=
+πZ∗F, Γc(Z, F) := πZ!F ∈ DM(k); we write Γ(c)(Z) := Γ(c)(Z, Q(0)Z). There is
+
+HEIGHT PAIRING AND NEARBY CYCLES
+13
+a Hodge realization functor DM(Z) → DH(ZC), F �→ FH, compatible with the
+six functors and the Verdier duality ∗. For a smooth Z of dimension d one has
+π!
+ZQ(0) = Q(d)Z[2d], and so M(Z) = Γc(Z)(d)[2d].
+The formalism of unipoteny nearby cycles in the setting of motivic sheaves was
+developed in §§3.4, 3.6 of [A2]. The motivic version of everything said in 3.1 holds
+except property (v) (for the t-structure is not available). The Hodge realization
+functor commutes with the nearby cycles functors.
+4.2. Notation: Notice that Hom(Q(i)[2i], Q(j)[2j]) is 0 if i ̸= j and Q for i = j,7
+and so every object M ∈ M(k) which is isomorphic to a direct sum of motives
+Q(i)[2i], i ∈ Z, can be written in a unique manner as ⊕i Vi(i)[2i] where Vi is a
+vector space (then Vi = Hom(Q(i)[2i], M)). Set τ ≤2aM := ⊕i≥−a Vi(i)[2i], etc.
+We are in the situation of 3.2 in the setting of k-varieties. As in loc.cit., I+ :=
+π∗Q(0)Y ∈ DM(X0) (so I+H is the corresponding Hodge sheaf from loc.cit.) Since
+Y is smooth and π is proper one has a natural self-duality I+∗ = I+(n−1)[2n−2].
+The t-structure in DM is not available, so we define the motivic intersection
+cohomology sheaf I using a motivic version of decomposition (3.2.1):
+Proposition. There is a natural orthogonal direct sum decomposition in DM(X0)
+(4.2.1)
+I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)
+whose Hodge realization is (3.2.1)
+Proof. It repeats the proof in 3.2 (minus its last paragraph). Namely, we first define
+a natural orthogonal decomposition
+(4.2.2)
+Γ(Zα) = Hn−2
+prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)
+in DM(xα) = DM(kxα) whose Hodge realization is (3.2.2).8 The construction in
+loc.cit. uses only basic six functors functoriality, so we can repeat it literally in the
+motivic setting. Then we proceed to define (4.2.1) as in loc.cit.
+□
+Set B := ⊕α ixα∗Hn−2
+prim(Zα)[1 − n] ∈ DM(X0). The self-dualities of Γ(Zα) and
+of I+, and the above orthogonal decompositions yield natural self-dualities
+(4.2.3)
+B∗ ∼
+→ B(n − 2)[2n − 2],
+I∗ ∼
+→ I(n − 1)[2n − 2].
+4.3. Lemma. (i) The adjunction χ : Q(0)X0 → π∗Q(0)Y = I+ takes values in
+I ⊂ I+.
+(ii) One has Cone(χ : Q(0)X0 → I) = B[1].
+Proof. (i) Follows since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = Hom(Q(0), τ [2,2n−4]Γ(Pα))
+= 0.
+(ii) Since χ|U = idQ(0)U the cone Cone(χ) is supported on {xα}. Now i∗
+xαCone(χ) =
+7This follows since M(Pn) = ⊕i∈[0,n]Q(i)[2i] and End(M(Pn)) = CHn(Pn × Pn) = Q[0,n].
+8So Hn−2
+prim(Zα) is a notation for a motive whose Hodge realization is the primitive cohomology
+of Zα; its definition does not involve any cohomology. To construct it explicitly, choose a k-point
+z in Pα ∖ Zα. Let πz : Zα → Pn−2 be the corresponding projection; this is a finite map of degree
+dα. Then Hn−2
+prim(Zα) is the kernel of the projector d−1
+α πt
+zπz acting on M(Zα)(2 − n)[4 − 2n].
+
+14
+A. BEILINSON
+Cone(i∗
+xα(χ)) equals Hn−2
+prim(Zα)[2 − n] by (4.2.2) and the construction of I, q.e.d.
+□
+Remark. Since Exti(Q(0)X0, Q(0)∗
+X0(1−n)[2−2n]) = Exti(Q(0), M(X0)(1−n)[2−
+2n]) = CHn−1(X0, −i) we see that Ext0 = Zn−1(X0) and Ext̸=0 = 0, i.e., one has
+Hom(Q(0)X0, Q(0)∗
+X0(1 − n)[2 − 2n]) = Zn−1(X0) = Zn−1(U).
+Example. One has χ∗χ = ǫ where ǫ : Q(0)X0 → Q(0)∗
+X0(1 − n)[2 − 2n] is the map
+that corresponds to the sum of irreducible components cycle (it is enough to check
+the assertion on U where it is obvious).
+4.4. We are in the situation of 3.3 in the setting of k-varieties. Consider the functor
+ψun
+tf : DM(X ∖ X0) → DM(X0). There is a canonical morphism ι : i∗
+X0 → ψun
+tf v∗
+of functors on DM(X) and its Verdier dual ι∗ : ψun
+tf v∗ → i!
+X0. Therefore we have
+a motivic sheaf R := ψun
+tf Q(0)X∖X0 equipped with a natural self-duality R∗
+∼
+→
+R(n − 1)[2n − 2] and mutually dual maps Q(0)X0
+ι→ R
+ι∗
+→ Q(0)∗
+X0(1 − n)[2 − 2n]
+that are isomorphisms over U.
+Let ∂ : B → Q(0)X0 be the boundary map of the triangle from 4.3(ii).
+Set
+αR := ι∂ : B → R, and let βR be α∗
+R combined with the self-duality identifications
+for R and B, so we have
+(4.4.1)
+B
+αR
+−→ R
+βR
+−→ B(−1).
+Lemma-construction. The composition βRαR is homotopic to zero.
+In fact,
+there is a canonical up to a homotopy κR such that d(κR) = βRαR.
+Proof. By Remark and Example in 4.3 one has βRαR = ∂∗ι∗ι∂ = ∂∗ǫ∂ = ∂∗χ∗χ∂ =
+(χ∂)∗χ∂. Notice that χ∂ is homotopic to 0; choose a homotopy λ, d(λ) = χ∂. Now
+set κR := λ∗χ∂.
+Independence of κR up to a homotopy from the choice of λ: if λ′ is another
+homotopy as above, i.e., d(λ) = d(λ′), then κ′
+R = λ′∗χ∂ = κR + (λ′ − λ)χ∂ =
+κR + d((λ − λ′)λ).
+□
+Remark. Our κR is self-dual up to homotopy: Indeed, one has κ∗
+R = (χ∂)∗λ =
+κR + d(λ∗λ).
+4.5. Below we use the notation from 2.1, 2.2. We have defined (αR, βR, κR) ∈
+DM(X0)(2). It yields the objects ER := E(αR, βR, κR) ∈ DM(X0) and (αI, βI, κI)
+:= σ(αR, βR, κR) ∈ DM(X0)(2). As follows from Remark in 4.4 and the defini-
+tions, the above three objects are naturally self-dual.
+Proposition. There is a homotopy equivalence θ : I
+∼
+→ ER such that the maps
+βIθ : I → B[1], θ−1αI : B(−1)[−1] are a morphism of the triangle in 4.3(ii) and
+its dual. Our θ is unitary, i.e., θ∗ = θ−1.
+Proof. Recall that we have a natural homotopy equivalence (λ, χ) : Cone(∂ : B →
+Q(0)X0)
+∼
+→ I (see 4.3(ii)), and ER is the direct sum B[1] ⊕ R ⊕ B(−1)[−1] with
+(αR, −κR, βR) added to the differential (see 2.1). Our θ is the composition I
+∼
+←
+Cone(∂)
+θ′
+→ ER where θ′ is the next morphism: its restriction to B[1] ⊂ Cone(∂)
+identifies it with the first summand in ER, and its restriction to Q(0)X0 ⊂ Cone(∂)
+is (0, ι, −λ∗χ).
+
+HEIGHT PAIRING AND NEARBY CYCLES
+15
+One has θ∗θ = idI: we need to check that θ′∗ρθ′ = (λ, χ)∗(λ, χ) : Cone(∂) →
+Cone(∂∗) where ρ : ER
+∼
+→ E∗
+R(1 − n)[2 − 2n] is the self-duality for ER. As follows
+from Remark in 4.4, ρ is the matrix with the self-dualities for R and B’s on the
+diagonal and the only non-zero off-diagonal entry being λ∗λ : B → B∗(1−n)[2−2n].
+The rest is an immediate calculation.
+The assertion that βIθ is the morphism of the triangle in 4.3(ii) means that
+βIθ′ is the projection Cone(∂) → B[1] which is evident from the construction. The
+assertion that αIθ is dual to βIθ follows from the unitarity of θ once we know that
+θ is a homotopy equivalence. Let us check it.
+Our θ′ is a morphism Cone(B → Q(0)X0) → Cone(B → Cone(βR)[−1]) com-
+patible with the projections to B, and so it is enough to check that the map
+(ι, −λ∗χ) : Q(0)X0 → Cone(βR)[−1] is a homotopy equivalence. Since ι is a homo-
+topy equivalence on U, it is enough to check our claim after applying i∗
+xα.
+The story of section 3.3 uses only the six functors formalism and basic facts
+from 3.1, so it remains literally true in the motivic setting. Consider the canonical
+homotopy equivalence a : i∗
+xαR
+∼
+→ Γ(Pα ∖ Zα) of (3.3.1). By the Verdier dual
+assertion to the lemma in 3.3, a identifies i∗
+xα(βR) with minus the residue map
+r : Γ(Pα ∖ Zα) → Hn−2
+prim(Zα)(−1)[1 − n] ⊂ Γ(Zα)(−1)[−1]. By (4.2.2) we have a
+split exact triangle Q(0) → Γ(Pα ∖ Zα)
+r→ Hn−2
+prim(Zα)(−1)[1 − n], so a identifies
+i∗
+xαCone(βR)[−1]) with Q(0) ⊂ Γ(Pα∖Zα). It follows directly from the construction
+of a that ai∗
+xα(ι) coincides with the latter embedding, and we are done.
+□
+4.6. Proof of the theorem in 1.9.
+We have (αI, βI, κI) ∈ DM(X0)(2), hence
+Γ(αI, βI, κI) ∈ DM(2). For two Bloch cycles A, B of classes clA, clB ∈ Hom(Q(0),
+Hn−2
+prim(Zα)(m)) we have (cl∗
+A, clB∗)Γ(αI, βI, κI) ∈ EM(Γ(I)(m − 1)[1 − n]) =
+EM(Γ(I+)(m − 1)[1 − n]) = EM(M) where M := M(Y )(−m)[−1 − 2m]. By the
+construction the Hodge realization embedding EM(M) ֒→ EH(M) = EH(Hm(Y ))
+identifies it with Eψ
+A,B from 1.6, and we are done.
+□
+References
+[A1]
+J. Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents
+dans le monde motivique (I), Ast´erisque 314, SMF, 2007.
+[A2]
+J. Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents
+dans le monde motivique (II), Ast´erisque 315, SMF, 2007.
+[B]
+A. Beilinson, Height pairing between algebraic cycles, K-theory, Arithmetic and Geom-
+etry, Yu. I. Manin (Ed.), Lect. Notes in Math. 1289, Springer, 1987.
+[Bl1]
+S. Bloch, Height pairings for algebraic cycles, Journal of Pure and Applied Algebra 34
+(1984), 119–145.
+[Bl2]
+S. Bloch, Cycles and biextensions, Contemporary Mathematics 83 (1989), 19–30.
+[BlJS]
+S. Bloch, R. de Jong, E. Can Sert˜oz, Heights on curves and limits of Hodge structures,
+arXiv:2206.01220 (2022).
+[CD]
+D.-C. Cisinski, F. D´eglise, Triangulated categories of mixed motives, Springer Mono-
+graphs in Mathematics, Springer, 2019.
+[G]
+S. Gorchinskiy, Notes on the biextension of Chow groups, Motives and algebraic cycles,
+Fields Institute Commun., vol. 56, Amer. Math. Soc., 2009, pp. 111–148.
+[Il]
+L. Illusie, Sur la formule de Picard-Lefschetz, Algebraic geometry 2000, Azumino,
+Advanced Studies in Pure Math, vol. 36, Mathematical Society of Japan, 2002, pp. 249–
+268.
+
diff --git a/79E0T4oBgHgl3EQfwQGR/content/tmp_files/load_file.txt b/79E0T4oBgHgl3EQfwQGR/content/tmp_files/load_file.txt
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@@ -0,0 +1,791 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf,len=790
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='02630v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='AG]  15 Aug 2022  HEIGHT PAIRING AND NEARBY CYCLES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Beilinson  To Yuri Ivanovich Manin with deepest gratitude  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We prove that, as was conjectured by Spencer Bloch, the Hodge period  of some limit Hodge structures equals the height pairing of algebraic cycles on the  resolution of singularities of the singular fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Introduction: the theorem and the idea of the proof  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The Hodge period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Suppose we have a Q-Hodge structure E with weights  in [−2, 0] equiped with isomorphisms ι0 : grW  0 E = Q(0), ι−2 : grW  −2E = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  One defines the Hodge period ⟨E⟩ = ⟨E, ι0, ι−2⟩ ∈ R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Consider the  R-Hodge structure E ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since the weight filtration on any R-Hodge structure  with two consequitive weights (canonically) splits one has E ⊗ R = G ⊕ grW  −1E ⊗ R  where G is an extension of R(0) by R(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our ⟨E⟩ is the class of this extension in  Ext1(R(0), R(1)) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One computes ⟨E⟩ explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let ER be E ⊗ R viewed a plain  R-vector space, EC be its complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let 1F 0 ∈ F 0 ⊂ EC be any lifting of  ι−1  0 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then ⟨E⟩ is the image of 1F 0 in (ER + W−1EC)/(ER + (F 0 ∩ W−1EC))  ∼  ←  W−2EC/W−2ER = C/2πiR  ∼  ← R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' A geometric example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let Y be a smooth proper equidimensional algebraic  variety over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We denote by Hi(Y ) the homology of Hi(Y (C), Q) seen as an object  of the category of Q-Hodge structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ditto for relative homology, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let Zm(Y )  be the group of algebraic m-cycles on Y with Q-coefficients, Zm(Y )0 := Ker(cl :  Zm(Y ) → H2m(Y )(−m)) be the subgroup of cycles homologically equivalent to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For a closed subset P ⊂ Y let Zm(P) ⊂ Zm(Y ) be the subgroup of cycles  supported on P, Zm(P)0 := Zm(P) ∩ Zm(Y )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For an m-cycle A on Y we denote  by |A| its support (which is a closed subset of Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Suppose m + m′ = dim Y − 1 and we have A ∈ Zm(Y )0, B ∈ Zm′(Y )0 such that  |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set E|A|,|B| := H2m+1(Y ∖ |B|, |A|)(−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notice that E|B|,|A| =  E∗  |A|,|B|(1) by the Poicar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' E|A|,|B| has weights in [−2, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has grW  −2E|A|,|B| = Zm′(|B|)∗  0(1),  grW  −1E|A|,|B| = H2m+1(Y )(−m), grW  0 E|A|,|B| = Zm(|A|)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notice that H2m(|A|)(−m) = Zm(|A|) and H>2m(|A|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By the Poincar´e  duality Hi(Y, Y ∖|B|)(− dim Y ) = H2 dim Y −i(|B|)∗, hence H2m+2(Y, Y ∖|B|)(−m) =  1991 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Primary 14C25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Secondary 14D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' height pairing, nearby cycles, Hodge periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Typeset by AMS-TEX  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  (H2m′(|B|)(−m′))∗(1) and H<2m+2(Y, Y ∖ |B|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Now use the long exact ho-  mology sequences for (Y ∖ |B|, |A|) and (Y, Y ∖ |B|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Denote by EA,B the Hodge structure obtained from H|A|,|B| by pullback by A  and pushforward by B:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  Zn(|B|)∗  0(1)  ֒→  E|A|,|B|  ։  Zm(|A|)0  B ↓  ↑ A  Q(1)  ֒→  EA,B  ։  Q(0)  Our EA,B is as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1, so we have ⟨EA,B⟩ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The height pairing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' [B], [Bl1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let k be a subfield of C and suppose that  Y comes from a variety Yk over k, Y = Yk ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let Zm(Yk) ⊂ Zm(Y ) be the  group of algebraic cycles with Q-coefficients on Yk, Zm(Yk)0 := Zm(Yk) ∩ Zm(Y )0,  and let CHm(Yk)0 ⊂ CHm(Yk) be their quotients modulo the rational equivalence  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One checks (see §2) that if A, B as above are cycles on Yk then the class  of ⟨EA,B⟩ in R/Q log |k×| depends only on linear equivalence classes of A and B,  and so one has a bilinear height pairing  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  ⟨ , ⟩Yk : CHm(Yk)0 ⊗ CHm′(Yk)0 → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Namely ⟨a, b⟩Yk = ⟨EA,B⟩ where A, B are any cycles on Yk of classes a, b such that  |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If k = Q and we assume some motivic rationality conjectures (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3) of [B]) then ⟨EA,B⟩ can be corrected (by adding a finite sum of corrections  log(p)⟨EA,B⟩p where p is a prime, ⟨EA,B⟩p is defined using the Gal(Qp)-action on  EA,B ⊗ Qℓ) so that the resulting real number depends only on rational equivalence  classes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' In this manner (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1) lifts naturally to an R-valued pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Finding elements of Chow groups that are homologically equivalent to zero  is an art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Spencer Bloch described one situation where they naturally arise, and  conjectured that the height pairing of his cycles can be computed in s different way,  namely, as Hodge periods of some nearby cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We start with preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let X be a smooth variety over C of pure dimension n ≥ 2, S be a smooth curve,  0 ∈ S be a closed point, and f : X → S be a proper map which is smooth otside  a finite subset {xα} of the fiber X0 = f −1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let Zα be the projectivized tangent  cone to X0 at xα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' this is a hypersurface in the projectivization Pα := P(TxαX) of  the tangent space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' denote by dα its degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We assume the next condition:  (∗) All hypersurfaces Zα are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let π : Y → X0 be the blowup of X0 at {xα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Condition (∗) implies that Y is  a smooth variety, and Zα are pairwise disjoint divisors on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set Z := ⊔Zα and  K := Ker(Hn−2(Z) → Hn−2(Y )) = Im(Hn−1(Y, Z) → Hn−2(Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If n = 2 then let  K0 ⊂ K be the subgroup of those elements A = ΣAα that deg Aα = 0 for every  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has a natural map Hn−1(X0) → Hn−1(Y, Z) defined as the composition  Hn−1(X0) → Hn−1(X0, {xα})  ∼  ← Hn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) The map Hn−1(X0) → Hn−1(Y, Z) is an isomorphism if n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If  n = 2 it is injective and its image equals the preimage of K0 in Hn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) Hn−1(Y, Z) has weights 1 − n and 2 − n, and grW  2−nHn−1(Y, Z) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The map  HEIGHT PAIRING AND NEARBY CYCLES  3  Hn−1(Y ) → Hn−1(Y, Z) has image W1−nHn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If n is even then Hn−1(Y )  ∼  →  W1−nHn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) Replace Hn−1(Y, Z) by Hn−1(X0, {xα}) and use the long exact homology  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (ii) The first assertions follow from the exact homology sequence and  purity of weights on H·(Y ), H·(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The last one comes because Hn−1(Z) = 0 if n  is even (since Zα are hypersurfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider a variation of Q-Hodge structures V on S ∖ {0} with fibers Vs =  Hn−1(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has a nondegenerate intersection pairing ( , ) : V ⊗ V → Q(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Choose a parameter t at 0 ∈ S and consider the limiting (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' nearby cycles)  Hodge structure ψtV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let ψun  t V be its direct summand where the monodromy acts  unipotently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since ψun  t  commutes with duality, ( , ) yields self-duality pairing on  it that we denote again by ( , ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has the log of monodromy morphism N =  NV : ψun  t V(1) → ψun  t V and the specialization morphism sp : ψun  t V → Hn−1(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let (ψun  t V)N := Coker(NV) be the monodromy coinvariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The next assertion  follows from the local invariant cycles theorem, see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5 for a detailed proof:  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' sp factors through the isomorphism (ψun  t V)N  ∼  → Hn−1(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ψun  t V has weights in [−n, 2 − n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has grW  2−nψun  t V = K if n > 2  and grW  2−nψun  t V = K0 if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By self-duality, grW  −nψun  t V = (grW  2−nψun  t V)∗(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  If n is even then grW  1−nψun  t V = Hn−1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since ψun  t V is self-dual and N is nilpotent, the claim follows from the propo-  sition and the lemma in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Bloch cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' suppose n is even, n = 2m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let  A = ΣAα be an m-cycle on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We say that A is a Bloch cycle if it is homologically  equivalent to zero on Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', cl(A) lies in K(−m) ⊂ Hn−2(Z)(−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If m = 0 then  we demand, in addition, that cl(A) ∈ K0 ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If A is a Bloch cycle then each cl(Aα) ∈ Hn−2(Zα)(−m) is primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The composition Hn−2(Z)(−m) → Hn−2(Y )(−m) → Hn−4(Zα)(−m + 1),  where the second arrow is the pullback by Zα ֒→ Y , sends any class c = Σcα to  cα ∩ c1(O(−1)) (for O(−1) is the normal bundle to Zα in Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' This composition  kills cl(Aα) since the first arrow does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  If A, B are two Bloch cycles then we denote by Eψ  A,B = Eψ  A,B,t the Hodge struc-  ture obtained from ψun  t V(−m) by pullback by cl(A) and pushforward by cl(B)∗:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  K∗(m + 1)  →  ψun  t V(−m)  →  K(−m)  cl(B)∗ ↓  ↑ cl(A)  Q(1)  ֒→  Eψ  A,B  ։  Q(0)  Our Eψ  A,B is as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1 so we have ⟨Eψ  A,B⟩ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the case when we have single singular point x0 ∈ X0 of  f and the singularity at x0 is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then the monodromy action on ψtV is  unipotent, the only possible Bloch cycle is the difference A of the rulings of the  quadric Z0, and it is actually a Bloch cycle if and only if the monodromy action on  ψtV is nontrivial or, equivalently, the Hodge structure on Hn−1(X0) is not pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) If m = 0 then the curve X0 can have either 1 or 2 irreducible compo-  nents, and A is a Bloch cycle if and only if X0 is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) If X/S is a family of quadratic hypersurfaces in Pn then A is not a Bloch cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) If X/S is a family of hypersurfaces of degree d on a given smooth projective  variety P then A is a Bloch cycle if d is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (ii) follows since the global monodromy for quadratic hypersur-  faces is ±1, and so it can’t contain non-trivial unipotent local monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) Consider the corresponding map r : S → B := {hypersurfaces of degree  d on P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since X is smooth r is transversal to the locus D ⊂ B of degenerate  hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Replacing S by a germ of another transversal to D that intersects  D near r(0) would not change the topology of X over a small disc around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' So we  can assume that S is a Zariski open subset of the base of a Lefschetz pencil on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Then, since local monodromies of a Lefschetz pencil are all conjugate, triviality of  one local monodromy amounts to triviality of the global monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Thus A is a  Bloch cycle if and only if the global monodromy on V is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us check  that this happens for large enough d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  If R ⊂ P is the axis of our pencil then H·(X) = H·(P) ⊕ H·−2(R)(−1), and so  hn−1,0(P) = hn−1,0(X) which equals hn−1,0(Xs) if the global monodromy is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Thus the monodromy is not trivial when hn−1,0(Xs) > hn−1,0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' To finish the  argument it remains to notice that hn−1,0(Xs) ≥ dim(H0(P, Ωn  P (d))/H0(P, Ωn  P )),  and so it tends to ∞ when d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Now suppose we have a subfield k ⊂ C and our  datum is defined over k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', there is Xk/Sk, a closed point 0 of Sk, a parameter t  on Sk at 0, and Bloch cycles A, B on Zk such that X/S, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', come by base change  k → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let a ∈ CHm(Yk)0, b ∈ CHm(Yk)0 be the classes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The next  result was conjectured by Spencer Bloch:  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has ⟨a, b⟩Yk = ⟨Eψ  A,B⟩ mod Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  In case n = 1 the theorem was proven in [BlJS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Suppose we are in the situation of Remark in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' If ⟨Eψ  A,B⟩ is corrected  in the same way as was discussed there, then the theorem lifts to an equality of real  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The proof does not change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' we will not discuss it below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Reformulation of the theorem that discards Hodge periods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' the idea of the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let A′, B′ be cycles on Yk of classes a, b such that |A′| ∩ |B′| = ∅ (no-  tice that they are, most probably, not supported on Zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We want to show that  ⟨EA′,B′⟩ = ⟨Eψ  A,B⟩ (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us compare the Hodge structures E = EA′,B′  and Eψ = Eψ  A,B themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Their weights lie in [−2, 0], and one has a canonical  identification grW  E = grW  Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Indeed, grW  0 E(ψ) = Q(0), grW  −2E(ψ) = Q(1) by the  constructions, and grW  −1E = H2m+1(Y )(−m) = grW  −1E(ψ) by the lemma in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2, and  the one in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4 combined with the corollary in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' This identification lifts (uniquely)  to W−1E = W−1Eψ and E/W−2E = Eψ/W−2Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Indeed, the classes of extensions  0 → H2m+1(Y )(−m) → E(ψ)/W−2E(ψ) → Q(0) → 0 both equal Deligne cohomol-  ogy class clD(A) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Griffiths’ Abel-Jacobi periods) of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' by duality, the classes  of (the duals to) extensions 0 → Q(1) → W−1E(ψ) → H2m+1(Y )(−m) → 0 both  equal to clD(B) (see loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  HEIGHT PAIRING AND NEARBY CYCLES  5  Now suppose we have a Q-Hodge structure H of weight −1 and two classes  a ∈ Ext1(Q(0), H), b ∈ Ext1(H, Q(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Consider the set EH  a,b = EH(H)a,b of  all Hodge structures E with weights in [−2, 0] and equipped with identifications  grW  0 E = Q(0), grW  −1E = H, grW  −2E = Q(1) such that the extensions E/W−2E and  W−1E have classes a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The group Ext1(Q(0), Q(1)) = C× ⊗ Q acts on EH  a,b by  the Baer sum action, and EH  a,b is a C× ⊗ Q-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notice that for q ∈ C× one has  ⟨q · E⟩ = log |q| + ⟨E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Applying this format to H = H2m+1(Y )(−m), a = clD(A),  b = clD(B) and EA′,B′, Eψ  A,B ∈ EH  a,b we get EA′,B′ − Eψ  A,B ∈ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Now the  theorem in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='8 follows immediately from the next result (notice that the Hodge  periods and the height pairing play no role here):  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has EA′,B′ − Eψ  A,B ∈ k× ⊗ Q ⊂ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The theorem would be an immediate corollary of the motivic formalism if all  the above constructions could be spelled in motivic world: Indeed, we would have  then a motivic version EM of EH which is an Ext1  M(Q(0), Q(1)) = k× ⊗ Q-torsor  equipped with the Hodge realization embedding EM ֒→ EH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' our EA′,B′, Eψ  A,B  would come from elements of EM, and so their difference lies in k× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The only  problem is that in the present day formalism of motives, due to Voevodsky, Ayoub,  and Cisinski-D´eglise, the t-structure is not available, so we do not have the motivic  version of separate homology groups like Hi(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The actual proof is an exercise in  spelling out the constructions in a way that makes the t-structure redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  I am very grateful to Spencer Bloch for explaining me his conjecture and stimu-  lating discussions (pity Spencer refused to coauthor the article), to Volodya Drinfeld  for valuable comments and discussions, and to Luc Illusie for calling my attention  to the construction of [I] which helped to clearify and simplify the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The height pairing and the construction of EM  a,b ∈ EM  a,b ⊂ EH  a,b  This section is a variation on the theme of [Bl2] and [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let C be a stable dg category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It yields two other dg categories C(1) and C(2)  constructed as follows:  An object of C(1) is a closed morphism α : M → N of degree 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has  Hom((M, N, α), (M ′, N ′, α′))i = Hom(M, M ′)i ×Hom(N, N ′)i ×Hom(M, N ′)i+1 ⊂  Hom(Cone(α), Cone(α′))i, and the differential is defined so that the latter embed-  ding is a morphism of complexes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' the composition of morphisms is defined in a sim-  ilar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There are three dg functors C(1) → C which send (M, N, α) to M, N, and  Cone(α) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We can view C(1) as the category of distinguished triangles,  and the rotation yields an autoequivalence ρ : C(1) → C(1) which sends α : M → N  to ρ(α) : N → Cone(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' the inverse autoequivalence is ρ−1(α) : Cone(α)[−1] → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  An object of C(2) is a datum (P, M, Q, α, β, κ) where P, M, Q are objects of C,  α ∈ Hom(P, M)1, β ∈ Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1 is such  that d(κ) = βα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' we sometimes abbreviate it to (α, β, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One can assign to such a  datum an object E = E(α, β, κ) ∈ C which equals P ⊕ M ⊕ Q with α, β, and −κ  added as the components to the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1 There is a filtration Q ⊂ Cone(β :  M[−1], Q) ⊂ E, and morphisms in C(2) are the same as morphisms between the  1Thus E = Cone((α, κ) : P [−1] → Cone(β : M[−1] → Q)) = Cone((κ, β) : Cone(α : P [−2] →  M[−1]) → Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  corresponding objects E that preserve this filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We have two dg functors  C(2) → C(1) which send to (α, β, κ) to α : P[−1] → M and β : M → Q[1], and  six dg functors C(2) → C which send (α, β, κ) to P, M, Q, Cone(α : P[−1] → M),  Cone(β : M[−1] → Q), and E(α, β, κ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The dg category C(3) carries a natural involution σ which sends (P, M, Q, α, β, κ)  to the object (Q[−1], E(α, β, κ), P[1], ασ, βσ, 0) where ασ and βσ are the evident  embedding and projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One can view an object (α, β, κ) ∈ C(2) as an object of C equipped with  a 3-step filtration in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Namely, this could be E(α, β, κ) equipped  with an evident filtration with successive quotients Q, M, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Or this could be  M equipped with a filtration whose successive quotients are P[−1], E(α, β, κ), and  Q[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The involution σ exchanges the two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For C as above we denote by C× the ∞-groupoid of its homotopy equivalences,  by C×τ the corresponding 1-truncaded plain groupoid, and by HC the homotopy  category of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For S, T ∈ C set Exti(S, T ) := HiHom(S, T ) = HomHC(S, T [i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Denote by Ext(S, T ) the plain Picard groupoid of extensions that corresponds to  the two-term complex τ [0,1]Hom(S, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  For M, N ∈ C let C(1)×  M,N be the ∞-groupoid of collections (α′ : M ′ → N ′, ιM, ιN)  where (α′ : M ′ → N ′) ∈ C(1) and ιM : M → M ′, ιN : N → N ′ are homotopy equiv-  alences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It is equivalent to the Picard ∞-groupoid that corresponds to the complex  τ ≤0Hom(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The 1-truncated plain Picard groupoid C(1)×τ  M,N  corresponds to the  two-term complex τ [−1,0]Hom(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Similarly, for three objects P, M, Q ∈ C we have the ∞-groupoid C(2)×  P,M,Q whose  objects are data (P ′, M ′, Q′, α′, β′, κ′, ιP , ιM, ιQ) where (P ′, M ′, Q′, α′, β′, κ′) ∈ C(2)  and ιP : P → P ′, ιM : M → M ′, ιQ : Q → Q′ are homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The 1-truncated plain groupoid C(2)×τ  P,M,Q contains a normal subgroup Ext0(P, Q) =  HomHC(P, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let by E = E(M) = E(P, M, Q) be the quotient groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  It  is equivalent to the groupoid of triples (α, β, κ) where α ∈ Hom(P, M)1, β ∈  Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1/d(Hom(P, Q)0) is such that  d(κ) = βα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' a morphism (α, β, κ) → (α′, β′, κ′) in E is a pair (φ, ψ) where φ ∈  Hom(P, M)0/d(Hom(P, M)−1), ψ ∈ Hom(M, Q)0/d(Hom(M, Q)−1) are such that  α′ − α = d(φ), β′ − β = d(ψ), κ′ − κ = βφ + ψα + ψd(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The projection C(2)  P,M,Q → C(1)  P [−1],M × C(1)  M,Q[1] yields a map of plain groupoids  E(P, M, Q) → C(1)×τ  P [−1],M × C(1)×τ  M,Q[1] = Ext(P, M) × Ext(M, Q), (α, β, κ) �→ (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The group Ext1(P, Q) acts on E by translations of κ, and non-empty fibers Eα,β  are Ext1(P, Q)-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' E(P, M, Q) is naturally functorial with respect to P and Q: every pair  of closed morphisms µ : P1 → P and ν : Q → Q1 yields a map E(P, M, Q) →  E(P1, M, Q1), (α, β, κ) �→ (αµ, νβ, νκµ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' is compatible with the Ext1(P, Q)-action  via the map (µ∗, ν∗) : Ext1(P, Q) → Ext1(P1, Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Suppose Ext2(P, Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Then Eα,β are non-empty, and the addition maps  Eα1,β × Eα2,β → Eα1+α2,β, Eα,β1 × Eα,β2 → Eα,β1+β2 define on E the structure of  an Ext1(P, Q)-biextension of (Ext(P, M), Ext(M, Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' In our first example C is the dg category whose homotopy category is the  bounded derived category DH of the category H of Q-Hodge structures, and  HEIGHT PAIRING AND NEARBY CYCLES  7  P = Q(0), Q = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We denote the corresponding E by EH = EH(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then  Ext̸=1  DH(P, Q) = 0 and Ext1  DH(P, Q) = C× ⊗ Q, so EH is a C× ⊗ Q-biextension of  (Ext(Q(0), M), Ext(M, Q(1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let Ext1  0(Q(0), M) ⊂ Ext1(Q(0), M), Ext1  0(M, Q(1)) ⊂ Ext1(M, Q(1)) be the  subgroups of those elements a, b that the maps H0a : Q(0) → H1M, H−1b :  H−1M → Q(1) vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let Ext0(Q(0), M) ⊂ Ext(Q(0), M), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', be the Picard  groupoids of such extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Suppose that Hom(Q(0), H0M) = Hom(H0M, Q(1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (i) The restriction of EH to (Ext0(Q(0), M), Ext0(M, Q(1))) descends to the C×⊗Q-  biextension of (Ext1  0(Q(0), M), Ext1  0(M, Q(1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) EH is naturally functorial with respect to M: if ϕ : M → M ′ is a morphism,  and we have a′ ∈ Ext1  0(Q(0), M ′), b′ ∈ Ext1  0(M ′, Q(1)) with ϕ∗(a) = a′, ϕ∗(b′) = b  then there is a canonical identification EH(M)a,b = EH(M ′)a′,b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) The isomorphisms Ext1  0(Q(0), M)  ∼  → Ext1(Q(0), H0M), Ext1  0(M, Q(1))  ∼  → Ext1  (H0M, Q(1)) which assign to an extension its zero cohomology, lifts naturally to an  isomorphism of biextensions H0 : EH(M)  ∼  → EH(H0M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has EH0 = H0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us prove (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' the rest is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We need to check that for every closed α ∈  Hom1  0(Q(0), M), β ∈ Hom1  0(M, Q(1)) the action of Aut(α)×Aut(β) = Hom(Q(0), M)  ×Hom(M, Q(1)) on EH  α,β is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Since H has homological dimension 1 our M is isomorphic to the direct sum of its  homologies and so Aut(α) = Ext1(Q(0), H−1M), Aut(β) = Ext1(H1(M), Q(1)) by  the condition on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The action of (e, h) ∈ Ext1(Q(0), H−1M)×Ext1(H1(M), Q(1))  on EH  α,β is the translation by H−1(β)e + hH0(α) which is 0 since α, β ∈ Ext1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Suppose that H0M is pure of weight −1 (which implies the condition  of the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then the function EH(M) → R, (α, β, κ) �→ ⟨E(α, β, κ)⟩ :=  ⟨H0E(α, β, κ)⟩, see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1, is a natural trivialization of the R-biextension log |EH(M)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Everything said in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 works for the category HR of R-Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The  extension of scalars functor H → HR, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' �→?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ⊗ R, yields a morphism of our biex-  tensions EH(M) → EHR(M ⊗ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The map Ext1(Q(0), Q(1)) → Ext1(R(0), R(1))  equals log | | after the standard identifications of the Ext groups with, respectively,  C× ⊗ Q and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Since Ext1(R(0), H0M ⊗ R) = Ext1(H0M ⊗ R, R(1)) = 0 by  the condition on M, one has EHR(M ⊗ R) = EHR(H0M ⊗ R) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The map  EH(M) → EHR(M ⊗ R) = R is ⟨ ⟩ of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let k ⊂ C be a subfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Denote by DM(k) the dg category of geometric  Voevodsky Q-motives over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We have the Hodge realization dg functor DM(k) →  DH, M �→ M H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Consider the story of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2 for C = DM(k) with P = Q(0),  Q = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' As before one has Ext̸=1  DM(k)(Q(0), Q(1)) = 0, and there is a canonical  identification Ext1(Q(0), Q(1)) = k× ⊗ Q such that the Hodge realization map  between the Ext1’s is the embedding k× ⊗ Q ֒→ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' So for any M ∈ DM(k)  we get a k× ⊗ Q-biextension of (Ext1(Q(0), M), Ext1(M, Q(1))) together with the  Hodge realization morphism EM(M) → EH(M) := EH(M H) of the biextensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since the homomorphism k× ⊗ Q ֒→ C× ⊗ Q is injective, the maps of  torsors EM(M)α,β → EH(M)α,β := EH(M)αH,βH are injective too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We define Ext1  0(Q(0), M) ⊂ Ext1  0(Q(0), M) and Ext1  0(M, Q(1)) ⊂ Ext1(M, Q(1))  as preimages of the Ext1  0 subgroups of the Hodge setting by the Hodge realiza-  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  tion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Assume that H0M H is pure of weight −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Then (i) and (ii) of  the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 remain true in the DM(k) setting (with C× replaced by k×):  this follows from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' by Remark above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Thus we have a k× ⊗ Q-biextension  EM(M) of (Ext1  0(Q(0), M), Ext1  0(M, Q(1))) together with a map of biextensions  EM(M) → EH(M), so the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4 provides a natural trivialization of the  R-biextension log |EM(M)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The image of EM  a,b in R/Q log |k×| depends only on  a, b ∈ Ext1  0(M, Q(1)) × Ext1  0(Q(0), M), and we denote it by ⟨a, b⟩M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It is clearly  biadditive with respect to a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2 We have defined a canonical height pairing  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  ⟨ ⟩M : Ext1  0(Q(0), M) × Ext1  0(M, Q(1)) → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We return to the situation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 and set M := M(Yk)(−m)[−1 − 2m] where  M(Yk) is the motive of Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has Ext1(Q(0), M) = CHm(Yk), Ext1(M, Q(1)) =  CHm′(Yk) by the Poincar´e duality, and Ext1  0 are the subgroups CH(Yk)0 of cycles  homologically equivalent to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Therefore we get a k× ⊗ Q-biextension EM of  (CHm(Yk)0, CHm′(Yk)0), the map of biextensions EM → EH, the trivialization of  log |EM|, and the height pairing ⟨ , ⟩M : CHm(Yk)0 × CHm′(Yk)0 → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  By (iii) of the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 one has H0 : EH(M)  ∼  → EH(H2m+1(Y )(−m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For  a ∈ CHm(Yk)0, b ∈ CHm′(Yk)0 pick, as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3, cycles A, B that represent them  such that |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 Let us construct (a, b, κA,B) ∈ EM  a,b such that the Hodge  realization EH  A,B of EM  A,B := E(a, b, κA,B) (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1) has zero cohomology H0EH  A,B  equal to the Hodge structure EA,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' This would imply that for our M the  height pairing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1) equals (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The composition of the maps M(|A|)  α→ M(Yk)  β→ M(Yk, Yk ∖ |B|) is naturally  homotopic to 0: indeed, M(Yk, Yk ∖ |B|) := Cone(M(Yk ∖ |B|) → M(Yk)), and the  homotopy κ|A|,|B| is M(|A|) → M(Yk ∖|B|) ⊂ Cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Thus we have (α, β, κ|A|,|B|) ∈  DM(2) (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notice that E(α, β, κ|A|,|B|) = M(Yk ∖ |B|, |A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  One has Ext−2m(Q(m), M(|A|)) = Zm(|A|) := the group of m-cycles supported  on |A| (recall that dim |A| = m), and Ext2m+2(M(Yk, Yk ∖ |B|), Q(m + 1)) =  Zm′(|B|) by the Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Therefore we have (αA, Bβ, Bκ|A|,|B|A) =  (Q(m)[2m+1], M(Yk), Q(m)[2m+2], αA, Bβ, Bκ|A|,|B|A) ∈ DM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The promised  (a, b, κA,B) ∈ EM  a,b is (αA, Bβ, Bκ|A|,|B|A)(−m)[−1 − 2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The fact that H0EH  A,B  equals the Hodge structure EA,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 follows from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The unipotent nearby cycles in the Hodge setting  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' A nearby cycles reminder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' In this section we play with algebraic varieties over  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For an algebraic variety X we denote by H(X) the abelian category of perverse  Hodge Q-sheaves of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Saito on X, by DH(X) its bounded derived category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It sat-  isfies the usual Grothendieck six functors formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Below ∗ is the Verdier duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Every object of H(X), hence of DH(X), carries a canonical weight filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  For F ∈ DH(X) let Γ(X, F), Γc(X, F) ∈ DH be the complex of chains, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  chains with compact support, with coefficients in F equipped with the natural  Hodge structure, H·  (c)(X, F) := H·Γ(c)(X, F) ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' set Γ(c)(X) := Γ(c)(X, Q(0)X),  H·  (c)(X) := H·  (c)(X, Q(0)), and denote by ( , ) the Poincar´e duality pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Simi-  larly for a closed subvariety A ⊂ X we set ΓA(X) := ΓA(X, Q(0)) ∈ DH, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  2Indeed, a morphism from a biextension by a trivial group to a trivialized biextension amounts  to a biadditive pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  3Recall that |A|, |B| ⊂ Yk are supports of the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  HEIGHT PAIRING AND NEARBY CYCLES  9  Let g : X → A1 be a function on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' set X0 := g−1(0), and let v : X ∖ X0 ֒→ X,  iX0 : X0 ֒→ X be the open and closed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has the unipotent nearby  cycles functor ψun  g  : DH(X ∖ X0) → DH(X0) that carries a natural logarithm  of monodromy morphism N = Ng = NF : ψun  g (F)(1) → ψun  g (F) where F ∈  D(X ∖ X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It has ´etale local origin with respect to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For sheaves on X there  is a natural morphism of functors ι : i∗  X0 → ψun  g v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  There are basic canonical  identifications:  (i) Compatibility with Verdier duality: One has ψun  g (F∗) = (ψun  g F)∗(1)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) Compatibility with proper direct images: Suppose h : X → T is a proper map  and t is a function on T such that g = th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' then one has ψun  t h∗F = h∗ψun  g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) One has Cone(NF) = i∗  X0v∗F(1)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iv) For every n > 0 one has ψun  gnF  ∼  → ψun  g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  These identifications are mutually compatible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) and (ii) are compatible with  the action of N, and (iv) identifies Ngn with nNg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Finally, one has  (v) ψun[−1] is t-exact for the perverse t-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Suppose that X is smooth of dimension n and F = Q(0)X∖X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then  F∗ = F(n)[2n] hence ψun  g (F)∗ = (ψun  g F)(n − 1)[2n − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (a) If g is smooth then ιQ(0)X : Q(0)X0  ∼  → ψun  g F, NF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (b) Suppose g is semi-stable and X0 has two irreducible components Y and Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By  (a) one has natural morphisms jY ′∖Y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='QY ′∖Y → ψun  g F → jY ∖Y ′∗QY ∖Y ′ compatible  with the N-action (we take it that on the left and right object N acts trivially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  They form an exact triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' its Verdier dual is the same triangle with Y and Y ′  interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4 and follow the notation there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let j : U := X0 ∖ {xα} ֒→ X0 ←֓ {xα} : ⊔ixα be the complementary open  and closed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let I be the intersection cohomology sheaf j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='∗Q(0)U =  τ ≤n−2j∗Q(0)U 4 on X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' set I+ := π∗Q(0)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has natural self-duality isomor-  phisms I∗ = I(n − 1)[2n − 2], I+∗ = I+(n − 1)[2n − 2] (recall that Y is smooth of  dimension n − 1 and π is proper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The decomposition theorem for π is easy and explicit:  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There is a natural orthogonal direct sum decomposition  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)  compatible with the self-dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has a natural orthogonal direct sum decomposition  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2)  Γ(Zα) = Hn−2  prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)  defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the embedding Zα ֒→ Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The pullback and Gysin  maps Γ(Pα) → Γ(Zα) → Γ(Pα)(1)[2] are mutually dual for the Poincar´e duality  pairings, and their composition in either direction equals to the multiplication by  c1(O(dα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5 Thus the composition of τ ≤2n−4Γ(Pα) → Γ(Zα) → τ ≥0(Γ(Pα)(1)[2]) is  an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' This yields a direct sum decomposition Γ(Zα) =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='⊕τ ≤2n−4Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4Below τ is the usual truncation, pτ is the perverse one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  5Since O(dα) is the normal bundle to Zα in Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  Since multiplication by c1(O(dα)) preserves the direct sum decomposition, the only  nonzero cohomology of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' is Hn−2  prim(Zα) ⊂ Hn−2(Zα), q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Consider the embeddings of smooth divisors iZα : Zα ֒→ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ZαQ(0)Y =  Q(−1)[−2]Zα, i∗  ZαQ(0)Y = Q(0)Zα, and the composition of the adjunction maps  iZα∗i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ZαQ(0)Y → Q(0)Y → iZα∗i∗  ZαQ(0)Y equals the multiplication by c1(O(−1))  map Q(−1)[−2]Zα → Q(0)Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6 Apply π∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' then i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαI+ = Γ(Zα)(−1)[−2], i∗  xαI+ =  Γ(Zα) by base change, and the composition of the adjunctions ixα∗i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαI+ → I+ →  ixα∗i∗  xαI+ is multiplication by c1(O(−1)) map ixα∗Γ(Zα)(−1)[−2] → ixα∗Γ(Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Composing the maps τ ≤2n−6Γ(Pα) ֒→ Γ(Zα) and Γ(Zα) ։ τ [2,2n−4]Γ(Pα) that  come from decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2) from the left and from the right with the latter ad-  junctions, we get the maps ixα∗(τ ≤2n−6Γ(Pα))(−1)[−2] → I+ → ixα∗τ [2,2n−4]Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Their composition is an isomorphism, which yields a decomposition I+ = I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ⊕  ixα∗τ [2,2n−4]Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since the adjunctions are mutually dual, the decomposition is  orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2) one has i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' = Hn−2  prim(Zα)(−1)[−n] ⊕ Q(n − 1)[2 − 2n], i∗  xαI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' =  Hn−2  prim(Zα)[2 − n] ⊕ Q(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Thus I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' [n − 1] is a perverse sheaf which equals Q(0)[n − 1]U on U and has no  subquotients supported on {xα}, and so I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) The adjunction map Q(0)X0 → π∗Q(0)Y = I+ takes value in I ⊂ I+  since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) Set B := ⊕ixα∗Hn−2  prim(Zα)[1 − n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By the formula for i∗  xαI at the end of the  previous paragraph, one has an exact triangle Q(0)X0 → I → B[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' As in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5, t is a local coordinate at 0 ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' shrinking S we can assume that  t is defined and invertible on S ∖ {0}, so X0 = (tf)−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the functor  ψun  tf : DH(X ∖ X0) → DH(X0) (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set R := ψun  tf Q(0)X∖X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(i) one  has a canonical self-duality identification R∗ = R(n − 1)[2n − 2] and the mutually  dual maps Q(0)X0  ι→ R  ι∗  → Q(0)∗  X0(1 − n)[2 − 2n] which are isomorphisms over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The next result is due to Illusie [Il];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' we will need it in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The reader can skip  it at the moment and jump directly to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For every critical point xα one has canonical isomorphisms  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαR = Γc(Pα ∖ Zα),  i∗  xαR = Γ(Pα ∖ Zα)  interchanged by the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The N-action on i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαR, i∗  xαR is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (a) The claim is local at xα, so for the proof we remove from X the rest  of critical points, and still call it X by the abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let S♭ → S be the  covering of degree dα obtained by adding t♭ = t1/dα to the sheaf of functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' its  Galois group is µdα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set X♭ := X×SS♭ and let f ♭ : X♭ → S♭ be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our  X♭ is a hypersurface {(x, t♭) : (tf)(x) − t♭dα = 0} in X × A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' its only singular point  is (xα, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The projectivized tangent cone Qα of X♭ at (xα, 0) is a hypersurface in  P +  α := P(T(xα,0)X × A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The Galois group µdα acts on X♭ hence on Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (b) Let us check that Qα is a µdα-covering of Pα completely ramified along Zα  and ´etale over its complement, and Qα is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' To see this, consider the leading  term [tf]dα(x) (of the Taylor expansion) of tf at xα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' then the leading term of  6Since O(−1) is the normal bundle to Zα in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  HEIGHT PAIRING AND NEARBY CYCLES  11  (tf)(x) − t♭dα at (xα, 0) is [tf]dα(x) − t♭dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The zeros of [tf]dα is Zα ⊂ Pα, of  [tf]dα(x) − t♭dα is Qα ⊂ P +  α , and so the projection Qα → Pα (x, t♭) �→ x, is as  claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The smoothness of Qα follows from that of Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (c) Let π+ : X+ → X♭ be the blowup of X♭ at (xα, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By (b) X+ is smooth  and the map f + := f ♭π+ : X+ → S♭ has semistable reduction at 0 ∈ S♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The  fiber X+  0 has two irreducible components: one equals Y and the other Qα, and  their intersection equals Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The action of µdα on X♭ yields one on X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The  µdα-action on X+  0  fixes Y and acts on Qα as described in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The projection  π+  0 : X+  0 → X♭  0 = X0 contracts Qα to xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Set R+ := ψun  tf +Q(0)X+∖X+  0 , R♭ := ψun  tf ♭Q(0)X♭∖X♭  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' These are sheaves on X+  0  and X♭  0 = X0 respectively that are naturally µdα-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(ii) (with  h = π+) one has a natural identification π+  0∗R+ = R♭ compatible with the µdα-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since the projection p : X♭ → X is a µdα-torsor over X ∖ X0 one has  Q(0)X∖X0 = (p∗Q(0)X♭∖X♭  0)µdα , and so, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(ii) with h = p, one has R = R♭µdα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Therefore R = (π+  0∗R+)µdα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (d) By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(iv) with g = t♭f +, n = dα, one has ψun  tf + = ψun  t♭f +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our t♭f + is semi-  stable, so we have the exact triangle jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='QY ∖Zα → R+ → jQα∖Zα∗QQα∖Zα  as in Example (b) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Applying π+  0∗ we get an exact triangle j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='QU → R♭ →  ixα∗Γ(Qα∖Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Passing to µdα-invariants we get, by (b), an exact triangle j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='QU →  R → ixα∗Γ(Pα ∖ Zα);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' here we use the identification Γ(Qα ∖ Zα)µdα  ∼  → Γ(Pα ∖ Zα)  defined as the composition Γ(Qα ∖ Zα)µdα ⊂ Γ(Qα ∖ Zα)  tr  → Γ(Pα ∖ Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Thus  we get the isomorphism i∗  xαR  ∼  → Γ(Pα ∖ Zα) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The second isomorphism  there comes in the dual manner from the dual exact triangle jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='QQα∖Zα →  R+ → jY ∖Zα∗QY ∖Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since π+  0∗ commutes with duality, the two isomorphisms are  mutually dual, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Let αR be the composition B  ∂→ Q(0)X0  ι→ R where ∂ is the boundary map of the  triangle from Remark (ii) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2, so I = Cone(∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us compute the map i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xα(αR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Consider the standard triangle Hn−2  prim(Zα)[1−n]  δ→ Γc(Pα ∖Zα)  tr  → Q(1−n)[2−2n]  that comes from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' −i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xα(αR) equals the composition δR of the maps Hn−2  prim(Zα)[1 − n]  δ→  Γc(Pα ∖ Zα)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  =  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the exact triangle  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2)  jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='Q(0)Qα∖Zα ⊕ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='Q(0)Y ∖Zα → Q(0)X+  0 → Q(0)Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let (δQ, δY ) : Q(0)Zα[−1] → jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='Q(0)Qα∖Zα ⊕ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='Q(0)Y ∖Zα be the bound-  ary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Its composition with the map to Q(0)X+  0 , and hence with the further  composition with Q(0)X+  0  ι→ R+, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Therefore the sum of the compositions  Q(0)Zα[−1]  δQ  −→ jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ι→ R+ and Q(0)Zα[−1]  δY  −→ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ι→ R+ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Ap-  ply i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xαπ+  ∗ and consider the restriction of our compositions to Hn−2  prim(Zα)[1 − n] ⊂  Γ(Zα)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For the first one it is δR, for the second one it is i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  xα(αR), and we are  done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set P := R[n − 1] = ψun  tf Q(0)X∖X0[n − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' this is a perverse sheaf on X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' one  has a canonical self-duality identification P∗ = P(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the perverse  sheaves PN := Ker(N : P → P(−1)), PN := Coker(N : P(1) → P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) Q(0)X0[n − 1] is a perverse sheaf of weights n − 1 and n − 2 with  grW  n−1 = I[n − 1], grW  n−2 = ⊕α ixα∗Hn−2  prim(Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) One has PN = Q(0)X0[n − 1], PN = (Q(0)X0[n − 1])∗(1 − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) P has weights in [n − 2, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has Wn−1P = Q(0)X0[n − 1], P/Wn−2P =  (Q(0)X0[n − 1])∗(1 − n), grW  n−2P = ⊕α ixα∗Hn−2  prim(Zα), grW  n−1P = I[n − 1], grW  n P =  (grW  n−2P)∗(1 − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) The exact triangle from Remark (ii) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2 amounts to an exact triangle  ⊕ixα∗Hn−2  prim(Zα) → Q(0)X0[n − 1] → I[n − 1], and we are done since its left and  right terms are pure perverse sheaves of weights n − 2 and n − 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) For any sheaf A on X one has a canonical exact triangle i∗  X0A → i∗  X0v∗v∗A →  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  X0A[1]: Indeed, the map v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='v∗A → v∗v∗A factors as composition v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='v∗A → A →  v∗v∗A, and so one has an exact triangle Cone(v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='v∗A → A) → Cone(v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='v∗A →  v∗v∗A) → Cone(A → v∗v∗A) which is supported on X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The promised exact  triangle is its restriction to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Now take for A the perverse sheaf Q(0)X[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The first term of the triangle is  Q(0)X0[n] which is perverse sheaf shifted by 1, its third term is (Q(0)X0[n−1])∗(−n)  which is a perverse sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Therefore they equal, respectively, pH−1 and pH0 of  i∗  X0v∗v∗Q(0)X[2n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', of Cone(N : P → P(−1)) by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(iii), and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (iii) Since N is nilpotent, the weights of P are bounded from below by the  minimum of weights of PN, which is n − 2 by (ii) and (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By self-duality of P they  are bounded then from above by n, and we have the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It implies that  Wn−2P ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The rest follows directly from (i), (ii), and self-duality of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Proof of the proposition in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We use the notation in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Injectivity of  sp : (ψun  t H)N → Hn−1(X0) follows from the local invariant cycles theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us  check the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(ii) applied to h = f (recall that f is proper) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1(v)  applied to ψun  t , one has ψun  t H = H0(X0, P)(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4 we have exact sequence of  perverse sheaves 0 → ⊕α ixα∗Hn−2  prim(Zα)(n−1) → P(n−1) → (Q(0)X0[n−1])∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Its left term has finite support, and so has no cohomology in degrees ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Therefore  the map H0(X0, P)(n − 1) → H0(X0, (Q(0)X0[n − 1])∗) = Hn−1(X0) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  This map equals sp, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The motivic setting and the construction of EψM  a,b  ∈ EM  a,b  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='8 so k ⊂ C is a subfield and we play with varieties  over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Changing slightly the notation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='8, for a variety Z = Zk we set  ZC := Z ⊗k C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The notation of §3 is preserved except that we equip from now on  all Hodge sheaves and Hodge structures met previously with extra upper index H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We play with motives (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' motivic sheaves) over varieties, see [A1] and [CD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  For a variety Z the category of constructible Q-motives over Z is denoted by  DM(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We use Grothendieck’s six functors formalism for DM as developed  in [CD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Recall that DM(Spec k) = DM(k) is the category of Voevodsky’s geo-  metric Q-motives over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  For a variety Z one has M(Z) = πZ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ZQ(0) where  πZ : Z → Spec k is the structure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For a motivic sheaf F on Z set Γ(Z, F) :=  πZ∗F, Γc(Z, F) := πZ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='F ∈ DM(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' we write Γ(c)(Z) := Γ(c)(Z, Q(0)Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There is  HEIGHT PAIRING AND NEARBY CYCLES  13  a Hodge realization functor DM(Z) → DH(ZC), F �→ FH, compatible with the  six functors and the Verdier duality ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For a smooth Z of dimension d one has  π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  ZQ(0) = Q(d)Z[2d], and so M(Z) = Γc(Z)(d)[2d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The formalism of unipoteny nearby cycles in the setting of motivic sheaves was  developed in §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6 of [A2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The motivic version of everything said in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1 holds  except property (v) (for the t-structure is not available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The Hodge realization  functor commutes with the nearby cycles functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notation: Notice that Hom(Q(i)[2i], Q(j)[2j]) is 0 if i ̸= j and Q for i = j,7  and so every object M ∈ M(k) which is isomorphic to a direct sum of motives  Q(i)[2i], i ∈ Z, can be written in a unique manner as ⊕i Vi(i)[2i] where Vi is a  vector space (then Vi = Hom(Q(i)[2i], M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Set τ ≤2aM := ⊕i≥−a Vi(i)[2i], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We are in the situation of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2 in the setting of k-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' As in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', I+ :=  π∗Q(0)Y ∈ DM(X0) (so I+H is the corresponding Hodge sheaf from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=') Since  Y is smooth and π is proper one has a natural self-duality I+∗ = I+(n−1)[2n−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The t-structure in DM is not available, so we define the motivic intersection  cohomology sheaf I using a motivic version of decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1):  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There is a natural orthogonal direct sum decomposition in DM(X0)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)  whose Hodge realization is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It repeats the proof in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2 (minus its last paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Namely, we first define  a natural orthogonal decomposition  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2)  Γ(Zα) = Hn−2  prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)  in DM(xα) = DM(kxα) whose Hodge realization is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='8 The construction in  loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' uses only basic six functors functoriality, so we can repeat it literally in the  motivic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then we proceed to define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1) as in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Set B := ⊕α ixα∗Hn−2  prim(Zα)[1 − n] ∈ DM(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The self-dualities of Γ(Zα) and  of I+, and the above orthogonal decompositions yield natural self-dualities  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3)  B∗ ∼  → B(n − 2)[2n − 2],  I∗ ∼  → I(n − 1)[2n − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) The adjunction χ : Q(0)X0 → π∗Q(0)Y = I+ takes values in  I ⊂ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) One has Cone(χ : Q(0)X0 → I) = B[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' (i) Follows since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = Hom(Q(0), τ [2,2n−4]Γ(Pα))  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  (ii) Since χ|U = idQ(0)U the cone Cone(χ) is supported on {xα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Now i∗  xαCone(χ) =  7This follows since M(Pn) = ⊕i∈[0,n]Q(i)[2i] and End(M(Pn)) = CHn(Pn × Pn) = Q[0,n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  8So Hn−2  prim(Zα) is a notation for a motive whose Hodge realization is the primitive cohomology  of Zα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' its definition does not involve any cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' To construct it explicitly, choose a k-point  z in Pα ∖ Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let πz : Zα → Pn−2 be the corresponding projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' this is a finite map of degree  dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Then Hn−2  prim(Zα) is the kernel of the projector d−1  α πt  zπz acting on M(Zα)(2 − n)[4 − 2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' BEILINSON  Cone(i∗  xα(χ)) equals Hn−2  prim(Zα)[2 − n] by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2) and the construction of I, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since Exti(Q(0)X0, Q(0)∗  X0(1−n)[2−2n]) = Exti(Q(0), M(X0)(1−n)[2−  2n]) = CHn−1(X0, −i) we see that Ext0 = Zn−1(X0) and Ext̸=0 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', one has  Hom(Q(0)X0, Q(0)∗  X0(1 − n)[2 − 2n]) = Zn−1(X0) = Zn−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' One has χ∗χ = ǫ where ǫ : Q(0)X0 → Q(0)∗  X0(1 − n)[2 − 2n] is the map  that corresponds to the sum of irreducible components cycle (it is enough to check  the assertion on U where it is obvious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We are in the situation of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 in the setting of k-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the functor  ψun  tf : DM(X ∖ X0) → DM(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There is a canonical morphism ι : i∗  X0 → ψun  tf v∗  of functors on DM(X) and its Verdier dual ι∗ : ψun  tf v∗ → i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Therefore we have  a motivic sheaf R := ψun  tf Q(0)X∖X0 equipped with a natural self-duality R∗  ∼  →  R(n − 1)[2n − 2] and mutually dual maps Q(0)X0  ι→ R  ι∗  → Q(0)∗  X0(1 − n)[2 − 2n]  that are isomorphisms over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Let ∂ : B → Q(0)X0 be the boundary map of the triangle from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Set  αR := ι∂ : B → R, and let βR be α∗  R combined with the self-duality identifications  for R and B, so we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1)  B  αR  −→ R  βR  −→ B(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Lemma-construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The composition βRαR is homotopic to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  In fact,  there is a canonical up to a homotopy κR such that d(κR) = βRαR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By Remark and Example in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 one has βRαR = ∂∗ι∗ι∂ = ∂∗ǫ∂ = ∂∗χ∗χ∂ =  (χ∂)∗χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notice that χ∂ is homotopic to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' choose a homotopy λ, d(λ) = χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Now  set κR := λ∗χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Independence of κR up to a homotopy from the choice of λ: if λ′ is another  homotopy as above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', d(λ) = d(λ′), then κ′  R = λ′∗χ∂ = κR + (λ′ − λ)χ∂ =  κR + d((λ − λ′)λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our κR is self-dual up to homotopy: Indeed, one has κ∗  R = (χ∂)∗λ =  κR + d(λ∗λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Below we use the notation from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' We have defined (αR, βR, κR) ∈  DM(X0)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It yields the objects ER := E(αR, βR, κR) ∈ DM(X0) and (αI, βI, κI)  := σ(αR, βR, κR) ∈ DM(X0)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' As follows from Remark in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4 and the defini-  tions, the above three objects are naturally self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' There is a homotopy equivalence θ : I  ∼  → ER such that the maps  βIθ : I → B[1], θ−1αI : B(−1)[−1] are a morphism of the triangle in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3(ii) and  its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our θ is unitary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=', θ∗ = θ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Recall that we have a natural homotopy equivalence (λ, χ) : Cone(∂ : B →  Q(0)X0)  ∼  → I (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3(ii)), and ER is the direct sum B[1] ⊕ R ⊕ B(−1)[−1] with  (αR, −κR, βR) added to the differential (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Our θ is the composition I  ∼  ←  Cone(∂)  θ′  → ER where θ′ is the next morphism: its restriction to B[1] ⊂ Cone(∂)  identifies it with the first summand in ER, and its restriction to Q(0)X0 ⊂ Cone(∂)  is (0, ι, −λ∗χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  HEIGHT PAIRING AND NEARBY CYCLES  15  One has θ∗θ = idI: we need to check that θ′∗ρθ′ = (λ, χ)∗(λ, χ) : Cone(∂) →  Cone(∂∗) where ρ : ER  ∼  → E∗  R(1 − n)[2 − 2n] is the self-duality for ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' As follows  from Remark in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='4, ρ is the matrix with the self-dualities for R and B’s on the  diagonal and the only non-zero off-diagonal entry being λ∗λ : B → B∗(1−n)[2−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The rest is an immediate calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The assertion that βIθ is the morphism of the triangle in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3(ii) means that  βIθ′ is the projection Cone(∂) → B[1] which is evident from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' The  assertion that αIθ is dual to βIθ follows from the unitarity of θ once we know that  θ is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Let us check it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  Our θ′ is a morphism Cone(B → Q(0)X0) → Cone(B → Cone(βR)[−1]) com-  patible with the projections to B, and so it is enough to check that the map  (ι, −λ∗χ) : Q(0)X0 → Cone(βR)[−1] is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Since ι is a homo-  topy equivalence on U, it is enough to check our claim after applying i∗  xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  The story of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3 uses only the six functors formalism and basic facts  from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1, so it remains literally true in the motivic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Consider the canonical  homotopy equivalence a : i∗  xαR  ∼  → Γ(Pα ∖ Zα) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By the Verdier dual  assertion to the lemma in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='3, a identifies i∗  xα(βR) with minus the residue map  r : Γ(Pα ∖ Zα) → Hn−2  prim(Zα)(−1)[1 − n] ⊂ Γ(Zα)(−1)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='2) we have a  split exact triangle Q(0) → Γ(Pα ∖ Zα)  r→ Hn−2  prim(Zα)(−1)[1 − n], so a identifies  i∗  xαCone(βR)[−1]) with Q(0) ⊂ Γ(Pα∖Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' It follows directly from the construction  of a that ai∗  xα(ι) coincides with the latter embedding, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Proof of the theorem in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  We have (αI, βI, κI) ∈ DM(X0)(2), hence  Γ(αI, βI, κI) ∈ DM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' For two Bloch cycles A, B of classes clA, clB ∈ Hom(Q(0),  Hn−2  prim(Zα)(m)) we have (cl∗  A, clB∗)Γ(αI, βI, κI) ∈ EM(Γ(I)(m − 1)[1 − n]) =  EM(Γ(I+)(m − 1)[1 − n]) = EM(M) where M := M(Y )(−m)[−1 − 2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' By the  construction the Hodge realization embedding EM(M) ֒→ EH(M) = EH(Hm(Y ))  identifies it with Eψ  A,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='6, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  □  References  [A1]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents  dans le monde motivique (I), Ast´erisque 314, SMF, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  [A2]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents  dans le monde motivique (II), Ast´erisque 315, SMF, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content='  [B]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Beilinson, Height pairing between algebraic cycles, K-theory, Arithmetic and Geom-  etry, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Manin (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' ), Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
+page_content=' 1289, Springer, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'}
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+ 
+A Verification Framework for Component 
+Based Modeling and Simulation 
+“Putting the pieces together” 
+ 
+Imran Mahmood 
+ 
+  
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+ 
+ 
+ 
+ 
+Doctoral thesis in Electronics and Computer Systems 
+Stockholm, Sweden 2013 
+ 
+
+KTH
+VETENSKAP
+OCHKONST
+6
+KTH Informations- och
+kommunikationsteknik   
+Page 
+2 
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+ 
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+ISBN 978-91-7501-628-3 
+TRITA-ICT/ECS AVH 13:01 
+ISSN 1653-6363 
+ISRN KTH/ICT/ECS/AVH-13/01-SE 
+ 
+ 
+KTH School of Information and 
+Communication Technology 
+SE-164 40 Kista 
+Sweden 
+ 
+ 
+ 
+Akademisk avhandling som med tillstand av Kungliga Tekniska högskolan 
+framlägges till offentlig granskning för avläggande av teknologie doktorsexamen 
+tisdagen den 26 feb 2013 kl 14.00 i Sal E, Forum, Isafjordsgatan 39, Kista.  
+ 
+ Imran Mahmood, February 2013. 
+ 
+Tryck: Universitetservice US AB  
+ 
+
+   
+Page 
+3 
+ 
+Acknowledgement 
+I would like to dedicate this manuscript to my loved ones, including some dignitaries, 
+my parents who recently passed away, my guardians Mr.  & Mrs. Sajid Latif who 
+raised me well to make me see this day and most important of all: my beloved wife 
+and my little daughter. Their sacrifice for being apart and for my long absence cannot 
+be compensated for anything.  
+ 
+I offer my deepest gratitude to my supervisor Professor Rassul Ayani for this 
+devotion and support. Instead of just giving me the directions he actually grabbed my 
+hand and took me to the destination like a true guide. I am honored to work under 
+his supervision.  I am thankful to Assoc. Professor Vladimir Vlassov who gave sound 
+advice and provided valuable contributions in my research. I offer my affectionate 
+tribute to the esteemed palace of knowledge, the Royal Institute of Technology, and 
+specially the school of Information and Communication Technology.  
+ 
+I am thankful for continuous support and encouragement from Dr. Farshad Moradi 
+from Swedish Defense Research agency (FOI). I am grateful for the constructive 
+critics I received from my opponent Dr. Gary Tan and the member of the evaluation 
+committee Dr. Oliver Dale. 
+ 
+I am very grateful for the Higher Education Commission of Pakistan to provide 
+entire financial support for my studies. I thank Mrs. Mumtaz Begum for her support. 
+I would like to offer special thanks to Mr. Awais Ali Sohrawardi and Dr. B. 
+Muhammad for their moral support during my study period. I thank all my 
+colleagues, friends and especially the cricket team for wonderful time.  
+ 
+Finally I thank Sweden for its hospitality, care and warm memories.  
+ 
+ 
+ 
+ 
+Imran Mahmood  
+January 2013, Stockholm 
+ 
+ 
+ 
+ 
+ 
+
+   
+Page 
+4 
+ 
+Abstract 
+ 
+The discipline of component-based modeling and simulation offers promising gains 
+including reduction in development cost, time, and system complexity. This 
+paradigm is very profitable as it promotes the use and reuse of modular components 
+and is auspicious for effective development of complex simulations. It however is 
+confronted by a series of research challenges when it comes to actually practise this 
+methodology. One of such important issue is Composability verification. In modeling 
+and simulation (M&S), composability is the capability to select and assemble 
+components in various combinations to satisfy specific user requirements. Therefore 
+to ensure the correctness of a composed model, it is verified with respect to its 
+requirements specifications.   
+There are different approaches and existing component modeling frameworks that 
+support composability. Though in our observation most of the component modeling 
+frameworks possess none or weak built-in support for the composability verification. 
+One such framework is Base Object Model (BOM) which fundamentally poses a 
+satisfactory potential for effective model composability and reuse. However it falls 
+short of required semantics, necessary modeling characteristics and built-in 
+evaluation techniques, which are essential for modeling complex system behavior and 
+reasoning about the validity of the composability at different levels. 
+In this thesis a comprehensive verification framework is proposed to contend with 
+some important issues in composability verification and a verification process is 
+suggested to verify composability of different kinds of systems models, such as 
+reactive, real-time and probabilistic systems. With an assumption that all these 
+systems are concurrent in nature in which different composed components interact 
+with each other simultaneously, the requirements for the extensive techniques for the 
+structural and behavioral analysis becomes increasingly challenging. The proposed 
+verification framework provides methods, techniques and tool support for verifying 
+composability at its different levels. These levels are defined as foundations of 
+consistent model composability. Each level is discussed in detail and an approach is 
+presented to verify composability at that level. In particular we focus on the 
+Dynamic-Semantic Composability level due to its significance in the overall 
+composability correctness and also due to the level of difficulty it poses in the 
+process. In order to verify composability at this level we investigate the application of 
+three different approaches namely (i) Petri Nets based Algebraic Analysis (ii) Colored 
+Petri Nets (CPN) based State-space Analysis and (iii) Communicating Sequential 
+Processes based Model Checking. All the three approaches attack the problem of 
+verifying dynamic-semantic composability in different ways however they all share 
+the same aim i.e., to confirm the correctness of a composed model with respect to its 
+requirement specifications. Beside the operative integration of these approaches in 
+our framework, we also contributed in the improvement of each approach for 
+effective applicability in the composability verification. Such as applying algorithms 
+for automating Petri Net algebraic computations, introducing a state-space reduction 
+technique in CPN based state-space analysis, and introducing function libraries to 
+perform verification tasks and help the modeler with ease of use during the 
+composability verification. We also provide detailed examples of using each approach 
+with different models to explain the verification process and their functionality. 
+Lastly we provide a comparison of these approaches and suggest guidelines for 
+
+   
+Page 
+5 
+ 
+choosing the right one based on the nature of the model and the available 
+information. With a right choice of an approach and following the guidelines of our 
+component-based M&S life-cycle a modeler can easily construct BOM based 
+composed models and can verify them with respect to the requirement specifications. 
+ 
+ 
+ 
+Keywords: 
+Modeling 
+and 
+Simulation, 
+Component-based 
+development, 
+Composability, 
+Semantic 
+Composability, Dynamic-Semantic Composability, Verification, Correctness, Petri Nets Analysis, 
+Algebraic Techniques, Colored Petri Nets, State-space Analysis, Communicating Sequential 
+Processes,  Model Checking. 
+ 
+ 
+
+   
+Page 
+6 
+ 
+Table of  Contents 
+ 
+Acknowledgement ..................................................................................................................... 3 
+Abstract...................................................................................................................................... 4 
+Table of Contents ...................................................................................................................... 6 
+List of Figures ........................................................................................................................... 9 
+List of Tables ............................................................................................................................ 11 
+List of Acronyms....................................................................................................................... 13 
+Chapter 1 ................................................................................................................................... 16 
+Introduction .............................................................................................................................. 16 
+1.1 
+Background and the opening perspective ......................................................................................... 16 
+1.1.1 
+Component based Software Engineering .............................................................................. 17 
+1.1.2 
+Component based Modeling & Simulation ........................................................................... 18 
+1.1.3 
+Modeling and Analysis using Petri Nets ................................................................................ 19 
+1.1.4 
+Modeling and Analysis using Process Algebra ..................................................................... 19 
+1.1.5 
+Model Verification ..................................................................................................................... 19 
+1.2 
+Summary of the opening perspective ................................................................................................ 20 
+1.3 
+Preliminaries ........................................................................................................................................... 20 
+1.3.1 
+Definition 1: Set of Components ............................................................................................ 20 
+1.3.2 
+Definition 2: Requirement Specification ............................................................................... 20 
+1.3.3 
+Definition 3: Composition & Composability Pattern ......................................................... 21 
+1.3.4 
+Definition 4: Satisfiability Operator........................................................................................ 21 
+1.4 
+Problem Statement ............................................................................................................................... 21 
+1.5 
+Approach ................................................................................................................................................ 22 
+1.5.1 
+Problem Domain ....................................................................................................................... 22 
+1.5.2 
+Solution Domain ........................................................................................................................ 22 
+1.5.3 
+Solution Statement ..................................................................................................................... 24 
+1.6 
+Scope of the Thesis ............................................................................................................................... 24 
+1.6.1 
+Correctness.................................................................................................................................. 24 
+1.6.2 
+Validation .................................................................................................................................... 24 
+1.6.3 
+Emergence................................................................................................................................... 25 
+1.6.4 
+Generalization ............................................................................................................................ 25 
+1.7 
+Summary of the Contributions ........................................................................................................... 25 
+1.8 
+Structure of the Thesis ......................................................................................................................... 26 
+Chapter 2 .................................................................................................................................. 28 
+Component Based Modeling and Simulation .......................................................................... 28 
+2.1 
+Composability in M&S ......................................................................................................................... 28 
+2.2 
+A Brief History of Composability and related work ....................................................................... 29 
+2.2.1 
+Initiation ...................................................................................................................................... 29 
+2.2.2 
+Theoretical evolution ................................................................................................................ 29 
+2.2.3 
+Standards & Frameworks ......................................................................................................... 29 
+2.2.4 
+Technological Advances ........................................................................................................... 29 
+2.2.5 
+Composability verification and Validation ............................................................................ 30 
+2.3 
+Theory of Composability ..................................................................................................................... 30 
+2.4 
+Concepts related to Composability .................................................................................................... 32 
+2.4.1 
+Composability vs. Reusability .................................................................................................. 32 
+2.4.2 
+Composability vs. Interoperability .......................................................................................... 33 
+2.5 
+Composability Levels ........................................................................................................................... 34 
+2.5.1 
+Syntactic level: ............................................................................................................................ 35 
+2.5.2 
+Static-Semantic level: ................................................................................................................. 35 
+2.5.3 
+Dynamic-Semantic level: .......................................................................................................... 35 
+2.5.4 
+Pragmatic level: ........................................................................................................................... 36 
+2.6 
+Composability frameworks.................................................................................................................. 36 
+2.6.1 
+Discrete Event System Specification (DEVS) ...................................................................... 37 
+2.7 
+Base Object Model (BOM) framework ............................................................................................. 38 
+2.7.1 
+Structure of BOM ...................................................................................................................... 39 
+2.7.2 
+BOM Assembly .......................................................................................................................... 41 
+
+   
+Page 
+7 
+ 
+2.7.3 
+Model Mapping and Object Model Definition ..................................................................... 42 
+2.7.4 
+Formal specification for the Compositon of BOM ............................................................. 42 
+2.7.5 
+Summary ...................................................................................................................................... 45 
+Chapter 3 .................................................................................................................................. 46 
+Executable Modeling Formalisms ........................................................................................... 46 
+3.1 
+Petri Nets ................................................................................................................................................ 46 
+3.1.1 
+PN Definitions and Concept ................................................................................................... 47 
+3.1.2 
+Petri net graph ............................................................................................................................ 47 
+3.1.3 
+Properties of PN ........................................................................................................................ 49 
+3.1.4 
+PN Analysis ................................................................................................................................. 51 
+3.1.5 
+PN Classes ................................................................................................................................... 58 
+3.2 
+Communicating Sequential Processes ............................................................................................... 62 
+3.2.1 
+Basic Concepts and Definitions .............................................................................................. 62 
+3.2.2 
+CSP Analysis Techniques ......................................................................................................... 64 
+3.2.3 
+Temporal Logics ........................................................................................................................ 65 
+3.2.4 
+Time CSP .................................................................................................................................... 66 
+3.2.5 
+Probabilistic Systems ................................................................................................................. 66 
+3.2.6 
+CSP Implementation Tools ...................................................................................................... 67 
+3.2.7 
+Process Analysis Toolkit (PAT) .............................................................................................. 67 
+3.3 
+Summary ................................................................................................................................................. 68 
+Chapter 4 .................................................................................................................................. 69 
+Verification and Analysis .......................................................................................................... 69 
+4.1 
+Some Basic Concepts in Modeling and Simulation ........................................................................ 70 
+4.1.1 
+Verification and Validation in a Modeling Process .............................................................. 71 
+4.2 
+The Principles of Top-Down Refinement ....................................................................................... 73 
+4.3 
+Verification techniques ........................................................................................................................ 74 
+4.3.1 
+Informal Techniques ................................................................................................................. 74 
+4.3.2 
+Static Analysis: ............................................................................................................................ 75 
+4.3.3 
+Dynamic Analysis:...................................................................................................................... 76 
+4.3.4 
+Formal Analysis .......................................................................................................................... 76 
+4.4 
+Summary ................................................................................................................................................. 77 
+Chapter 5 .................................................................................................................................. 79 
+Proposed Methodology and the Verification Framework ........................................................ 79 
+5.1 
+Component-based Modeling & Simulation life-cycle ..................................................................... 79 
+5.2 
+Inception ................................................................................................................................................. 80 
+5.3 
+Modeling ................................................................................................................................................. 81 
+5.4 
+Execution ............................................................................................................................................... 82 
+5.5 
+Analysis ................................................................................................................................................... 82 
+5.6 
+Composability Verification Framework ............................................................................................ 83 
+5.6.1 
+Discovery Matching and Composition (DMC) .................................................................... 83 
+5.6.2 
+Structural and Behavioral Evaluation ..................................................................................... 84 
+5.6.3 
+Static Analysis ............................................................................................................................. 84 
+5.6.4 
+Dynamic Analysis....................................................................................................................... 90 
+5.7 
+PN Algebraic Technique...................................................................................................................... 93 
+5.7.1 
+BOM to PNML Transformation ............................................................................................ 93 
+5.7.2 
+PN Algebraic computations ..................................................................................................... 93 
+5.7.3 
+Property Verification Method.................................................................................................. 95 
+5.8 
+CPN based State-Space Analysis Technique .................................................................................... 96 
+5.8.1 
+BOM Extension ......................................................................................................................... 97 
+5.8.2 
+E-BOM to CPN Component Transformation ..................................................................... 99 
+5.8.3 
+Verification of the composed CPN model ........................................................................ 105 
+5.9 
+CSP based Model Checking Technique ......................................................................................... 110 
+5.9.1 
+BOM Extension ...................................................................................................................... 110 
+5.9.2 
+E-BOM to CSP# Transformation ....................................................................................... 112 
+5.9.3 
+Verification of the composed CPN model ........................................................................ 113 
+5.10 
+Summary ........................................................................................................................................ 115 
+Chapter 6 ................................................................................................................................ 116 
+Composability Verification Process ........................................................................................ 116 
+6.1 
+Composability Verification Process ................................................................................................ 116 
+6.1.1 
+Formulation of Simuland, Requirements and Conceptual Model.................................. 126 
+
+   
+Page 
+8 
+ 
+6.1.2 
+Syntactic Matching Process ................................................................................................... 127 
+6.1.3 
+Static-Semantic Matching Process ....................................................................................... 127 
+6.1.4 
+State-machine Matching Process .......................................................................................... 127 
+6.1.5 
+Approach Selection for Dynamic-Semantic Composability Verification ..................... 127 
+6.1.6 
+PN Algebraic Technique ....................................................................................................... 128 
+6.1.7 
+State-Space Analysis Technique ........................................................................................... 128 
+6.1.8 
+Model Checking ...................................................................................................................... 128 
+6.2 
+Summary .............................................................................................................................................. 129 
+Chapter 7 ................................................................................................................................ 130 
+Fairness verification using PN Algebraic Techniques ........................................................... 130 
+7.1 
+Fairness ................................................................................................................................................ 130 
+7.2 
+Fairness Verification .......................................................................................................................... 131 
+7.3 
+Manufacturing system ....................................................................................................................... 132 
+7.3.1 
+Scenario I .................................................................................................................................. 132 
+7.3.2 
+Scenario II ................................................................................................................................ 138 
+7.4 
+Summary .............................................................................................................................................. 141 
+Chapter 8 ................................................................................................................................ 142 
+Model Verification using State-space Analysis techniques .................................................... 142 
+8.1 
+Combat Modeling .............................................................................................................................. 142 
+8.1.1 
+Situated Environment ............................................................................................................ 142 
+8.1.2 
+Moving ...................................................................................................................................... 142 
+8.1.3 
+Looking (or sensing) ............................................................................................................... 143 
+8.1.4 
+Shooting .................................................................................................................................... 143 
+8.1.5 
+Communication: ...................................................................................................................... 143 
+8.2 
+Field Artillery ...................................................................................................................................... 144 
+8.2.1 
+Simuland ................................................................................................................................... 145 
+8.2.2 
+Field Artillery Model .............................................................................................................. 145 
+8.2.3 
+Requirement Specification..................................................................................................... 148 
+8.3 
+Verification of the FA model using CPN State-Space Analysis ................................................ 149 
+8.3.1 
+Static and Dynamic Analysis ................................................................................................. 149 
+8.3.2 
+BOM to E-BOM extension .................................................................................................. 149 
+8.3.3 
+E-BOM to CPN Transformation ........................................................................................ 153 
+8.3.4 
+Composition of CPN Components ..................................................................................... 158 
+8.3.5 
+State space Analysis ................................................................................................................ 159 
+8.4 
+State Space Reduction ....................................................................................................................... 162 
+8.5 
+Summary .............................................................................................................................................. 164 
+Chapter 9 ................................................................................................................................ 165 
+Model Verification using CSP based Model Checking Technique ........................................ 165 
+9.1 
+Field Artillery Scenario ...................................................................................................................... 165 
+9.2 
+Requirement Specification ................................................................................................................ 168 
+9.3 
+Verification using Model Checking ................................................................................................ 169 
+9.3.1 
+Static and Dynamic Analysis ................................................................................................. 169 
+9.3.2 
+BOM to E-BOM extension .................................................................................................. 169 
+9.3.3 
+E-BOM to CSP# Transformation ....................................................................................... 171 
+9.3.4 
+Model Checking of Field Artillery Model ........................................................................... 174 
+9.4 
+Summary .............................................................................................................................................. 176 
+Chapter 10 ............................................................................................................................... 177 
+Summary and Conclusion....................................................................................................... 177 
+10.1 
+Guidelines for choosing an approach ....................................................................................... 181 
+10.1.1 
+PN Algebraic Technique .................................................................................................. 181 
+10.1.2 
+CPN based State-Space analysis Technique ................................................................. 182 
+10.1.3 
+CSP based Model Checking Technique ........................................................................ 182 
+10.2 
+Thesis Contributions.................................................................................................................... 183 
+10.3 
+Conclusions ................................................................................................................................... 185 
+10.4 
+Future Directions ......................................................................................................................... 186 
+References .............................................................................................................................. 187 
+ 
+ 
+ 
+
+   
+Page 
+9 
+ 
+List of  Figures 
+Figure 1: A model as computable function (acquired from [34]) .................................................................. 30 
+Figure 2: Sequence of executions (acquired from [50]) .................................................................................. 31 
+Figure 3: Composed Model  (acquired from [50]) ........................................................................................... 31 
+Figure 4: Generic vs. Specific component design ............................................................................................ 32 
+Figure 5: Black Box, Glass Box, White Box ..................................................................................................... 33 
+Figure 6 Syntactic vs. Semantic Composability (acquired from [38]) ........................................................... 34 
+Figure 7: Ping-Pong DEVS [Wikipedia] ............................................................................................................ 38 
+Figure 8: BOM structure ...................................................................................................................................... 39 
+Figure 9: BOM Assembly ..................................................................................................................................... 41 
+Figure 10: (a) PingPong BOM in BOM Works ................................................................................................ 41 
+Figure 11: Composed BOM ................................................................................................................................ 44 
+Figure 12: Transition firing sequence  (acquired from [68]) .......................................................................... 49 
+Figure 13: Petri Net Analysis Techniques ......................................................................................................... 51 
+Figure 14: Producer Consumer Example .......................................................................................................... 53 
+Figure 15: M0 to M3 throguh firing sequece σ = t2, t1, t2 ............................................................................. 54 
+Figure 16: Seasons in a year (acquired from [68]) ............................................................................................ 54 
+Figure 17: (a) PN Model (b) Reachability Graph (acquired from [68]) ........................................................ 56 
+Figure 18: Producer Consumer PN Model and its Coverability Graph ...................................................... 56 
+Figure 19: A CPN Model ..................................................................................................................................... 59 
+Figure 20: Hierarchical Colored Petri Net ........................................................................................................ 60 
+Figure 21: Modeling Process (acquired from [108]) ........................................................................................ 71 
+Figure 22: Modeling Process (acquired from [29]) .......................................................................................... 72 
+Figure 23: Simulation study life-cycle (acquired from [28]) ........................................................................... 73 
+Figure 24: Verification Techniques .................................................................................................................... 74 
+Figure 25: CBM&S life-cycle ............................................................................................................................... 79 
+Figure 26: Simuland using UML Diagrams ....................................................................................................... 80 
+Figure 27: Implemenation and Simulation ........................................................................................................ 83 
+Figure 28: Discovery, Matching, Composition (DMC) .................................................................................. 84 
+Figure 29: Syntactic Matching ............................................................................................................................. 85 
+Figure 30: Some of the sub-classes of Data Type ontololgy .......................................................................... 88 
+Figure 31: Semantic Matching Technique ......................................................................................................... 88 
+Figure 32: Static-Semantic Matching .................................................................................................................. 90 
+Figure 33: SCXML format ................................................................................................................................... 91 
+Figure 34: State-machine Matching Process ..................................................................................................... 92 
+Figure 35: BOM to PN transformation ............................................................................................................. 93 
+Figure 36: PN Algebraic Technique ................................................................................................................... 96 
+Figure 37: Buffer Extended finite state-machine [120] ................................................................................... 97 
+Figure 38: BOM and E-BOM comparison ....................................................................................................... 99 
+Figure 39: CPN-CM represention of Queue component ............................................................................ 103 
+Figure 40: CPN State-space analysis ............................................................................................................... 106 
+Figure 41: State-space Analysis Technique .................................................................................................... 108 
+Figure 42: CSP based Model Checking Technique ....................................................................................... 115 
+Figure 43: Formulation of Simuland ............................................................................................................... 117 
+Figure 44: Syntactic Matching Process............................................................................................................ 118 
+Figure 45: Static-Semantic Matching Process ................................................................................................ 119 
+Figure 46: State-machine Matching Process .................................................................................................. 120 
+Figure 47: Approach Selection | PN Algebraic Technique ........................................................................ 121 
+Figure 48: PN Algebraic Technique (continued) .......................................................................................... 122 
+Figure 49: Implementation ................................................................................................................................ 122 
+Figure 50: State-Space Analysis Technique .................................................................................................... 123 
+Figure 51: State-Space Analysis Technique (continued) .............................................................................. 124 
+Figure 52: Model Checking ............................................................................................................................... 125 
+Figure 53: Model Checking (continued) ......................................................................................................... 126 
+Figure 54: Manufacturing System (acquired from [124]) ............................................................................. 132 
+Figure 55: Manufacturing System BOM based Composed Model ............................................................ 134 
+Figure 56: State-machine matching of manufacturing system .................................................................... 136 
+Figure 57: PN model of the manufacturing System ..................................................................................... 136 
+
+   
+Page 
+10 
+ 
+Figure 58: Modified manufacturing system composed BOM .................................................................... 139 
+Figure 59: Modified PN model of the manufacturing System ................................................................... 140 
+Figure 60: Activities of Combat Modeling ..................................................................................................... 144 
+Figure 61: Elements of Field Artliiery & Indirect Fire ................................................................................ 145 
+Figure 62: Field Artillery Composed BOM.................................................................................................... 148 
+Figure 63: State-machine Matching of Field Artillery Model...................................................................... 149 
+Figure 64: Observer CPN Component ........................................................................................................... 154 
+Figure 65: Field CPN Component .................................................................................................................. 155 
+Figure 66: BHQ CPN Component ................................................................................................................. 156 
+Figure 67: Battery CPN Component............................................................................................................... 157 
+Figure 68: FDC CPN Component .................................................................................................................. 158 
+Figure 69: Field Artillery CPN Composed Model ........................................................................................ 159 
+Figure 70: State space of Field Artillery CPN Model ................................................................................... 160 
+Figure 71: Reduced State-Space graph of Field Artillery Model ................................................................ 163 
+Figure 72: Field Artillery Composed Model .................................................................................................. 168 
+Figure 73: State-machine Matching of Field Artillery Model...................................................................... 169 
+Figure 74: Global code Block of Field Artillery Model ............................................................................... 171 
+Figure 75: CSP representation of Observer Component ............................................................................ 172 
+Figure 76: CSP representation of BHQ Component ................................................................................... 172 
+Figure 77: CSP representation of Battery Component ................................................................................ 173 
+Figure 78: CSP representation of Field Component .................................................................................... 173 
+Figure 79: Field Artillery Composed Model .................................................................................................. 174 
+Figure 80: Field Artillery Verificataion Assertions........................................................................................ 174 
+Figure 81: Verification Result of assertion 1 .................................................................................................. 175 
+Figure 82: Verification result of assertion 2 ................................................................................................... 175 
+Figure 83: Field Artillery Verificataion Assertions with TOT .................................................................... 175 
+Figure 84: Verification result of assertion 3 ................................................................................................... 176 
+ 
+ 
+ 
+
+   
+Page 
+11 
+ 
+List of  Tables 
+Table 1: Entity A .................................................................................................................................................... 43 
+Table 2: Entity B .................................................................................................................................................... 44 
+Table 3: Composed BOM .................................................................................................................................... 44 
+Table 4: Incidence Martic A ................................................................................................................................ 53 
+Table 5: State equation .......................................................................................................................................... 53 
+Table 6: Informal Verification Techniques ....................................................................................................... 75 
+Table 7: Static Analysis Techniques ................................................................................................................... 75 
+Table 8: Dynamic Analysis Techniques ............................................................................................................. 76 
+Table 9: Formal Analysis Techniques ................................................................................................................ 77 
+Table 10: Mandatory constraints in composability verification..................................................................... 81 
+Table 11: Semantic Matching Algorithm ........................................................................................................... 89 
+Table 12: State-machine Matching algorithm ................................................................................................... 91 
+Table 13: Incidence Matrix Calculation ............................................................................................................. 94 
+Table 14: Place-Invariants .................................................................................................................................... 95 
+Table 15: Transformation Rules....................................................................................................................... 102 
+Table 16: Compositional State-space generation algorithm ........................................................................ 109 
+Table 17: Time functions in E-BOM .............................................................................................................. 111 
+Table 18: Probability Distribution Functions ................................................................................................ 111 
+Table 19: E-BOM to CSP# transformation rules......................................................................................... 113 
+Table 20: Some examples of PAT Assertions ............................................................................................... 114 
+Table 21: Formal definition of Machine1 Base-BOM ................................................................................. 133 
+Table 22: Formal definition of Machine2 Base-BOM ................................................................................. 133 
+Table 23: Formal definition of Robot Base-BOM........................................................................................ 134 
+Table 24: Formal definition of Manufacturing System composed BOM ................................................. 134 
+Table 25: Syntactic Matching ............................................................................................................................ 135 
+Table 26: Static-Semantic Matching ................................................................................................................ 135 
+Table 27: Initial Marking and Incidence Matrix (Scenaro I) ....................................................................... 137 
+Table 28: P-Invariants and T-Invariants (Scenaro I) .................................................................................... 137 
+Table 29: B-Fairness Verification .................................................................................................................... 138 
+Table 30: Formal definition of Controller Base-BOM ................................................................................ 139 
+Table 31: Manufacturing System composed BOM....................................................................................... 139 
+Table 32: Initial Marking and Incidence Matrix (Scenaro II)...................................................................... 140 
+Table 33: P-Invariants and T-Invariants (Scenaro II) .................................................................................. 140 
+Table 34: Observer Basic-BOM ....................................................................................................................... 146 
+Table 35: Field Basic-BOM............................................................................................................................... 146 
+Table 36: BHQ Basic-BOM .............................................................................................................................. 147 
+Table 37: FDC Basic-BOM .............................................................................................................................. 147 
+Table 38: Battery (1,2,3) Basic-BOM .............................................................................................................. 147 
+Table 39: Field Artillery Composed BOM ..................................................................................................... 147 
+Table 40: Observer E-BOM ............................................................................................................................. 150 
+Table 41: Field E-BOM ..................................................................................................................................... 151 
+Table 42: BHQ E-BOM .................................................................................................................................... 152 
+Table 43: FDC E-BOM ..................................................................................................................................... 152 
+Table 44: Battery E-BOM ................................................................................................................................. 153 
+Table 45: Reduction Statisitics .......................................................................................................................... 163 
+Table 46: Observer Basic-BOM ....................................................................................................................... 166 
+Table 47: Field Basic-BOM............................................................................................................................... 166 
+Table 48: BHQ Basic-BOM .............................................................................................................................. 167 
+Table 49: Battery (1,2,3) Basic-BOM .............................................................................................................. 167 
+Table 50: Field Artillery Composed BOM ..................................................................................................... 167 
+Table 51: Observer E-BOM ............................................................................................................................. 169 
+Table 52: Field E-BOM ..................................................................................................................................... 170 
+Table 53: BHQ E-BOM .................................................................................................................................... 170 
+Table 54: BHQ E-BOM .................................................................................................................................... 171 
+Table 55: Kinds of properties that can be verified ....................................................................................... 178 
+Table 56: Type of the models that can be verified ....................................................................................... 179 
+Table 57: Scalability ............................................................................................................................................ 179 
+
+   
+Page 
+12 
+ 
+Table 58: Infinite Model Verification .............................................................................................................. 179 
+Table 59: Usability .............................................................................................................................................. 180 
+Table 60: Automation ........................................................................................................................................ 180 
+ 
+ 
+
+   
+Page 
+13 
+ 
+List of  Acronyms 
+ABV 
+Assertion-based Verification 
+Ac 
+Communicating Arcs 
+ACP 
+Algebra of Communicating Processes 
+ALSP 
+Aggregate Level Simulation Protocol 
+AOI 
+Area-of-Interest 
+AP 
+Atomic propositions 
+API 
+Application programming interface 
+ARC 
+Adelaide Refinement Checker 
+ASV 
+State-variable arc 
+AT 
+Transiting arc 
+BB 
+Basic BOM 
+BDD 
+Binary Decision Diagram 
+BHQ 
+Battalion Headquarters 
+BHQSM 
+Battalion Headquarters State-machine 
+BID 
+Battery ID 
+BL 
+Behavioral Layer 
+BOM 
+Base Object Model 
+CB 
+Composed BOM 
+CBM&S 
+Component Based Modeling and Simulation 
+CBSE 
+Component Based Software Engineering 
+CBT 
+Composable Behavioral Technologies 
+CCA 
+Common Component Architecture 
+CCP 
+Color set of communicating port 
+CCS 
+Milner's Calculus of Communicating Systems 
+CL 
+Communication Layer 
+CM 
+Conceptual Model 
+CODES 
+Composable Discrete-Event scalable Simulation 
+COST 
+Component Oriented Simulation Toolkit 
+CP 
+Communicating Port 
+CPN 
+Colored Petri Nets 
+CPN-CM 
+Colored Petri Nets Component Model 
+CPN-ML 
+ML scripting language for Colored Petri Nets 
+CSP 
+Hoare's Communicating Sequential Processes 
+CSV 
+Color set of State variable 
+CTL 
+Computation Tree Logic 
+DEDS 
+Discrete Event Dynamic Systems 
+DES 
+Discrete Event Systems 
+DEVS 
+Discrete Event System Specification 
+DIS 
+Distributed Interactive Simulation   
+DMC 
+Discovery, Matching & Composition 
+DOT 
+DOT file format 
+EC 
+Event Controller 
+EFSM 
+Extended Finite State-machine 
+EIC 
+DEVS input port couplings 
+EOC 
+DEVS output port couplings 
+EXPR 
+Expression 
+FA 
+Field Artillery 
+FD 
+Field Data 
+FDC 
+Fire Direction Center 
+FSM 
+Finite State-Machine 
+HLA 
+High Level Architecture 
+HPC 
+High Performance Computing 
+IC 
+DEVS Internal Coupling 
+IDE 
+Integrated Development Environment 
+
+   
+Page 
+14 
+ 
+INT 
+Integer 
+ISV 
+Initialization function of State-variable 
+JCSP 
+Java based Communicating Sequential Process 
+JSIMS 
+Joint Simulation System 
+JUNG 
+Java Universal Network Graph library 
+LCIM 
+Levels of Conceptual Interoperability 
+LTL 
+Linear Temporal Logic 
+LVC 
+Live, virtual, or constructive Simulation 
+MBSC 
+Model based simulation composition 
+MCT 
+Model Coupling Toolkit 
+MDF 
+Matrix Definitional Form 
+MDP 
+Markov Decision Processes 
+MOCCA 
+Component based Grid Environment 
+MPD 
+Markov decision processes 
+MUSCLE 
+A Multi-scale Coupling Library and Environment 
+NET 
+Network 
+OMT 
+High Level Architecture Object Model Template 
+OSA 
+Open Simulation Architecture 
+OWL 
+Web Ontology Language 
+PAT 
+Process Analysis Toolkit 
+PIPE 
+Platform Independent Petri Net Editor 
+PLTL 
+Probabilistic Linear Temporal Logic 
+PN 
+Petri Net 
+PNML 
+Petri Net Markup Language 
+POI 
+BOM Pattern Of Interplay 
+RS 
+Requirement Specifications 
+SAT 
+Boolean Satisfiability 
+SCT 
+Semantic Composability Theory 
+SCXML 
+State Chart extensible markup language 
+SE 
+Software Engineering 
+SIMNET 
+Simulation Networking 
+SISO 
+Simulation Interoperability Standards Organization 
+SL 
+Structural Layer 
+SM 
+Syntactic Matching 
+SML 
+Scripting language 
+SMM 
+State-Machine Matching 
+SSM 
+Static-Semantic Matching 
+SV 
+State Variable 
+SVIN 
+Input State Variable 
+SVOUT 
+Output State Variable 
+TCSP 
+Timed Communicating Sequential Processes 
+TENA 
+Test and Training Enabling Architecture 
+TOT 
+Time On Target 
+UML 
+Unified Modeling Language 
+VCP 
+Communication Port Variable 
+V&V 
+Verification and Validation 
+VVA 
+Verification, Validation and Accreditation 
+VVT 
+Verification, Validation and Testing 
+XML 
+Extensible Marking Language 
+XMSF 
+Extensible Modeling and Simulation Framework 
+XT 
+Firing Vector 
+ 
+ 
+ 
+ 
+ 
+
+   
+Page 
+15 
+ 
+Part I 
+Episteme 
+ 
+ 
+ 
+ 
+ 
+ 
+Epistêmê in Greek means “to know”. It is the theoretical knowledge; a principled system of 
+understanding; fundamental body of ideas and collective presuppositions that determine the 
+knowledge which is intellectually certain at any particular period of time; Pure-Science; episteme deals 
+with “what” and “why” of the subject. 
+ 
+ 
+Part-I covers the epistemology of the research under discussion where the theory, 
+concepts, principles, paradigms, philosophy and rationale of the problem domain and 
+the solution domain are sketched. In essence Part-I contains theoretical knowledge 
+and the background information required to understand the problem and proposed 
+solution discussed in the second part.  
+ 
+ 
+ 
+ 
+ 
+“If you can't explain it simply, you don't understand it well 
+enough”.   
+- Albert Einstein 
+ 
+ 
+
+   
+Page 
+16 
+ 
+Chapter 1 
+Introduction 
+ 
+This chapter provides the opening statement and general information about the research presented in 
+this thesis. It outlines background, history, the formal definition and the basic philosophy of the 
+problem under question and covers the motivation, goals and scope of the research and the 
+contributions of the thesis. In the end, a section on the thesis organization is rendered. 
+ 
+1.1 
+Background and the opening perspective  
+Over the last fifty years, there has been a revolutionary development influencing 
+almost all of the sciences. This progress is mainly instigated by the astonishing 
+growth of the use of the digital computers and the subsequent rise of the age of 
+computer simulations [1]. It is the emergence and widespread availability of 
+computing power and resources that have made possible the new dimension of 
+experimentation with complex models and their simulations [2]. Computer 
+simulations are now widely used in various scientific disciplines and application 
+domains. They are used for studying complex systems and gaining insight into the 
+operation of an existing system without disturbing the actual system. Furthermore 
+they are used for testing new concepts of the systems before implementation, 
+visualizing and predicting behavior of a future system. Besides, they are used for 
+analyzing and solving problems, drawing conclusions and aiding the process of 
+crucial decision making [3]. Therefore computer simulation is regarded as third 
+branch of science [4] and stands alongside of the first two branches namely theory and 
+experimentation. 
+Modeling and Simulation (M&S) is a discipline with its own body of knowledge, 
+theory, and research methodology [4]. The goals of M&S are aligned with the 
+systems theory, and include modeling & analysis, design & synthesis, control, 
+performance evaluation and optimization of a real system. The M&S community has 
+demonstrated a longstanding focus on providing support for these goals. With the 
+advent of the net-centric era of methods and technologies in designing complex 
+simulation systems, the focus of M&S industry has been driven by the most 
+recognized potential benefits of reduced development cost, time  and system 
+complexity [5]. This is because M&S development process is costly, time consuming, 
+resource intensive. Models can be large, complex and require a great deal of time, 
+resource and domain specific expertise to develop. Beside this, an enormous effort is 
+required to evaluate that the model is correct and meets its requirements. Therefore 
+M&S community has taken a deep interest in the quality design principles and their 
+underlying supportive theories to alleviate these challenges. It has been realized that 
+constructing a model from scratch each time it is needed is inefficient. Instead, the 
+practice of model reuse has been increasingly appreciated and is inspired from the 
+vision of software reuse, which was originally introduced in 1968 [6]. Apparently this 
+approach looks very appealing however it poses many obstacles in implementing, 
+such as lack of flexibility and adaptability in design, difficulty of integration, 
+mismatched interface, incomplete specification etc. [7]. These obstacles are 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+17 
+ 
+considered elusive research challenges and are now the primary research interests of 
+the software engineering and M&S communities [8]  
+1.1.1 Component based Software Engineering 
+Component-based software engineering (CBSE) has been identified as a key enabler 
+in the construction of complex systems by combining software components that are 
+already developed and prepared for integration [8].  
+Software Component 
+A software component is defined as a unit of composition which is independently developed 
+and can be combined with other components to build larger units. It must have clearly 
+specified interfaces to communicate with its environment while the implementation must be 
+encapsulated in the component and is not directly reachable from the environment [9], and 
+therefore can be easily used by the third party without having to know implementation details 
+[8], [10].  
+ 
+Building software from components contributes to a major paradigm shift in 
+software engineering. The basic philosophy behind the idea of component-based 
+development is to carry out the software development process by (quickly) 
+producing software applications through assembling prefabricated software 
+components and to archive these interoperable software components in form of an 
+increasingly large repository for further reuse [11]. CBSE promotes the principle of 
+modularity. That essentially helps to master the complexity of the reality by 
+decomposing it into parts [12] and enables the designer to use and reuse appropriate 
+parts for different purposes. These parts are the sub-systems built in a component-
+based fashion. These subsystem components may have been separately developed by 
+different teams. They may also have been developed for different purposes unrelated 
+to the current context of the usage. CBSE has many advantages, such as effective 
+management of complexity, logical separation, reduced time and cost, increased 
+productivity, improved quality, a greater degree of consistency, increased 
+dependability, and a wider range of usability. In addition, the growing connectivity of 
+real world problems is reflected in the requirement to compose cross domain 
+solutions [13], and therefore support knowledge sharing to a wider user community. 
+CBSE is therefore a discipline of software engineering that deals with the 
+composition of components to construct software systems which are capable of 
+performing functions according to the user’s requirements [14].  
+In CBSE, component integration and component composition are two distinguished 
+terms. Component integration is merely the task of connecting components together 
+whereas composition also includes reasoning about the semantic behavior of the 
+resulting assembly [14]. With the advent of component technology the integration 
+problems are becoming a difficulty of the past. Instead more crucial problems of 
+predicting the emergent behavior of assemblies and the problem of reasoning about 
+how well components will play together are now in debate. Component composition 
+supports this type of reasoning and provides a foundation for fundamental reasoning 
+to justifying validity of the resulting assemblies, their run-time compatibility and 
+emergent behavior. The main reason for the difference between integration and 
+composition is due to the fact that component interfaces do not provide enough 
+information to determine how well the composed components will play together 
+[14]. An interface can only help to determine if the component can be connected to 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+18 
+ 
+some other component but cannot supports reasoning about emergent properties of 
+the assemblies [14], [15]. Component composition promises such rationale; however 
+is still a subject of open research. 
+1.1.2 Component based Modeling & Simulation 
+Inspired by the discipline of component based software engineering, M&S 
+community has also started to develop simulation models by reusing previously 
+developed and validated “simulation components”, and composing them in a new 
+simulation model according to the desired user objectives [16], [17], [18], [19], [20]. 
+The basic and effective strategy for tackling any large and complex simulation 
+problem is “divide and conquer.” One major idea in component-based simulation 
+development is to create models that are themselves self-contained and 
+independently deployable. Thus different simulationist will be able to work on 
+different components independently, without needing much communication among 
+each other, (and particularly without the need to share the classified domain 
+knowledge) and yet the components will work together seamlessly. In addition, 
+during the maintenance phase, it is possible to modify some of the components 
+without affecting all of the others [21].  
+In simulation community the research on component based development falls under 
+the rubric of composability [22], where simulation models are considered to be the 
+building blocks and are referred as “model-components”.  
+Model Component1 
+A model component is an independent element of a simulation model that conforms to certain 
+component standard, has well-defined functionalities (inputs/outputs) and behaviors, 
+presented through its interface describing its communication with other components and a 
+formalized description of its internal behavior. A model component is not a stand-alone 
+component, but can be independently deployed, and it is subject to third-party composition 
+with or without modification [19]. 
+ 
+In component based development, some basic reusable model components are 
+composed together to create complex and sophisticated simulations, without 
+building them from scratch. The model components can be composed if their inputs 
+and outputs physically match each other however it is difficult to say whether this 
+combination is meaningful. Besides it cannot be said for sure if it will perform 
+according to the desired requirements unless the correctness of the composability is 
+checked.  
+Composability is the property of the models, as it essentially contends with the 
+alignment of issues on the modeling level [13], where it is viewed as creation of 
+complex models by selection and integration of basic reusable model-components. A 
+set of components can be integrated if their inputs and outputs are compatible, but 
+in order to guarantee that their combination is valid in the required executable 
+scenarios, we study the degree of composability.   
+With a slightly greater number of components, which are somewhat complex in 
+nature, the composability becomes an increasingly challenging problem. In the 
+                                                 
+1The term Model component should be differentiated from the term Component Model, which in 
+text refers to the underlying technology being used by the component based software engineering 
+platforms such as CORBA, EJB etc. 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+19 
+ 
+presence of functional and non-functional application requirements it poses severe 
+implications on the effort involved in verifying the requirements, and increasing 
+dynamism. Even though, the individual components are pre-verified; their 
+verification is usually done in a limited context, with assumptions that may not hold 
+after composition. As a result, the complexity of system verification grows 
+exponentially with the number of applications [23]2.  
+1.1.3 Modeling and Analysis using Petri Nets 
+Petri nets (PN) is a mechanism of modeling complex systems, in which states and 
+events can be manipulated according to certain rules and explicit conditions. PN 
+formalism was introduced by Carl Adam Petri in 1962. It provides an elegant and 
+useful graphical and mathematical formalism for modeling concurrent systems and 
+their behaviors [24].  
+PN graphs are quite suitable for representing Discrete Event Systems (DES) in 
+which operations depend on potentially complex control schemes. PN graphs are 
+intuitive and capture a lot of structural and behavioral information about the system. 
+Another motivation for considering PN for the modeling of DES is the body of 
+analysis techniques that have been developed for over three decades and are used for 
+reasoning about structural and behavioral properties of PN models. These 
+techniques include reachability analysis, state-space analysis, and model-checking as 
+well as linear-algebraic techniques [25].  
+The PN research has been developed in two directions for the past three decades: (i) 
+PN theory that focused on the development of basic tools, techniques and concepts 
+needed for the PN application; (ii) Applied PN theory which is mainly concerned 
+with the PN application for the modeling of systems and their analysis. Successful 
+work in this direction requires good knowledge of the application area in which PN 
+are applied and PN theories and techniques [26].  
+1.1.4 Modeling and Analysis using Process Algebra 
+ 
+Process Algebra is an algebraic approach for the modeling and analytical study of 
+concurrent processes. It has a diverse family of algebraic formalisms for modeling 
+concurrent systems. These formalisms comprise of algebraic language for the 
+specification of processes and provide calculi in form of algebraic laws that allow 
+process descriptions to be manipulated and analyzed, and permit formal reasoning 
+about their correctness and equivalence [27]. The main Process algebraic formalisms 
+are: 
+ CCS, Milner's Calculus of Communicating Systems  
+ CSP, Hoare's Communicating Sequential Processes 
+ ACP, Algebra of Communicating Processes 
+ LOTOS, Language Of Temporal Ordering Specification 
+1.1.5 Model Verification 
+In M&S, verification is concerned with building the model right. It is typically 
+defined as a process of determining whether the model has been implemented 
+correctly [28] and whether it is consistent with its specifications [29]. In principle, 
+                                                 
+2 Even though the referred text corresponds to the electronic components which are physically 
+composable, however the problem of composability complexity is the same and is mutually 
+understood by different communities. 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+20 
+ 
+verification is concerned with the accuracy of transforming the model’s requirements 
+into a conceptual model and the conceptual model into an executable model [29]. 
+The distinction of a conceptual model and executable model is of great importance 
+and is a fundamental principle in M&S. A conceptual model is abstract description of 
+a real system [30], captured based on given requirements and modeling objectives. 
+This is later refined and implemented into a more concrete executable model. In 
+these terms, conceptual modeling is a subset of model design [31]. Conceptual 
+modeling is about moving from a problem situation, through model requirements to 
+a definition of what is going to be modeled, and is independent of its implementation 
+details [30], which are later addressed in form of an executable model.  
+1.2 
+Summary of the opening perspective 
+In essence, component-based approach is highly favored in M&S community for 
+building large and complex models. But to ensure that the model is correct and 
+meets its requirement specifications, a substantial effort is required to evaluate its 
+degree of composability. In M&S community, the discipline of Model Verification 
+provides basic concepts and fundamental principles for the compressive study of the 
+degree of composability and reasoning its correctness with respect to the given 
+specifications. However the existing component-based simulation frameworks offer 
+limited built-in extensive verification techniques or none at all. Therefore third party 
+approaches such as PN analysis techniques and process algebra are considered for 
+the thorough examination of composed models at various levels of depth.  
+The sub-topics: (i) Component-Based Modeling & Simulation, (ii) PN /CSP Analysis 
+and (iii) Model-Verification are the elementary pillars and theoretical foundations of 
+this thesis and are expanded in details in chapter 2, 3 & 4 respectively. 
+1.3 
+Preliminaries 
+Based on the previous discussion, the formal definition of the problem of this thesis 
+is furnished in this section. In order to define the problem statement, following 
+definitions are used:  
+1.3.1 Definition 1: Set of Components 
+Let C = {c1, c2, c3 …, cn} be a given set of components discovered and selected from 
+a component repository R, as per the abstraction of the real-system. 
+1.3.2 Definition 2: Requirement Specification 
+The Requirement specification of the system model is defined as a tuple:   
+RS = 〈O, S〉 where: 
+O = {o1, o2, o3 …, on} is a set of objectives (or goals) and 
+S = {s1, s2, s3 …, sn} is a set of system constraints (or system properties). 
+Objective: 
+An objective oi ∈ O can be defined as a reachable “final-state” of the 
+composed model or an aggregated desirable output (a data value or event) 
+produced by the composed model which cannot be produced by individual 
+components. 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+21 
+ 
+System Constraint:  
+In modeling terms, a system constraint si ∈ S is defined as a system property 
+that must be satisfied; for instance a good state; which must be reached or a 
+bad state; which must be avoided (never be reached) during the execution.  
+The notions of constraints are different from Objectives, because they can be 
+necessary requirements but not the ultimate goals. E.g., a manufacturing system 
+should not only produce the desired products (objective) but also fulfill safety 
+requirements (constraints).  
+1.3.3 Definition 3: Composition & Composability Pattern  
+Let CM〈c1, c2, c3 …, cn〉 be a composition of a set of given components C, composed 
+using a particular composability pattern P. A pattern describes how the components 
+are attached to each other, i.e., the topology of the components. And provide 
+important information for composability verification. A pattern of composability can 
+be sequential, parallel, fork, join, iterative, or composite.  
+1.3.4 Definition 4: Satisfiability Operator 
+For each element in the requirement specification RS, a Satisfiability operator╞ maps 
+a given composed model CM to a Boolean (True or False) formally described as 
+follows: 
+• CM〈c1, c2, c3 …, cn〉    ╞ i  oi∈O → true | false 
+• CM〈c1, c2, c3 …, cn〉    ╞ j  sj∈S → true | false 
+For each relation ╞ i we define a verification function (algorithm or theorem) based 
+on which the satisfiability operator maps the resultant value. This verification 
+function determines whether a given composed model satisfies a required property.  
+1.4 
+Problem Statement 
+Based on the above definitions the problem statement is defined as follows: 
+“Given a composed model CM, composed from a set of components C using 
+a pattern P, and the requirement specification RS, can we verify that CM fulfills 
+all the objectives and satisfy all the constraints given in the requirement 
+specification”. 
+ 
+Formally: 
+ 
+This problem statement is considered as an initial point and basis of the research 
+presented in this thesis proposal. In this work it will be shown how a modeler can 
+correctly compose component models and verify the composition at different levels 
+through utilization of our proposed verification framework.  
+CM=Compose ([c1, c2, c3 …, cn], P) ∧ RS=〈O, S〉  → {CM ╞𝒊 ∀oi ∈ O ∧  CM ╞𝒋 ∀sj ∈ S} 
+(1.1) 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+22 
+ 
+1.5 
+Approach 
+In this section an overview of the approach and methodology is presented. Based on 
+the software engineering principle, this section is divided into two main parts (i) 
+Problem Domain and (ii) Solution Domain.  
+1.5.1 Problem Domain 
+Problem domain (or problem space) is an engineering term referring to all 
+information that defines the problem and its constraints. It includes the goals 
+that the problem-owner 3 wishes to achieve, the context within which the 
+problem exists, and all rules that define required essential functions or other 
+aspects of any solution product. It represents the environment in which a 
+solution will have to operate [Wikipedia]. 
+ 
+All the information provided in this thesis related to modeling & Simulation, 
+component-based model development, conceptual modeling, model components, 
+composability, model-verification and the problem of composability correctness 
+correspond to the problem-domain. In particular, Chapter 2 covers the main aspect 
+of the problem domain where the component based modeling and simulation is 
+discussed in detail. Following sub-sections briefly describe the selected method of 
+specification of the problem domain. 
+Base Object Model (BOM) as Composability Framework 
+BOM is selected in this thesis as a component specification standard which can be 
+used as a foundation for developing model components at conceptual level. They are 
+composed and are subjected to the composability verification process to evaluate 
+that they satisfy given requirements, hence represent component framework of the 
+problem domain. 
+Requirement Specification Template 
+A “Requirement Specification Template” is defined which is used to formulate 
+requirement specifications. It essentially contains a set of objectives and constraints 
+(of standard or scenario-specific properties), which are required to be satisfied for the 
+proof of correctness of the composed model. 
+1.5.2 Solution Domain 
+Solution domain (or solution space) is a term referring to all information that 
+defines the proposed solution of the problem. It includes the concepts, 
+principles, methods, techniques, algorithms, programs, software architects, 
+frameworks, processes and recommended practices, which help in solving the 
+problem under study. 
+ 
+Following sub-section gives a brief overview of the approach used in this thesis: 
+                                                 
+3  A problem owner can be the customer, solution buyer, organization or a prospective target 
+community. A problem owner sees the problem as an opportunity, whereas the solution engineer sees 
+the problem for which he/she has to provide a solution. 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+23 
+ 
+Multi-tier Composability Verification 
+The composed model undergoes multiple iterations for composability verification at 
+different levels. Each level corresponds to a tier in the verification process. When the 
+composability at a particular level is successfully verified, next level is iterated. When 
+all the levels are completed, the components are said to be fully composable. These 
+levels are discussed in detail in chapter 2. The verification of these levels is discussed 
+in chapter 5.  
+PN Formalism 
+PN formalism (and specially the Colored Petri Nets extension) is chosen for creating 
+executable models of the BOM based conceptual models. The proposed framework 
+automatically transforms BOM components in form of an executable PNML 4 or 
+CPN-based component which can be executed or undergo a verification process 
+using the corresponding PN execution environment.  
+CSP Formalism 
+CSP5 formalism with an extension of Timed-CSP is picked as another executable 
+modeling language for BOM based conceptual models. The proposed framework 
+transforms BOM components into executable CSP process components and 
+composed for execution and verification. 
+Automatic Transformation Tools 
+ 
+An automatic transformation tool is proposed, which transforms a BOM component 
+model into the selected executable modeling formalism such as PNML, CPN based 
+or CSP based executable model. It may also be required to provide additional details, 
+which cannot be modeled or represented by BOM. 
+Dynamic Analysis Approach 
+Three main dynamic analysis approaches are selected for composability verification 
+of BOM base composed models at dynamic-semantic composability level: 
+Algebraic analysis approach 
+This approach is used to transform a BOM composition into a classical PN model 
+using PNML format and verifies the properties using PN algebra. 
+State-space analysis approach 
+This approach is based on using Colored Petri Nets and State-Space analysis. CPN 
+tool is a strong simulation and verification tool. State-space analysis is a very accurate 
+correctness reasoning technique; however it is costly in terms of computational 
+power and memory. Therefore a reduction technique is also proposed to reduce a 
+state-space graph of a composed model, in order to avoid state-space explosion. 
+Model Checking approach 
+CSP based model checking is used for the formal verification of BOM based 
+composed model. In formal logic, model checking designates the problem of 
+determining whether a formula or a correctness property ϕ defined using LTL, CTL6 
+or similar property specification formalism, evaluates to true or false in an 
+                                                 
+4 Petri Net Markup Language 
+5 Communicating Sequential Process 
+6 Linear Temporal Logic, Computational Tree Logic 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+24 
+ 
+interpretation of a system K, written as K ╞ ϕ. Efficient algorithms are selected to 
+determine whether K ╞ ϕ holds [32].  
+In summary, these three approaches are extensively being used in formal verification 
+for over a couple of decades and therefore equipped with rich theoretical 
+foundations and practical tools and techniques.  We however believe that they are 
+being considered in this thesis for the composability verification of BOM based 
+models (or for that matter any M&S composition framework) for the first time and 
+will prove to be very promising and effective. A basic foundation is built using these 
+approaches in this thesis, and their usage are shown though appropriate examples. 
+Also necessary guidelines are provided for developing new verification methods 
+using these approaches and tools, in order to address various verification issues.  
+This research aims to propose a multi-tier verification life cycle for defining, 
+development, 
+archiving, 
+discovering, 
+matching, 
+selection 
+and 
+composing, 
+transforming, executing, verifying and finally reasoning about the correctness of the 
+composed models. This life-cycle extensively relies on the integrated component 
+development, composition and verification framework that is being proposed in this 
+research. This life-cycle follows our proposed process to perform verification of a 
+composed model at different levels. This life-cycle can be adapted by M&S 
+practitioners for rapid model construction, analysis, refinements and reuse and thus it 
+will boost the process of modeling and simulation of complex dynamic systems. 
+ 
+1.5.3 Solution Statement 
+Based on the proposed approach the solution statement is described as follows: 
+“A verified composed model guarantees that the selected components are 
+composable at all composability levels, and they meet the requirement 
+specification by satisfying given objectives and fulfilling the required 
+constraints”.  
+A correctly composed model, promotes reuse of base components thus support 
+rapid model development and can be reused as yet another component later on.  
+1.6 
+Scope of the Thesis 
+In this section, the scope and the boundaries of the thesis are outlined.  
+1.6.1 Correctness 
+In this thesis, “Correctness” is the main focus of the research. The approach, 
+methods, process and framework mainly deal with the correctness issue of the 
+composability verification. The other issues such as performance, efficiency and cost 
+estimation of the solution are currently beyond the scope of this thesis and 
+considered as future work.  
+1.6.2 Validation 
+Validation is a vital part of model evaluation and always goes hand in hand with 
+verification. However it is beyond the scope of this thesis. Although we believe that, 
+our framework is flexible and open-ended. Therefore it can accommodate necessary 
+extensions to support validation with a minor effort. 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+25 
+ 
+1.6.3 Emergence 
+Emergent behavior due to composition of sub-systems is an important and open 
+research topic in the composability domain. We however do not address this issue in 
+this thesis and consider it a future work. 
+1.6.4 Generalization 
+Currently, the proposed approach is based on Base Object Model, only as a 
+demonstration of how our approach can be applied on an existing component 
+standard. However the framework presented in this thesis is open-ended and can be 
+generalized to accommodate any other component standard. Furthermore, 
+heterogenic composability can also be supported. We however do not address 
+generalization issues in this thesis. 
+1.7 
+Summary of the Contributions 
+The existing work in the area of component based modeling and simulation is 
+fragmentary in nature, especially when the verification of component composability 
+of model at a conceptual level is concerned. Furthermore, even though different 
+composability verification approaches exist, but they have not been studied in depth 
+at different granular levels.  
+In this research, composability of BOM based model is studied in depth, focusing 
+mainly on the different levels. A multi-tier component based verification life-cycle is 
+proposed that tackles key issues of such as model development, discovery, selection, 
+matching, composition, requirement specification, transformation, implementation, 
+execution, analysis and most importantly verification.  
+In terms of verification, the major contributions of this thesis include development 
+of a composability verification framework, which integrates different methods, and 
+techniques to support different tasks in the composability verification process of a 
+composed model. These different tasks are categorized in different phase of a 
+proposed component based modeling and simulation (CBM&S) life-cycle. We 
+propose methods for evaluating structural and behavioral consistency of the 
+composed BOMs. For structural evaluation we propose a set of static analysis 
+techniques to verify that the components can be correctly connected and their 
+communication is semantically consistent, meaningful and is understood as intended.  
+For behavioral consistency of the composition we suggest a state-machine matching 
+technique. It verifies that the components can correctly interact with each other in a 
+right causal order to reach final states. For the further evaluation of the behavioral 
+composability our framework incorporates three main approaches: (a) PN Algebraic 
+technique (b) CPN-based State-space analysis technique and (c) CSP based model 
+checking. For each approach we develop automatic transformation tool that 
+transforms a BOM based composed model into the executable model of the 
+corresponding approach. We present three different case studies for the proof of 
+concept and for the evaluation of our verification framework.  
+We also suggest various extensions in each approach to suit the needs of 
+composability verification. For instance we propose algorithms for automation of the 
+PN algebraic approach. Also a CPN based component model is proposed for the 
+State-space algebraic approach in order to describe a BOM component (or any other 
+simulation component) in form of an executable model that can be executed using 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+26 
+ 
+CPN execution environment. We also introduce a State-space reduction technique 
+for the CPN based state-space analysis approach to avoid the risk of state-space 
+explosion. For the CSP based model checking approach we propose an external 
+function library for methods to support various modeling tasks such as definition of 
+probability distribution functions for probabilistic system models.  
+ 
+1.8 
+Structure of the Thesis  
+This thesis is divided into two main parts: 
+ 
+Part I Episteme: This part mainly covers the theoretical concepts, principles and 
+discussions. It comprises of chapters 1, 2, 3 & 4. 
+ 
+Chapter 1: Introduction:  
+Chapter 1 gives a bird’s eye view of the research presented in this thesis. It addresses 
+the concept, historical background and the basic philosophy of composability. The 
+problem is defined and the approach is briefly introduced. A section on the scope of 
+the thesis and main contributions are also presented.  
+ 
+Chapter 2: Component Based Modeling and Simulation: 
+Chapter 2 introduces and discusses component based modeling and simulation in 
+details, as it is the foundation of the problem domain. This chapter mainly covers the 
+theory, issues, different levels, framework and the formalism of model composition. 
+It also introduces Base Object Model (BOM) in details as a choice of Model 
+composition standard of this thesis.  
+Chapter 3: Executable Modeling Formalism 
+This chapter provides introduction, theory, basic definition and classification of PN 
+and CSP as executable modeling formalisms and regarded as solution domain. It also 
+describes basic concepts of the analysis techniques that are used later in this thesis. 
+Chapter 4: Verification and Analysis 
+Chapter 4 discusses theory and principles of verification. It also categorizes some of 
+the important verification techniques that are used in this thesis. 
+ 
+ 
+ 
+Part II Techne: This part contains practical aspects including approaches, methods, 
+tools, development frameworks and lifecycle. It also contains examples related to our 
+proposed solutions for the proof of concept. It comprises of chapters 5, 6, 7, 8 & 9. 
+ 
+Chapter 5: Proposed Approach and Framework  
+Chapter 5 is the center of the thesis as it provides the most important details of our 
+contributions. It describes the proposed verification framework and verification life-
+cycle. It covers our proposed methods, techniques, algorithms, procedures as our 
+
+Chapter 1  
+ 
+Introduction 
+   
+Page 
+27 
+ 
+contributions at different phases of composability verification process. These phases 
+and their concerning activities are outlined as composability verification life-cycle.  
+ 
+Chapter 6: Composability Verification Process 
+This chapter presents the proposed composability verification process. It provides 
+essential guidelines of how to use our proposed composability verification 
+framework (discussed in chapter 5). It uses work flow diagrams to describe the 
+overall process and gives necessary guidelines to the modeler at each step. 
+ 
+Chapter 7: Fairness verification using PN Algebraic Technique  
+Chapter 7 describes a case study of a manufacturing system as an example to explain 
+how the proposed framework helps to verify fairness property in a composed 
+system. The purpose of this chapter is to practically demonstrate algebraic 
+verification method. 
+ 
+Chapter 8: Model verification using State-space analysis technique 
+Chapter 8 covers an example of the verification of a Field Artillery Model. It 
+practically demonstrates how state-space analysis is used to verify a composed 
+system. The field artillery model is introduced in detail along with requirement 
+specifications and it is shown how the proposed approach can help to verify its 
+composability.  
+ 
+Chapter 9: Model Checking  
+This chapter demonstrates an example of verification using CSP based Model 
+Checking. The field artillery model discussed in chapter 8 is modified into a real-time 
+probabilistic system and is verified using CSP based model checking.  
+ 
+ 
+ 
+Chapter 10: Conclusion and Future work 
+This chapter provides summary and conclusion, discussion and future work of the 
+thesis. 
+ 
+
+   
+Page 
+28 
+ 
+Chapter 2 
+Component Based Modeling and 
+Simulation  
+ 
+Composability is an important quality characteristic and an effective means to achieve several benefits 
+in M&S discipline, but in reality, it is a challenging and daunting problem. The community has 
+conducted active research on its theoretical and practical intricacies. In theory, composability is 
+studied under various facets and views primarily distinguished, by its different “layers” or “levels” as 
+identified by different research groups. Whereas in practice, various practical challenges associated 
+with it are investigated. Most important of these issues are component specification, development, 
+integration, composability verification and validation, collectively referred to as phases of a 
+Component based life-cycle. In this chapter both theoretical and practical aspects of composability are 
+discussed in detail. 
+ 
+2.1 
+Composability in M&S 
+In M&S applications, composability has been defined in different ways. Much of 
+these definitions have been collected by A. Tolk in his article [13]. Harkrider and 
+Lunceford defined composability as:   
+The ability to create, configure, initialize, test, and validate an exercise by logically assembling 
+a unique simulation execution from a pool of reusable system components in order to meet a 
+specific set of objectives [33]. 
+Kasputis and Ng defined composability as:  
+The ability to compose models across a variety of application domains, levels of resolution, and 
+time scales [16] 
+Petty and Weisel recommended the following definition in their article on theory 
+of composability, which later was appended by P. K. Davis: 
+Composability is the capability to select and assemble simulation components in various 
+combinations into valid simulation systems to satisfy specific user requirements, meaningfully 
+[17] [34]. 
+ 
+ 
+It has been realized that composing models is more difficult than composing general 
+software components. This argument is predicated on the assumptions that models 
+are more complex; they are developed for particular purposes, and they depend on 
+context-sensitive assumptions [8] [17]. Model development is a hard design task, 
+mainly due to the complexity involved in the process. Nowadays this complexity is 
+increasing to levels in which the utilization of pre-defined models is considered very 
+useful to cut short the development time. Thus model composition is a paradigm, 
+where existing components are the building blocks for the construction of new larger 
+and more sophisticated models. When a model is composed, it must be evaluated in 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+29 
+ 
+terms of correctness with respect to its requirements. In short the predictability of 
+guaranteeing the correctness of model composition is called Composability.  
+2.2 A Brief History of Composability and related work 
+2.2.1 Initiation  
+Composability in M&S has primarily been investigated by the defense research 
+sector. The earliest uses of the term composability within the context of defense 
+simulation dates back to the Composable Behavioral Technologies (CBT) project 
+during the mid-1990s [35]. Later on the Joint Simulation System (JSIMS) project 
+investigated composability as a system objective [36]. In 1998, a project on model 
+based simulation composition (MBSC) was started in which a prototype composition 
+environment for JSIMS was developed. In 1999 Page and Opper investigated the 
+composability problem from a computability and complexity theoretic perspective 
+[35]. Composability became a key system objective for OneSAF project in 1999 [22].  
+2.2.2 Theoretical evolution 
+Later on a series of numerous articles were published which addressed various issues 
+of and methodologies of composability and became the theoretical foundations for 
+further research. Important works in this series include: Kasputis and Ng [16]; Davis 
+et al. [37]; Petty & Weisel [38]. Petty and Weisel extended the work of Page and 
+Opper, provided a broad survey of the uses of the term composability, and examined 
+the composite validation problem within the context of automata theory and 
+computable functions. Later a comprehensive report was published by Davis and 
+Anderson in 2003 [17] that provides a broad survey of the composability and 
+suggests its applications for the DoD7 in this area.  
+2.2.3 Standards & Frameworks 
+Later on, the research on composability remained focused on the development of 
+standard composition frameworks and its practical application in various domains of 
+modeling and Simulation. In 2005 the Extensible Modeling and Simulation 
+Framework (XMSF) was initiated by the Naval Postgraduate School to develop a 
+web-based simulation environment [39]. Advances in M&S technologies, gave rise to 
+different distributed simulation standards and protocols such as Simulation 
+Networking (SIMNET), Distributed Interactive Simulation (DIS), Aggregate Level 
+Simulation Protocol (ALSP) and the High Level Architecture (HLA). The details of 
+these standards are well documented by Moradi [19]. Due to the complex nature of 
+the standards, and distributed simulation itself, different composability frameworks 
+were introduced to co-op with these requirements. More general-purpose 
+frameworks such as the Discrete Event System Specification (DEVS) [40], the Open 
+Simulation Architecture (OSA) [41], the Base Object Model (BOM) [42], and the 
+Component Oriented Simulation Toolkit (COST) emerged and contributed to 
+various issues of composability in different ways.  
+2.2.4 Technological Advances 
+Due to the technological advances in computer engineering, many approaches 
+emerged with the aim to address issues and high end requirements of modeling and 
+                                                 
+7 United State Department of Defense 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+30 
+ 
+simulation such as representation of Complex, Dynamic and Adaptive Systems; 
+integration of large interdependent Systems; multi-resolution and multi-scale 
+modeling [43], and much more. In this period, many tools and techniques were 
+developed using composability paradigm. Model Coupling Toolkit (MCT) was 
+developed to support and simplify the construction of parallel coupled models [44]. 
+MUSE is another composable simulation environment for astrophysical applications 
+in which different simulation models of star systems are incorporated into a single 
+framework [45]. Some frameworks such as Common Component Architecture 
+(CCA) [46] and Component based Grid Environment (MOCCA) [47], were 
+proposed to be used in high-performance computing, where scientific components 
+are directly connected by their Users and Providers ports. A Multi-scale Coupling 
+Library and Environment (MUSCLE) provided a software framework for building 
+composable simulations according to the complex automata theory [48]. Compo-
+HLA is an environment proposed for supporting HLA component [49]. 
+2.2.5 Composability verification and Validation 
+Most of these frameworks lack strong built-in composability evaluation support. 
+Therefore some third-party composition, verification and validation frameworks 
+were developed by individual research teams such as Composable Discrete-Event 
+scalable Simulation (CODES) [20] and Semantic Web-based BOM composition 
+framework [19], where verification and validation of composability are strongly 
+focused.  
+2.3 Theory of Composability 
+The formal theory of composability was pioneered by Petty and Weisel [34], [38], 
+[50] in an initiative developed at the Virginia Modeling, Analysis & Simulation Center 
+(VMASC).  It was also called “semantic composability theory” (SCT). The aim of the 
+SCT is to check and prove the semantic composability of components using formal 
+descriptions and reasoning. A model is defined as a computable function: y = ƒ(x), 
+where function is calculable by a finite procedure and relates each input to a unique 
+output, as shown in Figure 1 
+ 
+ 
+Figure 1: A model as computable function (acquired from [34]) 
+ 
+A simulation is a sequence of executions of a model ƒ(x), where the output from 
+each execution step is the input to the next step of the execution: 
+ 
+Where i = input value; m=memory value; o=output value and n=current iteration, as 
+shown in Figure 2 
+(mn, on) = ƒ(mn-1, in-1) 
+(2.1) 
+
+xeX
+J(x)e Y
+Domain
+Codomain
+X
+YChapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+31 
+ 
+ 
+Figure 2: Sequence of executions (acquired from [50]) 
+ 
+The composition is defined as output of one function to be the input of another: 
+ 
+Figure 3 shows the representation of a composed model, which is developed through 
+composing other models (f1, f2 & f3). A composed model as a whole has also a set 
+of inputs, outputs, current states and next-states as shown in Figure 3. 
+ 
+Figure 3: Composed Model  (acquired from [50]) 
+The composition of models in SCT is in fact the composition of functions. Since a 
+set of computable functions is closed under composition any set of models can be 
+composed if the composition exists, but there is no guarantee that the resultant will 
+be a useful model. Thus focus of SCT is semantic composability, the question of 
+whether the model composition is meaningfully valid or not.  
+Validity 
+A model is defined as valid, if it is an accurate representation of the real-world with respect to 
+the intended use. For formal validation, the simulation of a composition is represented as 
+Labeled Transition System where nodes are model states, edges are function executions, and 
+labels are model inputs. A composition is valid if and only if its simulation is close to the 
+simulation of a perfect model. 
+Perfect Model 
+A model is perfect with respect to a natural system N 8 if and only if it represents a system of 
+perfect observations of the natural system [50]. 
+                                                 
+8 A natural system N is a real or imagined system. 
+h(x) = ƒ(g(x)) 
+(2.2) 
+
+io
+i,
+is
+i3
+mo
+111
+1112
+1113
+m4
+01
+02
+03
+04
+2
+3
+4i1
+i2
+i
+X3.2Xg.1
+m
+X3.3
+Y3.1
+→ mner.!
+X1.1
+mz
+X1.2
+Ji,1
+X3.4
+Y3.2
+→> Mno2
+Ji
+X22X2.!
+m3
+X1.3
+J1,2
+X2,3
+J2.1
+X3.s Js Y33
+→ mer3
+Ji.4Ji.3
+fn y2?
+tu
+X3.6
+J3.4
+J2.+Y23
+m,
+X.7
+Y3.s
+> mnoxt's
+J3.8 J3.7 J3.6
+个个
+03
+So to
+% l %Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+32 
+ 
+ 
+For details of different classes of models, their equivalence relations, formal 
+theorems and proofs of equivalence, interested readers should refer to [50]. The basic 
+concepts of a formal theory of semantic composability include formal definitions for 
+model, simulation, validity, and composition. A theory of composability can facilitate 
+the convenient reuse of simulation components, which holds the potential to the 
+time and cost of simulation development [34] [38] [50]. 
+2.4 Concepts related to Composability 
+In this sub-section, some of the concepts and idea related to composability are 
+discussed.  
+2.4.1 Composability vs. Reusability 
+Composability is differentiated from reusability in many aspects. Balci et al. define 
+Reusability as the degree to which an artifact, method, or strategy is capable of being 
+used again or repeatedly [5]. Robinson et al. on the other hand suggest that the term 
+simulation model reuse can be taken to mean various things from the reuse of small 
+portions of code, through component reuse, to the reuse of complete models [51]. 
+Composability offers means to achieve reusability, but reusability might not always 
+be the ultimate objective of model composition. For instance, in a particular 
+situation, a set of modular components are purpose-fully built and composed to 
+construct a large model, but they cannot be reused in a different project, due to their 
+highly specific design.  To be widely reusable, a component must be sufficiently 
+general, scalable, and adaptable. A requirement for reusability may lead to another 
+development approach, for example, a design on a more abstract level [9]. The 
+comparison between usability and reusability of composable components poses a 
+tradeoff between them being very specific in function and behavior so that they can 
+be used in a particular case to satisfy specific user’s requirements or them being very 
+generic and abstract so that they can be reused in different situations again and again.  
+ 
+Figure 4: Generic vs. Specific component design 
+Figure 4 illustrates a component is often more reusable if it has a generic design 
+and less reusable if it has functionally specific design.  
+ 
+Both use and reuse of composable components share three levels of transparency. 
+[7]. A component can be seen as a box, which contains the interfaces and internal 
+implementation. Three levels of composability transparency are defined: 
+
+Generic
+Functionally
+specific
+design
+More reusable
+Less reusableChapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+33 
+ 
+Black Box Composition 
+In black box composition, the user sees the interface, but not the implementation of 
+the component. The user documentation is provided that contains the details of the 
+inputs and outputs, requirements and restrictions of the component. All the 
+implementation details are hidden. The clients will get what the contract promises. 
+The changes are not feasible at the deployment end. The advantage of black-box 
+composition is that the testing done at the development side is persevered and there 
+is no need of further testing at the deployment side. 
+Glass Box Composition 
+In glass box composition the inside structure of a component can be viewed, but it is 
+not possible to modify. This solution has an advantage when compared to black box 
+reuse, as the modeler can understand the box and its use better. However it is not 
+possible to make any changes in the implementation. The advantage of this level 
+remains the same as that of black-box composition however an additional benefit is 
+that the user can gain knowledge of the internal implementation and can understand 
+the mechanics of the component.  
+White Box Composition 
+In white box composition it is possible to see and change the inside of the box as 
+well as its interface. A white box can share its internal structure and implementation 
+with another box through inheritance or delegation. The advantage of this level is 
+greater flexibility due to the provision of modifications. However this level incurs an 
+extra burden of testing at the deployment end. 
+ 
+Figure 5: Black Box, Glass Box, White Box 
+Figure 5 illustrates difference between black box, glass box and white box 
+composition.  
+ 
+2.4.2 Composability vs. Interoperability 
+Bearing in mind the definition of composability mentioned previously, the IEEE 
+definition of interoperability is: 
+The ability of two or more systems or components to exchange information and to use the 
+information that has been exchanged 
+The concept of interoperability is mainly about inter-connecting systems of various 
+types developed for different purposes; for different platforms, and about their 
+syntactically and semantically agreed upon communication [13]. In the context of 
+Internals 
+not known 
+Internals 
+known 
+Internals 
+known 
+No 
+modification 
+No 
+modification 
+Modifiable 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+34 
+ 
+modeling and simulation, interoperability is the ability of different simulations 
+connected in a distributed system to collaboratively simulate a common scenario [19]. 
+Page et al. [52] distinguishes composability and Interoperability as follows: 
+Composability contends with the alignment of issues on the modeling level. The underlying 
+models are purposeful abstractions of reality used for the conceptualization being implemented 
+by the resulting systems; whereas Interoperability contends with the software and 
+implementation details of interoperations; this includes exchange of data elements via 
+interfaces, the use of middleware, mapping to common information exchange models. 
+2.5 Composability Levels 
+Petty and Weisel emphasized on two basic types of composability: syntactic and 
+semantic in their theory of composability [38] [50]. According to which the syntactic 
+composability requires that the composable components should be constructed with 
+compatible implementation details such as parameter passing mechanisms, external 
+data accesses, and timing assumptions. The question of syntactic composability is 
+whether the components can be connected.  In contrast, semantic composability is a 
+question of whether the models can be meaningfully composed to form a composed 
+simulation system and whether the combined computation is semantically valid. It is 
+possible that two components may be syntactically linked, so that one can pass data 
+to the other, but they can be semantically invalid. Figure 6 represents the difference 
+between syntactic and semantic composability metaphorically.  
+ 
+Figure 6 Syntactic vs. Semantic Composability (acquired from [38]) 
+Composability is studied in more depth under different levels, as identified by 
+different research groups. Several levels of understanding and agreement are required 
+between the models in order for them to be meaningfully composed—that is, for 
+their composition to produce meaningful results [17]. 
+Davis recommended five distinctions of levels namely: syntax, semantics, pragmatics, 
+assumptions, and validity to study composability [43]. He describes these levels as 
+different consistencies of composability, which all together are examined for the 
+correctness of model composability. Petty & Weisel have suggested nine levels of 
+composability in terms of composition units. These levels are: Application, Federate, 
+Package, Parameter, Module, Model, Data, Entity and Behavior [38]. Tolk described a six 
+layered model called Levels of Conceptual Interoperability (LCIM) to study 
+composability and interoperability. This model includes: technical layer, syntactic 
+layer, semantic layer, pragmatic layer, dynamic layer, and the conceptual layer [13].  
+Similarly Medjahed & Bouguettaya introduced a composability stack in which the 
+composability of semantic web services is checked at four levels: Syntactic, Static 
+Semantic, Dynamic Semantic and Qualitative level [53].  First three levels of 
+
+AF
+Syntacticcomposability
+Semantic composabilityChapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+35 
+ 
+Medjahed & Bouguettaya’s composability stack were adopted by Moradi, et al. to 
+study the degree of composability of Base object Model (BOM) components [54]. In 
+this thesis, these levels are considered as fundamental benchmarks for the evaluation 
+of model composability. The notion of model composability and its correctness 
+strongly depend on the consistency of these levels as explained in the following 
+subsections.  
+2.5.1 Syntactic level:  
+At this level, the structure of the components is studied to know if they can fit 
+together i.e., the output of one can be read as an input to the other and that the 
+syntactic information of the connected components, such as message name, mode of 
+action and number of parameters match each other e.g., A “passenger airplane” 
+component will be a syntactic misfit in a military training simulation, where a “fighter 
+jet” component is required whose input will be a signal from “ground station” 
+component to engage a target and output will be an airstrike on the “target” 
+component. A passenger plane can neither take a target designation as input, not it 
+can fire on a ground target. So this component is not composable at syntactic level.  
+2.5.2 Static-Semantic level:  
+It is concerned with the meaningful interaction of the composed components. Static-
+Semantic level of composability involves in having a concise and mutual 
+understanding of the data exchanged by the components participating in the 
+composition. At this level, it is ensured that all the components possess the same 
+understanding of the terms, parameters, data types and units, so basically this level 
+deals with the interpretation of same meaning of concepts for the information 
+exchanged between the composed components. For instance, if two components 
+being composed interpret units of quantities in a different way, they will incorrectly 
+process data values during the information exchange and thus result in a situation not 
+intended by the user e.g., if a integer data value is intended to be the bearing of a 
+target (in degrees) but interpreted as target distance (in Km) by the other component 
+then it is a semantic mismatch.  
+The term “static” is prefixed, because all the information that is required to evaluate 
+this level is static and does not change during the entire component interaction.  
+2.5.3 Dynamic-Semantic level:  
+Dynamic Semantic Composability implies that the components are dynamically 
+consistent, i.e., they have suitable state-full behavior, necessary to reach the desired 
+goals and subsequently satisfy user requirements. The dynamic level of composability 
+ensures in having a behavioral consistency and coherency among the participating 
+components in achieving the common goals. The dynamic semantic composability 
+can only be achieved if the components are at the right states during their interaction. 
+Also they should possess required behavior to make a collective progress. E.g., in a 
+composed model of a restaurant, a waiter component may have two different 
+behaviors (i) Classical restaurant where a waiter takes order from customer, serves 
+food and then collects payment or (ii) Fast food restaurant where waiter takes order, 
+collects payment and then serves food. The selection of the correct behavior and the 
+correct customer component (the one who can correctly interact with the classical 
+restaurant waiter or fast food waiter) will affect the overall composability of the 
+model. This example presents how the components should be at right states to make 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+36 
+ 
+progress. A customer (expecting classical treatment) will wait forever for the (fast 
+food waiter) to serve food and vice versa.  
+Even if the components are at the right states, but their behavior is not correct, the 
+composition may not reach its goals. E.g., in a manufacturing system two machine 
+components produce two different parts that are later combined to make a finished 
+good, and they share a single robot component for input of raw material, it is required 
+that the robot component should be fair so that both machines get more or less equal 
+chance to proceed. If the robot is not fair the proportion of good produced will be 
+unbalanced and therefore the system will fail to meet its objectives even though the 
+components are at right states and continue to progress. 
+The term dynamic is prefixed, because the information such as current state of 
+components changes dynamically during component interaction.  
+2.5.4 Pragmatic level:  
+Consistency of meaning is not always straightforward because the same word means 
+very different things depending on context [43]. Pragmatic consistency refers to a 
+context based meaningful composition of the components. In linguistics the study of 
+the relations between linguistic phenomena and aspects of the context of language use is called 
+pragmatics whereas Context is defined as something that consists of the ideas, situations, events, 
+or information that relates to it and makes it possible to fully understand it [55]. 
+The pragmatic level of composability evaluates the difference of actual effect of the 
+messages with the intended effect of messages during communication [43]. The 
+research of pragmatic level of composability involves in-depth study of 
+computational linguistics, cognitive technologies and contextual computing [55]. An 
+important issue at this level is pragmatic ambiguity. Pragmatic ambiguity arises when 
+the message is not specific, and the context does not provide the information needed 
+to clarify the statement, and due to which the components do not interact according 
+to the desired objectives. An example of pragmatic ambiguity is the story of King 
+Croesus and the Oracle of Delphi (derived from [56]): 
+    "King Croesus consulted the Oracle of Delphi before warring with Cyrus of Persia.  The Oracle 
+replied that, "If Croesus went to war with Cyrus; he would destroy a mighty kingdom".  Delighted, 
+Croesus attacked Persia, and Croesus’ army and kingdom were crushed.  Croesus complained 
+bitterly to the Oracle’s priests, who replied that the Oracle had been entirely right.  By going to war 
+with Persia, Croesus had destroyed a mighty kingdom – his own." 
+In essence, a set of components can possibly fit together (syntactically), and their 
+communication is meaningful and understood (semantically), but unless all 
+components preserve essential behavior (dynamically) in order to reach the desired 
+composition goals, and they share the correct contextual knowledge (pragmatically), 
+the composability cannot be qualified as correct with respect to given requirement 
+specifications. 
+2.6 Composability frameworks 
+Composability essentially relies on a suitable composition framework that can 
+provide accurate reasoning of its correctness and support means to be able to 
+leverage certain component standard. Various component standards and their 
+respective frameworks have been developed for M&S to support composability. 
+Some of these frameworks contribute to conceptual modeling by providing the 
+needed formalism and influence the ability to develop and compose model 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+37 
+ 
+components at conceptual level, while others support model composition at 
+executable level. These frameworks practically support composability, as they usually 
+offer features such as model specification, development, and execution. A brief 
+description of some of the composability frameworks is provided below: 
+2.6.1 Discrete Event System Specification (DEVS) 
+DEVS [57] is a component based formalism based on dynamic systems theory. It 
+was developed for the purpose of describing the structure and behavior of systems. 
+It supports the concept of hierarchical and modular model construction through 
+coupling of components [19]. DEVS is basically a model specification formalism 
+however it incorporates different implementation frameworks such as DEVS-Java, 
+DEVS-C++ and DEVS-Sharp which are used to implement DEVS models into 
+executable form.  
+ 
+Two types of DEVS models exist, namely, atomic and coupled [20].  
+An atomic DEVS is a tuple M = 〈X, S, Y, δint, δext, λ, τ〉 where: 
+X = {(p, v) | p ∈ InPorts, v∈Xp} is the set of input ports and values 
+Y = {(p, v) | p ∈ OutPorts, v∈Yp} is the set of output ports and values 
+S is the set of states 
+δint : S →S is the internal transition function 
+δext: Q × X→S is the external transition function, where 
+       Q = {(s, e) | s ∈S, 0 ≤ e ≤ τ(s)} is the total state set 
+        e is the time elapsed since last transition 
+ λ : S →Y is the output function 
+τ : S →R0,∞
++  0, ∞ is the time advance function 
+ 
+A DEVS atomic component has inputs X, outputs Y, and a set of S states. At a given 
+moment, a DEVS model is in a state s∈S. In the absence of external events, it 
+remains in that state for a lifetime defined by τ(s). When τ(s) expires, the model 
+outputs the valueλ(s) through a port y ∈ Y, and it then changes to a new state given 
+by δint(s). A transition that occurs due to the consumption of time indicated by τ(s) is 
+called an internal transition. On the other hand, an external transition occurs due to 
+the occurrence of an external event. In this case, the external transition function 
+determines the new state, given by δext (s, e, x), where s is the current state, e is the 
+time elapsed since the last transition, and x∈X is the external event that has been 
+received. The time advance function can take any real value between 0 and ∞. A 
+state for which τ(s)=0 is called a transient state (which will trigger an instantaneous 
+internal transition). In contrast, if τ(s)=∞, then s is said to be a passive state, in 
+which the system will remain perpetually unless an external event is received.  
+ 
+ 
+ 
+ 
+ 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+38 
+ 
+A coupled DEVS is a tuple: M = (X, Y, D, {Md | d ∈ D}, EIC, EOC, IC, 
+Select) where: 
+X = {(p, v) | p ∈ InPorts, v∈Xp} is the set of input ports and values 
+Y = {(p, v) | p ∈ OutPorts, v∈Yp} is the set of output ports and values 
+D is the set of component names 
+Md is a DEVS model with 
+Xd = {(p, v) | p ∈ InPortsd, v ∈ Xp} 
+Yd = {(p, v) | p ∈OutPortsd, v ∈ Yp} 
+EIC is the set of input port couplings 
+EIC  ⊆ {((N, ipN), (d, ipd)) | ipN ∈ InPorts, d ∈ D, ipd ∈ InPortsd} 
+EOC is the set of output port couplings 
+EOC ⊆ {((d, opd), (N, opN)) | opN ∈ OutPorts, d ∈ D, opd ∈ OutPortsd} 
+IC is the set of internal couplings 
+IC ⊆ {((a, opa), (b, ipb)) | a, b ∈ D, opa ∈ OutPortsa, ipb ∈ InPortsb} 
+Select is the tie-break function 
+ 
+A system modeled using DEVS can be described as a composition of atomic and 
+coupled components. A coupled model comprises a set of input and output ports, a 
+set of component names D, a set of DEVS components Md, input port EIC and 
+output port EOC couplings, and, a set of internal couplings IC connecting internal 
+components with each other. The tie-break function decides which component to 
+proceed when two or more components have internal transitions scheduled at the 
+same time.  
+Figure 7 describes a DEVS example. In this example two atomic component A & B 
+are coupled together. Both components have two states Send τ(s)=0.1 and Wait 
+τ(s)=∞. Input port: ?receive and Output port: !send are defined and connected to each 
+other in coupled DEVS. 
+ 
+Figure 7: Ping-Pong DEVS [Wikipedia] 
+2.7 Base Object Model (BOM) framework 
+The SISO 9 standard BOM is defined as, “a piece part of a conceptual model 
+composed of a group of interrelated elements, which can be used as a building block 
+in the development and extension of simulations and simulation environments” [58].  
+ 
+BOM provides a simulation standard that allows model developers and simulation 
+engineers to create modular conceptual models in form of composable objects, 
+                                                 
+9 Simulation Interoperability Standards Organization 
+
+Ping-Pong
+AY
+B
+Send,0.1
+Send,0.1
+Ised
+?receive
+Isend
+!send
+?leceive
+?feceive
+Wait, inf
+Wait, inf
+?receive
+IsendChapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+39 
+ 
+which can be used as the basis for a simulation or simulation environment [59], [60]. 
+The concept of BOM is based on the assumption that components of models, 
+simulations, and federations can be reused as building blocks in the development of a 
+new simulation or a federation [54].  
+BOMs are unique because they provide a means to represent aspects of a conceptual 
+model that captures structural and behavioral descriptions of items abstracted from 
+the real system (simuland). Then they allow these conceptual models to be mapped 
+to one or more class definitions, which may be used by a software design, variety of 
+programming languages, or distributed simulation architectures such as HLA or 
+TENA10 [61], [62]. 
+BOM standard also offers a general purpose modeling architecture for defining 
+components to be represented within a live, virtual, or constructive (LVC) simulation 
+environment. It is well suited for characterizing models including the structural and 
+anticipated behavior of interacting systems, individuals, and other entities. Primarily 
+BOMs framework poses a satisfactory potential for effective composability of 
+conceptual models at syntactic and semantic levels, resulting in a framework for the 
+assembly of a system (i.e. simulation) or system of systems (i.e. distributed simulation 
+environment) [62].   
+In spite of these reasonable qualities, BOM framework still falls short of required 
+behavioral semantics and necessary built-in evaluation techniques, which are essential 
+for modeling complex system behavior and 
+reasoning about the correctness of the 
+composability at each of its different level. 
+Therefore it becomes a most suitable candidate 
+and a preferred choice of a composition 
+framework (in this thesis) for studying model 
+composability in depth and applying proposed 
+methods on BOM based compositions to 
+explain the approach.  
+2.7.1 Structure of BOM  
+A BOM is constituted of elements specifying 
+metadata information, conceptual model and 
+the class structure information defined using 
+HLA OMT constructs [59]. Figure 8 presents 
+different parts of BOM, explained as follows: 
+Model Identification  
+Model Identification associates the metadata 
+information with the BOM. Its purpose is to 
+document certain key identifying information 
+within the BOM description. It provides a 
+minimum but sufficient degree of descriptive 
+information about a BOM 
+                                                 
+10 Test and Training Enabling Architecture 
+Figure 8: BOM structure 
+(acquired from [59]) 
+
+Model Identification (Metadata)
+ Conceptual Model Definition
+Pattern of Interplay
+State Machine
+Entity Type
+Event Type
+Model Mapping
+Entity Type Mapping
+Event Type Mapping
+Object Model Definition
+HLA Object Classes
+HLA Object Classes
+HLA Object Class Attributes
+HLA Interaction Classes
+HLA Interaction Classes
+HLA Interaction Class Parameters
+HLA Data Types
+Notes
+Lexicon (definitions)Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+40 
+ 
+Conceptual Model Definition 
+From the composability point of view, this is the most important part of BOM and 
+therefore the main focus of this thesis. To understand this part, the definition of a 
+conceptual model should first be considered: 
+Conceptual Model 
+A Conceptual model is an abstract description or an appropriate simplification of a real (or 
+proposed) system, which is later, refined and implemented in to a more concrete executable 
+model (or simulation model). In these terms, conceptual modeling is a subset of model design 
+which is formed through an iterative process according to the objectives of system modeling 
+[63], [64]. 
+The term conceptual model is used in different ways in the literature. A conceptual 
+model could be a specific diagram like UML class diagram or it could be 
+documentation of a particular aspect of the simuland11 [29]. To better understand the 
+concept of BOMs, consider the home construction analogy. When a new house is to 
+be built the conceptual understanding of features of the building is captured in 
+architectural drawings, which is analogous to a conceptual model (BOM) [60]. BOM 
+Conceptual Model definition consists of following parts: 
+Pattern of Interplay (POI) 
+POI models a specific purpose or capability and is represented by one or more 
+pattern actions. For each pattern action, one or more senders and receivers are 
+specified to provide a means for understanding and the behavioral relationship 
+among conceptual entities. POI is represented by UML sequence diagram [60]. 
+State Machine  
+The state machine is used to model the behavior of a BOM’s conceptual entity. The 
+state machine is specified by a set of states where each state may transit to a 
+subsequent state called next state, upon an exit action, which is identified in a pattern 
+of interplay. UML state-machine diagram is used to represent BOM’s state-machine 
+[60]. 
+Entity Type  
+A conceptual entity is an abstraction of a real world entity. It defines a relationship 
+with other entities within a pattern of interplay and acts as a sender or receiver of the 
+events [60]. 
+Event Type 
+Conceptual events include information about the source, target, and content 
+(parameters) of a message or trigger. The difference between a trigger and a message 
+is that a trigger is used to broadcast information whereas the messages are directed 
+exchanges of information where the sender knows about the intended receiver of the 
+message [60].  
+ 
+Entities and Events represent data about the real world objects and their interaction 
+(physical description), whereas the pattern of interplay and state-machine collectively 
+represents the dynamic behavior of the component. 
+                                                 
+11 A simuland is the real world system of interest. It is the object, process, or phenomenon to be 
+simulated [29]. 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+41 
+ 
+2.7.2 BOM Assembly 
+The BOM concept provides a mechanism for combining BOMs and creating High-
+Level BOMs, called BOM Assemblies, as shown in Figure 9. A BOM Assembly 
+representing a composition of BOMs, is built in a hierarchical manner and includes 
+information about composed BOMs, which in turn is used to identify a composite 
+interface, and represent a federate, federation within the simulation space12. Typically 
+a developer of a simulation would search a BOM repository for suitable BOM 
+candidates for use in a simulation and combine those into a BOM Assembly (i.e. a 
+simulation model), which is then used to create the actual simulation [19].   
+A BOM assembly contains Model Identification, and pattern of the interplay among 
+conceptual entities being represented, which is provided through the association of 
+BOMs to the various Pattern Description actions that the BOM Assembly identifies, 
+within the Conceptual Model view [60].  
+ 
+Figure 9: BOM Assembly 
+BOM models can be created using XML script. But for constructing BOM models 
+graphically, a free IDE tool called BOM Works [65] is available. Figure 10 represents 
+an example developed using BOM Works. It is similar to the DEVS example shown 
+in Figure 7, to compare the difference.  
+ 
+Figure 10: (a) PingPong BOM in BOM Works  (b) POI (c) State-machineA (d) State-machineB 
+(e) EntityA (f) EntityB (g) EventA (h) EventB 
+ 
+ 
+ 
+                                                 
+12 Although use of HLA is not a mandatory subsequent step, it is likely that BOM assemblies are 
+intended to support an HLA based federation [59]. 
+
+BOM1
+BOM
+BOM
+BOM2
+Assembly
+Repository
+Discovery
+Composition
+BOM3PingPong
+OModelIdentification
+旦
+ Sending
+ActionA
+Waiting
+ActionA
+eAuthor
+Conceptual Model
+@PatiternsofInterplay
+ActionA
+百@PingPong
+V
+@ActionA
+ActlonB
+ActionB
+@ActionB
+Waiting
+ActionB
+Sending
+ State Machines
+BOA
+Sending
+(b)
+(c)
+(d)
+Waiting
+ Entity Type
+日 Entity Type
+B
+Waiting
+name
+A
+name
+B
+Sending
+semantics
+semantics
+idtag
+id2
+idtag
+Entity Types
+id3
+OA
+ characteristic
+name
+ characteristic
+name
+OB
+Message
+Message
+白?
+Event Types
+ID
+ID
+?EventA
+(e)
+(f)
+?EventB
+Model Mapping
+ Event Type
+ Event Type
+Entity Type Mappings
+name
+EventA
+name
+EventB
+EventType Mappings
+triggerCondition
+triggerCondition
+Object Model Definition
+semantics
+idtag
+semantics
+@ objects
+id1
+idtag
+tp!
+Interactions
+日 sourceCharacteristic
+name
+ sourceCharacteristic
+name
+由DataTypes
+A.ID
+B.ID
+@ Notes
+?targetCharacteristic
+name
+? targetCharacteristic
+name
+B.ID
+A.ID
+日contentCharacteristic
+name
+contentCharacteristic
+name
+A.Message
+B.Message
+(a)
+(g)
+(h)Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+42 
+ 
+2.7.3 Model Mapping and Object Model Definition 
+The model mapping provides a mechanism for mapping between the elements of the 
+conceptual model and the class structure elements of the Object Model Definition 
+that are described using HLA OMT 13 specification constructs. The object model 
+definition defines the structure of an object and interaction class, and their associated 
+attributes and parameters. HLA Object classes include HLA attributes and HLA 
+interaction classes include HLA parameters. These parts of BOM are not used in this 
+thesis, however interested readers can find more details in [58], [59], [60], [61], [62]. 
+2.7.4 Formal specification for the Compositon of BOM 
+Unlike DEVS, BOM does not have a graphical and mathematical formalism for 
+specifying how components are composed (even though parts of BOM such as state-
+machine and POI can be represented in UML and BOM documents can be 
+described using XML). This initiates a need for a graphical and formal representation 
+of BOM composition.  
+In this section, we introduce a formal and graphical specification of BOM14. We 
+define two types of BOM: (i) Basic BOM and (ii) Composed BOM. A basic BOM is 
+an undividable atomic BOM component, with an assumption that it represents only 
+one conceptual entity at the most. A composed BOM is a hierarchical combination 
+of basic and other composed BOM.  
+Basic BOM  
+We propose that a basic BOM (BB) can formally be defined as: 
+ Where: 
+ EnT is an entity type. We assume that a basic BOM has only one entity. EnT is 
+defined as: 
+     EnT =  Name {Characteristic: Type} 
+Where Name is the name of an entity uniquely defined by an identifier15 and characteristic is a set of 
+attributes of an entity. Each characteristic is uniquely defined by an identifier and has a type16 
+ 
+ EvT is a set of event types, each with sender, receiver and content 
+ 
+Evt = {(Name, Sender, Receiver, {Content: Type}) | Name ∈ Identifier, Sender 
+& Receiver ∈ EnT, Content∈ Identifier: Type ∈ type} 
+ 
+ S is a set of states, each has an exit-condition and a next state: 
+     S = {(Name, ExitCondition{Action, NextState})} | Name ∈ Identifier, Action ∈ 
+Act, NextState ∈ S 
+ 
+ 
+                                                 
+13 High Level Architecture Object Model Template 
+14 These concepts are not new and exist in literature for other component-based approaches [21]. In 
+this thesis, their application in BOM is intended for facilitating specification and ease of understanding 
+15 An identifier is a unique sequence of letters & digits, starting with a letter. 
+16 Type := Integer | String | Double | Complex 
+BB = 〈 EnT, EvT, AcT,  S 〉 
+(2.3) 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+43 
+ 
+ AcT is a set of actions, each has name, sender, receiver and an associated event: 
+     AcT= {(Name, Sender, Receiver, Event) | Name ∈ Identifier, Sender & Receiver 
+∈ EnT, Event ∈ EvT} 
+ 
+ 
+Composed BOM  
+A composed BOM (CB) can formally be defined as: 
+Where: 
+ AcTIN is a set of input actions that are received from other BOM. This set can be 
+empty if the Composed BOM is closed. 
+    AcTIN = {(Name, Sender, Receiver, BOM) | Name ∈ Identifier, Sender &              
+Receiver ∈ EnT, BOM ∈ File} 
+ 
+ AcTOUT is a set of input actions that are sent to other BOM. This set can also be 
+empty if the Composed BOM is closed. 
+AcTOUT = {(Name, Sender, Receiver, BOM) | Name ∈ Identifier, Sender & Receiver 
+∈ EnT, BOM ∈ File} 
+ 
+ POI is the pattern of interplay that defines how basic or composed BOMs are 
+connected to each other (through actions). It maps a list of send actions to a list 
+of receive actions. ‘ ! ’ symbol means send and ‘ ? ’ symbol means receive. 
+POI = {({!AcTSEND} , {?AcTRECV})} | AcTSEND & AcTRECV ∈AcT 
+ 
+ 
+ 
+Example 
+As an example, BOMs from Figure 10 can formally be represented as: 
+BB0 = 〈 EnT, EvT, AcT, S 〉 where: 
+EnT = EntityA {C0(Message:String)} 
+EvT = {E0(EventA, BB0, BB1, BB0.C0), { E1(EventB, BB1, BB0, BB1.C0)} 
+Act = { A0(ActionA, BB0, BB1, E0), A1(ActionB, BB1, BB0, E1)} 
+S = { S0(Sending, A0, S1), S1(Waiting, A1, S0)} 
+ 
+Table 1: Entity A 
+ 
+ 
+ 
+ 
+CB = 〈 AcTIN,  AcTOUT ,  POI 〉 
+(2.3) 
+
+Chapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+44 
+ 
+ 
+BB1 = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = EntityB {C1(Message:String)} 
+EvT = {E2(EventA, BB0, BB1, BB0.C0), { E3(EventB, BB1, BB0, BB1.C1)} 
+Act = { A2(ActionA, BB0, BB1, E2), A3(ActionB, BB1, BB0, E3)} 
+S = { S2(Waiting, A2, S3), S3(Sending, A3, S2)} 
+ 
+Table 2: Entity B 
+ 
+Similarly a composed BOM CB0 can be formally described as: 
+CB0 = 〈 AcTIN,  AcTOUT ,  POI 〉where: 
+AcTIN = ∅ (since there is no incoming actions from any other BOM) 
+AcTOUT = ∅  
+POI = {I/O0(!A0 , ?A2), I/O1(!A3, ?A1)} 
+ 
+Table 3: Composed BOM 
+ 
+We propose a graphical notation for representing basic BOM and their composition 
+shown in Figure 11. In this figure two basic BOM EntityA and EntityB are composed.  
+Figure 11: Composed BOM 
+The general information of a component such as entity name, characteristics, actions 
+and states are defined in the main block. In the lower block the states and their 
+transitions (with blue arrow) are shown. Each transition is mapped with actions (in 
+red arrow) with parameter labels (the IDs of characteristics).  The direction of the 
+arrow shows the type of the associated action (send or receive). The composition of 
+BOMs is shown through connectors (in green color).  
+ 
+
+A
+Action Connector
+EntityA
+EntityB
+S
+State Connector
+Characteristics:
+Characteristics:
+C0 - Message1 String
+C1 - Message2 : String
+Initial State
+Actions:
+Actions:
+Exit condition
+A - ActionA
+A2 - ActionA
+A1 - ActionB
+A3 - ActionB
+ State Transistion
+States:
+States.
+ SO=Sending
+S2=-Waiting
+Input/Output
+ S1-Waiting
+ S3-Sending
+connection
+A0
+1
+Waiting
+Sending
+Sending
+WaitingChapter 2  
+ 
+Component Based Modeling and Simulation 
+   
+Page 
+45 
+ 
+2.7.5 Summary 
+In this thesis, we harness the capability of BOM as a conceptual modeling 
+framework, because it provides a component standard using an XML specification; 
+gives guidelines for the further development of the executable model and helps 
+determine the appropriateness of the model or its parts for model reuse; and most 
+importantly due to its strong support for syntactic and semantic composability. It will 
+be shown, how BOM with its existing potential can be facilitated by composability 
+evaluation for accurate and rapid construction and modification of its corresponding 
+federates in HLA based simulations and hence brings forth an improvement in the 
+distributed simulation community. 
+ 
+ 
+ 
+
+   
+Page 
+46 
+ 
+Chapter 3 
+Executable Modeling Formalisms 
+ 
+In this chapter two popular model description formalisms are discussed namely Petri Nets and 
+Communicating Sequential Processes (CSP)17, which are normally used for modeling, execution (or 
+simulation) and verification of concurrent systems. This chapter provides an introduction, theory, 
+properties, classification, modeling methods and analysis techniques of PN and CSP. PN and CSP 
+are both considered as a part of solution domain in this thesis, because of their impressive 
+accumulation of knowledge in concurrency modeling and analysis techniques. These aspects are 
+imported in this thesis and used for composability verification.  
+ 
+PN and CSP formalisms are relatives since they are used to model same class of 
+systems called concurrent systems. Unlike other systems such as transitions systems 
+or automata, the formalisms of concurrent systems are strongly based on 
+concurrency theory. One of the major contributors of concurrency theory are: Carl 
+Adam Petri who initiated concept of interacting sequential processes and introduced 
+Petri Nets; C. A. R. Hoare who focused on developing programming language (CSP) 
+for concurrent systems; and Robin Milner who introduced Calculus of 
+Communicating System (CCS) and  π-Calculus. These are variants of approaches for 
+formally modeling concurrent systems and are the member of the family of 
+mathematical theories of concurrency known as process algebras, or process calculi. 
+CSP is also a member of process algebra. The main difference between PN and CSP 
+is that the former are based on graphs, while the latter are based on a textual 
+description. However both offer strong formal semantics for modeling executable 
+systems and share a broad pool of knowledge of theoretical principles and practical 
+techniques for the analysis and verification of models of complex behavior. In this 
+thesis, we propose using these two formalisms to model executable form of 
+components and study their composability.  
+3.1 
+Petri Nets 
+PN were introduced by Carl Adam Petri (and named after him) in 1962. They 
+provide an elegant and useful graphical and mathematical formalism [24]. With PN 
+the main idea is to represent states of subsystems separately. In this way, the 
+distributed activities of a system can be represented very effectively. PN are widely 
+used for modeling and control in a variety of the sorts of systems. Particularly, in 
+Discrete Event Dynamic Systems (DEDS) 18 in which many properties such as 
+synchronization, sequentiality (producer-consumer problem), concurrency and 
+                                                 
+17 The "Sequential" word of the CSP name is now something of a misnomer, since modern CSP 
+allows component processes to be defined both as sequential processes, and parallel [Wikipedia]. 
+18 Examples of DEDS are air traffic control systems; automated manufacturing systems; computer 
+and communication networks; embedded and networked systems; and software systems etc. The 
+activity in these systems is governed by operational rules designed by humans and their dynamics is 
+often driven by asynchronous occurrences of discrete events [67]. 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+47 
+ 
+conflict (mutual exclusion) concurrency, and choices can be well presented and 
+analyzed using PN [66]. Their structural and behavioral properties have been 
+successfully exploited for solving various problems of complex and dynamic systems. 
+Significant progress in these directions was made over three decades. Most essential 
+features of PN are the principles of locality, concurrency, graphical and algebraic 
+representation. They can be used not only for the specification and analysis of the 
+structural system design but also for design of the system behavior. [66], [67].  
+PN present two interesting characteristics. Firstly, they make it possible to model and 
+visualize systems with complex behaviors including parallelism, concurrency, 
+synchronization and resource sharing. Secondly the properties of these nets, their 
+analysis and theorems have been extensively studied [68]. 
+ 
+3.1.1 PN Definitions and Concept 
+In PN, two basic elements of modeling are places and transitions. Events are 
+associated with transitions which occur when some conditions are satisfied. 
+Information related to these conditions is contained in places. There are two types 
+of places namely: Input places and Output places. Input places are associated with 
+the conditions required for this transition to occur. Output places are associated with 
+conditions that are affected by the occurrence of this transition [25]. Transitions, 
+places, and certain relationships between them define the basic components of a 
+Petri net graph. A PN graph has two types of nodes, places and transitions, and 
+arcs connecting these. It is a bipartite graph in the sense that arcs cannot directly 
+connect nodes of the same type; rather, arcs connect place nodes to transition nodes 
+and transition nodes to place nodes [25]. 
+3.1.2 Petri net graph 
+Mathematically a PN is a 5 tuple: PN = 〈P, T, F, W, M0〉 where: 
+ P is a finite set of places P = {p1, p2… pm} represented as oval shaped node in the 
+PN graph 
+ T is a finite set of transitions T = {t1, t2… tn} represented as a line or a rectangular 
+shaped node in the graph  
+ F is a flow function such that F ⊆ (P ×T)∪(T×P) →N 19  
+ W: F →N + where N∈{1, 2, 3…} is arc weight function. 
+ M0: P→N is a function called the initial marking, where each element M0(p) has 
+N number of tokens20 initially in place p where N is a set of non-negative integers. 
+ For each transition t∈T a set of input places denoted as •t are those places which 
+are connected to t through incoming arcs: 
+ Similarly, for each transition t∈T a set of output places denoted as t• are those 
+places to which t is connected through outgoing arcs: 
+                                                 
+19Such that P∩T= ∅ (i.e. P&T are disjunctive sets) and P ∪ T≠∅ (i.e. neither P nor T are isolated).  
+Also an arc can be connected from place to transition (input arc) or from transition to place (output 
+arc) but not to the node of same type. 
+20 In classical PN, tokens are represented as black dots. They are assigned to, and can be thought to 
+reside in, the places of a Petri net. 
+•t = {pi | (pi, t) ∈F} 
+(3.1) 
+t• = {pi | (t, pi)∈F} 
+(3.2) 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+48 
+ 
+Definition: Marking 
+A marking is an assignment of tokens to the places of a PN. The number and 
+position of tokens defines a system state, and it may change when the tokens move. 
+This movement of tokens due to the firing of transitions causes the execution of a 
+PN [26]. The marking M can be defined as an n-vector, M = (m0, m1, m2 … mn), 
+where n = |P| (no. of places), and each mn ∈ N, i = 1...n.  The vector M gives for 
+each place pi in a PN the number of tokens in that place.  
+Definition: PN State-space 
+The state of a PN model is defined by its marking. The firing of a transition 
+represents a change in the marking of the net. The state space of a PN with n places 
+is the set of all markings.  State-space will be discussed in detail later in this chapter. 
+Definition: Enabling of a Transition 
+A transition t in a given PN is called enabled or fire-able by a marking Mi  iff for each 
+input place p∈•t its marking is equal or greater than the weight of the arc from it to t, 
+(or t has no input place). Mathematically, a transition t is fire-able iff 
+Definition: Firing of a Transition 
+If a transition t is enabled, it may fire by removing W(p, t) number of tokens from 
+each input place p and putting W(t, p’) tokens in each output place p’, due to which 
+a new marking Mn+1 is generated. 
+Mn+1 is immediately reachable from Mn. Mn is reachable from M0 if firing a sequence 
+σ = t1, t2 … tk of enabled transitions leads M0 to Mn, written as M0
+σ→ Mn 
+ 
+ 
+Example21 
+ 
+ 
+Consider the PN model PN = 〈P, T, F, W, M0〉 as shown in Figure 12 where: 
+P = {p1, p2, p3, p4, p5} and T = {t1, t2, t3, t4},  
+Let W = 1 for all arcs  
+Initial marking M0 = [1 0 0 0 0] 
+                                                 
+21 This example is inspired from [68] 
+∀p ∈ •t | M(p)≥W(p, t) ∨ •t=∅ 
+(3.3) 
+Mn+1 
+𝑡→ Mn | M(p’) = M(p) - W(p, t) +  W(t, p’) ∀p∈P 
+(3.4) 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+49 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+Figure 12: Transition firing sequence  (acquired from [68]) 
+ 
+σ1: M0 = [1 0 0 0 0]  
+𝑇1
+��  M1 = [0 1 0 1 0]  
+𝑇2
+��   M2  [0 0 1 1 0]  
+𝑇3
+��    M4  [0 0 1 0 1]   
+𝑇4
+��   
+M4  [1 0 0 0 0] 
+ 
+ 
+σ2: M0 = [1 0 0 0 0]   
+𝑇1
+��  M1 = [0 1 0 1 0]  
+𝑇3
+��   M3  [0 1 0 0 0]  
+𝑇2
+��   M4  [0 0 1 0 1]   
+𝑇4
+��   
+M4  [1 0 0 0 0] 
+ 
+In this example there are two possible transition firing sequences σ1= T1, T2, T3, T4 
+and σ2 = T1, T3, T2, T4 
+3.1.3 Properties of PN 
+Just like other models, PN are constructed from informal requirement specifications, 
+which is not a trivial task, and requires a great deal of modeling experience. If a 
+system being modeled is very complex, a PN model may differ considerably from its 
+original specification. A model can only be useful if it is logically correct with respect 
+to its specifications [69]. Different concepts of correctness exist. A system is said to 
+be correct when two aspects, namely the specification and the implementation, are 
+equivalent, or when the system satisfies a set of desirable properties. These desirable 
+properties allow the system designer to identify the correctness of the system [69].  
+In PN literature a “basic kit of PN properties” is referred to a set of properties that 
+are related to frequently occurring problems or the key issues related to the logical 
+structure and behavior of complex systems, therefore they are classified into two 
+main categories namely (i) Structural Properties and (ii) Behavioral Properties. It is 
+important to note that fulfillment of these properties answer many questions of 
+
+m1m2mChapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+50 
+ 
+system correctness, therefore they contribute in the analysis of PN models. Some of 
+the selected behavioral PN properties are listed and briefly discussed informally22 
+below.  
+Reachability 
+Reachability is a fundamental property for studying the dynamic behavior of a 
+system. In PN, reachability property is studied to analyze if a particular system state 
+(in terms of markings) can be reached or not. A marking Mn is said to be reachable 
+from an initial marking M0 if there exists a sequence of firings that transforms M0 to 
+Mn. In reachability analysis, a set of all possible firing sequences from M0 are 
+populated in a reachability graph R(N, M0) and the reachability problem for PN is the 
+problem of finding if a given marking Mn ∈ R(N, M0) [70]. 
+Boundedness 
+In classical systems theory, a state variable that is allowed to grow to infinity is 
+generally an indicator of instability in the system [25]. Therefore it is desirable that a 
+system holds boundedness. A PN is said to be bounded (or k-bounded) if the 
+number of tokens in each place does not exceed a finite number k for any marking 
+reachable from initial marking, i.e., M(p) ≤ k for every place p and every marking Mn 
+∈ R(N, M0) [70]. 
+Deadlock-free and Liveness 
+A PN is said to be deadlock-free if from any reachable marking at least one transition 
+can always occur. A stronger condition than deadlock-freeness is liveness. A 
+transition is live if it is potentially fire-able in all reachable markings. In other words, 
+a transition is live if it never loses the possibility of firing. A net is live if all 
+transitions are live [69]. 
+Reversibility 
+A PN is said to be reversible if, from each marking Mn, the initial marking M0 is 
+reachable. Thus, in a reversible net one can always get back to the initial marking or 
+state [70].  
+Fairness 
+Fairness has different meanings and understanding in literature. In specific terms, 
+fairness means to give some contenders an equal number of chances, such that no 
+one proceeds for more than “k-times” without letting the others to take their turn. In 
+PN s, two transitions tl and t2 are said to be in a bounded-fair (or B-fair) relation if 
+the maximum number of times that either one can fire while the other is not firing is 
+bounded. A PN is said to be a B-fair net if every pair of transitions in the net are in a 
+B-fair relation [70].  
+Mutual Exclusion 
+This property captures constraints such as the impossibility of a simultaneous access 
+of a critical section (resource) by two or more processes. In PN, mutual exclusion 
+can be defined in terms of places or transitions. Two places p and q are mutually 
+                                                 
+22 In literature these properties are discussed in detail with mathematical definitions and proofs [70]. In 
+this chapter they are only discussed for background concept. Some of these properties are used later 
+in this thesis. 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+51 
+ 
+exclusive in a PN if their token counts cannot be both positive in the same marking, 
+i.e., ∀m ∈ RS m(p)·m(q) = 0. Similarly, two transitions in a PN are mutually 
+exclusive if they cannot be both enabled in any marking [71]. 
+ 
+Some of the important structural properties of PN are defined below: 
+Controllability: 
+A PN is said to be completely controllable if any marking is reachable from any other 
+marking [70]. 
+Conservativeness:  
+A PN N is said to be (partially) conservative if there exists a positive integer y(p) for 
+every place p such that the weighted sum of tokens, is a constant, for every marking. 
+Given a PN model, we are often required to ensure conservation with respect to 
+certain weights representing the fact that resources are not lost or gained. 
+Persistence 
+A PN is said to be persistent if, for any two enabled transitions, the firing of one 
+transition will not disable the other. A transition in a persistent net, once it is enabled, 
+will stay enabled until it fires [25]. 
+ 
+3.1.4 PN Analysis 
+The major strength of PN is the modeling of systems that exhibit concurrency. 
+However modeling by itself is of little use. It is necessary to be able to analyze the 
+modeled system. The analysis leads to important insights into the structure and 
+behavior of the modeled system [26].  There are many techniques available for the 
+analysis of PN models and can be employed for verification depending upon the 
+nature of the model. Each technique may also have different variants. In this section 
+two of the most commonly used techniques for the analysis of a PN model are 
+discussed:  
+ 
+Figure 13: Petri Net Analysis Techniques 
+ 
+These techniques provide solutions and mechanism for verifying the properties 
+mentioned in the previous section. In this thesis, these techniques are selected for 
+composability verification and their application is shown in Part II with suitable 
+examples. In this chapter, they are briefly explained and discussed, with their 
+advantages and limitations. 
+Petri Net  
+Analysis Techniques 
+Algebraic Method 
+State-Space 
+Analysis 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+52 
+ 
+Algebraic Method 
+This technique is also called Linear-Algebraic Technique (or Linear Invariant due to 
+its abundant use of invariants).  In the framework of using algebraic techniques for 
+reasoning about PN, solving a PN problem is reduced to finding a solution for an 
+algebraic equation associated with the PN [24]. Due to the nature of this technique, 
+the method is in general efficient (and in most cases, polynomial in the size of the 
+PN). The dynamic behavior of PN models can be described by algebraic equations. 
+In order to work with Algebraic method, the following basic concepts are applied: 
+Matrix Definitional Form (MDF)  
+A PN model has a Matrix Definitional Form (MDF) that consists of three n×m2F23 
+matrices:  
+(i) Output matrix A+ 
+ 
+i.e., if pj is connected to the output of ti then 𝒂𝒊𝒋
++ is equal to the weight of output arc; 
+0 otherwise [70].  
+ 
+(ii) Input matrix A- 
+i.e., if pj is connected to the input of ti then 𝒂𝒊𝒋
+− is equal to the weight of output arc; 0 
+otherwise [70].  
+ 
+(iii) Incidence matrix A 
+In the incidence matrix A, each entry aij represents the change of tokens in place j 
+when transition i fires once [70].  
+ 
+Firing Count Vector 
+A marking Mk is an m × 1 column vector. The jth entry of Mk denotes the number of 
+tokens in place j after the kth firing in some firing sequence. An n×1 column vector X 
+of nonnegative integers is called firing count vector, where the ith entry of X denotes 
+the number of times transition t must be fired to transform Mk-1 to Mk [70]. 
+State Equation  
+State equation for a PN is written as: 
+Where:  
+Mk-1 is the current marking 
+                                                 
+23 n×m refers n transitions and m places. 
+A+ = [𝒂𝒊𝒋
++] n×m, where 𝒂𝒊𝒋
++ = w(ti, pj); if pj ∈ ti•, and i ∈ n; j ∈ m 
+(3.5) 
+A- = [𝒂𝒊𝒋
+−] n×m, where 𝒂𝒊𝒋
+− = w( pj , ti); if pj ∈•ti 
+(3.6) 
+A =  A+ - A- , where [𝒂𝒊𝒋] = [𝒂𝒊𝒋
++ − 𝒂𝒊𝒋
+−]  
+(3.7) 
+Mk =  Mk-1 + A.X  
+(3.7) 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+53 
+ 
+Mk is the new marking 
+A is incidence matrix  
+X is the firing count vector  
+Example 
+An example of a Producer-Consumer PN model is shown in Figure 14. 
+ 
+Figure 14: Producer Consumer Example 
+ 
+Using equation 3.7 the incidence matrix A of this model is calculated as follows: 
+ 
+A+ T1 
+T2 
+T3 
+T4 
+P1 
+1 
+0 
+0 
+0 
+P2 
+0 
+1 
+0 
+0 
+P3 
+0 
+1 
+0 
+0 
+P4 
+0 
+0 
+0 
+1 
+P5 
+0 
+0 
+1 
+0 
+ 
+ 
+ 
+ 
+ 
+- 
+ 
+A- 
+T1 
+T2 
+T3 
+T4 
+P1 0 
+1 
+0 
+0 
+P2 1 
+0 
+0 
+0 
+P3 0 
+0 
+1 
+0 
+P4 0 
+0 
+1 
+0 
+P5 0 
+0 
+0 
+1 
+ 
+ 
+ 
+ 
+ 
+= 
+ 
+A 
+T1 
+T2 
+T3 
+T4 
+P1 
+1 
+-1 
+0 
+0 
+P2 
+-1 
+1 
+0 
+0 
+P3 
+0 
+1 
+-1 
+0 
+P4 
+0 
+0 
+-1 
+1 
+P5 
+0 
+0 
+1 
+-1 
+ 
+ 
+ 
+ 
+ 
+Table 4: Incidence Martic A 
+ 
+ 
+In this model, the initial marking is [1 0 0 1 0]. With a firing sequence σ = t2, t1, t2 
+the firing count vector will be [1 2 0 0]. Using the state equation, the marking Mx can 
+be generated as follows: 
+ 
+ 
+M0 
+P1 1 
+P2 0 
+P3 0 
+P4 1 
+P5 0 
++ 
+ 
+A 
+T1 
+T2 
+T3 
+T4 
+P1 
+1 
+-1 
+0 
+0 
+P2 
+-1 
+1 
+0 
+0 
+P3 
+0 
+1 
+-1 
+0 
+P4 
+0 
+0 
+-1 
+1 
+P5 
+0 
+0 
+1 
+-1 
+ 
+ 
+ 
+ 
+ 
+. 
+ 
+X 
+T1 1 
+T2 2 
+T3 0 
+T4 0 
+ 
+ 
+ 
+= 
+ 
+Mx 
+P1 0 
+P2 1 
+P3 2 
+P4 1 
+P5 0 
+Table 5: State equation 
+ 
+ 
+ 
+Figure 15 graphically illustrates, how a firing sequence of σ = t2, t1, t2 can lead M0 to 
+M3. Green color highlights the firing of a transition. It can be noted that the marking 
+M3 in the lower right corner matches the marking generated by matrix state-equation 
+in Table 5. 
+
+P4
+P3
+T
+2.
+T4
+P2
+P5Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+54 
+ 
+ 
+Figure 15: M0 to M3 throguh firing sequece σ = t2, t1, t2 
+State equation alone can only help to algebraically compute a future marking. In 
+order to analyze the model algebraically, some more concepts are used, such as PN 
+Invariants.  
+ 
+PN Invariants 
+Occurrences of transitions transform the token distribution of a net, but they 
+often respect some global properties of markings, regarded as Linear Invariant 
+Laws. Invariants are very useful for analyzing structural and behavioral properties 
+of PN. From an initial marking, the marking of a PN can evolve by the firing of 
+transitions (and if there is no deadlock) the number of firings is unlimited. 
+However, not just any marking can be reached, all the reachable markings have 
+some properties in common; a property which does not vary when the transitions 
+are fired is said to be invariant. Similarly, not just any transition sequence can be 
+fired; some invariant properties are common to the possible firing sequences. 
+Hence, invariants enable certain properties of the reachable markings and firable 
+transitions to be characterized, irrespective of the evolution.  
+Figure 16 illustrates a PN model of different seasons in a year. It can be seen that, 
+regardless of the change of seasons, there will always be one and only one token 
+for all 4 places. Thus at all times, M(p1) + M(p2)  + M(p3)  + M(p4)  = 1. This 
+invariant property has an obvious meaning that at all time there is one and only 
+one season [68]. It also means that the net is structurally bounded. 
+ 
+Figure 16: Seasons in a year (acquired from [68]) 
+ 
+
+Spring
+T1
+Summer
+P
+J
+T4
+T3
+Winter
+AutumnTChapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+55 
+ 
+ 
+There are two important types of invariants of PN: 
+P-Invariant 
+Place Invariants formalize invariant properties regarding places in PN, e.g., if in a set 
+of places the sum of tokens remains unchanged independently of any firing, then this 
+set can define a place invariant. They are useful to evaluate structural properties of 
+PN. In simple words, a place belonging to a P-invariant is bounded [24], [70]. 
+A P-invariant exists in a PN if 
+Where y is an m × 1 column vector of integers such that ∃ y = (y1, y2 … yn) > 0 i.e., 
+has at least one positive non-zero entry [71]. It means the firing of any transition does 
+not change the weighted sum of tokens in the PN. More generally, a vector y is called 
+P-Invariant if  
+A . y =  0 
+It is easy to see that if there is a P-invariant, for all p ∈ P, then the PN is guaranteed 
+to be structurally bounded. Hence, place invariants can be used for reasoning about 
+structural boundedness [24]. P-invariant is a P-semi-flow if every element of it is 
+non-negative [67]. 
+T-Invariants 
+Transition Invariants on the other hand formalize properties regarding transition 
+firing sequences applicable to a PN. They are useful to evaluate behavioral properties 
+such as liveliness and fairness [24], [70]. 
+A n × 1 firing count vector X, is called a T-Invariant if 
+A . X =  0 
+i.e., firing each transition the number of times specified in X, brings the PN back to 
+its initial marking M0 [24]. T-invariant is a T-semi-flow if every element of J is non-
+negative [67]. 
+A T-Invariant X is a minimal T-invariant, if there is no other T-invariant X′ such that 
+x′i ≤ xi for all i∈T. There can be multiple T-invariants for a PN. A minimal T-
+Invariant is called the Reproduction vector of the net. 
+The intrinsic difference between P- and T-invariants are the facts that all places in a 
+PN if covered by P-invariants is a sufficient condition for boundedness, whereas the 
+existence of T- invariants is only a necessary condition for a PN model to be able to 
+return to a starting state, because there is no guarantee that a transition sequence 
+with transition count vector equal to the T- invariants can actually be fired [71].  
+Advantages and Disadvantages 
+The advantage of algebraic analysis is that the net structure is much less than the 
+number of reachable markings and therefore there is no risk of state-space explosion. 
+Various properties of PN consequently can be proven using linear algebraic 
+techniques. However the weakness of this method is that it only entertains limited set 
+of properties and provides only sufficient or necessary conditions. Also this method 
+� 𝑚 . 𝑦𝑝 = � 𝑚0 . 𝑦𝑝
+𝑛
+𝑝=1
+ 
+𝑛
+𝑝=1
+ ∀m ∈ R(N, m0)  
+(3.8) 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+56 
+ 
+involves complex underlying mathematical theorems, each one different for different 
+property verification and thus cannot be generalized for automated reasoning.  
+State-Space Analysis 
+State space analysis is one of the most prominent approaches for conducting formal 
+analysis and verification. In contrast to algebraic techniques, it is relatively simpler 
+approach for analyzing the behavior of a model. The basic idea in this approach is to 
+calculate all possible system states and the events which cause the change of states 
+and represent them in a directed graph. When the graph is completely constructed, 
+different search techniques can be applied to analyze the model.  
+In PN terms, this method is also commonly known as Reachability graph analysis. 
+The state-space analysis of a PN model is performed by exhaustively generating all 
+the reachable markings from a given initial marking, and then reasoning about the 
+PN properties of the model by examining the structure of the reachability graph. 
+The reachability graph consists of vertices which correspond to reachable markings 
+and of arcs corresponding to firing of transitions resulting in the passing from one 
+marking to another. A simple example of reachability graph is shown in Figure 17. 
+ 
+Figure 17: (a) PN Model (b) Reachability Graph (acquired from [68]) 
+ 
+In some cases, the construction of reachability graphs becomes infinite if the PN or 
+some of its parts are repetitive and the net is unbounded, or in other words the PN 
+has infinite number of reachable markings. Therefore instead of keep on 
+constructing nodes of the graph infinitely, an alternative technique is used, in which a 
+finite graph is constructed by abstracting out certain details and inserting the symbol 
+ω (the symbol of “infinity”) to representing the marking of an unbounded place. This 
+is called cover-ability graph. The coverability graph of the Producer-Consumer PN 
+model is shown in Figure 18 
+ 
+Figure 18: Producer Consumer PN Model and its Coverability Graph 
+ 
+It can be seen that the markings in which place P3 is unbounded contain ω symbols.  
+
+P
+0
+2
+[]}[]
+m2
+mo
+m1
+0
+1
+m3
+(a)
+(b)P4
+P3
+T
+2.
+T4
+P2
+P5(1,0,0,1,0)
+t1
+(0,1,0,1,0)
+t3
+t4
+ti
+(0,1,1,1,0)
+(0,1,0,0,1)
+(1,0,0,0,1)
+t1
+t3
+t2
+(1,0,0,1,0)
+(1,0,0,0,1)
+(0,1,0,0,1)
+t4
+t,
+t2
+ti
+(0,1,0,1,0)Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+57 
+ 
+A constructed state space can help in answering a large set of analytical questions 
+concerning the structure and behavior of the model such as verifying deadlock-
+freedom, absence of live-locks; presence of liveness, the possibility of being able to 
+reach good states, and impossibility of reaching bad states and the guarantee of 
+fulfilling the objectives.  Following are some examples of how state space analysis 
+help in model verification:  
+Boundedness 
+The problem of boundedness is easily solved using a coverability tree with an 
+assumption that a PN is bounded if the symbol ω never appears in its coverability 
+tree. Since ω represents an infinite number of tokens in some place, therefore its 
+absence can guarantee that the PN is structurally bounded [25]. 
+Deadlock freedom 
+A deadlock freedom problem is solved, if there is no node in the graph (which is not 
+a final node), and yet it does not have an outgoing arc meaning there is no further 
+enabled transition. Existence or one or more such nodes shows that the model has 
+possibility of deadlock and can also help to find out the exact cause of it. 
+Live lock freedom 
+Similarly, a live-lock can be detected using state space analysis. For concurrent 
+systems, a process is tasked to perform some particular actions [72]. These actions 
+are normally intended to make progress and are called progress actions. A live lock is 
+detected, if there exists a cycle within the reachability graph, in which no progress 
+action is being executed.  
+State Reachability 
+Reachability of good states (or bad states) can be guaranteed using state space 
+analysis. A state is reachable if there is a valid firing sequence that leads to that state 
+from the initial marking. (In graph, there exists a path from the initial node to the 
+corresponding node of the desired state). There could be multiple paths in a graph 
+that reach the desired state. A shortest path analysis can be useful to analyze the 
+minimum number of steps required to reach that state.  
+For details on how state space analysis are conduced, interested readers are 
+recommended to refer to a very informative step by step tutorial on PN state space 
+analysis [73]. 
+ 
+Advantages and Disadvantages 
+The main advantage of state space method is that it is a way to explore all the 
+possible states of the system. Also it provides counter examples as to why an 
+expected property does not hold. Furthermore, the automatic calculation and 
+generation of state-space provides an ease of use, due to the fact that the computer 
+tool hides a large portion of the underlying complex mathematics from the user, who 
+is only required to formulate the property which is to be investigated and a suitable 
+query function to evaluate it [74]. 
+ 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+58 
+ 
+The main disadvantage of using state spaces is the state explosion problem. The 
+construction of the reachability graph is very expensive and intensive from a 
+computational point of view. This is because the size of the state space may grow 
+exponentially with respect to the size of the PN model (measured, for example, by 
+the number of places). Even relatively small systems may have an astronomical or 
+infinite number of reachable states. This problem escalates severely, when the models 
+includes time. A lot of effort has been invested in the development of reduction 
+methods to alleviate this problem. Reduction methods represent the state space in a 
+compact form. The reduction should not affect the properties of the system and they 
+should be preserved and can still be derived from the reduced state space. However, 
+due to the complexity and diversity in verification, there is no single reduction 
+method which works well in all situations. Therefore the choice of a reduction 
+method completely depends on the nature of the system being verified [74]. Some of 
+the important reduction methods are Sweep line method [75], Hash Compaction 
+Method [76], Symmetry Method [77] and Equivalence Method [78].  
+In this thesis, we propose another reduction method which suits our need 
+(Composability verification) and can help to alleviate the state explosion problem, if 
+the model under consideration becomes large and resource intensive. 
+ 
+3.1.5 PN Classes 
+The computational power of basic or classical PN is weak as it has been shown that 
+PN are not as expressive as Turing machines, making them inadequate for modeling 
+certain real-world systems. To overcome this shortcoming, a number of extended 
+PN have been introduced to enhance the expressive capabilities of PN. There are 
+different ways to classify PN. In structural sense, they can be classified into three 
+main categories [79]: 
+ 
+Level-1 PN: are characterized by 'Boolean tokens', i.e. places are marked with at most 
+one unstructured token. 
+ 
+Level-2 PN:  are characterized by 'Integer tokens', i.e. places are marked with several 
+unstructured tokens - they represent counters. 
+ 
+Level-3 PN:  are characterized by high-level tokens, i.e. places are marked with 
+structured tokens where information is attached to them. 
+ 
+There are many extensions of PN formalism. In this section we only discuss some of 
+the extensions of PN, which are used in this thesis. 
+Colored Petri Nets (CPN) 
+CPN is a level-3 extension of PN, in which places are marked with structures token 
+representing data. CPN is a graphical language for constructing models of concurrent 
+systems and analyzing their properties. CPN is a general purpose discrete event 
+language which combines the capabilities of PN, as a foundation of the graphical 
+notation and a programming language (CPN ML), which is based on Standard ML 
+[80] functional programming language, that provides the primitives for the definition 
+of data types and for specifying data manipulation routines [78].  
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+59 
+ 
+CPN is formally defined by the tuple [81]: 
+CPN = (P, T, A, Σ, V, C, G, E, I) where: 
+P is a finite set of places  
+T is a finite set of transitions such that:  P ∩ T = ∅ 
+A ⊆ P×T ∪T ×P is a set of directed arcs. 
+Σ is a finite set of non-empty color sets. 
+V is a finite set of typed variables such that: Type[v] ∈ Σ for all variables v ∈ V 
+C: P→Σ is a color set function that assigns a color set to each place. 
+G: T → Expression is a guard function that assigns a guard to each transition t 
+E: A→ Expression is an arc expression function that assigns an arc expression to 
+each arc a  
+I: P → Expression is an initialization function that assigns an initialization 
+expression to each place p. 
+Tokens of an ordinary PN have no types. With CPN it is possible to define token 
+using data types and complex data manipulation i.e., each token has attached a data 
+value called the token color. The token colors can be investigated and modified by 
+the occurring transitions [81]. 
+“CPN Tools” is a software package for the editing, simulation, state space analysis, 
+and performance analysis of CPN models [82]. The tool acts as an integrated 
+development environment (IDE) for the construction of CPN models. It provides a 
+canvas for creating PN graphs, offers features for writing CPN ML code with a 
+facility of incremental syntax checking. It also comes along with a bundled simulator 
+that efficiently handles the execution of untimed and timed nets. The most important 
+feature of CPN tool from our point of view is the generation and analysis of state 
+spaces. The analysis of state space includes various built-in state-space querying 
+functions, and support for creating analysis report which altogether greatly 
+contributes to the verification process. For further details of CPN formalism and its 
+application [78], [81] are referred.  
+ 
+Figure 19: A CPN Model 
+ 
+Figure 19 shows a basic example of a CPN model. The nodes A and B in oval shape 
+represent places. The place is initialized with three tokens of String type. The 
+rectangular shaped node represents transition. An input arc connects Place A with 
+the transition with an arc variable v of type String (to carry tokens of the same type). 
+
+1'"Token1"++
+1'"Token2"++
+1'"Token3"
+[v="Token2"]
+V
+Trans
+A
+B
+STRING
+STRING1'"Token1"++
+"Token3"
+[v="Token2"]
+1.
+1'"Token2"++
+V
+Trans
+V
+A
+B
+STRING
+STRINGChapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+60 
+ 
+Similarly an output arc connects transition to place B. The transition has a guard 
+expression that checks the token value. If the expression is true only then the 
+transition can be fired. The second part of the Figure 19 shows the result of the 
+firing of transition, i.e., the token “Token2” being deposited to place B. 
+Hierarchical CPN 
+CPN model can be organized as a set of modules; where modules can be seen as 
+black boxes which make it possible to work at different abstraction levels, 
+concentrating on one at a time.  
+Substitute Transitions 
+CPN tools offer facility to construct hierarchical CPN models. In hierarchical nets a 
+transition can represent an entire piece of net structure. Such a transition is called 
+substitution transition [82].  
+Sub-page /Super-page 
+A page that contains a substitution transition is called a super-page. When a CP-net 
+uses a substitution transition the logic that the transition represents is kept on a page 
+called a subpage [82]. 
+Ports and sockets 
+Super-pages and sub-pages are connected by ports and sockets. A socket is a place in 
+the super-page that has at least one arc between a substitution transition and a 
+socket. A port on the other hand is a place in a subpage, marked with one of the 
+port-type tags: (i) In-Port (ii) Out-Port or (iii) In/Out-Port. It is bound with a socket 
+in the main page using Port & socket assignment. This relationship is used to define how 
+a subpage should be connected with the surroundings of its super-page. Some of the 
+assignment rules are as follows: 
+• A port with an In-tag must be assigned to a socket which is an input arc of the 
+substitution transition.  
+• An Out-tag indicates that the port must be related to a socket which is an output 
+arc,  
+• I/O-tag indicates that the socket must be both an input and output arc [82]. 
+ 
+ 
+Figure 20: Hierarchical Colored Petri Net 
+ 
+Figure 20 presents an example of hierarchical CP-net. In the super-page (above), a 
+substitute transition Process is shown which represents a sub-module (below). A 
+
+1""Token1"++
+1'"Token2"++
+1""Token3"
+Process
+B
+STRING
+Process
+STRING
+Stage1
+Stage2
+Stage3
+In
+Out
+STRING
+STRING
+Q
+R
+STRING
+STRINGChapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+61 
+ 
+process has three stages, and input and an output marked with In and Out ports 
+which are connected with A and B socket places in the super-page.  
+Timed Petri Nets 
+PN with timing dependencies can be classified according to the way of specifying 
+timing constraints. These constraints can be timing intervals or single numbers, or 
+elements of the net these constraints are associated with i.e., places, transitions or 
+arcs [83]. The next criterion is an interpretation of the timing constraints. When 
+associated with a transition, the constraint can be viewed as  
+ 
+(i) Firing time  
+A transition consumes the input tokens when it becomes enabled, but does not 
+create the output tokens until the delay time associated with it has elapsed [83]. 
+ 
+(ii) Holding time  
+When the transition fires, the actions of removing and creating tokens are performed 
+instantaneously, but the tokens created are not available to enable new transitions 
+until they have been in their output places for the time specified as the duration time 
+of the transition which created them [83]. 
+ 
+(iii) Enabling time  
+A transition is forced to be enabled for a specified period of time before it can fire, 
+and tokens are removed and created in the same interval [83]. 
+Timed extensions are known also for high-level PN. One of them is timed Colored 
+Petri nets [78], in which the time concept is based on introducing a global clock used 
+to represent the model time. Tokens are equipped with time stamps, which describe 
+the earliest model times at which they can be used to fire a transition. Stamps are 
+modified according to expressions associated either with transitions, or with their 
+output arcs. Timing intervals can be interpreted as periods of non-activity of tokens, 
+and the transitions are fired according to the strong earliest firing rule [78]. 
+Formally a time PN is a tuple: 
+ 
+ 
+N = (P, T, F,m0,Eft, Lft) 
+ 
+Where: 
+(P, T, F, m0) is a PN,  
+Eft = Earliest firing time for each t∈T 
+Lft = Latest firing time for each t∈T 
+ 
+ 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+62 
+ 
+3.2 Communicating Sequential Processes 
+CSP is the second formalism that is selected in this thesis for the modeling of 
+executable components. CSP is a language developed by Sir Charles Antony Richard 
+Hoare [84]. It aimed to be used for specification and reason about the concurrent 
+interaction of the system processes. The idea of CSP was conceived for the study of 
+concurrent processes using formal notation with required expressive power and 
+algebraic laws. The formal notation and the associated algebraic laws allow the 
+process models to be controlled and analyzed. They also enable formal reasoning 
+about their correctness and prove equivalences between the processes. They also 
+provide sufficient theoretical foundations for the development of the necessary tools 
+for these purposes.  
+3.2.1 Basic Concepts and Definitions 
+The main primitives of CSP formalism are (i) Processes (ii) Events and (iii) Algebraic 
+Operators. 
+Process 
+In CSP terms, a process is an independent, self-contained, modular description of an 
+entity and a basic unit to capture behavior. A process has particular interface, 
+captured by events that are used to interact with the environment which itself is a 
+process, called the universe of the system (Σ). The environment can be viewed as a 
+system of concurrently evolving processes. In any run a process performs a sequence 
+of events. A process has a name, list of parameters and expression which determines 
+its computational logic: 
+Process (parameters) = Expression 
+Expression is behavior of a process which can be described as an occurrence of an 
+event or the sequence of some events, known as a trace. A process can only perform 
+a finite number of events in any finite time, and thus all traces have finite length [85]. 
+Events 
+The ultimate unit in the behavior of a process is an event [85]. Events characterize 
+communications or interactions. Events are abstraction of observations. Each event 
+forms an interaction between the process and its environment. If the interaction does 
+not occur then the process is blocked. Event can be defined with no data or data 
+with typed values. A set of all events of a Process P are called Alphabet of P (αP). 
+The following line describes a simple vending machine which takes in a coin and 
+dispatches a coffee every time [84]. 
+VM() = insert-coin → coffee → VM(); 
+Where VM() is a process (with no parameters) and its expression contains a sequence 
+of atomic events: insert-coin and coffee and then the process is self-referenced 
+(recursion). Events can be written in compound form, i.e., with parameters as shown 
+in the following line: 
+VM() = insert-coin.10 → coffee.1 → VM(); 
+Also there could be data operations using statement blocks inside the event body: 
+VM() = insert-coin.10{Balance= Balance +10} → coffee.1{coffee--} → VM(); 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+63 
+ 
+A statement block could be a complete sequential program contains assignment 
+statements, if-then-else clauses, for or while loops and math functions etc. 
+Input/output Channels 
+Processes may also communicate through channels. Channels are special type of 
+events, called communication events. Usually a communication on a channel results 
+from an input and output occurring in parallel. The input channel is represented by 
+‘?’ symbol whereas the output channel is represented by ‘!’ symbol. The channel 
+parameters can be send or received using the form: c ! x or c ? y 
+Algebraic operators 
+There are many different useful operators that are used to represent different notions 
+of process behavior and their compositions [85]. Some are described as follows: 
+ Prefix a → P 
+The prefix operator combines an event and a process to produce a new process. 
+ 
+ Sequential composition  P ; Q 
+It composes two processes P and Q in a sequential order i.e. the latter only starts 
+when the former terminates  
+ 
+ Deterministic Choice P � Q 
+The deterministic (or external) choice operator allows choosing between two 
+component processes, and allows the environment to resolve the choice 
+externally. 
+ 
+ Non-deterministic Choice P ⊓ Q 
+The nondeterministic (or internal) choice operator allows a choice between two 
+component processes, but does not allow the environment any control over 
+which one of the component processes will be selected. 
+ 
+ Conditional Choice if cond  P  else  Q  
+The choice depends on the evaluation of a condition to choose between P or Q. 
+ 
+ Interleaving P |||  Q 
+The interleaving operator represents completely independent concurrent activity 
+between the processes P and Q i.e., without barrier synchronization. 
+ 
+ Parallel Composition P || Q 
+The parallel composition operator represents concurrent activity between P and Q 
+that requires barrier synchronization between the component processes. If an 
+event is in the alphabet of both P and Q, then it can only occur when both 
+processes are able to engage in that event. 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+64 
+ 
+3.2.2 CSP Analysis Techniques  
+Many techniques have been developed for the analysis of CSP models however 
+Model Checking has surpassed them all in many aspects and is commonly favored by 
+most of the CSP based modeling environments. In this section Model Checking 
+technique is briefly described. 
+Model Checking 
+“Model checking is an automated technique that, given a finite-state model of a system 
+and a formal property, systematically checks whether this property holds for that model 
+[86]” 
+The instigation and rapid advancements of model checking methods is one of the 
+towering achievements in the area of model based software verification, especially 
+with the advent of difficulties faced by the computing communities when the 
+struggle of sequential program verification was followed by even more daunting 
+exertion of verifying concurrent programs [87]. The growing difficulty in error 
+tracing of such programs is due to the increase of complexity of the system behavior 
+and the arbitrariness of large portion caused by emergent system states which cannot 
+be easily tacked by ordinary testing and debugging methods.  
+Starting from late 70’s Model checking and other similar algorithmic and automata 
+theoretic approaches are the result of efforts of notable researchers who pioneered 
+different standards that can be marked as a collective foundation of principles that 
+shaped the modern model checking techniques [87].  
+Model checking became successful in different communities due to following 
+reasons: 
+ Unlike traditional testing methods it is an exhaustive approach that provides an 
+in-depth analysis of a system model to certify absence of bugs (instead of just 
+finding few of them through debugging). 
+ Model 
+checking 
+returns 
+answers 
+— 
+either 
+successful 
+outcomes 
+or 
+counterexamples showing the exact trace of errors and their causes 
+ Improvements in model checking techniques have effectively alleviated the risk of 
+state-space explosion problem [87]. 
+ Model Checking has a sound and mathematical underpinning and is based on 
+theory of graph algorithms, data-structures, and logic [86]. 
+ Model checking support formalism both for the specification of the input models 
+(such as FSM, PN, CSP or others) and the specification of system properties 
+being verified (which are mostly in the form of LTL or CTL or their extensions). 
+Therefore any 3rd party community can use a model checker as a black box 
+without knowing the insights and complexity of the process. 
+Beside its various strengths some of the weaknesses include: 
+ Most model checkers require the models to have reduced details using compact 
+and less expressive states and without specifying enumerations due to the risk of 
+state-explosion. Therefore the reduction in the system expressiveness may cost 
+extra effort and possibly lead to overlooking important features and getting 
+inadequate verification results. 
+ Despite the development of several very effective methods and improved data-
+structures to combat the state-explosion problem, models of realistic systems may 
+still be too large to verify. 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+65 
+ 
+Types of Model Checking  
+Model checking approaches are classified into two types: (i) Explicit and (ii) Symbolic 
+based on how they enumerate states [88]. 
+Explicit model checking techniques store the explored states in a hash table, 
+where each entry corresponds to a single system state. For just a few hundred states 
+the nodes in the state space graph becomes as large as ~1011 [88]. On the other hand 
+explicit model checkers support state-enumeration that gives detailed expressiveness 
+of the system states. 
+Symbolic model checking techniques store sets of explored states symbolically by 
+using efficient data structures represented by canonical structures such as Binary 
+Decision Diagrams (BDDs) [89], and traverse the state-space symbolically by 
+exploring a set of states in a single step. The use of these BDD-based methods has 
+greatly improved scalability in comparison to explicit state enumeration techniques, 
+yet they have performance degradation because  BDDs constructed in the course of 
+symbolic traversal grow extremely large, and BDD size is critically dependent on 
+variable ordering. This causes a newer trend of research towards separating Boolean 
+reasoning and representation. Hence Boolean Satisfiability (SAT) [90] has been 
+studied and explored for Boolean reasoning and efficient semi-canonical 
+representations which results in  the development of SAT-solvers which are efficient 
+and have compact representation compared to BDDs. SAT, together with efficient 
+representation, have become a viable alternative to BDDs for model checking 
+applications [88]. 
+Bounded model Checking is a model checking approach where the number of 
+steps in forward traversal of the state space are bounded and checks whether a 
+property violation can occur in k or fewer steps [88]. The approach reports either 
+“violation found” or “no violation possible within the bounded depth (i.e., k steps), 
+which can be incremented to look ahead for possible violation of the property. This 
+method is promising because it does not cause state-space explosion or at least let 
+the user control its possibility.  
+In this thesis all three model checking approaches are accompanied by the tools 
+selected for composability verification of CSP based models. 
+ 
+3.2.3 Temporal Logics 
+Logic provides formal languages containing formulas for the representation of the 
+statements and their logical reasoning within some area of application [91]. Generally, 
+a logical language is given by an alphabet of different symbols and the definition of 
+the set of formulas which are strings over the alphabet [91]. In logic, the term 
+temporal logic is used for representing and reasoning about propositions qualified in 
+terms of time. Temporal logic has found an important application in formal 
+verification, where it is used to specify system requirements. Linear Temporal Logic 
+(LTL) and Computational Tree Logic (CTL) are its two main variants. LTL formulas 
+are interpreted on computation paths. Let A and B be atomic predicates and ¬ , ∧ , 
+∨ , ↔ and True be the operators of classical logic, whereas 
+, 
+, 
+ and U are 
+the operators of linear temporal logic called Next, Always and Eventually and Until.   
+ 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+66 
+ 
+The intuitive meanings of some LTL statements are: 
+• ¬ A : A does not hold 
+• A ∧ B : Both A & B hold 
+• 
+A: A holds at the next state 
+• 
+A: A holds in all states 
+• 
+A: A will eventually hold 
+• A U B: A will hold until B holds. 
+ 
+In CTL there are additional path quantifiers ‘∃’ and ‘∀’ denoting ‘there exists a path’ 
+and ‘for all paths’, respectively. CTL formulas are interpreted on computation trees. 
+With respect to a tree the intuitive meanings of the formulas mentioned above are: 
+• ∃ 
+A: There exists a path in which A holds at the next state 
+• ∀ 
+A: For all paths A always holds in all states 
+ 
+3.2.4 Time CSP 
+CSP has been in evolution for decades. One of the major extensions of CSP is 
+devised with timing primitives, denoted as TCSP, to support time sensitive process 
+modeling [92].  In TCSP, each of the untimed CSP operators is interpreted in a timed 
+context, and two primitive timing operators are added: (i) timeout and (ii) interrupt, 
+with a Newtonian Time assumption (i.e., that all the processes have a single global 
+clock with same progress rate). 
+ Timeout P ⊳d Q   
+Timeout operator can be used to introduce delay in the processes.  
+ Timed Interrupt P △e Q 
+Interrupt is used if the process is permitted to run for no more than a particular 
+length of time. 
+ 
+The concept of TCSP is used later in this thesis to model and perform verification of 
+real-time systems. 
+ 
+3.2.5 Probabilistic Systems 
+Systems that exhibit probabilistic aspects essential for designing randomized 
+algorithms, modeling unreliable or unpredictable behavior or specifying model-based 
+performance evaluation are called probabilistic systems [86]. In order to model 
+random phenomena in such systems, transition systems are enriched with 
+probabilities. Probabilistic systems can be specified in different ways. Two very 
+popular ways are: (i) Markov chains (MC) and (ii) Markov decision processes (MPD). 
+In this thesis, we considered MPDs as specification formalism for probabilistic 
+systems because they support both nondeterministic and probabilistic choices and 
+unlike MC they can model the interleaving behavior of the concurrent processes in 
+an adequate manner [86].  
+ 
+ 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+67 
+ 
+A Markov Decision Process is a tuple 〈S, Act, P, linit, AP, L 〉 [86]. 
+Where: 
+S = Set of states 
+Act = set of actions 
+P: S × Act × S → [0, 1] is the transition probability function such that for all states 
+s∈S and actions α∈ Act: 
+� P(s, α, s′)∈{0,1}
+s′∈S
+ 
+ 
+linit: S → [0, 1] is the initial distribution such that:  
+� 𝑙𝑖𝑛𝑖𝑡(s) = 1
+s′∈S
+ 
+AP is a set of atomic propositions 
+L: S → 2AP is a labeling function 
+The concept of MDP is used later in this thesis to model and perform verification of 
+probabilistic systems. 
+3.2.6 CSP Implementation Tools 
+There are a variety of implementation support tools and languages for developing 
+CSP models such as CTJ (Java), CSP++ (C++), CSP.NET, PyCSP (Python), JCSP 
+(Java) and CSP# (C-Sharp) [93].  
+ 
+Similarly various techniques exist for CSP analysis such as: 
+• FDR2 model checker is developed by Formal Systems Europe Ltd [94]. 
+• ARC, the Adelaide Refinement Checker, is a CSP verification tool [95]. 
+• ProB is an animator and model-checker and support refinement checking and 
+LTL model-checking of CSP [96]. 
+• PAT is a model checker, simulator and refinement checker for CSP [97]. 
+ 
+In this thesis we selected PAT model checker because of its user friendly 
+environment for modeling CSP models, fast simulator and model checker and above 
+all its support for CSP extensions such as Real-Time CSP, Probabilistic CSP and 
+Real-time Probabilistic CSP. 
+3.2.7 Process Analysis Toolkit (PAT) 
+PAT is an established tool developed by National University of Singapore in 
+concurrent system verification and has been used in real-world industrial projects. 
+PAT is designed to develop, compose, simulate and analyze event-based system 
+models using an extension of CSP formalism called CSP-Sharp (or CSP#24). This 
+extension comprises of some additions such as shared variables and asynchronous 
+message passing. Moreover it supports using complex data types (such as Set, Queue, 
+and Stacks) and functions from external libraries written in C# therefore allow to 
+                                                 
+24It uses C# like syntax for the specification of CSP processes 
+
+Chapter 3  
+ 
+Executable Modeling Formalisms 
+   
+Page 
+68 
+ 
+model complex process behaviors. PAT also supports automated refinement 
+checking and model checking of LTL extended with events [98]. 
+PAT is an appropriate modeling, composition, simulation, verification and reasoning 
+framework of CSP based process models. These models can be of different nature 
+such as concurrent, real-time and probabilistic systems. The main strength of this 
+framework is that it implements various model checking techniques and provide 
+verification support for different properties. That includes general system properties 
+such as deadlock-freeness, divergence-freeness or reachability and user specific 
+properties defined in terms of LTL assertions. It also includes refinement checking, 
+model checking of real-time and probabilistic systems. To achieve good 
+performance, advanced optimization techniques are also implemented in PAT, such 
+as partial order reduction using BDD, symmetry reduction and parallel model 
+checking [97]. 
+ 
+3.3 Summary 
+In this chapter we have discussed two executable modeling formalisms namely: (i) 
+Petri Nets and (ii) Communicating Sequential Processes and their associated 
+concepts, tools and techniques. Both formalisms are used in this thesis for describing 
+executable models. The conceptual background of both PN and CSP is required to 
+understand the approach presented later in this thesis.  
+ 
+ 
+
+   
+Page 
+69 
+ 
+Chapter 4 
+Verification and Analysis 
+ 
+Verification and Validation are important aspects of any software engineering expedition. They are 
+independent procedures with different characteristics that are used to check that a program, service, 
+model or a system is correct, meets requirements specifications and that it fulfills its intended purpose. 
+They are critical constituents for achieving the necessary levels of quality assurance, and are essential 
+prerequisites for a credible and reliable use of the delivered product. The main focus of this chapter is 
+on Verification and its different analysis techniques. The aim of this chapter is to outline basic 
+concepts, principles, issues and different approaches of software verification. This chapter can be 
+viewed as a manual to understand the verification process being proposed later in this thesis. 
+ 
+The correctness of a program is a relative concept, meaning that the program is 
+doing no less than prescribed by its specification [99]. Verification, Validation and 
+Testing (VVT) in combination is a broader and more complex discipline of system 
+engineering. In M&S the combination of Verification, Validation and Accreditation 
+(VVA) is generally referred where “Accreditation” is the formal certification that a 
+model or simulation is acceptable to be used for a specific purpose [100]. 
+Nevertheless the goal is to assure the quality of the product and the impetus behind 
+this assurance is intensified when the systems are highly critical, either because they 
+are very expensive to produce, such as land rovers investigating outer planets, or 
+because human lives depend on them, such as computers controlling airplanes and 
+cars, and life assisting real-time systems in hospitals [101]. These systems need to be 
+correct, because their failure can lead to loss of human lives or enormous economic 
+losses. Moreover correct systems can be used in a wrong manner which can also 
+results in a failure. This is a general problem when systems are designed in a modular 
+fashion, and are implemented with assumptions on a new environment. A similar 
+case caused a drastic failure at the launch of Ariane-5 expendable rocket launch 
+system, because a software module was reused from Ariane-3 with certain 
+assumptions that did not hold for Ariane-5 which self-destructed just because one 
+single variable of 64 bit floating point value was erroneously converted to a 16 bit 
+integer causing the system to crash [102]. So for critical systems it is worth the effort 
+to have a guarantee that they are correct and have no errors. 
+Verification and validation aim to increase the credibility of models and simulation 
+results by providing evidence and indication of correctness and suitability. 
+Verification in particular deals with the correctness of the model perceived from a 
+real-system, whereas validation deals with the suitability or fitness of the model with 
+respect to its real-system. Testing on the other hand aims to uncover incorrectness in 
+the system. In the following section, definitions and concepts of these inter-related 
+terms are discussed.  
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+70 
+ 
+4.1 
+Some Basic Concepts in Modeling and Simulation 
+The first applied technical discipline that began to struggle with the methodology and 
+terminology of V&V was the operations research (OR) community, also referred to 
+as systems analysis or modeling and simulation (M&S) community [103]. 
+ 
+Verification 
+According to the Department of Defense (DoD) Defense Modeling and Simulation 
+Office verification is defined: as a process of determining that a model implementation 
+accurately represents the developer’s conceptual description and specification [104]. 
+ 
+In general verification refers to an evaluation process that determines whether a 
+product is consistent with its specifications or compliant with applicable regulations. 
+In M&S, verification is typically defined as the process of determining if a model is 
+consistent with its specification [29]. Verification deals with the model correctness 
+and is concerned with building the model right [28], i.e., a model which works 
+correctly and has no bugs. In principle, verification is concerned with the accuracy of 
+transforming the model’s requirements into a conceptual model and the conceptual 
+model into an executable model [29]. 
+For the sake of clarity the notions of correctness are defined as follows: 
+Correct: Free from error; accurate; in accordance with the fact, truth, or reason; Conforming to the 
+acknowledged standards of a method, routine or behavior [Oxford Dictionary] 
+Correctness 
+The degree to which a program, model or a system as a whole is free from defects in its specification, 
+design, and implementation [105] 
+The ability of a software product (or a simulation model) to perform the exact task, as defined by its 
+specification [106]. 
+We define a composed model to be correct if its structure and behavior matches its 
+specification. Correctness of a composed model is therefore relative to its 
+specifications. A software entity can exist in three apparent states of correctness 
+namely: (i) correct when it has been established correct against its specification; (ii) 
+defective when it has been established incorrect against its specification and (iii) 
+unknown when its correctness has not been established against a specification [107]. 
+In SE a software entity's specification is the sum of all its passing unit-tests [107]. We 
+define specification to be a set of goals (or objectives) and property constraints (see 
+1.3.2) that must be fulfilled by the composed model to be established as correct. 
+ 
+Validation 
+According to the Department of Defense (DoD) Defense Modeling and Simulation 
+Office validation is defined: as a process of determining the degree to which a model is an 
+accurate representation of the real world from the perspective of intended uses of the model [104]. 
+Model validation on the contrary, deals with building the right model, i.e., the model 
+which is an accurate representation of the real system [28]. Model validation is usually 
+defined to mean “substantiation that a computerized model within its domain of 
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+71 
+ 
+applicability possesses a satisfactory range of accuracy consistent with the intended 
+application of the model [108]. 
+Testing 
+Model Testing on the other hand, ascertains whether inaccuracies or errors exist in 
+the model. The objective of testing is to show that the model (or system) is incorrect 
+(rather than proving that it is correct). Testing can only find errors but cannot 
+guarantee the absence of errors; therefore it is more of an ad-hoc and inexpensive 
+method of necessity, where the correctness is established merely on the fact that all 
+tests have passed, which is insufficient and unreliable. When the test fails, it succeeds 
+in revealing an error. When a test is passed, it fails to detect an error. If a number of 
+tests fail to detect a bug, they increase a confidence level in the system even if the 
+correctness cannot be guaranteed [99]. 
+ 
+4.1.1 Verification and Validation in a Modeling Process 
+A Modeling Process has been defined by Sargent [108] as shown in Figure 21. In this 
+process Verification is referred to as an activity which ensures that the computer 
+programming and implementation of the conceptual model is correct. 
+ 
+Figure 21: Modeling Process (acquired from [108]) 
+ 
+Whereas validation is defined in three perspectives: 
+Conceptual model validity is defined as determining that the assumptions 
+underlying the conceptual model are correct and that the model representation of the 
+problem entity (simuland) is “reasonable” for its intended purpose.  
+Operational validity is defined as determining that the model’s output behavior has 
+sufficient accuracy for the model’s intended purpose.  
+Data validity is defined as ensuring that the data necessary for the model execution 
+and model experiments to solve the problem are adequate and correct [108].  
+ 
+Mike Petty in his article [29] also clarifies the difference between the two terms at 
+different stages of model evaluation process as illustrated in Figure 22. 
+
+Problem
+Entity
+Conceptual
+Operational
+Validity
+Model
+Analysis
+Validity
+Experimentation
+and
+Modeling
+Data
+Validity
+Computerized
+Computer Programming
+Conceptual
+Model
+Model
+and Implementation
+Computerized
+Model
+VerificationChapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+72 
+ 
+ 
+Figure 22: Modeling Process (acquired from [29]) 
+ 
+A simuland is the real system that is to be simulated whereas a model is a 
+representation of the simuland, developed with its intended application in mind and 
+therefore captures only the necessary abstractions of the simuland and omit others. 
+The requirements are driven by the intended application. Conceptual models 
+document those aspects of the simuland including the structural and behavioral 
+aspects such as objects, entities, events, functions, environmental phenomena etc. 
+The executable model is the computer program that can be executed and is intended 
+to simulate the simuland as detailed in the conceptual model. Therefore the 
+conceptual model can be viewed as a design specification for the executable model. 
+The results are the output produced by a model during a simulation. 
+Figure 22 presents Verification and Validation as activities that compare one thing to 
+another. Verification compares the requirements with the conceptual model. In this 
+comparison, verification seeks to determine if the conceptual model satisfies the 
+given requirements. The second comparison is between the conceptual model and 
+the executable model, where the goal is to determine if the implemented executable 
+model is consistent with respect to the conceptual model. Validation compares the 
+simuland with the conceptual model to determine if the simuland has been accurately 
+described in the conceptual model. The second comparison is between the simuland 
+and the results which determine if the output of the simulation is sufficiently accurate 
+with respect to the actual behavior of the simuland [29].  
+ 
+Another comprehensive VV&T model is presented by Balci [28] in the form of a 
+simulation study life-cycle as shown in Figure 23. The phases are shown by oval 
+symbols. The dashed arrows describe the processes which relate the phases to each 
+other. The solid arrows refer to the credibility assessment stage. Every phase of the 
+life-cycle has an associated VV&T activity. Problem Formulation (or problem 
+definition) is the process of formulating a problem which is sufficiently well-defined 
+to enable specific research action and the investigation of suitable solution 
+techniques. The output of system investigation results in the System and objective 
+definition which further aids in model formulation. Model formulation is the process 
+of defining a conceptual model which abstracts or envisions the real system under 
+study. The conceptual model is further represented inform of a Communicative 
+Model which is a model representation and can be communicated to other designers 
+and can be compared against the system and the study objectives. It is further 
+
+Requirements
+analysis
+Requirements
+Simuland
+Accreditation
+Modeling
+Validation
+Talidation
+Verification
+Conceptual
+Results
+model
+Execution
+Implementation
+Verification
+Transformation
+Executable
+Comparison
+modelChapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+73 
+ 
+transformed into an executable model through the process of programming. An 
+Experimental Model is the programmed model incorporating an executable 
+description of operations along with the design of experiments, for experimenting 
+with the simulation model with a specific purpose. The process of experimentation 
+produces the Simulation Results, which are presented for decision makers for their 
+acceptance and implementation or undergo refinements if required. 
+ 
+Figure 23: Simulation study life-cycle (acquired from [28]) 
+The model-evaluation life-cycles shown in Figure 21, Figure 22 and Figure 23 have been 
+considered as guidelines and they are used as inspiration for the verification life-cycle 
+proposed and presented later in this thesis.  
+4.2 The Principles of Top-Down Refinement 
+The principle of top-down refinement has been appreciated in the area of model 
+verification. Constructing a highly detailed model that satisfies all levels of 
+correctness in one attempt is very difficult. Instead it is easy to construct a less 
+detailed abstract model at first. Let S1 be an initial model. To get from S1 to the final 
+shape of the model, the Top-Down Refinement paradigm advocates the derivation 
+
+COMMUNICATED
+PROBLEM
+Problem
+Formulated Problem
+Formulation I
+VV&T
+FORMULATED
+PROBLEM
+Investigation of
+Feasibility Assessment
+Solution Techniques I
+of Simulation
+DECISION MAKERS
+PROPOSED SOLUTION
+Acceptability of
+TECHNIQUE
+Simulation Results
+(Simulation)
+INTEGRATED
+DECISION
+System
+ System and Objectives
+SUPPORT
+Investigation !
+Definition VV&T
+SYSTEMAND
+OBJECTIVES
+DEFINITION
+ Model Formulation
+Simulation Results
+Presentation VV&T
+Presentation of
+ Model
+Qualification
+CONCEPTUAL
+MODEL
+Communicative
+Model
+Model VV&T
+Representation
+SIMULATION
+Experimental
+Data
+COMMUNICATIVE
+RESULTS
+Model VV&T
+VV&T
+MODEL(S)
+Programmed
+/ Programming
+Model VV&T
+PROGRAMMED
+MODEL
+Experiment
+Design VV&T
+EXPERIMENTAL
+ Design of Experiments
+MODELChapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+74 
+ 
+of an (ordered) sequence S1, S2…Sf of models of S. For i = 1...f, model Si+1 is a 
+refinement of its immediate predecessor model Si if the following conditions are met: 
+ 
+(i) Si+1 is more expressive than Si  
+ 
+(ii) Si+1 is less abstract than Si  
+ 
+(iii) It is relatively easy to evaluate Si+1 on the basis of verified Si 
+ 
+Consequently, the last model in the refinement sequence should be correct by 
+construction. The following are some consequences of the top-down refinement 
+paradigm. First, Si+1 is harder to understand than Si and therefore harder to prove on 
+its own; it is precisely the refinement step that allows the verification of Si+1 under 
+the assumption that Si has already been proved correct [99]. 
+In this thesis the proposed verification process is based on this fundamental principle 
+where the verification is performed iteratively and on a relatively refined shape of the 
+model. 
+4.3 Verification techniques 
+There exist a large variety of verification methods. The diversity is due to the range 
+of different simulation project types, different subjects (simuland), and different 
+types of data. Most of the verification methods are inspired from software 
+engineering domain, because the executable models in simulation projects are almost 
+always realized as software [29].  
+In literature, Verification techniques are generally classified into four main categories 
+as show in Figure 24. 
+ 
+Figure 24: Verification Techniques 
+ 
+ 
+4.3.1 Informal Techniques 
+These techniques are most commonly used. They are called informal because the 
+tools and methods used rely heavily on human reasoning and inspection without any 
+underlying mathematical formalism [28]. These techniques are well structured and are 
+conducted with proper guidelines by following standard policies and procedures, 
+however these techniques are tedious and not very much effective [109].  
+ 
+ 
+Verification  
+Techniques 
+Informal 
+Techniques 
+Static Analysis 
+Dynamic 
+Analysis 
+Formal Analysis 
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+75 
+ 
+ 
+Some of the commonly used informal methods are shown in Table 6. 
+Audit 
+An audit is undertaken to assess how adequately the system study is 
+conducted with respect to established plans, policies, procedures, 
+standards and guidelines [28].  
+Desk 
+checking 
+Desk checking or self-inspection is a thorough examination 
+performed by an individual as a first step. In this method syntax 
+checking, specification comparison, code, control flow graph 
+analysis are performed [28].   
+Inspections 
+Inspections are conducted by a team and performed at different 
+phases of developments such as problem definition, conceptual 
+modeling, executions etc. Inspections are conducted to find and 
+document faults [28].   
+Turing Tests 
+Turing test is performed by domain experts (of the system under 
+study). They are presented with two sets of output data obtained 
+one from the model and one from the specification (without 
+identifying which one is which) and are asked to differentiate both 
+and based on their feedback model corrections are made [28].  
+Table 6: Informal Verification Techniques 
+4.3.2 Static Analysis:  
+These techniques are applied to assess the static model design and the 
+implementation (source code), without executing the model. They aim at checking 
+the structure of the model, the dataflow and control flow, the syntactical accuracy, 
+and the consistency. Some of the commonly used static analysis methods are shown 
+in Table 7. 
+Structure 
+Analysis 
+Structure Analysis is used to examine the model structure. It is 
+conducted by constructing a control flow graph of the model 
+structure [28].  
+Data 
+Analysis 
+It involves data dependency tests and data flow analysis to ensure 
+that data used by the model is properly defined and proper 
+operations are applied to data objects [28].  
+Cause-
+Effect 
+Graphing 
+Cause-Effect graphing assists model correctness evaluation by 
+answering “what causes what” questions in the model representation.  
+It is performed by identifying causes and effects in the model and 
+checking if they are reflected accurately in the specification [28]. 
+Syntactic 
+Analysis 
+Syntactic analysis is usually performed by the compiler of the 
+simulation language being used. Syntactic analysis can also be 
+performed using a set of rules applied on the model representation to 
+verify if it satisfies given specification. 
+Semantic 
+Analysis 
+This technique is used to determine the modeler’s intent and verify 
+that the true intent is accurately reflected in the model representation 
+[28].  
+Table 7: Static Analysis Techniques 
+ 
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+76 
+ 
+4.3.3 Dynamic Analysis:  
+Dynamic analysis techniques are based on the execution of the model in order to 
+evaluate its behavior. They do not simply examine the output of an execution but 
+also observe the model as it is being executed. The insertion of additional code into 
+the model called instrumentation is needed to collect or monitor the behavior during its 
+execution [109]. Table 8 presents some of the important dynamic analysis verification 
+techniques. 
+ 
+Assertion 
+Checking 
+An assertion is a statement that should be true during the 
+execution of a model. Assertions are placed in various parts of 
+the model and monitored during execution [28]. 
+Bottom up 
+Checking 
+This technique is used in conjunction with the bottom up 
+model development strategy. The sub models are checked 
+individually. Then the parents at the higher level are checked 
+[28].  
+Fault/Failure 
+insertion  
+This approach is used to insert a fault or a failure in the model 
+and observe whether the expected incorrect behavior is 
+produced. This approach is effective to detect unexplained 
+behavior and hence uncover errors [28].  
+Functional 
+Testing 
+This technique is used to assess the accuracy of model input-
+output transformation, to evaluate how accurately a model 
+transforms a given input into a set of output data [28]. 
+Sensitivity 
+Analysis 
+Sensitivity analysis is performed by changing the values of 
+model input variables and parameters over some range of 
+interest and observing the effect on model behavior. 
+Unexpected effects may reveal errors [28]. 
+Table 8: Dynamic Analysis Techniques 
+ 
+ 
+4.3.4 Formal Analysis 
+Formal analysis refers to mathematical analysis of proving or disproving the 
+correctness of a system with respect to a certain unambiguous specification or 
+property. The methods for analysis are known as formal verification methods, and 
+unambiguous specifications are referred as formal specifications. Formal verification 
+can provide complete coverage on an abstract model of the system, modeled using 
+finite state machines, PN or any other specification formalism. However it should be 
+noted that formal verification can ensure the correctness of a design only with 
+respect to certain properties that it is able to prove [88]. There are many formal 
+analysis techniques, which we classify in four main groups: 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+77 
+ 
+Equivalence 
+Checking 
+It is also called Reference Model Checking, which is widely 
+used verification technique that allows two behavioral models 
+to be compared with each other. In general, one of the two is 
+taken as the reference model and represents the so-called 
+golden model (or perfect model). It verifies that the behavior of 
+two models is the same for the exercised scenarios. This 
+technique has limitation that it does not actually verify that the 
+design is bug free, and provides proof of relative correctness 
+[109]. 
+Theorem 
+Proving 
+This method involves verifying the truth of mathematical 
+theorems that are postulated or inferred throughout the design 
+using a formal specification language. The procedure involves 
+two main components: (i) proof checker (which can be 
+completely automated in most cases) and (ii) an inference 
+engine (which may require occasional human guidance) [109].  
+Property 
+Verification 
+Formal properties specify the requirements of the correct 
+system design. The objective of this method is to check 
+whether an implementation satisfies these requirements. Static 
+Assertion-based Verification (ABV) and dynamic [110]. 
+Model 
+Checking  
+Model checking establishes a solid confidence in a reliable V&V 
+process. Model checking is an automated and comprehensive 
+verification technique that can be used to verify whether the 
+properties specified (usually using Temporal Logic) for a given 
+design or its components are satisfied for all legal design inputs. 
+Model checking also faces a limitation, since it suffers from the 
+well-known state explosion problem. In a worst-case scenario, 
+the state space of the design may grow exponentially large with 
+the number of state variables. Model checking can be fully 
+automated for design verification and can yields results much 
+more quickly than theorem proving [109].  
+Table 9: Formal Analysis Techniques 
+ 
+Some of these techniques have been adopted in our proposed verification 
+framework.  
+4.4 Summary 
+In this chapter, different concepts of verification, validation and testing are discussed 
+as they collectively contribute to proving the correctness and accuracy of a model. 
+Some existing model development processes (devised mainly by M&S community) 
+are also discussed, since they are the bases of the proposed verification life-cycle 
+presented later in this thesis. The proposed framework essentially focuses on 
+Verification (however its design is also open to adopt validation techniques). 
+Different verification techniques are classified into four main groups and some of the 
+selected techniques are briefly explained, as they will be used later in this thesis. 
+ 
+
+Chapter 4  
+ 
+Verification and Analysis 
+   
+Page 
+78 
+ 
+ 
+ 
+ 
+ 
+Part II 
+Techne 
+ 
+ 
+ 
+ 
+ 
+Technê in Greek is translated as craftsmanship or craft or art. In science it is the practice of 
+knowledge; Techne resembles Epistēmē in the implication of knowledge of principles, although techne 
+differs in that its intent is making or doing, as opposed to "in-depth understanding"; Applied-
+Science; It deals with “How” of the subject. 
+ 
+ 
+Part-II covers the technology of the research under discussion, where the theoretical 
+concepts provided in Part I are applied, and technically discussed under an integrated 
+framework of methods, techniques, algorithms and processes and their practical 
+implications are provided in the form of a proposed solution. 
+ 
+ 
+“Without knowledge the practice is useless, and without practice 
+the knowledge is useless” 
+– Ali bin Usman Hajvery 
+(Kashaf-Almahjoob) 
+ 
+ 
+
+   
+Page 
+79 
+ 
+Chapter 5 
+Proposed Methodology and the 
+Verification Framework 
+ 
+This chapter renders the core of the solution framework proposed in this thesis.  In this chapter, a 
+collection of methods, techniques, algorithms, sub-processes, activities and approaches are presented, 
+as proposed solution to various issues in the composability verification of BOM based model 
+components. All these contributions are integrated into a unified framework which we refer to as: 
+Composability Verification Framework. 
+ 
+The proposed verification Framework consists of different methods, techniques, 
+algorithms, sub-processes, activities and approaches which all together encompass 
+the component based modeling & simulation (CBM&S) life-cycle.   
+5.1 
+Component-based Modeling & Simulation life-cycle 
+CBM&S life-cycle is inspired by different modeling architectures proposed by 
+Sargent, Petty and Balci and discussed in section 4.1.1. It is extended with our 
+proposed contributions at its different stages. The proposed CBM&S life-cycle is 
+mainly divided into four main quadrants: (i) Inception (ii) Modeling (iii) Execution 
+and (iv) Analysis. Each quadrant has different phases and in each phase there are 
+multiple activities (or cycle of activities). Each activity consists of methods and 
+techniques pertinent to its respective phase. These phases are revisited iteratively 
+during the life-cycle; where each iteration represents a tier; hence the entire CBM&S 
+life-cycle is a multi-tier process; whilst each tier results into a refinement of the 
+solution of the problem under investigation; as it follows the principle of top down 
+refinement, discussed in section 4.2. All the above mentioned features of the 
+CBM&S life-cycle are shown in Figure 25 divided into four quadrants:  
+ 
+Figure 25: CBM&S life-cycle 
+ 
+
+ANALYSIS
+INCEPTION
+Phase-I
+Refinement
+Simuland
+ Phase-V
+Requirements Engineering
+Analysis Technique
+ Phase-II
+Analysis
+Requirements
+Modeling
+Abstract Level Execution
+ Phase-IV
+Phase-III
+Executable Model
+Conceptual Model
+Fransformation
+Activity
+Formal Model
+EXECUTION
+MODELINGChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+80 
+ 
+The following sub-sections provide microscopic details of each quadrant along with 
+their associated inside activities, methods and techniques. 
+5.2 Inception 
+The first quadrant of the CBM&S life-cycle called “Inception” initiates the process. 
+At first the abstraction of a real-system is accumulated as simuland. A simuland can 
+be ingested in the form of UML diagrams (Figure 26) or using any other formal or 
+informal representation. 
+ 
+Figure 26: Simuland using UML Diagrams 
+The basic idea is to gather the body of knowledge so that the modelers can envision 
+the real system under a certain frame of reference i.e., the context under which the 
+system is being studied. When the simuland is ingested into the framework, it is used 
+(i) to gather requirements, through the process of requirement engineering and (ii) to 
+search and discover suitable components from a BOM repository for the 
+construction of a composed model. If a required component does not exist in the 
+repository then it is built from scratch and added in the repository. The outcome of 
+the requirement engineering activity results in formulation of requirements 
+specifications. The requirement specification formalism (as defined in section 1.3.2) 
+is used to express formal requirements for this framework:  
+RS = 〈O, S〉 
+Where  
+ 
+O = {o1, o2, o3 …, on} is a set of objectives or goals that must ultimately be fulfilled. 
+These goals are usually defined in the context of the scenario of the modeling 
+domain. Therefore the properties expressed as goals or objectives may be scenario-
+specific and not the standard system properties e.g. in a restaurant model the 
+objectives could be that the customers are served food and payments are collected, 
+and not that the model should be deadlock free (which however might be a necessary 
+condition).  
+ 
+S = {s1, s2, s3 …, sn} is a set of system constraints (system properties or scenario-
+specific safety/liveness properties). Deadlock freedom (or other similar system 
+properties) could be the required constraints necessary to fulfill the above objectives 
+and therefore must be satisfied. We propose to define the following mandatory (or 
+default) constraints in the requirements specification of the composability 
+verification framework:  
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+81 
+ 
+ 
+S1 = All the interacting components25should be composable at Syntactic level 
+S2 = All the interacting components should be composable at static-semantic level 
+S3a = State-machines of the interacting components should match each other such that they can 
+continue to progress until they reach the final or goal states26. 
+S3b = If the conceptual model is transformed into an executable model, the latter should correctly 
+represent the structure and behavior of the former.  
+Table 10: Mandatory constraints in composability verification 
+We assert that [S1 ∧ S2 ∧ (S3a ∧ S3b)] is a necessary condition for the overall 
+composability verification. S1 and S2 ensure that the composed model is structurally 
+consistent. Whereas S3a confirms that the behavior of the composed model is 
+coherent for reaching given objectives. The satisfaction of S3b obeys the definition 
+of Model verification (see section 4.1) in the sense that it confirms the second part of 
+the definition that is: “the accuracy of transforming the conceptual model into an executable 
+model” and therefore the overall success of the verification process depends on the 
+satisfaction of S3b constraint. The conjunction of these default constraints impose 
+the three C’s of requirements namely (i) Consistency, (ii) Completeness, and (iii) 
+Correctness [111]. Consistency is required for the evenness in the input and output 
+connections of the composed components. Completeness is required for the totality 
+of the information of the components being composed to check that the 
+composition does not lack required inputs for making progress. Correctness is 
+needed to confirm that the composed components interact in a correct way as they 
+are supposed to.  
+If all the objectives are fulfilled and all the constraints are satisfied and then we say 
+that the model is composable at all levels and is verified with respect to its 
+specifications. The overall objective of our proposed framework is to provide 
+environment and tool support to assess this postulation. 
+The outcome of discovery results in a set of candidate BOMs and their matching 
+with the simuland and the requirements results in a selection of BOMs suitable for 
+the composition. This selection is composed to form a conceptual model. 
+5.3 Modeling 
+In the Modeling quadrant, a BOM based composed model is taken as an input and 
+the conceptual model is formed. Also a formal model and its graphical notation (as 
+proposed in section 2.7.4) are produced for the purpose of documentation of the 
+conceptual model26F27. Considering that BOM itself is a conceptual framework and is 
+used to model passive components which cannot undergo any form of execution 
+therefore the conceptual model is subjected to a series of extensions and refinements 
+                                                 
+25  In a composed model it is not necessary that every component interacts with every other 
+component for instance A, B and C are composed such that A interacts with B and B interacts with C 
+but A does not interact with C.  
+26 If there are no final-states defined in a model and the model is non-terminating then we assume that 
+certain important states called goal-states are present in the model, reachability of which confirms that 
+the goals are fulfilled.  
+27 This step is optional but beneficial if different teams are working on different phases of the 
+development life-cycle. This documentation makes it easy to understand the structure and behavior of 
+basic components and their composition. 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+82 
+ 
+using external input and our proposed model transformation algorithms so that it 
+can be implemented into executable forms and sent to the “Execution” quadrant 
+(Figure 25) for abstract level execution. Our proposed extensions and refinements are 
+listed as follows: 
+• BOM State-machines to State Chart XML (Transformation) 
+• Composed-BOM to Petri Net –PNML (Transformation) 
+• Basic-BOM to Extended-BOM (Extension) 
+• Extended-BOM  (E-BOM) component to Colored Petri Net (CPN) Component 
+Model (Transformation) 
+• Basic-BOM to Extended-BOM with Time (Extension) 
+• Basic-BOM to Extended-BOM with probabilistic factors (Extension) 
+• BOM to CSP based Process Model (Extension & Transformation) 
+In the later section these extensions, refinements and transformations will be 
+explained in detail. It is important to note that each time the conceptual model is 
+extended or refined the Modeling quadrant is revisited in iteration. 
+5.4 Execution 
+As previously discussed this quadrant is mainly for the abstract-level execution 
+activities. It takes following implemented and executable forms of the conceptual 
+model from the Modeling quadrant as input: 
+• State Chart XML (SCXML)  
+• Petri Net –PNML  
+• Colored Petri Net (CPN) Composed Component Model 
+• Communicating Sequential Process (CSP) based Component Processes 
+In the later section these executable forms and their abstract level execution 
+processes will be discussed in detail.  
+5.5 Analysis 
+The outcome of an execution process yields some results. These results are analyzed 
+in the Analysis quadrant.  Our verification framework supports different analysis 
+techniques listed as follows: 
+• State-machine matching Analysis 
+• Petri Nets based Algebraic Analysis 
+• Colored Petri Net based State-Space Analysis 
+• Model Checking Analysis 
+ 
+These analysis techniques will be discussed in later section. When all the necessary 
+steps in the composability verification are complete and the composed model under 
+investigation is said to be verified with respect to the given requirement specification 
+then the CBM&S life-cycle proceeds to the further steps for implementation and 
+simulation as shown in Figure 27. The details of these steps are out of the scope of 
+this thesis. 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+83 
+ 
+ 
+Figure 27: Implemenation and Simulation 
+5.6 Composability Verification Framework 
+In this section different method, techniques, procedures, algorithms and modules of 
+our proposed composability verification framework are discussed in detail and 
+considered as building blocks in the CBM&S life-cycle and will be connected to its 
+different phases. These details are necessary to understand the composability 
+verification process being presented in chapter 6.  
+5.6.1 Discovery Matching and Composition (DMC) 
+In component based development, it is a normal practice to construct reusable 
+components and store them in a library or repository so that they can be reused later 
+as required. To reuse an existing component, a Discovery, Matching, Composition 
+(DMC) paradigm [19] is used. We assume that a library of BOM components is 
+maintained in a repository. Using the information given in the simuland a modeler 
+attempts to search and discover BOM components from the repository. If a 
+collection of candidate components is retrieved, they are filtered through matching 
+process. A matching process matches the candidate components from the simuland 
+and requirement specifications and results in a selection of components suitable for 
+the composition. The aspects of syntactic and semantic matching during the 
+discovery and selection of BOM components are proposed and discussed in detail in 
+[54]. In this article a set of discovery rules are presented which must be fulfilled while 
+matching a candidate selection from the simuland. We apply these rules for the 
+syntactic and semantic matching of the candidate selection with the simuland. We 
+further suggest matching the candidate selection with given requirements, because a 
+selection may match with its respective simuland but if it does not match with its 
+requirements then the composability verification will fail. We implement the concept 
+of DMC process in our framework as shown in Figure 28. It is also assumed that if a 
+required component does not exist in the repository, then it is constructed from 
+scratch and is added in the repository for reuse. The result of DMC process is a 
+BOM-based composed model. This composed model is taken as input in the 
+Modeling quadrant and considered as a conceptual model of the system. It is 
+recommended that the modelers also use our proposed formal specification and 
+graphical notation presented in section 2.7.4 to construct a formal model. This 
+formal model can be used for documentation and shows how the components are 
+composed. It is however an optional step and is not considered as a phase in our 
+
+IMPLEMENTATION
+Composed Model
+Code Generation
+Simulation Model
+Successful Completion of
+the Composability
+Decision Support
+Design of Experiment
+Verification Process
+Experimental
+Model
+Simulation
+Simulation ResultsChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+84 
+ 
+CBM&S life-cycle. In chapter 7 & 8 the formal models of the examples are also 
+described for reader’s understanding.  
+  
+Figure 28: Discovery, Matching, Composition (DMC) 
+5.6.2 Structural and Behavioral Evaluation 
+The conceptual model ingested in the Modeling quadrant requires structural and 
+behavioral evaluation so that we can confirm that the model is consistent, complete 
+and correct. And it is suitable for thorough verification at different levels of 
+composability. Checking the structure and behavior of the conceptual model before 
+subjecting it to the deeper levels of composability verifications is useful. If the model 
+is structurally and behaviorally consistent then the confidence level is increased based 
+on which different useful assumptions can be made later during the in-depth 
+verification. 
+If there are discrepancies in the structure or behavior of the model then we can skip 
+further steps, save time and computational resources and perform necessary design 
+refinements before the entire process is repeated.  This setup obeys the principle of 
+top-down refinement as discussed in section 4.2. The structure of the model is 
+analyzed using static analysis techniques (see section 4.3.2), whereas the behavior of 
+the model is evaluated using dynamic analysis techniques.  
+ 
+5.6.3 Static Analysis 
+We propose two types of Static analysis procedures (i) Syntactic Matching and (ii) 
+Static-Semantic Matching. These procedures are used to evaluate the structure and 
+verify composability at syntactic and static-semantic levels. They are called static 
+analysis because they are evaluated based on pre-defined rules and do not require any 
+form of execution and the information on which these rules are applied is static. 
+
+Phase-I
+Simuland
+Reguirements
+Component Search
+Engineering
+Phase-II
+Reguirements
+BOM
+Repository
+Matching
+Discovery
+Candidate
+Matching
+BOMs
+Selection
+Modeling
+12N
+Phase-I1
+Composition
+Conceptual
+Model
+Formal
+Model
+TransformationChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+85 
+ 
+Syntactic Matching (SM) 
+This module is responsible for evaluating BOM composability at syntactic level based 
+on the following rules. The outcome of this module verifies that the components can 
+be correctly connected to each other syntactically. These rules were introduced in a 
+BOM matching technique presented in [54]. 
+SM-Rule 1: 
+The name of each event28 exchanged between the two components should be same i.e., 
+the send-event should have the same name as the receive-event.  
+ 
+A send-event is defined in the BOM’s event types where the sender is the BOM itself 
+and the receiver is some other BOM (in the composition) whereas a receive-event is 
+the definition of an event in the BOM event types, where the sender is some other 
+BOM (in the composition) and the receiver is the BOM itself. 
+SM-Rule 2: 
+Each send-event should have at least one corresponding receive-event and vice-versa i.e., 
+the send/receive pair should be complete.  
+ 
+SM-Rule 3:  
+The number of parameters (content characteristics of event types) of the send-events 
+should be the same as the number of parameters of the receive-events. 
+The satisfaction of Syntactic Matching rule1, rule2 and rule3 fulfills the default 
+constraint S1 (see Table 10) which is a necessary condition for the overall 
+composability verification. Figure 29  shows different steps in the syntactic matching 
+activity.  
+ 
+Figure 29: Syntactic Matching 
+ 
+                                                 
+28 It is assumed that in the BOM construction the events and their corresponding actions are given the 
+same name 
+
+Phase-I
+Refinement
+Simuland
+Reguirements
+Engineering
+Phase-V
+Phase-II
+Analysis
+Constraint Sl satisfied.
+Requirements
+Technique
+Static Analysis Technique
+Rule evaluation
+Analysis
+Satisfy
+Rule3
+Rule2
+Rulel
+Modeling
+Abstract Level
+/Violate
+BOM
+Execution
+Components
+Phase-I1I
+ Phase-IV
+Conceptual
+Executable
+Model
+ModelChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+86 
+ 
+Static-Semantic Matching (SSM) 
+This module is responsible for evaluating BOM composability at static-semantic level 
+based on certain rules. The outcome of this module verifies that the composition of 
+the components is meaningful and the communication between the components is 
+understood as intended. In order to certify these facts we propose static-semantic 
+matching at two levels: (i) Operational Level matching and (ii) Message level [53]: 
+(i) 
+Operational Level matching 
+In BOM-based composed models Operations are described by Pattern-of-Interplay 
+(POI). POI is formed by a collection of actions from the basic BOMs being 
+composed. In operational-level semantic matching, it is ensured that the composed 
+components share the same “domain of interest” and they are composed for the 
+same purpose (or aim) so that we can guarantee that the composition is (static) 
+semantically meaningful and without any pragmatic ambiguity. Even with the same 
+domain of interest, the component may serve for varied purposes e.g., in Military 
+domain a Battalion Head Quarter (BHQ) component may have many purposes and 
+can take part in many different operations.  Therefore it is also important that the 
+purpose of the selected components should be clear for a meaningful outcome. 
+In order to ensure semantic consistency at operational-level we propose to specify 
+following semantic-attributes 29 in the definition of actions at the time of the 
+construction of Basic BOMs and in the POI when the basic BOMs are being 
+composed. In the static-semantic matching these attributes are used to compare 
+that the correct actions are involved in the BOM composition. 
+ 
+o Area-of-Interest: It describes the area or the domain of interest of the system that 
+is being modeled using the components and the operation. We propose to define 
+“Area-of-Interest” as a semantic-attribute in each action of Basic BOM and also in 
+the POI. This attribute will confirm that all the components share the same domain 
+knowledge. If of some general purpose components that may belong to multiple-
+domains (e.g., Queues etc.) we propose to construct a specialization of the 
+component and make it a member of the selected area-of-interest. E.g., In a 
+restaurant composed model a generic queue component can be specialized into a 
+restaurant-queue with actions JoinRestaurantQueue() and ServeCustomer() instead of 
+Put() and Get() actions. 
+ 
+o Purpose: Purpose describes the aim or goal of the entire operation. In BOM 
+composition, POI represents a single operation being performed by the composed 
+components. However it is also possible that one or more composed components 
+may be designed to serve multiple purposes; and in a given scenario only some part 
+of the multi-purpose components is involved in the composition. e.g., a Customer 
+component could be generic and can have multiple purposes whereas a Restaurant 
+waiter component is specific to a restaurant scenario, so it is important that if a 
+Customer component is selected in a Restaurant scenario then its purpose should 
+be aligned with the other components in this scenario. Hence we define “purpose” 
+as a semantic-attribute of actions in the basic BOM (with multiplicity  ≥ 1). 
+ 
+                                                 
+29 In BOM the conceptual modeling elements (Entities, Events States and Actions) support semantic 
+fields [65] 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+87 
+ 
+(ii) 
+Message Level matching 
+BOM represent event driven components and function by sending or receiving 
+events (messages). At the message level it is required that the communication 
+between composed components is meaningful and semantically understood by the 
+receivers as intended by the senders. At this level we propose to match Data-Types 
+and Units of measurements of the parameters of send-events and receive-events [53] 
+[54]. 
+ 
+It is assumed that the BOM components have corresponding OWL attachments as 
+proposed in [54]. The BOM-OWL attachments are used to define semantic classes of 
+the domain ontology, their properties, data-types and the individuals and stored in 
+the BOM repository. In order to evaluate static-semantic matching at both 
+Operational and Message levels, we apply following rules: 
+ 
+SSM-Rule 1 
+The intersection of the “Area-of-Interest” attribute of all the actions (involved in an 
+operation) should be exactly the same as that of POI or should belong to an 
+equivalent class30 in the respective ontology: 
+�
+Acti. AOI
+n
+i=1
+ ≅ POI. AOI 
+ 
+ 
+SSM-Rule 2 
+The intersection of the “Purpose” attribute of all the actions should be exactly the 
+same as that of POI or should belong to an equivalent class in the respective ontology: 
+�
+Acti. purpose
+n
+i=1
+≅ POI. purpose 
+  
+SSM-Rule 3 
+Data types of each element in the event parameters of the send-event and receive-events 
+should be of the same class, equivalent class or should be in direct hierarchical 
+relationship i.e., the sender’s parameter data-type should belong to the direct child class 
+of the receiver’s parameter data-type (but not the inverse). 
+ 
+e.g., a send-event contains a parameter of type ‘second’, whereas the receive-event 
+expects a parameter of type ‘time’ which according to the rule it is a semantic match. 
+Figure 30 presents primitive data-types as an example. In real situations BOM 
+components will have more domain specific complex data-types. 
+                                                 
+30In OWL two classes can be marked equivalent if they have same semantic meanings and both classes 
+have the same individuals (instances) e.g., Healthcare and Medical are synonyms. We denote it as  ≅ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+88 
+ 
+ 
+Figure 30: Some of the sub-classes of Data Type ontololgy 
+SSM-Rule 4 
+The units of the measurements expressed in the event parameters should be same or 
+equivalent or should belong to a direct class hierarchy such that they are convertible 
+without (or with acceptable) loss of information.  
+We assume that if two measurement units are in either of the direct relationship i.e., 
+parent or child then their conversion loss will be acceptable e.g., a send-event has a 
+parameter with unit m/s (meter per second) to express speed whereas the receive-
+event expects Km/hr (Kilometer per hour). This is a valid semantic match because 
+the quantities are convertible without loss.  
+Semantic Matching Technique 
+In order to match two elements we propose a semantic matching technique as shown 
+in Figure 31. This technique uses OWL-API [112], a semantic reasoning engine 
+(FaCT++, Pellet, or HermiT) and an OWL ontology document to process a query of 
+any two elements A & B and outputs their semantic relationship as one of the 
+following: 
+1. Exact (A = B) 
+2. Equivalent (A ≅ B i.e., A and B belong to equivalent classes)  
+3. Direct-Parent (A is a direct parent of B) 
+4. Direct-Child (A is a direct child of B) 
+5. Indirect (A and B are not in direct contact but belong to same hierarchy) 
+6. No relationship (A and B are not related) 
+ 
+Figure 31: Semantic Matching Technique 
+This technique is used to evaluate Static-Semantic Matching Rules 1, 2, 3 & 4 using 
+the algorithm31 given in Table 11.  
+                                                 
+31 The Pseudo-code conventions and format of the algorithms provided in this thesis, for most parts, 
+follows the guidelines set by [132]. 
+OWL 
+Doc 
+A 
+Reasoner 
+OWL-API 
+B 
+Query 
+Relation 
+Result 
+
+ODay
+OYear
+Binary
+ODate
+OMonth
+OInteger
+Number
+ODataType
+ODouble
+Time
+Minute
+Text
+Language
+OHour
+Second
+OCharacter
+O StringChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+89 
+ 
+Algorithm: Semantic Matching 
+Input: {Actions}, POI, BOM-OWL 
+Output: TRUE, FALSE 
+1 Owl ←  Load Ontology(BOM-OWL) 
+2 {CommonAOI} ← ⋂
+𝑎𝑖 
+𝑛
+𝑖=0
+∈ Actions.AOI  ⊳ Gives a set of common area of interest of all actions 
+3 for caoi ∈ {CommonAOI} do 
+4 
+   SR1 ← Get-Semantic-Relation(caoi, POI.AOI, Owl) ⊳ It is assumed that Get-Semantic-Relation()  
+5 
+    function is implemented using semantic matching technique shown in Figure 31 ⊲ 
+6 
+if SR1 = “Exact” or “Equivalent” then  ⊳ Rule1 satisfy...continue 
+7 
+next 
+8 
+else 
+9 
+Return FALSE 
+10 
+end if 
+11 end for 
+12  
+13 {CommonP} ← ⋂
+𝑎𝑖 
+𝑛
+𝑖=0
+∈ Actions.purpose  ⊳ Gives a set of common purpose of all actions 
+14 for cp ∈ { CommonP } do 
+15 
+SR2 ← Get-Semantic-Relation(cp, POI.purpose, Owl) 
+16 
+if SR2 = “Exact” or “Equivalent” then  ⊳ Rule2 satisfy...continue 
+17 
+next  
+18 
+else 
+19 
+Return FALSE 
+20 
+end if 
+21 end for 
+22  
+23 {Events} ←  Get-Events(Actions) ⊳ gets corresponding Events of Actions 
+24 
+for  e ∈ Events  do 
+25 
+if e=Send-Event then 
+26 
+f ← Get-Receive-Event(e, Events) ⊳ gets corresponding Receive Event of e 
+27 
+{PE} ← e.Parameters ⊳ Set of parameters of send-event e 
+28 
+{PF} ← f.Parameters  ⊳ Set of parameters of receive-event f 
+29 
+⊳ No. of parameters of e and f must be same because of SM-Rule3 
+30 
+for pe∈PE & pf ∈PF do 
+31 
+SR3 ← Get-Semantic-Relation(pe.Type, pf.Type, Owl) ⊳ Compare Parameter types 
+32 
+if SR3 = “Exact” or “Equivalent” or “Direct-Child” then   
+33 
+⊳ Rule3 satisfy…continue to rule4 
+34 
+SR4 ← Get-Semantic-Relation(pe.Unit, pf.Unit, Owl) ⊳ Compare Units 
+35 
+if SR3 = “Exact” or “Equivalent” or “Direct-Parent” or “Direct-Child” then   
+36 
+Return TRUE ⊳ Static-Semantic Matching Successful 
+37 
+else 
+38 
+Return FALSE 
+39 
+end if 
+40 
+else 
+41 
+Return FALSE 
+42 
+end if 
+43 
+end for 
+44 
+else 
+45 
+next 
+46 
+⊳ Goes to the next send-event and need not to check receive-events (because SM-Rule2) 
+47 
+end if 
+48 
+end for 
+Table 11: Semantic Matching Algorithm 
+ 
+ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+90 
+ 
+The semantic matching algorithm takes a set of actions (parsed from Basic BOMs 
+which are being composed); the pattern of interplay (POI) which specifies how the 
+actions are connected to each other and the corresponding OWL ontology document 
+as input. The output of this algorithm is TRUE if the static-semantic matching is 
+successful otherwise FALSE if any of the rule is violated. Figure 32 shows steps in 
+the verification of BOM composability at Static-Semantic level. 
+ 
+Figure 32: Static-Semantic Matching 
+If the semantic matching is successful, it will fulfill the default constraint (S2) of the 
+requirement specification (see Table 10) which is a necessary condition for the overall 
+composability verification.  
+5.6.4 Dynamic Analysis 
+We use Dynamic Analysis technique (see section 4.3.3) to evaluate the behavior of 
+the conceptual model. At first the components undergo a state-machine matching 
+process for the evaluation of the behavior consistency. When this evaluation is 
+successful, we proceed with the in-depth verification at the dynamic-semantic 
+composability level, choosing one of the different proposed set of dynamic analysis 
+techniques. These analyses are called dynamic analysis because they require execution 
+at different abstract levels as mentioned in section 5.4 
+State-Machine Matching (SMM) 
+State-machines represent behavior of the components and are the essential dynamic 
+part of BOM components. In the verification of BOM composability at dynamic-
+semantic level, it is important that the behavior of the composed components should 
+be coherent with each other i.e., their interactions are consistent in order to make 
+progress towards composition goals. To ensure this fact we assert (as a necessary 
+condition) that the state-machines of the composed components should match each 
+other. BOM state-machines are event driven in nature and make progress by 
+exchanging events. In order to ensure that the state-machines of the composed BOM 
+components match each other they are required to be executed at an abstract level. 
+Therefore we proposed a technique in [113] which transforms each BOM state-
+machine to SC-XML (State-Chart XML) [114] format. A sample of SCXML is shown 
+in Figure 33.  
+
+Phase-I
+Simuland
+Refinement
+Reguirements
+Engineering
+Phase-V
+Phase-II
+Analysis
+Constraint S2 satisfied
+Requirements
+Technique
+Static-Semantic Analysis
+Technigue
+Analysis
+OWL
+OWL API
+Doc
+Ouery
+Rulel
+Rule2
+Reasoner
+Rule3
+Modeling
+Abstract Level
+Rule4
+Execution
+BOM
+Phase-III
+Phase-IV
+Violate
+Satisfy
+Components
+Conceptual
+Executable
+Model
+Model
+XChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+91 
+ 
+ 
+Figure 33: SCXML format 
+We develop a runtime environment using SCXML API for the execution. This 
+environment parses SCXML files (transformed BOM state-machines) and creates 
+instances. Then it initializes all the state-machines to their initial states and simulates 
+sending and receiving of the events to observe state-machine transitions until they 
+reach their final state. The state-machine matching process is based on the following 
+algorithm: 
+Algorithm: State-Machine Matching 
+Input: {SM} ∈ BOM State-Machines, {Actions} 
+Output: TRUE, FALSE 
+1  {SCXML} ←  TransformSMtoScXML(SM)   
+2 ⊳ Transform all BOM-Statemachines in SCxml format ⊲ 
+3  
+4 Create and Initialize EventController: EC 
+5 ⊳ Event Controller controls sending and receiving of events ⊲ 
+6  
+7 for scxml ∈ { SCXML } do 
+8 
+   SC  ← Parse(scxml)  ⊳ Parse scxml document  
+9 
+    Create and Initialize  SCXMLExecutor(SC)  
+10 
+⊳Instantiate SCXMLExecutor thread for each state-machine ⊲ 
+11  
+12 
+Done ← FALSE 
+13 
+while  (Done =FALSE) do 
+14 
+CurrentState ← GetCurrentState() ⊳ SCXMLExecutor returns current state 
+15 
+if CurrentState.IsFinal = TRUE then 
+16 
+Done ← TRUE 
+17 
+end if 
+18 
+⊳Get Next Action to send or receive ⊲ 
+19 
+{NextActions}← CurrentState.GetActions() 
+20 
+for next ∈ NextActions  do 
+21 
+if next.Type = “Send” then 
+22 
+EC.Put(next) ⊳ Simulate sending of next action 
+23 
+SCXMLExecutor.Trigger(next) ⊳Transit from the current state to next state  
+24 
+else 
+25 
+EC.Get(next) ⊳ Simulate recieving of next action 
+26 
+SCXMLExecutor.Trigger(next) ⊳Transit from the current state to next state 
+27 
+end if 
+28 
+end for 
+29 
+end while ⊳Due to either of the send or receive actions the state-machine will  
+30 
+transit to the next state and therefore the current state will be updated.  
+31 
+If the final state is reached then the state-machine matching will be  
+32 
+terminated successfully⊲ 
+33 end for 
+Table 12: State-machine Matching algorithm 
+ 
+
+<scxml initialstate= STATE1>
+<state id = STATE1 final = True>
+<transition event=EVENT NAME
+target=STATE2/>
+</state>
+<state id = STATE2 final = false>
+<transition event=EVENT NAME
+target=STATE1/>
+</state>
+</scxml>Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+92 
+ 
+Figure 34 shows the state-machine matching process. It takes BOM state-machines 
+as modeling objects, automatically transforms it into a SCXML executable format 
+and perform state-machine matching using abstract level execution environment.  
+A successful run of this routine implies that all the state-machines match each other, 
+which satisfies a necessary (but not sufficient) condition of BOM composability i.e. 
+constraint S3a of the requirement specification. The fulfillment of S3a certifies 
+consistency and completeness of the behavioral design of the composed 
+components. Consistency is due to the fact that the components are in correct causal 
+order and Completeness, because their inputs and outputs (send and receive-events) 
+are complete to reach their final states. However we still cannot guarantee 
+correctness the 3rd C of requirements, unless the composition satisfies its requirement 
+specification i.e., all the assigned objectives and required constraints. Also the state-
+machine matching approach may result in reaching final-states but it does not 
+explore all possibilities of the behavioral interaction of the composed components. 
+So it is required to analyze the model at a greater depth using an appropriate dynamic 
+analysis approach. 
+ 
+Figure 34: State-machine Matching Process 
+Therefore for deeper evaluation we propose to utilize the modeling and analytical 
+strength of Petri Net and CSP formalism and incorporate three analysis approaches 
+in our verification framework as introduced and discussed chapter 3. The selection of 
+a suitable approach for the composability verification at dynamic-semantic level 
+depends on the nature of the model. In the following subsections, each of these 
+approaches is discussed in detail. 
+
+Phase-I
+Refinement
+Simuland
+Reguirements
+Engineering
+Phase-V
+Phase-II
+Analysis
+X
+Necessary condition of Constraint S3 satisfied
+Reguirements
+Technique
+Fail
+Success
+4
+Put(
+Analvsis
+Is Final
+state
+1
+8
+SM-1
+SMIM
+reached?
+0155
+BOM
+Components
+Event
+Putl!
+2
+BOM-SM to
+Contro ler
+SCXML
+SM-2
+Get
+04
+Transformation
+Abstract Level
+Modeling
+Action
+Execution
+Lookup
+SM-N
+Table
+Puto
+Phase-I1I
+ Phase-IV
+sc.XvL Executors
+BOM
+Conceptual
+Executable
+Execution
+Model
+Model
+Transformation
+TransformationChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+93 
+ 
+5.7 PN Algebraic Technique 
+The basic idea of this technique is to transform BOM into Petri Net format and 
+verify the properties given in the requirement specifications using algebraic methods. 
+In the verification framework, following steps are proposed to conduct algebraic 
+analysis: 
+5.7.1 BOM to PNML Transformation 
+In the first step, BOM components are transformed into Petri Net Markup Language 
+PNML format [115] which is an XML based form to specify Place/Transition Nets. 
+At first BOM state-machines of all components are parsed and each state is 
+transformed as a Place in the PN model. Similarly each event (send or receive event) 
+is transformed into a Transition in PN with no duplication. An outgoing arc is 
+connected from a place-P to a transition-t if the corresponding state-S (of the sender) 
+has a corresponding event-t as its exit condition and next state S′. An incoming arc is 
+connected from transition-t to another place-P′ which represents the next state S′. 
+Similarly state-R (of the receiver) is transformed into place-Q and the next state R′ 
+into Q′. The incoming and outgoing arcs are connected to t. The sender and receiver 
+entities (of BOM) are represented as tokens in the places. Figure 35 shows how part 
+of a sender and receiver state-machine is transformed into a PN. The place P and Q 
+have tokens showing the current state (or marking) of the composed model. When 
+transition t is fired (meaning event t is sent by P and received by Q) the tokens are 
+transported to P′ and Q′ showing the next marking of composed model. 
+ 
+Figure 35: BOM to PN transformation 
+The transformation process is complete, when all the states and events of every state-
+machine in BOM are plotted in the PN model such that no element is duplicated, 
+and each place or transition is connected so that there are no broken links.  
+5.7.2 PN Algebraic computations 
+In this step the PN incidence matrix and Place/Transition invariants are calculated. 
+To perform this step we use Platform Independent Petri Net Editor (PIPE) API 
+[116]. PIPE is a java based open source API for performing different Petri Net 
+related operations. It offers API functions to automatically compute algebraic 
+resources of a PN model such as Incidence matrix and Place/Transition invariants.  
+Incidence Matrix 
+An incidence matrix of a PN model is calculated by subtracting A- from A+ 
+incidence matrices: 
+
+R
+PChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+94 
+ 
+Algorithm: Incidence Matrix Calculation 
+Input: PN Model (P-places × T-transitions) 
+Output: m × n Matrix A 
+1 Initialize a Matrix Aminus of size m × n such that m=|P| and n=|T| 
+2 for i=0 to m do 
+3 
+for j=0 to n do 
+4 
+if pi ∈ P is connected to tj ∈ T  then   ⊳ i.e., p is the input place of t 
+5 
+A[i][j] ← arc weight  ⊳ arc weight is always ≥ 1 
+6 
+else 
+7 
+A[i][j] ← 0 
+8 
+end if 
+9 
+end for 
+10 end for 
+11  
+12 Initialize a Matrix Aplus of size m × n such that m=|P| and n=|T| 
+13 for i=0 to m do 
+14 
+for j=0 to n do 
+15 
+if tj ∈ T is connected to pi ∈ P  then ⊳ i.e., p is the output place of t 
+16 
+A[i][j] ← arc weight  ⊳ arc weight is always ≥ 1 
+17 
+Else 
+18 
+A[i][j] ← 0 
+19 
+end if 
+20 
+end for 
+21 end for 
+22  
+23 Initialize a Matrix A of size m × n 
+24 for i=0 to m do 
+25 
+for j=0 to n do 
+26 
+A[i][j] ←  Aplus[i][j] - Aminus[i][j]  
+27 
+end for 
+28 end for 
+29 Return A 
+Table 13: Incidence Matrix Calculation 
+Lines 10 calculate the A- matrix. Lines 12-21 calculate A+ matrix and lines 23-28 
+calculate the final incidence matrix. 
+ 
+Place and Transition Invariants 
+The methods for calculating P-Invariants and T-Invariants of a PN model have been 
+extensively studied. The basic principle to compute the fundamental set of P-
+invariants and T-Invariants is based on Farkas Method [117]. The algorithm for 
+finding P-Invariant is presented as follows. The input of the procedure is the 
+Incidence Matrix A and an Identity matrix B, both of size m × n. The output is a 
+matrix C whose rows are the fundamental set of P-Invariants.  
+ 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+95 
+ 
+Algorithm: P-Invariant Calculation 
+Input: Incidence Matrix A, Identity Matrix B 
+Output: Matrix C (rows of C = P-Invariants) 
+1 C ← A | B                 ⊳ Augmentation of A with m × n identity matrix B  
+2 for i=1 to n  do      ⊳ n = |T| 
+3 
+for each pair of rows c1, c2 in C[i-1] where c1[i]  and c2[i]  have the opposite signs  do 
+4 
+c ← |c2[i]|. c1 + |c1[i]|. c2 
+5 
+c´ ← c/g.c.d of each element of row c   ⊳ g.c.d =Greatest common divisor 
+6 
+augment matrix C[i-1]with row c´ 
+7 
+end for 
+8 
+Delete all rows of C[i-1] whose ith component is non-zero, the result is C 
+9 end for 
+10 Return C 
+Table 14: Place-Invariants 
+ 
+The same procedure is used to find T-invariants by taking the transpose of the 
+Incidence Matrix A. Details and a discussion about the improvement of this 
+algorithm are presented in [118]. These algorithms are implemented in PIPE API and 
+can be used in form of function calls. 
+ 
+5.7.3 Property Verification Method 
+The outcome of algebraic analysis technique is the satisfaction or violating of a 
+property with respect to a PN model. There are different methods to perform 
+property verification however there is usually certain theorems behind the reasoning 
+of necessary and sufficient conditions for the fulfillment of a property. In Petri Net 
+literature many solutions (proofs) for the property proving theorems are contributed 
+and can be applied to prove different properties when required. Using these 
+theorems and the available algebraic resources a property verification method 
+(algorithm) is developed which evaluates the conditions given in the theorem on the 
+PN model and results in satisfaction or violation of the required property. Figure 36 
+presents the mechanism of algebraic verification technique in the verification 
+framework: 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+96 
+ 
+ 
+Figure 36: PN Algebraic Technique 
+ 
+To explain our approach we present the theorems and an example property 
+verification method for the analysis of fairness property in a PN model in chapter 7. 
+PNML Execution and State-space Graph 
+It should be noted that PIPE library also offers an execution environment which can 
+be used to run the transformed PNML model. If the tokens (each representing a 
+BOM entity) eventually reaches its final state (place) then the execution is successful. 
+This asserts that the model is correctly transformed and it correctly represents the 
+behavior of its source i.e. the conceptual model. PIPE library also offers a function 
+to generate and visualize state-space graph of the PNML model. This can be useful 
+to find deadlocks and verify other system properties through graph reachability.  
+5.8 CPN based State-Space Analysis Technique 
+The second approach proposed for the dynamic semantic composability verification 
+is based on Colored Petri Nets and State-space analysis technique. This approach 
+effectively utilizes the potential of Colored Petri Net formalism, CPN modeling and 
+programming language, its execution environment and supporting tools in order to 
+verify a composed model at dynamic-semantic level with respect to the requirement 
+specifications. The unique feature of this approach is its data-centric nature. As 
+discussed in section 3.1.5 CPN supports level-3 PN modeling where tokens are 
+structured and can represent data objects. Also the transitions cover greater details of 
+the system behavior. Therefore the structure and the behavior of the system can be 
+modeled with greater details. In order to exploit the data-centric nature of our 
+approach we proposed the following stages: 
+
+ Phase-I
+Refinement
+Simuland
+Reguirements
+Engineering
+Phase-V
+All constraints satisfied
+Phase-II
+Analysis
+Reguirements
+Technique
+Fail
+Success
+Constraints as System Properties
+Property
+Analvsis
+Properties
+Proving
+to prove
+SM-1
+Algebraic
+Theorems
+Verification
+BOM
+Components
+Property
+PN
+Verification Method
+Model
+BOM to PNMIL
+S1I-2
+Transformation
+Abstract Level
+PIPE
+Modeling
+Execution
+Function
+SM-N
+BOM
+Phase-Ill
+ Phase-IV
+State-machines.
+Execution of the
+Conceptual
+Executable
+Werification
+Model
+Model
+method
+Transformation
+TransformationChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+97 
+ 
+5.8.1 BOM Extension  
+The current BOM standard lacks certain structural and behavioral semantics which 
+are essential for modeling complex system behavior therefore we require 
+specification of additional modalities that can help in capturing the structure and 
+behavior of a system at a greater detail [119]. We therefore propose to extend the 
+BOM conceptual model specification by applying the concept of Extended Finite State-
+Machines (EFSM), which is introduced and discussed with detail in [120]. An 
+Extended Finite State Machine (EFSM) is defined by the tuple: 
+M = (Q, I, Σ1, Σ2, V, Λ) where: 
+Q (≠∅) is a finite set of states.  
+I ⊂ Q is the set of initial states 
+Σ1 is a finite set of (send or receive) events. 
+Σ2 is a finite set of actions (Actions are the instructions to be executed and should 
+not be confused with the BOM actions, which are used in pattern of interplay). 
+V is the set of state variables.  
+Λ is a set of transitions; each transition λ ∈ Λ 
+ 
+Where 
+q and q′ ∈ Q 
+e ∈ Σ1 is an event 
+g is a condition (or guard) 
+a ∈ Σ2 is an action.  
+ 
+It means if the system is at a state q, an event e occurs, and the guard g is satisfied, 
+then action a will be executed and the system will transit to the next state q′. During 
+the firing of transition λ ∈ Λ the variables {vin} are used as input and the variables 
+{vout} are used as output. 
+Example: 
+This example is a modified version of an extended finite state-machine of a queue 
+discussed in [120] and is intended to explain the notions of EFSM. A queue 
+component is either empty or nonempty, and in which insertions are done at the rear of 
+the queue and deletions are done at the front of the queue. Also the queue has a 
+maximum size. Two events put and get are used to update the states of the queue. 
+ 
+Figure 37: Buffer Extended finite state-machine [120] 
+ 
+e [g] / a 
+λ = q    
+q′ 
+{vin} | {vout} 
+
+1
+empty
+nonEmpty
+4 
+3Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+98 
+ 
+The EFSM model of the buffer is: M = (Q, I, Σ1, Σ2, V, Λ) where 
+Q= {empty, nonempty} 
+Σ = {put(string obj), get} 
+q0: empty 
+V = {front, rear, M, Data} 
+Λ: Transition Specifications: 
+ 
+Transition 1 allows Queue to transit from empty state to non-empty when put 
+event is received. During this transition the variable rear is incremented. Also 
+the parameter “Obj” of Put event is stored in Data at the rear location. 
+ 
+ 
+ 
+Transition 2 lets Queue to revisit non-empty state when put event is received if 
+rear is less than the maximum size. During this transition the variable rear is 
+incremented. Also the parameter “Obj” of Put event is stored in Data at the 
+rear location.    
+ 
+ 
+Transition 3 lets Queue to revisit non-empty state. It is fired if rear variable is 
+greater or equal to front+1 and less than the maximum size. It will send Get 
+event with data at the front location is sent as parameter. During this transition 
+the variable front is incremented.  
+ 
+ 
+ 
+Transition 4 allows Queue to return back to empty state when if front+1 
+reaches the maximum size. It will send Get event with Data at front location. 
+During this transition both front and rear variables are reset to zero. 
+ 
+We apply the concept of EFSM to the BOM conceptual model, so that we can 
+introduce state-variables and extended representation for transitions (events, guards, 
+actions), to a form, which we name: Extended BOM or E-BOM. There are several 
+advantages in the BOM extension:  
+The usage of variables (or state-variables) in BOM state-machines allows to model 
+the attributes of a component (structure) and their effects caused due to the change 
+of states and occurrence of transitions (behavior). And values of these attributes can 
+Put(obj) [ ] / action{ rear++; Data[rear]=obj;} 
+1: empty 
+nonempty 
+{rear} | {rear, Data} 
+Put(obj) [rear<M]/ action{ rear++; Data[rear]=obj;} 
+2: nonempty 
+ 
+nonempty 
+{M, rear} | {rear, Data} 
+Get(Data[front]) [rear≥front+1∧rear <M]/ action{ front++} 
+3: nonempty 
+ 
+nonempty 
+{M, front, rear, Data } | { front } 
+Get(Data[front])  [front+1=M]/ action{rear=0; front=0} 
+4: nonempty 
+ 
+empty 
+{M, front, Data} | { rear, front } 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 
+99 
+ 
+be used in the arithmetic or logical evaluation to assess trigger conditions of the 
+transitions and to transfer variable values from one model to another model, thus in 
+a composition, a value output by one component can be consumed by the other in 
+the composition. Similarly by introducing actions we can manipulate state-variables 
+of a component during an occurrence of a transition and thus we can model realistic 
+and complex behavior of a system which cannot be obtained through simple labeled 
+transitions (such as events/actions in standard BOM). 
+In the process of BOM to E-BOM extension some of the features are imported 
+from the existing BOM while others are provided by the modelers. Figure 38 provides 
+a comparison of which parts can be imported from BOM and how they are used in 
+E-BOM and which parts are externally provided by the modelers. A user input utility 
+is designed for the modeler’s input where the modeler can choose initial states, select 
+state-variables from the existing BOM entity characteristic or add new variables. 
+Assign data-types to these variables. Design transitions using events from existing 
+BOM and adding guards (conditional statements) and actions. Currently we allow the 
+modelers to use SML language for scripting guards and actions. This is because the 
+extended BOM will later be used in CPN environment which uses SML for code 
+scripts. When each BOM component is extended to the respective E-BOM we 
+proceed to the next stage. 
+ 
+Figure 38: BOM and E-BOM comparison 
+5.8.2 E-BOM to CPN Component Transformation 
+We propose a three layered Component Model based on CPN constructs 32 and 
+transform previously developed E-BOM into this component model using Meta-
+Model transformation rules (defined later in this section). Our proposed CPN 
+component Model is defined as: 
+CPN-CM = 〈SL, BL, CL〉 where: 
+SL = Structural layer 
+BL = Behavioral layer 
+CL = Communication layer 
+                                                 
+32  To understand this mechanism background knowledge of CPN formalism and CPN tools 
+environment is required. We recommend to consult the online documentation [82] 
+
+BOM
+E-BOM
+Conceptual Model
+States
+Entity Types
+Initial States
+M
+Characteristics -
+State Variables
+Pattern of Interplay
+SV Data types
+M
+Actions
+Transitions
+State-machine
+· Current State
+States
+· Event
+Exit Action
+Guard
+Next State
+Action
+Events
+Input Variables
+Source
+Output Variables +
+Target
+ Next State
+Contents
+-Events
+·Parameters
+Parametertypes -
+ Can be imported from BOM
++ Modeler will provide this
+in formationChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 100 
+ 
+Structural layer 
+The structural layer represents physical attributes or properties of the components. 
+The state-variables defined in E-BOM are transformed into the structural layer. In 
+CPN-CM these State-Variables are defined in form of CPN-places with distinct data-
+types represented by colors. They are manipulated by the firing of connected 
+transitions during the execution of the CPN model. The state-variables are connected 
+to the transitions if they are defined in their input or output sets (as discussed in 
+EFSM). The values of the state-variables are specified using CPN-tokens. The data-
+types of the tokens are the same as that of their respective state-variable places. The 
+formal definition of the structural layer is: 
+SL = 〈SV, ΣSV, CSV, VSV, ISV〉 where 
+• SV is a set of places (that represent state-variables of the Conceptual Model)  
+• ΣSV represents a set of color sets that represent state-variable data types  
+• CSV: SV→ΣSV is a function that assigns a color set to each state-variable  
+• VSV is a finite set of typed variables33, assigned to each member of SV.  
+• ISV: SV → EXPR is an initialization function that assigns an initial value 
+expression of type Type [SV] to each state-variable.  
+Behavioral Layer  
+The Behavioral Layer represents state-machine of an E-BOM component in form of 
+CP-Net, where each state from E-BOM component becomes a CPN place of integer 
+type, each E-BOM transition becomes a CPN transition, and the flow integer type 
+token(s), represent the change of component’s state. The state-variables of the 
+structural layer are connected to the transitions of this layer, if they are specified as 
+inputs or outputs in the E-BOM transition. The guards and actions from E-BOM are 
+transformed into CPN transition inscriptions. They should satisfy CPN-ML scripting 
+syntax otherwise the CPN-ML complier will give errors. Formally the Behavioral 
+layer is: 
+BL = 〈S, T, AT, ASV, AC, G, Act〉 where 
+• S is a set of places, called states. Each place is assigned a type INT. This set 
+represents the states of E-BOM state-machine 
+• T is a set of transitions (representing E-BOM transitions)  
+• AT is the set of arcs, called transiting-arcs. Each transiting-arc connects a state s 
+∈ S to a transition t ∈ T, and t to another state s′ ∈ S, if s → t → s′. Each arc 
+in AT is assigned a variable v of type INT. When a transition t is fired a token is 
+transferred from state-s to state-s′ to reflect a change of state in the state-
+machine. 
+• ASV is another set of arcs, called “sv-arcs” that connects state-variables (of the 
+structural layer) to the transitions. If a state-variable SV is assigned as an input 
+variable in the E-BOM transition, then it is connected to the transition with an 
+incoming arc. If a state-variable is assigned as an output variable in E-BOM 
+transition, then the transition is connected to the state-variable with an 
+outgoing arc. A variable v is assigned on each sv-arc whose type is same as that 
+of the attached state-variable. 
+                                                 
+33 Note that these variables are built-in CPN variables that are used to define arc bindings for 
+transporting tokens and are different from our notions of state-variables 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 101 
+ 
+• AC is another set of arcs, called “communication-arcs”, that connects 
+communication ports (CP) of the communication layer to the transitions. A 
+cp∈CP is connected to a transition t through a communication-arc if it 
+corresponds to a receiving event (as specified in E-BOM). Whereas a transition 
+t is connected to a cp∈CP through an arc if it corresponds to a sending event. 
+A variable v is assigned to each communication-arc whose type is same as that 
+of the attached CP. 
+• G: T→EXPR is a guard function that assigns a guard to a transition t (as 
+specified in E-BOM) such that Type[G(t)] = Bool  
+• Act: T→EXPR is an action and assigned to a transition t (as specified in E-
+BOM). The action can be a single executable statement or a complete 
+procedure. Usually the actions are used to read data from attached input state-
+variables or communication ports; to process the data and to produce an 
+output to the attached state-variables or communication-ports. Actions can 
+also be used to type-cast the data. 
+Communication Layer  
+The Communication Layer is responsible for communication with the other 
+components. It provides interface for connecting the inputs and outputs of the 
+components through “port places” and also provides information about the type of 
+data exchange i.e., the tokens of complex data-types that carry message parameters 
+contents, as described in E-BOM event types. The transitions of behavioral layer are 
+connected to the port places of this layer. The arc direction depends on the type of 
+the transition (send or receive). When a message (E-BOM event) is received at the 
+In-port of the communication layer it causes the transition in the behavioral layer to 
+fire as a result of which the state-machine (in the behavioral layer) progresses. This 
+may also happen when a transition causes an outgoing event at the Out port which is 
+expected by another component. Formally the Communication layer CL is: 
+〈CP, ΣCP, CCP, VCP, PT〉 
+• CP is a set of port-places called “communication-ports”. A communication-port 
+connects the inputs/outputs of the component to other components (see section 
+3.1.5). Each CP corresponds to an event (send or receive). 
+• ΣCP represents a set of color sets that represents communication ports data types 
+(as specified by the corresponding event parameters of the E-BOM). Each color 
+set is constructed by combining the data-types of all the parameters. If there are 
+no parameters then NULL type is assigned, which carries a blank token. 
+• CCP: CP→ΣCP is a function that assigns a color set to each port-place.  
+• VCP is a finite set of typed variables, assigned to each CP with Type[v]∈ 
+Type[CP].  
+• PT: CP→ {In, Out, I/O} is a port type function that assigns a port type to each 
+port place (as discussed in section 3.1.5).  
+ 
+ 
+ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 102 
+ 
+E-BOM to CPN-CM Transformation Rules 
+The rules of transformation from E-BOM to CPN-ML are summarized as follows: 
+ 
+ 
+E-BOM 
+→ 
+CPN Component Model 
+ 
+ 
+ 
+Structural Layer 
+Behavioral Layer 
+Comm. Layer 
+1 
+States 
+→  
+State-Places of Type[int] 
+ 
+2 
+Initial-States 
+→  
+Add token type[int], Value 
+= {0, 1…n} for each initial-
+state 
+ 
+3 
+State-Variables (SV) 
+→ 
+SV-Places of (type color-sets 
+defined in 4), Add tokens if SV 
+need initialization 
+ 
+ 
+4 
+SV Data Types 
+→ Define Color-Sets of SV-Places 
+ 
+ 
+5 
+Events 
+→  
+ 
+Add Communicating-Place 
+for each Event of type color-
+set defined in 6 
+6 
+Event Parameters and 
+their types 
+→  
+ 
+Add a color-set of type same 
+as event parameters. If there 
+are multiple parameters then 
+add color-set of type product, 
+where each event parameter 
+type is added in the product. 
+(If there is no parameter, use 
+Null type) 
+7 
+BOM-Action 
+→  
+Transitions  
+ 
+8 Exit-Condition (Action, 
+Next-State) 
+→  
+Connect a transiting arc 
+from a state-place to 
+transition and from 
+transition to next state-
+place, Declare v variable of 
+type[int] on each arc. 
+ 
+9 
+Guards of Transitions 
+→  
+Add guard on each 
+transition 
+ 
+10 Actions of Transitions 
+→  
+Add action script on each 
+transition 
+ 
+11 
+{Input-Variables} of 
+Transitions  
+→ 
+Connect SV-arc from each SV to 
+the transition, if it is added as 
+input variable in that transition. 
+Declare sv of Type SV on the 
+arc. 
+ 
+ 
+12 {Output-Variables} of 
+Transitions 
+→ 
+Connect SV-arc to each SV from 
+a transition if it is added as 
+output variable in that transition. 
+Declare sv of Type[SV] on the 
+arc.  
+ 
+ 
+13 
+Receive-Events ∈ 
+Events 
+→  
+ 
+Connect an arc from CP to 
+the transition if it has a 
+receive event. Declare cp 
+variable of type[CP] on the 
+arc. Mark the CP as in-port. 
+14 Send-Events ∈ Events 
+→  
+ 
+Connect an arc from 
+transition to the CP if it has a 
+send event. Declare cp 
+variable of type[CP] on the 
+arc, Mark the CP as Out port 
+Table 15: Transformation Rules 
+ 
+ 
+ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 103 
+ 
+Example 
+Let us consider the E-BOM of the Queue example discussed in the previous section: 
+ 
+Based on the above E-BOM the transformation will result in the CPN-CM model 
+shown in Figure 39. This model is completely executable in CPN environment and 
+can be composed with other components through communication layer. 
+ 
+Figure 39: CPN-CM represention of Queue component 
+ 
+The red area represents structural layer and consists of state-variables. The green area 
+represents behavioral layer and shows the state-machine. Blue area represents 
+communication layer and shows the Communication Ports (CPs). When some data is 
+received at CP-Put, the value of rear and Max variables are read as input. Guard 
+rear<M is satisfied. A token is taken from the empty state. Action is executed that 
+E-BOM: Queue Component 
+ States 
+{empty, nonempty} 
+  Initial States 
+{empty} 
+  State Variables 
+& 
+  their Data types 
+{front : integer, rear : integer, Max : integer, data: string[ ]} 
+Events 
+{Put (obj:String), Get(obj:String)} 
+Transitions 
+ 
+Put 
+Put 
+Get 
+Get 
+   Current State 
+empty 
+nonempty 
+nonempty 
+empty 
+   Event 
+Put(obj) 
+Put(obj) 
+Get(null) 
+Get(null) 
+   Guard 
+[ ] 
+[rear<M ] 
+[ rear≥front+1∧rear <M ] [ front+1=M  ] 
+   Action 
+rear++ 
+rear++ 
+front++ 
+rear=0; front=0 
+   Input Variables rear 
+rear, M 
+rear, front, M, data[front] 
+rear,  front, data[front] 
+   Output Variables rear, data(obj) rear, data(obj) front 
+rear, front 
+   Next State 
+nonempty 
+nonempty 
+nonempty 
+empty 
+ 
+ 
+ 
+ 
+ 
+Structural Layer 
+Behavioral Layer 
+Comm. Layer 
+
+Empty
+INT
+[rearAM]
+rea
+put
+Rear
+Put
+Put
+R
+Input (rear):
+In
+INT
+output (R):
+STRING
+rtar
+atn
+vet
+1H+
+al R
+(regpput)
+rear+1s
+n
+R
+end
+Max
+Data
+Nonempty
+INT
+M Item
+INT
+FonE
+ffront+1=rear
+then 1*0
+elseempty
+[rear>=front+]
+andalsorearM
+ant
+Front
+Get
+9
+Get
+Input(front):
+Out
+output (Fy:
+STRING
+ket
+val F=
+font+1
+In
+endChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 104 
+ 
+increments the rear variable then the transition Put is finally fired. After which a 
+token is produced at the nonempty state showing the state-transition. Also rear and 
+data variables are updated. Max variable retains the token (due to bi-directional arc). 
+If Put is fired again it will repeat the same process. If Get is fired provided the guard 
+is satisfied, then front, Max and data variables are read as input. The data (picked 
+from the front of the queue) will be sent to the out-CP. When the data is emptied the 
+token will be sent to the empty state. 
+ 
+Automated Transformation Tool 
+In order to automate the E-BOM to CPN transformation process, we develop a 
+transformation utility, which takes an E-BOM component as input and produces 
+CPN- code for all three layers of CPN component model automatically. The code 
+follows CPN-XML specifications. For each E-BOM component, a separate CPN 
+sub-page is generated (programmatically) and the necessary CPN elements (places, 
+transitions, arcs, color sets, variable declarations, initial markings multi-sets, guards, 
+actions, code segments, CPN ML functions, ports, ports-tag) are generated in one 
+CPN output file, which can be loaded in CPN tools. Once all the CPN models of the 
+BOM composition are generated, the modeler creates a main model and “manually” 
+combines the generated CPN-CM modules (using CPN hierarchical features). The 
+output of this step is a composed CPN model. The modeler is also required to 
+initialize each component with data (in form of token assignments i.e. the initial 
+values of the tokens of state-variables and initial states of the state-machine).  
+S3b Evaluation 
+The S3b constraint in the requirement specification requires that “If the conceptual 
+model is transformed into an executable model, the latter should correctly represent 
+the structure and behavior of the former” (see Table 10). Therefore we have to 
+compare each CPN component (executable model) with its respective BOM 
+(Conceptual Model) to check that its structure and behavior is preserved after the 
+transformation. To show that S3b holds after the transformation we rely on the 
+following assertions: 
+1. As BOM is extended to E-BOM hence BOM ⊂ E-BOM. Any information 
+added by the modeler in E-BOM cannot cause loss of structural information 
+of BOM. Therefore E-BOM structurally preserves BOM.  
+2. To check that the generated CPN component contains all the Events and 
+their parameters, States and their exit-conditions, Actions and their 
+senders/receivers we need at least one transformation rule that is responsible 
+to transform these elements: 
+a. Rules 6 & 7 (see Table 15) are responsible for transforming Events 
+and their parameters into CPN component.  
+b. Rules 1 & 8 are responsible for transforming states and their exit-
+conditions. 
+c. Rule 7 is responsible for specification of BOM-actions as transitions 
+in CPN model. Also rule 5 defines port-places which are used to 
+connect senders or receivers. 
+Existence of Rule 1, 6, 7 & 8 confirm that the structure of corresponding 
+BOM is preserved in the transformation. 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 105 
+ 
+3. CPN Tools provide a built-in compiler for the compilation of CPN models 
+and report if there is any syntax error in the model. The absence of error 
+confirms that the transformed model is structurally consistent and 
+behaviorally functional.  
+For the behavioral bi-similarity we propose an inspection technique. At first we 
+evaluate that all the generated components possess the same behavior as defined in 
+the conceptual model. So we test the functional output of each CPN model by giving 
+the needed inputs. If by giving correct inputs, the model produces desired output 
+then its functional behavior is correct. To perform functional testing, the modeler 
+initializes all the IN-type communicating-ports (CPs) with tokens of required 
+parameters. (See Figure 39 for an example where IN-CP “Put” is initialized with 
+tokens of type String). Then the model is executed. If the model produces desired 
+output on the corresponding Out-CPs (In Figure 39 the desired output should be a 
+token of type string retrieved at Get Out-CP), then the functional test is successful. 
+The modeler performs functional test on all generated CPN components.  
+In the second step, when all CPN components are composed (i.e. the socket-places 
+of the main model are connected to the Communicating-Port places of the CPN 
+components then the modeler is required to inspect that CPN components are 
+connect exactly according to the Pattern of Interplay of the BOM composition. Also 
+when the composed model is executed the sequence of sending and receiving events 
+from one component to another (which can be observed at the main model by 
+seeing the movement of tokens) follows the pattern of interplay. If the execution is 
+according to the pattern of interplay and the components make progress until they 
+reach their final states, then we say that the behavior of the transformed model is bi-
+similar to the conceptual model. This confirms the satisfaction of S3b constraint. 
+The execution can be automated or interactive. In automated mode the choices 
+between multiple progressive paths are randomly picked whereas in interactive model 
+the modelers can pick a path of his choice. Using this option the modeler can probe 
+paths that can lead to a successful execution scenario. During the execution CPN 
+tool also offers Data Collection Monitors for recording the data values, which are very 
+valuable for collecting statistics and results of the execution.  
+5.8.3 Verification of the composed CPN model 
+In the next step, the state-space analysis is performed. At first the state-space of the 
+composed CPN model is generated using CPN state space calculation tool. As 
+discussed in section 3.1.5 a state-space is a graph of nodes (of system-states or 
+markings) and arcs (transitions). When the state-space is generated, different query 
+functions can be used to explore the state space graph for various verification 
+questions. A query function is like an algorithm that explores the state-space graph. 
+These algorithms are based on theoretical concepts of Petri Nets state-space analysis 
+and are used to verify PN properties. Therefore we translate a system property given 
+in the requirement specification into a suitable PN property. There have been a lot of 
+contributions in the PN literature in specifying PN properties and methods of 
+reasoning of their satisfiability or violation. In CPN state-space analysis, the existing 
+methods can be utilized in developing query functions for their respective PN 
+properties. 
+CPN tools provide some built-in-functions for the common query tasks. We also 
+propose a library of additional functions to perform queries specific to our 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 106 
+ 
+composability verification framework and the requirement specifications. Figure 40 
+illustrates the state-space analysis of a composed CPN model using a query function.  
+We divide these query functions into two categories: 
+ 
+(i) General System Properties  
+This category includes commonly known system properties such as freedom of deadlock, 
+live lock, starvation, or existence of boundedness, mutual exclusion, fairness, sequentiality, time-
+synchronization etc. if any of these or similar system properties are included as a 
+constraint in the requirement specification then it is translated in CPN terms and a 
+suitable query function is selected from the Function library to perform verification 
+using state-space of the composed CPN model. 
+ 
+Figure 40: CPN State-space analysis  
+For instance a deadlock freedom property can be translated into CPN terms as:  
+“An absence of a marking with no outgoing arcs in the entire state-space graph” 
+So essentially we need to find such a node in the state-space graph that violates 
+above condition. If no such node is found then the model is said to be deadlock free. 
+A library function ListDeadMarking() returns a set of all those markings (if 
+any) which have no outgoing arcs. If the result of this query is an empty list, then we 
+assert that the model is deadlock free. Similarly there are other library functions that 
+deal with the evaluation of other system properties. 
+(ii) Scenario Specific Properties  
+These properties are specific to the scenario (of the real system) under which the 
+model is built. The objectives or goals from the requirement specification are usually 
+translated in form of scenario specific properties. In CPN terms a typical goal or 
+objective can be translated as a certain desirable marking, where the values of state-
+variables in structural layer evaluate to a particular criteria or reaching of particular 
+state(s) in behavioral layer is desired or certain data at the output port(s) of the 
+communication layer is looked-for. A goal or objective can be expressed in a 
+combination of all these possibilities too.  
+Scenario specific properties may also include certain safety or liveness assumptions, 
+which represent certain desirable (or un-desirable) situations that must (or must not) 
+occur in order to satisfy (or violate) the requirements. These properties are mostly 
+the CPN translations of the constraints defined in the requirement specifications. 
+Conceptual 
+Model 
+Requirement 
+Specification 
+ 
+System Property 
+ 
+Satisfie
+ 
+Violate
+ 
+State Space 
+System property   
+CPN Translation 
+Composed CPN 
+Model 
+Query function 
+(Algorithm) 
+Function 
+Library 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 107 
+ 
+Unlike general system properties, verifying scenario specific properties is not a 
+standard operation, and depends on the way they are defined. Most commonly, we 
+make use of our proposed library functions: IsEqual(), IsNotequal(), 
+IsBetween(), IsUpperBound() or IsLowerBound() to construct a 
+“predicate”, that serves as a condition evaluation criteria. Then we use 
+SearchNode(predicate)function to find those nodes, which satisfies the 
+predicate. If one or more nodes are found, then it is verified that the goal is reachable. 
+In cases, where it is important to know how a sequence of the occurrence of 
+transitions, leads to a particular situation when a property is satisfied (e.g., how an 
+objective or goal is reached) we use SearchArc()function with the predicate. 
+This tells us the path in the graph that leads to fulfillment of a property. We also 
+develop an export function, that creates a .DOT file of the entire state-space and can 
+be viewed in graph tools such as GraphViz or Gephi, for visualization and 
+performing further tests on the graph such as finding certain paths/shortest 
+paths/longest paths between two particular nodes. When, a CPN composed model 
+satisfies all the properties in the requirement specification, we say that it is verified at 
+dynamic semantic composability level. In chapter 8 we discuss a Field Artillery 
+Scenario as an example of CPN state-space analysis to explain our approach. 
+An example of translating a scenario-specific property in CPN terms is a restaurant 
+model where we assume that customers may leave the restaurant without paying the 
+bill because they have been waiting for a long time for the waiter to bring bill. This 
+act of the customers is known as “Balking” and is undesirable. Its translation in CPN 
+can be as follows: 
+ 
+“There should be no arc with the name “balk” that leads to any marking in the graph” 
+ 
+Arcs are generated due to firing of the transitions. Existence of balk arc means 
+somewhere in the model an incidence occurred when a customer balked (by firing 
+balk transition). So essentially we need to find that such arc is absent in the state-
+space graph. This can be done by using SearchArc()function. Note that this is a 
+simple example. There could be cases in which a sequence of transitions (called 
+traces) or cycles are searched to verify a property. Error! Reference source not 
+found. describes the overall process of state-space analysis technique.  
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 108 
+ 
+ 
+Figure 41: State-space Analysis Technique 
+ 
+ 
+State-Space Reduction Technique 
+In order to alleviate the well-known problem of state-space explosion we propose a 
+reduction technique called “compositional state space”. The main idea of this technique is that in 
+a hierarchical composition of CPN model, we propose to only consider the places in the 
+main model and treat all the composed components as black boxes. The inputs and outputs 
+of each component can be observed using the flow of Tokens and the data they carry. 
+Therefore in the state-space graph we only keep the markings in which any token is present 
+in the Main model (i.e., any of the place in the main model has at least one token) and delete 
+all other nodes in the state-space graph using the algorithm presented in Table 16. The 
+resultant graph will be a reduced form of the actual graph and only considers those markings 
+that reflect a compositional state-space. It is called compositional state-space because it only 
+represents a part of the actual state-space which is the result of interactions due to the 
+composition of components. In our experience this subset state-space of the whole state-
+space is sufficient to evaluate whether the objectives, goals and the constraints are satisfied 
+or not.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Refinement
+Phase-I
+Function
+Simuland
+Library
+Reguirements
+ Select
+Engineering
+Phase-V
+Phase-II
+Analvsis
+Query function
+(Algorithm)
+Technique
+RS completely satisfied
+Reguirements
+State Space Analysi:
+Analvsis
+CPN propert:
+Translate
+Property
+A...
+1.5
+CPN State Space
+Modeler's Input
+BOM to E-BOM
+Extension
+CPN
+Modehng
+Abstract Level
+Composition
+CPN-CM
+Automatbe
+E-BOM
+Execution
+E-BOM
+Transfomation
+Editor
+N
+Phase-IV
+Phase-Im
+CRepeat
+Conceptual
+Executable
+I to N
+BOM
+Model
+Model
+CPN Model Execution
+Trans formation
+Trans formationChapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 109 
+ 
+Algorithm: Compositional State-Space Generation 
+Input: Original State-Space Graph G  
+|              Output: Reduced State-Space Graph G 
+1 {Vertices}  ← Get-Vertices(G) ⊳ Retrieve all the nodes of the graph in a collection 
+2 for v∈to {Vertices} do 
+3 
+If  False ← Is-Filtered(v) then 
+4 
+G ← Remove-Vertex(G, v) 
+5       Else 
+6 
+next 
+7 
+end if 
+10 end for 
+11 Return G ⊳ Reduced state-space 
+12  
+13 Procedure Remove-Vertex(Graph G, Vertex v) 
+14 {Predecessors} ← Get-Predecessors(G, v) ⊳ Retrieve all the predecessor vertices of v in G 
+15 {Successors} ← Get- Successors (G, v) ⊳ Retrieve all the successors vertices of v in G 
+16 for p∈to { Predecessors } do 
+17 
+for s∈to { Successors } do 
+18 
+G ← Add-Edge(p, s, “DIRECTED”) ⊳ Add a directed arc from each predecessor  
+19 
+                                                                            to each successor ⊲ 
+20 
+G ← Delete-Vertex(v) 
+21 
+end for 
+22 end for 
+23 Return G 
+24  
+25 Procedure Is-Filtered (Vertex v) 
+26 
+If {places}← GetData(v) then ⊳ Each vertex is a marking, which contains data of all  
+27              ⊳ the places of the model with their names and their token values ⊲ 
+29            for p∈to { places } do 
+30 
+if p is a main place and it is not empty then 
+31 
+Return  TRUE ⊳ A valid marking with a non-empty Main place is found 
+32 
+else 
+33 
+next  
+34 
+end if 
+35 
+end for 
+36 
+⊳ if the loop is complete then there is no main place which is not empty  
+37 Return FALSE 
+Table 16: Compositional State-space generation algorithm 
+ 
+Using the Compositional state-space generation algorithm we can filter unnecessary 
+nodes and reduce the size of the graph. The current limitation of this approach is 
+that we first need to construct the actual state-space which is a bottleneck if the 
+model is too large. But this limitation is due to the fact that the process of CPN 
+state-space graph generation cannot be externally modified otherwise if the principle 
+of our reduction technique is applied to the state-space generation algorithm it will 
+directly generate the reduced graph. In chapter 8 we will present the results of our 
+reduction technique by applying it to the Field Artillery example model. 
+ 
+ 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 110 
+ 
+5.9 CSP based Model Checking Technique 
+The third approach proposed for the dynamic semantic composability verification is 
+based on Model Checking, which is widely accepted as formal technique for software 
+verification. In this approach we propose to use Communicating Sequential 
+Processes formalism as a model description language and Process Analysis Toolkit 
+(PAT) as its execution and verification environment in order to verify a composed 
+model at dynamic-semantic level with respect to the requirement specifications. The 
+strength of this approach is in its ability to answer a large variety of verification 
+questions due to the fact that the verification criteria can be specified using LTL, 
+CTL or any of the temporal logic extensions. The Model Checking technique is 
+becoming more promising and acceptable by many software verification users since 
+there is an abundance of improved algorithms, efficient data-structures and faster 
+techniques which are constantly being contributed by the model checking 
+community in order to manage large models with complex modeling requirements. 
+We propose to integrate CSP based model checking verification approach in our 
+composability verification framework. The following stages are proposed in order to 
+perform composability verification using model checking approach. 
+5.9.1 BOM Extension  
+The E-BOM extension for CSP based Model Checking approach is also inspired 
+from the concept of Extended Finite State-Machine as discussed in section 5.8.1. 
+The extended BOMs for CSP can also have state-variables but since CSP# 
+specification does not allow declaration of strings or higher data-types, the state-
+variable definitions are restricted to integer and Boolean34, which in our experience 
+are sufficient to model the behavior of BOM components using CSP (or otherwise it 
+is required to narrow it down to a less detailed version of the component, where only 
+the necessary behavioral details are specified). The transitions of E-BOM contain 
+current-state, event (with parameters), guard, actions and next states. However in this 
+case the action scripts are written in CSP# specification language instead of CPN-
+ML language. And instead of input and output variables, we have local variables 
+which are accessible only to the component and global variables which are accessible 
+to all the components of the composed model. Some additional information such as 
+time constraints and probability factors are further proposed to be included in the 
+BOM extension so that the behavior of complex systems such as real-time systems 
+and probabilistic systems can be modeled and verified.  
+Since Timed-CSPs support a number of timed behavioral patterns to capture 
+quantitative timing requirements, such as delay, timeout, deadline, therefore we 
+suggest using these patterns as time functions in the BOM extension, which helps in 
+the automatic transforming of E-BOM into Timed-CSP components. These time 
+functions are essentially assigned to the E-BOM transitions as explained in the 
+following table: 
+ 
+                                                 
+34 Generally the high level or user defined data types are not permissible in most of the model 
+checking description languages due to the economy of state size, and to avoid risk to state-space 
+explosion. However if the use of such type is inevitable, the PAT tool do provide mechanisms of 
+importing classes from external libraries. If this is the case then the modeler is required to program the 
+components in PAT manually, instead of relying on our automatic transformation tool.  
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 111 
+ 
+Time Function 
+Usage and Explanation 
+Wait[duration]  
+Wait is assigned to model the delay in an activity. An enabled 
+transition waits for the given duration before it is fired.  
+TimeOut[duration, next] When a timeout function is assigned to a transition, it waits for an 
+event to occur. If the event occurs before timeout, it transits to the 
+next state described in the transition definition; otherwise it transits 
+to the next state described in the timeout parameters.  
+Deadline[duration] 
+A transition is constrained to fire when the deadline is reached. The 
+difference between Wait[] and Deadline[] function is that the 
+former makes the system inactive, i.e., it cannot do anything but 
+wait, whereas when the latter is used the system is active and can 
+respond to events etc., until its deadline is reached. 
+Table 17: Time functions in E-BOM 
+Using any of these time functions during the BOM extension is useful to capture the 
+behavior of the real-time systems. In order to further capture the behavior of the 
+complex reactive systems, we also propose to introduce probabilistic factors in the 
+BOM extension. These probabilistic factors can either be used to model the system 
+behavior in form of Markov Decision Processes (MDP) as discussed in section 3.2.5. 
+Or the probability factors can be used to model random time delays, timeouts or 
+deadline, using a particular probability distribution function.  For modeling the MDP 
+behavior probability factors can be assigned to multiple transitions of a component’s 
+state using the following notation provided by the PAT tool: 
+Pcase { 
+[P1]: Transition 1 
+[P2]: Transition 2 
+… 
+[Pn]: Transition n 
+} ; 
+Where  ∑
+𝑃𝑖
+𝑛
+𝑖=1
+= 1 
+For randomizing time functions, we propose to assign the commonly used 
+probability distribution functions as parameters in the E-BOM: 
+Probability Distribution 
+Functions 
+Usage and Explanation 
+Normal[mean, variance]  
+Returns a random value from a normal distribution with a given 
+mean and variance. (Since PAT does not support higher types 
+so we have confined these functions to use integers). 
+Discrete[a, b] 
+Returns a random value from a discrete uniform distribution 
+between a and b (a and b included), such that a < b 
+Exponential [1/lambda] 
+Returns a random value from a an exponential distribution with 
+parameter 1/lambda 
+Table 18: Probability Distribution Functions 
+ 
+In order to implement these assignments we develop an external function library in 
+C# which can be imported and used in PAT. A call to these functions generates a 
+random number according to the specified probability distribution. Beside the time 
+functions, these functions can also be used to generate random values for global or 
+local variables, which can help in modeling different probabilistic system behaviors.  
+When each BOM component is extended to the respective E-BOM we proceed to 
+the next stage. 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 112 
+ 
+5.9.2 E-BOM to CSP# Transformation 
+At this stage, each E-BOM component is transformed into a CSP# process 
+component and composed into an executable system. The main idea of this 
+transformation is based on [121], which discusses the transformation of UML state 
+machines to CSP. We however extend this transformation with Communication 
+channels, Time-functions and probability factors to be able to use it for E-BOM 
+transformation. Table 19 shows the rules used in the transformation process: 
+E-BOM 
+ 
+CSP# Statement and Description 
+States 
+→ 
+State-Name() 
+Each state in E-BOM is defined as a CSP process. 
+ 
+Final-State() = Skip; 
+This statement defines a final state in CSP where Skip is a reserved word means the process terminates 
+successfully. If no such statement exists in any component of the composed model then it is said to be a 
+non-terminating model. 
+Component 
+→ 
+Component-Name = Initial-State() or Component-Name(i) = Initial-State(i) 
+An initial state is defined and assigned to the component. If a component has multiple instances it is 
+passed a parameter ‘i’ which represents the instance number. 
+Transitions 
+→ 
+Simple Transitions: 
+ 
+State() = [guard] event !/? parameters {action} → NextState(); 
+ 
+The transitions are defined using the above format, where State() is the current state of a component. 
+[guard] is a conditional statement. If it is true only then the transition will be enabled. Event is sent using 
+‘!’ symbol or received using ‘?’ symbol through an event channel. For each event in an E-BOM 
+component, we define a channel as follows: 
+channel event-name 0; 
+In CSP# “0” means the buffer size of the communication channel is zero, which further means that it is a 
+synchronous channel. Parameters are the values that are passed during an event exchange and a separated 
+using ‘.’ Actions are scripts that should be executed when the transition is fired. Usually these actions are 
+used to update local or global variables. NextState() is the new state which will be reached when the 
+transition is fired.  It must be defined within the CSP component body. 
+Transitions with Time functions: 
+ 
+Following statement represents a transition with timed-functions: 
+ 
+State() = [guard] Wait[d];  event !/? parameters {action} → NextState(); 
+ 
+State()=[guard] event!/?parameters {action} → NextState() deadline[d]; 
+ 
+State()=[guard]event?parameters{action}→NextState() timeout[d] NextState2 (); 
+ 
+Note that in case of timeout, the transitions should only be receiving an event.  
+Markov Decision Process style Transitions: 
+State() = pcase{ 
+  [Prob1]: [guard] event !/? parameters {actionA} → NextStateA() 
+ 
+  [Prob2]: [guard] event!/? parameters {actionB} → NextStateB() 
+};  
+ 
+Note the postfixes A and B in action or next states of the transitions. Using this CSP# code style multiple 
+transitions can be modeled with different probabilities for either creating a variation of the action which is 
+fired when one of these transitions is selected in a simulation run, or the next states (or both). 
+Transitions with Probability distribution functions: 
+For using probability functions, at first it is required to import our external probability function library 
+using: 
+ 
+#import "PAT.Lib.ProbabilityDistributionFunctions"; 
+ 
+ 
+Following are some examples of how the function calls can be made: 
+var x = call(Normal, 10, 4);   
+An integer variable is defined which will randomly be assigned a value using normal distribution with 
+mean=10 and variance=4 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 113 
+ 
+ 
+Wait[call(Exponential, 1/4)];   
+Delay function with an exponential distribution, where the inter-arrival rate is ¼. 
+State-
+variables 
+→ 
+var Variable-Name=Initial-Value; 
+or  
+#define Constant initial-value; 
+ 
+In CSP# weakly typed variables are used which means that while declaring a variable, the type is not 
+specified. The global variables can be accessed by all components whereas the local variables can only be 
+accessed by the component they belong.  
+Component 
+→ 
+Component-Name = Initial-State() or Component-Name(i) = Initial-State(i) 
+An initial state is defined and assigned to the component. If a component has multiple instances it is 
+passed a parameter ‘i’ which represents the instance number. 
+Composed 
+Model 
+→ 
+Composed-Model = Component1 ||| Component2 ||| … ComponentN; 
+The composed model (name) is defined as a composition of CSP process components with an 
+interleaving operator ‘|||’ between each other. However if there are broadcast events (i.e., one event is 
+sent to all components); or one to many; or many to one synchronization events are used then a parallel 
+operator ‘||’ is used to compose CSP process components.  
+Table 19: E-BOM to CSP# transformation rules 
+We develop a transform tool that takes all the E-BOMs as input, and outputs a single 
+composed model using CSP# description language. The generated CSP# composed 
+model can be opened in PAT tool and compiled for checking errors. If no errors are 
+found then the transformed model is said to be structurally consistent and 
+behaviorally functional and it is ready for simulation and verification. It can also be 
+directly compiled, executed and verified using command line operation. 
+S3b Evaluation 
+The S3b constraint in the requirement specification requires that “If the conceptual 
+model is transformed into an executable model, the later should correctly represent the structure and 
+behavior of the former” (see Table 10). In order to evaluate S3b, i.e., to check that the 
+structure and the behavior of the generated executable model (CSP composed 
+model) correctly represents its conceptual model (BOM composition), we propose 
+following steps: 
+1. For each CSP component, manually inspect that it contains all the states that exist 
+in its corresponding BOM component 
+2. Inspect that the exit condition(s) of each State in BOM correspond to a 
+transition(s) and a next state(s) in CSP. 
+3. Execute the generated CSP model in PAT simulator and observer that all the 
+components reach their final states (or in case of a non-terminating model each 
+component re-visit its initial state iteratively). 
+Step 1 & 2 confirms by inspection that the structure of the generated model correctly 
+represents its conceptual mode whereas step 3 confirms that it behavior is bi-similar 
+to the conceptual model and therefore satisfies S3b constraint. 
+5.9.3 Verification of the composed CPN model 
+At this stage, the CSP composed model undergoes composability verification using 
+PAT model checker. At first the requirement specification is translated into CSP# 
+property (or assertion) description language. This language is based on a mix of 
+classical Linear Temporal Logic (LTL) and its different extensions such as Real-Time 
+LTL and Probabilistic LTL and is used to construct assertions (verification 
+questions) of various types, such as reachability properties, safety properties, liveness 
+properties, deadlock freeness etc. We use the syntax of assertion specification 
+language of PAT to translate the objectives and constraints of given requirement 
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 114 
+ 
+specifications. Following are some generic examples of how to specify PAT 
+assertions: 
+ 
+1 
+#assert System deadlockfree; 
+This 
+assertion 
+checks 
+deadlock 
+freedom in the ‘System’ 
+2 
+#assert System reaches goal?0; 
+This assertion checks that whether 
+the ‘System’ can reach its goal (by 
+receiving a goal event with ‘0’ 
+parameters) 
+3 
+#assert System |= <>goal?0; 
+This is an equivalent LTL assertion 
+It checks if the goal is eventually 
+reachable. 
+4 
+#assert System |= []<>goal?0; 
+This LTL assertion checks if the 
+goal is always eventually reachable 
+by the system. Note that it is 
+different from assertion 2. 
+5 
+#assert System |= <>goal?0 deadline[50]; 
+This 
+assertions 
+verifies 
+goal 
+reachability with time constraint i.e., 
+its checks if the goal is reachable 
+within 50 time units or not? 
+6 
+#assert System |= <>goal?0 with prob; 
+This assertion checks the min and 
+max 
+probability 
+of 
+the 
+goal 
+reachability.  
+7 
+#define goal (Some-Variable == True); 
+#assert System reaches goal; 
+This is another way to verify goal 
+reachability, 
+where 
+the 
+goal 
+definition is based on some value of 
+a variable. 
+Table 20: Some examples of PAT Assertions 
+ 
+When an assertion is defined and its syntax is correct, we can verify it by running the 
+PAT model checker and select the desired assertion from the list. The model checker 
+will present the verification results with success, showing that the assertion is verified 
+or it will provide a counter example if the assertion is not satisfied. In chapter 9, an 
+example of field artillery is presented to show how a CSP composed model is verified 
+with requirement specifications defined as PAT assertions. Figure 42 describes the 
+overall process of state-space analysis technique.  
+
+Chapter 5  
+ 
+Proposed Methodology and the Verification Framework 
+   
+Page 115 
+ 
+ 
+Figure 42: CSP based Model Checking Technique 
+ 
+5.10 Summary 
+In this chapter the proposed composability verification framework is discussed in 
+details with its structural and functional specifications. Each activity, algorithm, 
+technique and the process is explained in the perceptive of Component based M&S 
+life-cycle. The composability verification is performed at three levels of 
+composability called static, semantic and dynamic-semantic composability. The main 
+objective of the proposed framework is to verify composability at these levels with 
+respect to requirement specifications. The first two levels are suggested to be 
+evaluated using static-analysis techniques whereas the third level is proposed to be 
+verified using dynamic analysis techniques. At first the behavior of the composed 
+components is evaluated using State-machine matching technique. If they pass this 
+step, they are subjected to one of the three proposed approaches called (i) Algebraic 
+Analysis Technique, (ii) State-space analysis technique or (iii) Model checking for 
+dynamic-semantic composability verification. The choice of these approaches 
+depends on the nature of the model. In chapter 10 we will present some guidelines 
+on how to choose an appropriate approach.  
+ 
+When the entire composability verification process is successful, it implies that the 
+BOM based composed model is structurally and behaviorally consistent, it is 
+composable at syntactic, semantic and dynamic-semantic level and is correct with 
+respect to the given requirement specifications.  
+ 
+ 
+
+Phase-I
+Refinement
+Simuland
+Reuirements
+Engineering
+Phase-V
+Phase-II
+Analysis
+Technique
+Rs completely satisfied
+Reguirements
+CSP Model
+Checking
+Analvsis
+GlobalVariables
+Modeler's Input
+and Modal Checker
+Translate
+BOM to E-BOM
+Extension
+Abstract Level
+CSP#
+Modeling
+Execution
+Automztis
+E-BOM
+E-BOM
+Transfomztion
+Edlitor
+Phase-I1I
+Assertions
+Executable
+Conceptual
+04
+Model
+Model
+CSP Model Execution
+Transformation
+Transformation   
+Page 116 
+ 
+Chapter 6 
+Composability Verification Process 
+ 
+Chapter 5 mainly presented the specification of our proposed composability verification framework 
+including details of different modules, their mechanics and the procedures they perform. In this 
+chapter we present how to use our framework. It can be used as a manual of our composability 
+verification framework. At the end of this chapter we also provide necessary recommendations for the 
+selection of appropriate approach based on the given inputs. 
+ 
+The description of shapes used in the following flow diagrams is as follows: 
+Object, 
+Data, 
+Model, 
+Component etc. 
+ 
+Any shape of this color 
+express a 3rd party tool 
+ 
+ List, Collection or Set of 
+objects, Data, Model 
+ 
+Stop 
+means 
+that 
+the 
+process has failed.  
+  
+Process or action 
+ 
+Go means it is successful, 
+therefore process with 
+the implementation phase 
+ 
+Iterative process.  
+ 
+Compare two objects. 
+ 
+Extension or Transformation 
+of object 
+ 
+Compare multiple objects 
+ 
+Extension or transformation 
+of many objects  
+ 
+ 
+ 
+Comments 
+ 
+ 
+ 
+Data, Process, Information 
+flow 
+ 
+ 
+ 
+Page connector 
+ 
+ 
+ 
+Repeat 
+process 
+(Go 
+to 
+previous step) 
+ 
+ 
+ 
+6.1 
+Composability Verification Process 
+Figure 43 to Figure 53 illustrate the composability verification process in the form of a 
+flow chart. (The illustrated steps are explained later in this section): 
+Stop 
+Go 
+X 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 117 
+ 
+ 
+Figure 43: Formulation of Simuland, Requirements and Conceptual Model 
+Abstraction 
+Real System 
+Requirement 
+Engineering 
+Simuland 
+UML 
+diagrams 
+(or 
+others) 
+representing 
+structure 
+and 
+behavior of the 
+system 
+Formal 
+or 
+Informal 
+BOM 
+Repository 
+Construct 
+components 
+from scratch and store 
+them in the repository 
+Search components that 
+match the requirements 
+  
+Candidate BOMs 
+Select suitable BOMs 
+Filter 
+BOMs 
+and select the 
+most 
+suitable 
+set of BOMs 
+that match the 
+requirements 
+Compose selected BOMs 
+  
+Conceptual Model 
+1 
+Matching and 
+Selection 
+1 
+2 
+3 
+4 
+5 
+6 
+7 
+Requirements 
+  
+Objectives 
+  
+Constraints 
+  
+Selected BOMs 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 118 
+ 
+                                                                         
+  
+ 
+ 
+ 
+If all candidate 
+BOMs 
+are 
+tried 
+without 
+success 
+then 
+repeat step 3 
+ 
+Syntactic Matching Process 
+Start 
+Rule 
+based 
+Syntactic 
+Matching 
+process to check that 
+the components can 
+correctly fit to gather 
+and their inputs and 
+outputs match each 
+other. 
+Sender BOM-i 
+Events 
+Entities 
+States 
+Actions 
+Receiver BOM j 
+ 
+Events 
+Entities 
+States 
+Actions 
+Check SM-Rule 1: 
+Name of the send-event and 
+receive-event 
+should 
+be 
+same 
+Send-Event  
+Rule-1 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5.  
+8 
+Yes 
+Proceed 
+to Rule-2 
+Check SM-Rule 2: 
+Each send-event should have at 
+least one corresponding receive-
+event and vice-versa 
+Sender BOM-i 
+Events 
+Entities 
+States 
+Actions 
+Receiver BOM j 
+ 
+Events 
+Entities 
+States 
+Actions 
+Equal 
+Names 
+Rule-2 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+Proceed 
+to Rule-3 
+Check SM-Rule 3: 
+The 
+number 
+of 
+parameters 
+(content characteristics of event 
+types) of the send-events should 
+be the same as the number of 
+parameters of the receive-events. 
+Sender BOM-i 
+Events 
+Entities 
+States 
+Actions 
+Receiver BOM j 
+ 
+Events 
+Entities 
+States 
+Actions 
+Equal no. of 
+parameters 
+Receive-Event  
+Rule-3 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+9 
+10 
+11 
+2 
+1 
+Figure 44: Syntactic Matching Process 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 119 
+ 
+ 
+ 
+  
+Static-Semantic 
+Matching Process 
+Check SSM-Rule 1: 
+“Area-of-Interest” attribute of 
+all the actions should be exactly 
+same as that of POI or should 
+belong to an equivalent class in 
+the respective ontology 
+Rule-1 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+12 
+Yes 
+Proceed 
+to Rule-2 
+13 
+3 
+2 
+Start Rule based Static-
+Semantic 
+Matching 
+process to check that 
+the 
+composition 
+is 
+meaningful 
+and 
+the 
+components 
+can 
+correctly 
+understand 
+each other.  
+Pattern of Interplay (POI) 
+BOM-1 
+Events 
+Entities 
+States 
+Actions 
+BOM N 
+ 
+Events 
+Entities 
+States 
+Actions 
+BOM-OWL 
+  
+  
+  
+  
+  
+AOI 
+Purpose 
+Data Type 
+Units 
+Check SSM-Rule 2: 
+“Purpose” attribute of all the 
+actions should be exactly same 
+as that of POI or should belong 
+to an equivalent class in the 
+respective ontology 
+Rule-2 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+Proceed 
+to Rule-3 
+Check SSM-Rule 3: 
+Data types of each element in the 
+event parameters of the send-event 
+and receive-events should be of same 
+class, equivalent class or should be in 
+direct hierarchical relationship 
+Rule-3 
+Passed
+? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+Proceed 
+to Rule-4 
+14 
+15 
+Figure 45: Static-Semantic Matching Process 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 120 
+ 
+ 
+ 
+ 
+  
+Check SSM-Rule 4: 
+The units of the quantities expressed 
+in 
+each 
+element 
+in 
+the 
+event 
+parameters of the send-event and 
+receive-events should be of same class, 
+equivalent class or should be in direct 
+hierarchical relationship 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+3 
+Yes 
+If Static-Semantic Matching 
+Rule 1, 2, 3 & 4 are satisfied 
+then we can confirm that the 
+communication among the 
+components is meaningful as 
+intended.    
+Perform State-machine 
+Matching Process 
+ 
+Rule-4 
+Passed
+? 
+15 
+16 
+Start 
+Dynamic-Semantic 
+Composability evaluation. 
+In the first step, the state-
+machines 
+of 
+all 
+the 
+composed 
+BOM 
+components are matched 
+to check their behavior 
+compatibility.   
+  
+Candidate BOMs 
+Transform  
+BOM State-machines     
+to SCXML 
+Abstract level execution 
+SCXML
+1 
+SCXML
+2 
+SCXML
+N 
+Is 
+Final? 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+The composed 
+components 
+are behaviorally 
+compatible! 
+If the SCXML instances of composed 
+BOM components are executed at an 
+abstract level, and they all reach their 
+final states then the state-machines are 
+said to be matched. 
+4 
+17 
+18 
+Figure 46: State-machine Matching Process 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 121 
+ 
+ 
+Figure 47: Approach Selection | PN Algebraic Technique 
+ 
+4 
+Select an appropriate approach for dynamic-
+semantic composability analysis 
+PN Algebraic 
+Technique 
+19 
+CPN State-Space Analysis 
+Model Checking 
+If 
+standard 
+BOM 
+components 
+are 
+composed 
+(with 
+no 
+information 
+available 
+for their extension) and 
+only general structure 
+and behavior is to be 
+analyzed 
+then 
+this 
+approach is suitable.  
+  
+  
+Conceptual 
+Model 
+Transform  
+BOM to PNML 
+PNML 
+Model 
+23 
+21 
+20 
+22 
+If details of the BOM 
+components are available 
+which are required to 
+extend them into to E-
+BOMs 
+and 
+functional 
+specification 
+of 
+the 
+components 
+is 
+to 
+be 
+evaluated 
+then 
+this 
+approach is suitable. 
+  
+5 
+Transform BOM-
+state 
+machines 
+into a single PN 
+model  
+If 
+the 
+model 
+requirements 
+contain time constraints and (or) 
+the 
+model 
+possess 
+non-
+deterministic behavior and (or) 
+the model has a large number of 
+state (but does not require 
+detailed enumerations) then we 
+propose to use this approach.  
+6 
+PNML Execution 
+24
+ 
+If 
+the 
+execution 
+is 
+successful such that it 
+leads to final states, 
+then the model satisfies 
+S3b. 
+Meaning 
+the 
+behavior 
+of 
+transformed 
+model 
+correctly represents its 
+conceptual model. 
+PIPE execution 
+Environment 
+No 
+Stop 
+and 
+repeat from 
+step 5. 
+Yes 
+Continue 
+Verification  
+7 
+Success 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 122 
+ 
+ 
+ 
+ 
+Figure 49: Implementation 
+ 
+  
+Requirements 
+Can
+did
+ 
+Can
+did
+ 
+Objectives 
+Can
+dida
+ 
+Can
+dida
+ 
+Constraints 
+Property Verification 
+Translate objectives 
+and Constraints into 
+PN properties 
+26
+ 
+25
+ 
+Next 
+property 
+Algebraic Computation 
+Resources 
+(Incidence Matrix, P-Invariants,     
+T-Invariants etc.) 
+ \ 
+Property-
+Proving 
+Theorem 
+  
+Property 
+Verification 
+Method 
+PIPE Function 
+library 
+Calculate algebraic computation 
+resources of the PNML model 
+using PIPE library functions 
+Construct a property verification 
+method using an appropriate PN 
+property proving theorem 
+RS 
+Satisfied? 
+No 
+Composability 
+verification failed. 
+Modifications in 
+the 
+conceptual 
+model 
+are 
+required. 
+Yes 
+Composability verification 
+is successful.  
+ 
+The conceptual model is 
+qualified 
+for 
+the 
+implementation phase 
+Go 
+Stop 
+PNML 
+Model 
+7 
+Composed 
+Model 
+Simulation 
+Model 
+Experimental 
+ Model 
+Code 
+Design of  
+Experiment 
+Simulation Results 
+Simulation 
+Go 
+Figure 48: PN Algebraic Technique (continued) 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 123 
+ 
+ 
+Figure 50: State-Space Analysis Technique 
+  
+\  
+Conceptual 
+Model 
+  
+BOM to E-BOM 
+Extension 
+27 
+Modeler’s input is 
+required here to 
+extend BOM into 
+E-BOM 
+  
+E-BOM 
+  
+Transform E-BOM to 
+CPN Component Model 
+Each 
+E-BOM 
+component 
+is 
+automatically 
+transform into CPN 
+component Model 
+  
+CPN   
+Component models 
+28 
+5 
+E-BOM 
+Extension Utility 
+CPN-CM is our proposed 
+component specification 
+based on CPN language 
+Structural Comparison 
+with the Conceptual 
+Model 
+Does the transformed 
+model contain all Events, 
+parameters, 
+Actions, 
+States, exit-conditions as 
+specified 
+in 
+the 
+conceptual Model? 
+Is the behavior of the 
+transformed model bi-
+similar to the conceptual 
+model? 
+ 
+Perform functional test 
+on each component to 
+check that the inputs 
+produce desired output 
+according 
+to 
+the 
+conceptual model. 
+ 
+Perform CPN execution 
+to compare that the 
+progress is made by all 
+the 
+components 
+according to the pattern 
+of interplay defined in 
+the conceptual model.  
+ 
+  
+Behavioral Comparison 
+(Functional Testing) 
+CPN Execution 
+Environment 
+Perform Model inspection and 
+check that all the elements of 
+Conceptual model are present 
+in the transform model 
+Perform functional testing: 
+Initialize 
+inputs 
+of 
+each 
+transformed 
+component 
+separately and execute the 
+component 
+in 
+the 
+CPN 
+execution 
+environment 
+to 
+check if it produces desired 
+output 
+according 
+to 
+the 
+conceptual l model 
+ 
+Is 
+Successful? 
+ 
+No 
+Stop 
+and 
+repeat from 
+step 27. 
+8 
+Yes 
+S3b 
+partially 
+satisfied 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 124 
+ 
+ 
+Figure 51: State-Space Analysis Technique (continued) 
+Requirements 
+Can
+did
+ 
+Can
+did
+ 
+Objectives 
+Can
+dida
+ 
+Can
+dida
+ 
+Constraints 
+Translate objectives 
+and Constraints into 
+CPN properties 
+Next property 
+Modeler is required to manually 
+compose all the generated CPN 
+components into a main CPN 
+model.  
+Compose CPN 
+Components 
+CPN 
+Composed 
+Model 
+Initialize and Execute 
+CPN Model  
+Successful 
+No 
+There are exceptions in the 
+execution.  Check and 
+Repeat step 29, or 28, 26 
+ 
+  
+29 
+30
+ 
+Generate State-Space 
+31 
+Yes 
+S3b completely 
+satisfied.  
+  
+Perform CPN Property 
+Verification 
+32 
+RS 
+Satisfied? 
+No 
+Composability 
+verification failed. 
+Modifications 
+in 
+the 
+model 
+are 
+required 
+Yes 
+Composability 
+verification 
+is 
+successful, go to implementation 
+phase  
+Go 
+Stop 
+8 
+CPN 
+Hierarchical 
+Modeling Tool  
+CPN Execution 
+Environment 
+CPN State-Space 
+Analysis Tool  
+Property Verification 
+Query Function 
+(written in CPN-ML)  
+CPN ML  
+Programming 
+Environment 
+CPN Query 
+Function library 
+Next 
+function 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 125 
+ 
+ 
+Figure 52: Model Checking 
+Continue from step 22 
+\ 
+ 
+Conceptual 
+Model 
+  
+BOM to E-BOM 
+Extension 
+33 
+Modeler’s input is 
+required here to 
+extend BOM into 
+E-BOM 
+  
+E-BOM 
+  
+Transform E-BOM to 
+CSP Process 
+Each 
+E-BOM 
+component 
+is 
+automatically 
+transform into CSP 
+Process model. 
+ 
+ 
+CSP  
+Process Components 
+34 
+6 
+E-BOM 
+Extension Utility 
+CSP process components 
+are 
+represented 
+using  
+PAT’s CSP# specification 
+Structural Comparison  
+Does the transformed 
+model contain all states, 
+and events as specified in 
+the conceptual Model? 
+Behavioral Comparison 
+PAT Simulation 
+Environment 
+Perform Model inspection and 
+check that all the elements of 
+Conceptual model are present 
+in the transform model 
+Perform behavioral similarity 
+evaluation by simulating the 
+CSP composed model in the 
+PAT simulation environment. 
+If it reaches final states then it 
+correctly 
+represents 
+the 
+behavior of its corresponding 
+conceptual model.  
+ Pass? 
+ 
+No 
+Stop 
+and 
+repeat from 
+step 33, 19 
+or 5 
+9 
+Yes 
+S3b satisfied 
+E-BOM with Time 
+constraints 
+and 
+probabilistic factors 
+Compose CSP 
+Components 
+CSP Composed Model 
+35 
+Compose all the generated 
+CSP components using 
+parallel operator 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 126 
+ 
+ 
+Figure 53: Model Checking (continued) 
+ 
+The Composability Verification Process is explained as follows: 
+ 
+6.1.1 Formulation of Simuland, Requirements and Conceptual 
+Model 
+In step 1, the Real system is studied and a suitable simuland is formulated. It can be 
+described formally or informally. We assume that UML diagrams are used to describe 
+the simuland. The system is also studied to gather requirements and formulate 
+requirements using our proposed formal requirement specification method (step 2). 
+With this information at hand, suitable components are searched in the BOM 
+repository, with an assumption that a composition of these components will form a 
+conceptual model that represents the simuland (step 3).  If a desired component is 
+Requirements 
+Can
+did
+ 
+Can
+did
+ 
+Objectives 
+Can
+dida
+ 
+Can
+dida
+ 
+Constraints 
+Translate objectives 
+and Constraints into 
+CSP assertions 
+Start PAT 
+Model Checking Tool 
+36 
+  
+Verify Assertion 
+37 
+RS 
+Satisfied? 
+No 
+Composability 
+verification failed. 
+Modifications 
+in 
+the 
+model 
+are 
+required 
+Yes 
+Composability 
+verification 
+is 
+successful, go to implementation 
+phase  
+Go 
+Stop 
+9 
+PAT Model 
+Checker 
+  
+Assertions  
+LTL, CTL, RT-
+LTL, PLTL 
+Next  
+Assertion 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 127 
+ 
+not found it is constructed form the scratch, added to the repository, and then used 
+in the current context (step 4).  
+The discovered components are called candidate components. Among these 
+candidates, most suitable ones i.e., those that best match the simuland and the given 
+requirements, are selected (step 5, 6).  
+These BOM components are composed and a conceptual model is constructed (step 
+7). We recommend that the modeler also creates a formal model of the conceptual 
+model using our proposed BOM formalism and graphical notation. This will help in 
+documentation and understanding details of conceptual model and its composition.  
+ 
+6.1.2 Syntactic Matching Process  
+When the verification process starts the Composed BOM model (conceptual model) 
+is passed through a rule-based static analyzer to verify the composability at syntactic 
+level (step 8-11). If this level is passed then the constraint S1 (as defined in Table 10) 
+is satisfied and only then the model is cleared for the next step (otherwise the 
+verification process is stopped and another candidate selection is picked, composed 
+and this step is revised).  
+ 
+6.1.3 Static-Semantic Matching Process  
+In the next step the components are analyzed at static-semantic level using the 
+semantic analyzer (step 12-15). When this step is passed then the constraint S2 (as 
+defined in Table 10) in is satisfied and the BOM composition is ready to be verified at 
+dynamic-semantic level.  
+ 
+6.1.4 State-machine Matching Process  
+At this level, the first step is to perform state-machine matching of BOM 
+components using State-machine checker (step 15 – 18). A successful state-machine 
+matching satisfies the constraint S3a (as defined in Table 10). 
+ 
+6.1.5 Approach Selection for Dynamic-Semantic Composability 
+Verification 
+In the next stage the verification framework offers three choices of verification 
+technique for the analysis of dynamic-semantic composability level. The modeler can 
+choose algebraic technique if there is no information available to extend the BOM 
+components into E-BOM. Therefore the conceptual model will be transformed into 
+PNML without requiring any extension. If the modeler has details and data available 
+to transform BOM into E-BOM and the model does not represent a real-time 
+system, then it is highly recommended that the second proposed approach (CPN 
+state-space analysis) should be chosen. If the model represents a real-time system and 
+it is stochastic in nature then the modeler should choose the third approach (Model 
+checking). These are general guidelines and are not concrete rules. The ultimate 
+choice of the approach depends on the nature of the model, nature of the 
+requirement specification properties and the available information. 
+ 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 128 
+ 
+6.1.6 PN Algebraic Technique 
+When Algebraic technique is selected, at first the conceptual model is transformed 
+into PNML model (step 23). This PNML model is executed in PIPE execution 
+environment to evaluate S3b (step 24). If successful then the requirement 
+specification properties are taken one by one and translated into a PN property (step 
+25). Thereafter a property proving theorem is selected that proves this PN property. 
+Based on this theorem a property verification method is constructed inform of an 
+algorithm. Running this algorithm proves or falsifies the requirements specification 
+property (step 26). If all the properties in the requirement specification are satisfied 
+then the model is successfully verified otherwise the process is stopped and model 
+refinements are made. 
+ 
+6.1.7 State-Space Analysis Technique 
+When the CPN state-space analysis technique is selected, at first each BOM 
+component is extended to E-BOM (step 27). This step requires modeler’s input and 
+can be delivered using the BOM-to-E-BOM extension utility. When the extension is 
+complete, each E-BOM is transformed into our proposed CPN component model 
+using our automatic transformation tool (step 28). The output of this step is a set of 
+CPN components. At this step it is required to conduct structural and behavioral 
+comparison between the generated components and the respective BOM using 
+inspection and functional testing methods. If the comparison is successful then S3b 
+constraint of the requirement specification is partially satisfied.  
+The modeler is then required to compose these generated components in a main 
+model using CPN hierarchical tool (step 29). (Binding IN-ports and OUT-ports of 
+each component using sockets in the main model). When the model is composed, it 
+is executed (step 30) using CPN execution environment to test that all components 
+correctly interact with each other and make necessary progress to reach the final 
+states. If the execution is successful then the constraint S3b is fully satisfied, 
+conforming that the structure and behavior of the executable model correctly 
+represents its respective conceptual model and therefore any verification operation 
+performed on the executable model will imply correctness of its conceptual model.  
+In the next step the CPN model is subjected to the state-space analysis (step 31). At 
+first a state-space graph of the model is generated. Then for each objective and 
+constraint in the requirement specification a verification query function is either 
+created or selected from the function library. The execution of this function is done 
+using CPN-ML program execution environment and the result of this function tells 
+if the property is satisfied or violated (step 32). If all the properties are satisfied we 
+say that the composability verification process is successful.  
+ 
+6.1.8 Model Checking 
+When the Model Checking technique is selected, at first each BOM component is 
+extended to E-BOM (step 33). This step requires modeler’s input and can be 
+delivered using the BOM-to-E-BOM extension utility. It is possible to assign Time 
+constraints and the probabilistic factors with the states or the transitions.  When the 
+extension is complete, each E-BOM is automatically transformed into CSP# process 
+specification (step 34) and composed (step 35). The composition of each CSP 
+
+Chapter 6  
+ 
+Composability Verification Process 
+   
+Page 129 
+ 
+process representing a BOM component be done using sequential operator ‘;’ parallel 
+operator ‘||’ interleaving operator ‘|||’ or (non-deterministic or user’s) choice 
+operator ‘[ ]’ depending upon the nature of the composed components. We suggest 
+composing each CSP in parallel so that each process executes in parallel and 
+synchronizes with each other by sending or receiving events at their respective 
+communication channels. The composed model can then be simulated using the 
+PAT simulator. A successful simulation run with at least one path leading to the final 
+state(s) shows that the behavior of the composed model correctly represents its 
+conceptual model and thus satisfies constraint S3b.  
+In the next step, the assertions defined in CSP format (using LTL, Real-Time LTL or 
+PLTL) are verified using PAT model checker which results in its satisfaction or 
+violation (step 36, 37). If all the assertions are satisfied we say that the composability 
+verification process is successful. 
+ 
+6.2 Summary 
+In this chapter, a flow diagram of composability verification process is presented. It 
+indicates different steps and forms of inputs and outputs of each step in the process. 
+This flow diagram can be used as a guideline to perform composability verification 
+using three different approaches. Some recommendations are also presented in 
+making a suitable choice. Once the verification process is completed successfully, the 
+composed model can undergo implementation phase where it is programmed and 
+simulated using a suitable simulation platform. Also the experimental model can be 
+constructed to perform different experiments on the implemented model and 
+simulation results are generated for study and decision making. The implementation 
+phase is out of the scope of this thesis. 
+ 
+ 
+ 
+
+   
+Page 130 
+ 
+Chapter 7 
+Fairness verification using PN 
+Algebraic Techniques 
+ 
+This chapter explains how algebraic techniques can help in verifying system properties of a Composed 
+Model, using an example of a manufacturing system in which fairness is selected to be the required 
+system characteristic. 
+7.1 
+Fairness 
+Fairness has been defined in section 3.1.3 in terms of a Petri Nets property. In this 
+chapter the concept of fairness is covered in detail. Intuitively, fairness is a liveness 
+property that means no component of the system which becomes possible (or 
+becomes enabled) sufficiently often should be delayed indefinitely. [122].  
+On the basis of the extent of sufficiency, fairness is generally categorized in the 
+following three types in literature: 
+Unconditional Fairness 
+Also called Impartial implies that every component in a system proceeds infinitely 
+often without any condition. The term “proceed” means to make progress, (e.g., 
+firing of a transition). Unconditional fairness is also known as non-deterministic 
+choice and is usually present among the components that are independent of each 
+other [122]. 
+Weak Fairness 
+Also called Just, implies that every component in a system that is enabled 
+continuously from some point onwards eventually proceeds. 
+Strong fairness 
+Every component in a system that is enabled infinitely often proceeds infinitely 
+often. A noticeable difference in weak and strong fairness is that weak fairness 
+involves persistent enabling of a component that wants to proceed, whereas strong 
+fairness is not persistently enabled.  
+Some important generalizations of fairness exist in literature [122]: 
+Equi fairness: means to give each component an equal chance to proceed. It can be 
+regarded as Justice. This type of fairness does not always apply in real world 
+scenarios because of priority policies or some other reasons. 
+Bounded fairness: means to give each component an equal number of chances such 
+that no component proceeds for more than “k-times” without letting the others to 
+take their turn 
+For instance there is a check-in service at the airport that serves two types of queues: 
+(i) Business class and (ii) Economy class at a time. It will be called fair, if it mostly 
+serves business class passengers but not more than (say) 10 times, without serving a 
+passenger from the economy class queue. 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 131 
+ 
+In Petri Nets, fairness can be viewed in two perspectives namely: Transition fairness 
+and Marking fairness. The former corresponds to fairness of choice of transitions, 
+and the latter deals with the fair reachability of states.  
+7.2 Fairness Verification 
+There are different ways to verify fairness of a model.  The focus of this chapter is to 
+discuss the technique for the verification of fairness property using PN Algebraic 
+analysis and provide the necessary and sufficient conditions for a PN model to be 
+fair. The evaluation of these conditions in a PN model involves theorems and linear 
+algebraic computations; therefore it is classified as an Algebraic technique. Based on the 
+theorems below, we propose an algorithm for automatic fairness verification.  
+In Petri Nets, fairness is mainly perceived in terms of occurrences (or firing) of 
+transitions. Two transitions t1 and t2 are said to be in a fair relation if there exists a 
+positive integer k such that for any reachable marking M and any firing sequence σ:  
+(The symbol #(t/σ) denotes the number of times a transition t occurs in a firing sequence σ)  
+In words, neither of the transitions should occur more than a finite number of times 
+(k) without letting the other to occur at least once. This is known as bounded fairness 
+(or B-Fairness) with upper bound = k. If every pair of transition is in a bounded fair 
+relation, then the entire net is said to be fair [123]. 
+For the algebraic verification of fairness property in a PN model the following 
+theorems are applied. Details and proofs of these theorems are discussed in [123].  
+Theorem I 
+Given a PN with an incidence matrix A, if there exists a firing-count vector X, such that:  
+A.X ≥ 0 and X≠0 
+Then a necessary condition for the PN to be fair is that each entry of X is positive.  
+ 
+Theorem II 
+If a Petri Net N is bounded for any initial marking M 0 then the condition in Theorem I is 
+necessary and sufficient for N to be fair.  
+Corollary: If there exists a P-Invariant Y of positive integers such that: A.Y=0 then the PN is 
+guaranteed to be structurally bounded.  
+ 
+Theorem III 
+A fair Petri Net PN has only one reproduction vector (i.e., a minimal T-Invariant) at the most.  
+ 
+Based on the above definition of the bounded fairness and theorems I, II and III a 
+PN is said to be fair if it satisfies two conditions: (i) There must exist a single T-
+Invariant X of a given PN model whose each entry is non-zero and the product    
+AX = 0 and (ii) There must be at least one P-Invariant, which means that the net is 
+structurally bounded. 
+# (t1/σ) = 0 ⇒ #(t2/σ) ≤ k  ∧ #(t2/σ) = 0 ⇒ #(t1/σ) ≤ k 
+(7.1) 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 132 
+ 
+7.3 Manufacturing system 
+In this section, a component based composed model of a manufacturing system is 
+presented. Using this composed model, it is shown how the proposed verification 
+framework is used to verify the required specification, which in this example consists 
+of fairness as an important quality constraint. Two different scenarios of this 
+example are discussed. In the first scenario the model is shown to be unfair as 
+verified by our verification framework. In the second scenario the model is modified. 
+It is then verified and it satisfies the fairness constraint.  
+ 
+7.3.1 Scenario I 
+It is assumed that the manufacturing system model is composed of two machines M1 
+& M2 and a shared Robot R as shown in Figure 54. The robot loads raw material on 
+the machines and operate on them for producing goods. The Robot is assigned 
+(“loaded”) to either of the machines at a time. When the Robot is loaded, it deposits 
+raw material on the machine and process it. When the good is produced the robot is 
+unloaded and is available for the other machine. 
+ 
+Figure 54: Manufacturing System (acquired from [124]) 
+The process of composability verification is initiated as follows: 
+ 
+Simuland and Requirement Specification 
+In the first step, the entities, events and the states of the simuland are perceived 
+according to Figure 54. The simuland and the requirement specifications are used to 
+construct an appropriate conceptual model according to the steps given in the 
+composability verification process described in Chapter 6. 
+ 
+We define Requirement speciation of the manufacturing system as: 
+RS0 = 〈O, S〉 where: 
+ 
+Objectives O = {o1, o2, o3} 
+o1: Machine1 should continuously produce product1 without any infinite delay 
+o2: Machine2 should continuously produce product2 without any infinite delay 
+o3: Both machines should produce products with a ratio of 1:1 
+
+Loading
+Processing
+A
+Unloading
+Finished
+Raw Material
+Product A
+Machine A
+Robot
+Loading
+Processing
+B
+Unloading
+Finished
+Product B
+Machine AChapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 133 
+ 
+ 
+System Constraints S = {s1, s2, s3, s4} 
+s1: Machine1, Machine2 and the Robot components should be composable at 
+syntactic level 
+s2: Machine1, Machine2 and the Robot components should be composable at static-
+semantic level 
+s3a: State-machine matching of the composed model should be successful. Since the 
+models are non-terminating so there are no final states, instead the goal-states: 
+“Machine1 completes production” & “Machine2 completes production” will be considered.   
+s3b: The transformed executable model correctly represents the structure and 
+behavior of the conceptual model. 
+s4: The shared robot should treat both machines with fairness (i.e., k-fairness; k=1). 
+ 
+Conceptual Model 
+The formal specification and graphical representation of each BOM model 
+participating in the manufacturing composed model are as follows. For the ease of 
+readability following color codes are used for different BOM elements in the formal 
+definition: 
+ 
+BB0 = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Machine1 {C0(Id:Integer)} 
+ 
+EvT = {E0(LoadingM1, Robot, Machine1, null), E1(UnloadingM1, Robot, Machine1, null), 
+E2(ResetM1, Robot, Machine1, C0)} 
+ 
+Act = { A0(LoadingM1, Robot, Machine1, E0), A1(UnloadingM1, Robot, Machine1, E1), A2(ResetM1, 
+Robot, Machine1, E2)} 
+ 
+S = {S0(M1Waiting, A0, S1),  S1(M1Processing, A1, S2), S2(M1Completed, A2, S0)}  
+Table 21: Formal definition of Machine1 Base-BOM 
+ 
+ 
+BB1 = 〈 EnT, EvT, AcT, S 〉 where: 
+EnT = Machine2 {C1(Id:Integer)} 
+ 
+EvT = {E3(LoadingM2, Robot, Machine2, null), E4(UnloadingM2, Robot, Machine2, null), 
+E5(ResetM2, Robot, Machine2, C1)} 
+ 
+Act = { A3(LoadingM2, Robot, Machine2, E3), A4(UnloadingM2, Robot, Machine2, E4), A5(ResetM2, 
+Robot, Machine2, E5)} 
+ 
+S = {S3(M2Waiting, A0, S4),  S4(M2Processing, A1, S5), S5(M2Completed, A2, S3)} 
+Table 22: Formal definition of Machine2 Base-BOM 
+ 
+ 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 134 
+ 
+BB2 = 〈 EnT, EvT, AcT, S 〉 where: 
+EnT = Robot {} 
+ 
+EvT = {E6(LoadingM1, Robot, Machine1,  null), E7(UnloadingM1, Robot, Machine1, null), 
+E8(LoadingM2, Robot, Machine2, null), E9(UnloadingM2, Robot, Machine2, null)} 
+ 
+Act = {A6(LoadingM1, Robot, Machine1, E6), A7(UnloadingM1, Robot, Machine1, E7), 
+A8(LoadingM2, Robot, Machine2, E8), A9(UnloadingM2, BB2, Machine2, E9)} 
+ 
+S = {S6(Idle, {A6, S7},{A8, S7} ),  S7(Busy, {A7, S6}, {A9, S6})} 
+ 
+Table 23: Formal definition of Robot Base-BOM 
+ 
+ 
+ 
+CB0 = 〈 AcTIN,  AcTOUT ,  POI 〉 where: 
+AcTIN = AcTOUT = ∅  
+ 
+POI = {POI0(!A6 , ?A0), POI1(!A7 , ?A1),  POI2(!A2, ?A2),  POI3(!A8 , ?A3), POI4(!A9 , ?A4), POI5(!A5, 
+?A5) } 
+ 
+Table 24: Formal definition of Manufacturing System composed BOM 
+ 
+Figure 55: Manufacturing System BOM based Composed Model 
+ 
+ 
+Figure 55 represents the BOM based Conceptual Model of the manufacturing system 
+which includes three BOMs, formally defined using our proposed graphical notation. 
+The figure shows how the characteristics, Events, Actions and states are mapped to 
+each other (using dotted red line). In machine 1 characteristic C0 is mapped to Event 
+E2 which means event uses characteristic C0 as parameter. Similarly Event E0 is 
+mapped to A0, E1 to A1 and E2 to A2 respectively which means the Actions uses their 
+mapped events. The mapping of actions to the states in the figure shows which 
+action will cause which state to transit to the new state (shown by blue arrow). The 
+
+Machine1
+Robot
+Machine2
+Characteristics:
+Characteristics:
+Characteristics:
+Co = Id : Integer
+C1 = Id : Integer
+Actions:
+Actions:
+A6=LoadingM1
+Actions:
+A0=LoadingM1
+A7=UnloadingM1
+A3=LoadingM2
+A1=UnloadingM1
+A8=LoadingM2
+A4=UnloadingM2
+A2=ResetM1
+A9=UnloadingM2
+A5=ResetM2
+States:
+States:
+States:
+S0=M1Waiting
+S6=ldle
+S3=M2Waiting
+S1=M1Processing
+S7=Busy
+S4=M2Processing
+S2=M1Completed
+S5=M2Completed
+M1Waiting
+Idle
+M2Waiting
+M2Processing
+A6
+S6
+A0
+SO
+S3
+S2
+A2
+S7
+S5
+A5
+M1Completed
+Busy
+M2CompletedChapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 135 
+ 
+basic BOM components are connected to each other using the formal definition 
+shown in Table 24, which describes the source (!) and destination (?) of an action 
+from one component to other. In Figure 55 this is shown using black arrow lines with 
+their input/output (I/O) label. This is called Pattern of Interplay (in BOM 
+specification). 
+Static Analysis  
+Rules 
+Machine1 
+Machine2 
+Robot 
+• Name of the send-event and 
+receive-event should be same 
+• Each send-event should have at 
+least one corresponding receive-
+event and vice-versa 
+• The number of parameters 
+(content characteristics of event 
+types) of the send-events should 
+be the same as the number of 
+parameters 
+of 
+the 
+receive-
+events. 
+?LoadingM1(null) 
+ 
+!LoadingM1(null) 
+?UnloadingM1(null)  
+!UnloadingM1(null) 
+ 
+?LoadingM2(null) 
+!LoadingM2(null) 
+ 
+?UnloadingM2(null) 
+!UnloadingM2(null) 
+Table 25: Syntactic Matching 
+ 
+It can be seen in Table 25 that the name of the send-event and receive-events are the 
+same. ( !=Send, ?=Receive). And they are in one-to-one relationship. Also the no. of 
+parameters of each event is equal to 1. Based on these facts the components are said 
+to be syntactically composable (S1 satisfied).  
+ 
+We assume that Machine1, Machine2 and Robot components have the semantic-
+attributes as shown in Table 26 which satisfy all the static-semantic matching rules. 
+The attributes highlighted in red color are semantically equivalent (Exact match) 
+therefore S2 is satisfied.  
+ 
+Machine1 
+Machine2 
+Robot 
+AOI = {Production, Manufacturing, 
+Production-line, Lathing} 
+AOI = {Production, Manufacturing, 
+Production-line, Polishing} 
+ 
+AOI = {Production, Manufacturing, 
+Conveyer, Automation} 
+Purpose = {Manufacture Product1, 
+Manufacture Product3} 
+Purpose = {Manufacture Product2, 
+Manufacture Product3} 
+Purpose = {Manufacture Product3} 
+Data Types of parameters= {null} 
+Data Types of parameters = {null} 
+Data Types of parameters = {null} 
+Units of Measurement = {} 
+Units of Measurement = {} 
+Units of Measurement = {} 
+Table 26: Static-Semantic Matching 
+ 
+Dynamic Analysis  
+The state-machine matching process is successfully conducted as both Machine1 and 
+Machine2 reach their goal-states namely: Mcompleted and M2-completed and satisfy S3a. 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 136 
+ 
+ 
+Figure 56: State-machine matching of manufacturing system 
+ 
+BOM to PNML Transformation 
+In the next step the components are subjected to PNML transformation process. 
+The output of the transformation process is a PN model shown in Figure 57. It can 
+be seen from the inspection that the States and their exit conditions, Events and 
+Actions all are present in the transformed model (as specified in the original 
+conceptual model). Also this PN model is executed in the PIPE runtime 
+environment. The execution is successful because the places P3 and P6 acquired 
+tokens (showing that these goal states were reached during the execution). This 
+satisfies S3b.  
+ 
+ 
+Figure 57: PN model of the manufacturing System 
+ 
+ 
+
+Machine1
+Robot
+Machine2
+M1
+Waiting
+LoadingM1
+Idle
+LoadingM2
+M1
+Processing
+UnloadingM1
+Busy
+M2
+Waiting
+M1
+Completed
+M2
+M1Reset
+Processing
+UnloadingM2
+M2
+Completed
+M2ResetM1Waiting
+P1
+P4
+M2Waiting
+T1
+T4
+LoadingM1
+LoadingM2
+Idle
+P7
+ResetM1
+ResetM2
+T3
+P2
+P5
+T6
+M1Processing
+P8
+M2Processing
+Busy
+UnloadingM1
+Robot
+UnloadingM2
+T2
+T5
+P3
+P6
+M1Completed
+M2Completed
+Machine1
+Machine2Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 137 
+ 
+Algebraic Resource Computation 
+At this step, the initial marking M0 and the Incidence Matrix A of the PN composed  
+model shown in Figure 57 are calculated using PIPE library functions as follows: 
+ 
+ 
+M 0 P1 P2 P3 P4 P5 P6 P7 P8 
+ 
+1 
+0 
+0 
+1 
+0 
+0 
+1 
+0 
+ 
+A P1 P2 P3 P4 P5 P6 P7 P8 
+T1 -1 
+1 
+0 
+0 
+0 
+0 -1 
+1 
+T2 0 
+-1 
+1 
+0 
+0 
+0 
+1 
+-1 
+T3 1 
+0 
+-1 
+0 
+0 
+0 
+0 
+0 
+T4 0 
+0 
+0 
+-1 
+1 
+0 -1 
+1 
+T5 0 
+0 
+0 
+0 
+-1 
+1 
+1 
+-1 
+T6 0 
+0 
+0 
+1 
+0 
+-1 0 
+0 
+ 
+Table 27: Initial Marking and Incidence Matrix (Scenaro I) 
+Note that the labels of rows and columns in A and elements in M0 correspond to 
+places and transitions in Figure 57. The matrix A is given as input to the Invariant 
+calculation module that calculates the following P-Invariants and T-Invariants in the 
+PN model of the Manufacturing System: 
+ 
+P1 P2 P3 P4 P5 P6 P7 P8 
+1 1 1 0 0 0 0 0 
+ 
+T1 1 
+T2 1 
+T3 1 
+T4 0 
+T5 0 
+T6 0 
+ 
+T1 0 
+T2 0 
+T3 0 
+T4 1 
+T5 1 
+T6 1 
+ 
+Table 28: P-Invariants and T-Invariants (Scenaro I) 
+ 
+Property Verification Function 
+In order to proceed with the verification, we have to translate the objectives and 
+constraints of the requirement specification into PN properties: 
+o1: Machine1 should continuously produce product1 without any infinite delay 
+o2: Machine2 should continuously produce product2 without any infinite delay 
+o3: Both machines should produce products with a ratio of 1:1 
+s4: The shared robot should treat both machines with fairness (i.e., k-fairness; k=1). 
+ 
+It is clear from the {o1, o2, o3 and s4} that if the robot serves both machines with 
+fairness (S4) only then both of them will be able to produce their respective products 
+continuously without indefinite delay (O1 & O2). And if the fairness is bounded 
+such that k=1, then both machines will produce products with equal ration 1:1.  
+Therefore the translation of the requirement specification in PN form is as follows: 
+ 
+PN Property P1 = “The model should be Bounded-Fair (with K=1) such that the robot serves 
+both machines alternatively”.  
+ 
+In order to verify bounded-fairness we consider the property proving theorems I, II 
+& III defined in section 7.2. Based on these theorems we construct a property 
+verification function using the following algorithm: 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 138 
+ 
+ 
+Algorithm: B-Fairness Verification  
+Input: {P-Invariants}, {T-Invariants}, A; Output: TRUE 
+1 If |{T-Invariants}| = 1 then ⊳ List of T-Invariants has exactly 1 invariant, meaning it is a 
+2 
+Reproduction vector , Theorem III⊲ 
+3 
+XT ← T-Invariants[0]        ⊳ Get the only T-Invariant from the list 
+4 
+if A.XT ≥ 0 and each element in XT >0 then   ⊳ Multiply XT with Incidence matrix and 
+5 
+Check that each element of T-invariant is  
+6 
+positive, Theorem I⊲ 
+7 
+if |{P-Invariants}|>0  then            ⊳ Check if there is any P-Invariant, meaning PN  
+8 
+        Model is bounded, Theorem II⊲ 
+9 
+Return TRUE 
+10 
+else 
+11 
+Return FALSE ⊳ Theorem II violated 
+12 
+end if 
+13 
+else 
+14 
+Return FALSE  ⊳ Theorem I violated 
+15 
+end if 
+15 else 
+17 
+Return FALSE ⊳ Theorem III violated 
+18 end if 
+ 
+Table 29: B-Fairness Verification 
+ 
+ 
+Based on this algorithm we perform property verification of the given PN model. It 
+is evident that the T-invariants (see Table 28) contain zero entries which violate 
+Theorem I. Also there is more than one T-invariant which violates Theorem III 
+therefore the net is said to be unfair. As the PN is unfair, it is impossible to guarantee 
+that objectives o1, o2, o3 will be satisfied because either of the machines may over 
+perform by acquiring robot multiple times without letting the other to get the robot 
+for at least once (failure of o1 & o2). Therefore either of the machines may face a 
+situation in which it is unable to produce enough number of products to meet the 
+required objectives i.e., the ratio 1:1 for producing products cannot be fulfilled 
+(failure of o3); consequently the composed model may fail to satisfy given 
+specifications. 
+ 
+ 
+ 
+7.3.2 Scenario II 
+In order to understand the fairness verification process, a counterexample is 
+presented. In this example another component is added to the composition called 
+Controller that can supervise the robot assignments. The job of the Controller is to 
+enforce fairness in the system.  The BOM model of the controller is defined as 
+follows: 
+ 
+ 
+ 
+ 
+
+Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 139 
+ 
+BB3 = 〈 EnT, EvT, AcT, S 〉 where: 
+EnT = Controller {} 
+ 
+EvT = {E10(LoadingM1, Controller, Robot , Machine1,  null), E11(LoadingM2, Controller, Robot, 
+Machine2, null)} 
+ 
+Act = {A10(LoadingM1, Controller, Robot, Machine1,  E0), A11(LoadingM2, Controller, Robot, 
+Machine2, E1)} 
+ 
+S = {S8(AssignM1, {A10, S9}),  S9(AssignM2, {A11, S8})} 
+ 
+Table 30: Formal definition of Controller Base-BOM 
+ 
+CB0 = 〈 AcTIN,  AcTOUT ,  POI 〉 where: 
+AcTIN = AcTOUT = ∅  
+POI = {POI0(!A10 , {?A0, ?A6}), POI1(!A7, ?A1),  POI2(!A2, ?A2), POI3(!A11 , {?A3, ?A8}),  POI4(!A9 , 
+?A4), POI5(!A5, ?A5) } 
+ 
+Table 31: Formal definition of Modified Manufacturing System composed BOM 
+The other components have the same definition except that the sender of Event E0 
+and E1 is BB3 (controller) and the receivers of event E0 are BB0 (Machine1) and BB2 
+(Robot); whereas the receivers of event E1 are BB1 (Machine2) and BB2 (Robot). 
+Figure 58 shows the composed BOM of modified manufacturing system.  
+ 
+Figure 58: Modified manufacturing system composed BOM 
+ 
+
+Machine1
+Robot
+Machine2
+Characteristics:
+Characteristics:
+Characteristics:
+CO = Id : Integer
+C1 = Id : Integer
+Actions:
+Actions:
+A6=LoadingM1
+Actions:
+A0=LoadingM1
+A7=UnloadingM1
+A3=LoadingM2
+A1=UnloadingM1
+A8=LoadingM2
+A4=UnloadingM2
+A2=ResetM1
+A9=UnloadingM2
+A5=ResetM2
+States:
+States:
+States:
+S0=M1Waiting
+S6=ldle
+S3=M2Waiting
+S1=M1Processing
+S7=Busy
+S4=M2Processing
+S2=M1Completed
+S5=M2Completed
+M1Waiting
+M1Processing
+Idle
+S6
+M2Waiting
+M2Processing
+SO
+S1
+A6
+40
+4.1
+S3
+S4
+A4
+S2
+A2
+A7
+S7
+A9
+S5
+A5
+M1Completed
+Busy
+M2Completed
+Controller
+Characteristics:
+A
+Action Connector
+Actions:
+A10=LoadingM1
+S
+Initial State
+A11=LoadingM2
+States:
+State
+S8=AssignM1
+S9=AssignM2
+Exit condition
+State Transistion
+AssignM1
+Input/Output
+A10
+S8
+connection
+S9
+A11
+AssignM2Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 140 
+ 
+When the verification process is started, the BOM components are transformed into 
+PN model as shown in Figure 59 where the controller component is attached to both 
+machines and the robot and controls the machine assignment to enforce Kfairness. 
+It is evident from the figure that when the robot is assigned to Machine1 once, it 
+cannot be reassigned (because of the lack of token in P9), and vice-versa. If the 
+number of tokens are increased to ‘n’, the same model can work for k=n fairness.  
+In the initialization phase, the initial marking M0 and Incidences Matrix A were 
+calculated as follows: 
+M0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 
+ 
+1 0 0 1 0 0 1 0 1 
+0 
+ 
+A P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 
+T1 -1 1 0 0 0 0 -1 1 -1 
+1 
+T2 0 -1 1 0 0 0 1 -1 0 
+0 
+T3 1 0 -1 0 0 0 0 0 0 
+0 
+T4 0 0 0 -1 1 0 -1 1 1 
+-1 
+T5 0 0 0 0 -1 1 1 -1 0 
+0 
+T6 0 0 0 1 0 -1 0 0 0 
+0 
+ 
+Table 32: Initial Marking and Incidence Matrix (Scenaro II) 
+ 
+ 
+ 
+Figure 59: Modified PN model of the manufacturing System 
+ 
+When the Invariant calculation module is executed, the following T-Invariant and P-
+Invariant were discovered for the model shown in Figure 59: 
+T1 1 
+T2 1 
+T3 1 
+T4 1 
+T5 1 
+T6 1 
+ 
+P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 
+1 1 1 0 0 0 0 0 0 
+0 
+ 
+Table 33: P-Invariants and T-Invariants (Scenaro II) 
+ 
+Having only one T-Invariant (and the only one) with non-zero entries and having a 
+P-Invariant (with some non-zero entries), satisfies the conditions (of Theorem I, II & 
+III) required for the model to be bounded fair.  
+
+Controller
+P1
+P4
+P9 AssignM1
+M2Waiting
+[个
+LoadingM1
+P10AssignM2
+LoadingM2
+T4
+T3
+Idle
+T6
+ResetM1
+P2
+P7
+P5
+ResetM2
+M1Processing
+M2Processing
+P8
+Busy
+UnloadingM1
+UnloadingM2
+T2
+Robot
+T5
+P3
+P6
+M1Completed
+M2Completed
+Machine1
+Machine2Chapter 7  
+ 
+Fairness verification using PN Algebraic Techniques 
+   
+Page 141 
+ 
+ 
+ 
+Based on these result from PN algebraic analysis technique, we can confirm that the 
+composed model satisfies given requirement specifications. Due to the supervised 
+controller, the Robot is bound to operate fairly between the two machines, which 
+results in fulfillment of the objectives O1, O2 and O3 and also satisfied required 
+constraint S1.  
+7.4 Summary 
+Fairness property becomes significant in the composability verification of a 
+composed model because it does not allow any component to dominate and 
+excessively proceed, while other components do not proceed even for once. As 
+illustrated by the example of the manufacturing system, fairness of Robot allocation 
+can ensure that both machines will perform to produce a required number of 
+products. If there is no fairness we cannot guarantee that this objective will be 
+reached. 
+Using the example of Fairness verification in the manufacturing system, we explain 
+how our Algebraic Verification Technique works. It is a notable fact that this 
+technique does not face state-space explosion because it does not involve reachability 
+graph construction and can work only by calculating incidence matrix and P/T 
+invariants. There are a lot of PN properties which can be verified using these PN 
+algebraic computation resources. On the other hand this approach can only be used 
+to verify a limited set of PN properties (for which suitable theorems exist).  
+ 
+ 
+
+   
+Page 142 
+ 
+Chapter 8 
+Model Verification using State-space 
+Analysis techniques 
+ 
+Colored Petri Nets and its analysis techniques are very useful for accurate and efficient verification as 
+it is one of the competitive formalisms in the specification of the concurrent systems. Its application in 
+the Composability verification proves to be very constructive, especially with a focus on the dynamic 
+semantic composability level. The analysis techniques contributed by the CPN community over a 
+couple of decades provide a significant improvement on efficient and accurate reasoning regarding the 
+model correctness. In this chapter a Field Artillery Model is presented as an example. It is shown 
+how the BOM based Field Artillery Model is transformed into our proposed Colored Petri Net 
+components and verified using state-space analysis.  
+ 
+Combat Modeling is about the models that describe or represent weapon systems 
+and combat situations. There are numerous types of combat models. These types are 
+distinguished by their modeling objectives. Some of the fundamental objectives of 
+combat modeling are training, war-games, weapon testing etc. 
+8.1 
+Combat Modeling 
+Combat modeling purposefully abstracts and simplifies combat entities, their 
+behaviors, activities, and interrelations to answer defense-related research questions. 
+There cannot be a general model that answers all questions however there is a 
+concept of a generic situated environment and four core activities that can be found 
+on every battlefield [125].  
+8.1.1 Situated Environment 
+Combat Modeling starts with analyzing the challenges to model the Situated 
+Environment. All modeled combat entities are situated in the environment, the virtual 
+battlefield. They perceive the environment including other entities, and map their 
+perception to an internal representation based on the knowledge and goals. They 
+communicate and act with other combat entities within the environment. The 
+environment contains all objects, passive ones like obstacles, as well as active ones 
+like enemy or friendly units [125].  
+8.1.2 Moving 
+Moving is the core activity of combat modeling that deals with the movement of 
+individual entities. These entities could be weapon, people etc. or aggregate models 
+that are used to model the movements of groups of entities. The models use patches 
+and grids; they use physical models for weapon systems and reference schemas for 
+unit movement [125]. 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 143 
+ 
+thermal, and optical sensors, can contribute to perceiving the environment and the 
+other entities as close to reality as possible. Intelligence, surveillance, and 
+reconnaissance operations contribute to similar requirements. In order to sense 
+special properties of an entity, each of these special properties needs to be modeled 
+explicitly. If it is modeled explicitly, it needs to make a difference in the 
+reconnaissance process. Furthermore, if a detail is important for the military decision 
+process, it needs to be part of the perception, and hence needs to be observed by 
+sensors, which requires that the respective things are modeled as properties of the 
+entities [125]. 
+8.1.4 Shooting 
+Modeling the outcomes of duels between weapon systems and battles between units 
+is still a topic of major interest. On the weapon system level, direct and indirect fires 
+are analyzed. Direct fire means that the target is in the line of sight of the shooter. In 
+case of indirect fire systems such as Artillery and other ballistic weapons, they do not 
+need to see the target and shoot at it straight. Their weapons follow a ballistic curve 
+being described by the term indirect fire. Many models have been developed to keep 
+up with the score. For instance a game based point systems that count how often and 
+where a target is hit and use “hit-and-kill” probabilities (which are based on real-
+world data) to simulate hitting or missing a target  [125].  
+ 
+8.1.5 Communication:  
+This core activity deals with the modeling of Communications, Command and 
+Control. It ties all the earlier activities together as command and control is situated in 
+the environment and commands the entities to shoot, move, observe, and 
+communicate. Several models of command and control in military headquarters are 
+discussed, as more and more simulation models have to come up with decisions 
+based on available information where until recently human decision makers had to be 
+involved. The better command and control is modeled the less military experts are 
+needed to provide a realistic training environment [125]. Based on these principles of 
+combat modeling an example model of Field Artillery is presented to explain the 
+approach of composability verification using state-space analysis. Figure 60 highlights 
+the activities of combat modeling.  
+ 
+Figure 60: Activities of Combat Modeling 
+ 
+Situated 
+Environment 
+Moving 
+Looking  
+or Sensing 
+Shooting 
+Communication 
+(Command & 
+Control) 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 144 
+ 
+8.2 Field Artillery 
+Field Artillery (FA) is one of the indirect fire systems35 that engage the opponent 
+without requiring line of sight between the shooting system and the target. Infantry 
+uses small, medium or heavy howitzers (artillery guns) that provide fire support for 
+combat units. Similarly Navy artillery provides fire power, where missiles can be fired 
+on land based or sea based direct or indirect targets. The general mission of FA is to 
+destroy, neutralize or suppress the enemy by cannon, rocket, and missile fires and to 
+help integrate all fire support assets into combined arms operations [126]. The field 
+artillery system provides close support to maneuver friendly forces, counter fire and 
+interdiction as required. These fires neutralize, canalize, or destroy enemy attack 
+formations or defenses; obscure the enemy’s vision or otherwise inhibit his ability to 
+acquire and attack friendly targets; and destroy targets deep in the enemy rear with 
+long-range rocket or missile fires [127].  
+FA weapons are usually located in defiladed areas in order to protect them from 
+enemy detection. This nature of FA gunnery makes it an indirect fire problem. 
+Observed fire (the technique that solves the indirect FA gunnery problem) is carried 
+out by the coordinated efforts of the Forward Observers, Head Quarter (HQ), the 
+Fire Direction Center (FDC), and firing sections of the firing unit (Batteries) [126]. 
+Figure 61 gives an overview of the essential elements of a field artillery and the 
+situation of an indirect fire, where a forward observer spots an enemy unit and 
+requests fire support from a nearby friendly unit. It should be noted that this 
+scenario is only assumed and simplified for the sake of an example, whereas the 
+today’s state of the art of field artillery systems is much more modernized and 
+technologically advanced.  
+ 
+Figure 61: Elements of Field Artliiery & Indirect Fire 
+8.2.1 Simuland 
+Based on Figure 61 an Indirect Fire Support scenario is considered. In this scenario 
+the enemy units are not in the line of sight of the firing units. A soldier (forward 
+observer) from the observation post observes the enemy field and detects potential 
+targets. When a target is spotted, he calls BHQ for fire support and provides the 
+target details. BHQ requests FDC to process the target tactically & technically. In 
+tactical terms, the target should be of high importance to gain tactical advantage. In 
+technical terms the target should be in the firing range of the supporting artillery. If 
+                                                 
+35 Although there are some exceptions, in which Field Artillery engages in direct fire mode 
+
+ForwardObserver
+Spotted Target
+FiringUnits
+BHQ&FDCChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 145 
+ 
+the target is valid FDC approves the request otherwise the request is denied. If the 
+request is approved BHQ assigns the target to the firing units (batteries). We suppose 
+that the target can be one of three types: light (e.g., camps, troops, and trucks), 
+medium (e.g., tanks, light guns) or heavy (e.g., artillery units, missile launchers). The 
+target is assigned to one, two or three batteries respectively. This is because medium 
+and heavy targets require the fire power of more than one battery for complete 
+destruction. Based on this assumption, BHQ assigns target to the batteries. Battery 
+components align themselves for correct orientation and elevation by computing the 
+target’s range and bearing (angle), load appropriate ammunition and fire the round. 
+When a Field component receives fire, and if the detonation is within a destruction 
+radius then the target is said to be destroyed otherwise it is missed, as will be 
+observed by the observer, who provides this information to the BHQ. This process 
+is restarted for other potential targets, until all the enemy-units are suppressed, which 
+is the ultimate goal. 
+ 
+8.2.2 Field Artillery Model 
+Based on the above informal description of the simuland a Field Artillery Model is 
+constructed. There could be multiple objectives of modeling field artillery including 
+training, exercises, weapon testing or operational optimization. The following BOM 
+based models were discovered, selected and composed with respect to the simuland. 
+ 
+Field Artillery Conceptual Model 
+The BOM based conceptual model of Field Artillery is formally defined as follows: 
+ 
+Observer = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Observer {C0(Id), C1(Loc), C2(CurrentTarget), C3(Result)} 
+ 
+EvT = {E0(ObserveField, Observer, Field, null), E1(TargetSpotted, Field, Observer, target), 
+E2(CallForFireSupport, Observer, BHQ, currtgt), E3(RequestApproved, FDC, BHQ, Observer, null), 
+E4(RequestDenied, FDC, BHQ, Observer, null), E5(Detonation, Field, Observer, detonation), 
+E6(TargetDestroyed, Observer, BHQ, null), E7(TargetMissed, Observer, BHQ, null)} 
+ 
+Act = {A0(ObserveField, Observer, Field, E0), A1(TargetSpotted, Field, Observer, E1), 
+A2(CallForFireSupport, Observer, BHQ, E2), A3(RequestApproved, FDC, Observer, E3), 
+A4(RequestDenied, FDC, Observer, E4), A5(Detonation, Field, Observer, E5), A6(TargetDestroyed, 
+Observer, BHQ, E6), A7(TargetMissed, Observer, BHQ, E7)} 
+ 
+S = {S0(ObserverReady, A0, S1),  S1(ObservingField, A1, S2), S2(RequestingFireSupport, A2, S3), 
+S3(WaitingForReponse, {A3, S4}, {A4, S0}) , S4(WaitingForImpact, A5, S5) , S5(EvaluateDamage, {A6, 
+S0}, {A7, S0}} 
+Table 34: Observer Basic-BOM 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 146 
+ 
+Field = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Field {C4(Id), C5(FD), C6(Impacts)} 
+ 
+EvT = {E8(ObserveField, Observer, Field, null), E9(TargetSpotted, Field, Observer, target), E10(Fire, 
+Battery1, Field, fire), E11(Fire, Battery2, Field, fire), E12(Fire, Battery3, Field, fire), E13(Detonation, 
+Field, Observer, Impacts), E14(UpdateField, BHQ, Field, update) } 
+ 
+Act = {A8(ObserveField, Observer, Field, E8), A9(TargetSpotted, Field, Observer, E9), A10(Fire, 
+Battery1, Field, E10), A11(Fire, Battery2, Field, E11), A12(Fire, Battery3, Field, E12), A13(Detonation, 
+Field, Observer, E13), A14(UpdateField, BHQ, Field, E14) } 
+ 
+S = {S6(FieldReady, {A8, S7}, {A10, S8}, {A11, S8}, {A12, S8}),  S7(BeingObserved, A9, S6), 
+S8(TakingFire, A13, S9) , S9(WaitingForUpdate, A14, S6)} 
+Table 35: Field Basic-BOM 
+ 
+BHQ = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT =BHQ {C7(Id), C8(Loc), C9(CurTarget), C10(TargetStatus) } 
+ 
+EvT = {E15(CallForFireSupport, Observer, Field, target), E16(ProcessRequest, BHQ, FDC, target), 
+E17(RequestApproved, FDC, BHQ, Observer, null), E18(RequestDenied, FDC, BHQ, Observer, null), 
+E19(AssignTarget, BHQ, Battery1, Battery2, Battery3, assign_target), E20(FiringCompleted, Battery1,  
+BHQ, null), E21(FiringCompleted, Battery2,  BHQ, null), E22(FiringCompleted, Battery3,  BHQ, null), 
+E23(TargetDestroyed, 
+Observer, 
+BHQ, 
+null), 
+E24(TargetMissed, 
+Observer, 
+BHQ, 
+null), 
+E25(UpdateField,  BHQ, Field, update) } 
+ 
+Act = {A15(CallForFireSupport, Observer, Field, E15), A16(ProcessRequest, BHQ, FDC, E16), 
+A17(RequestApproved, FDC, BHQ, Observer, E17), A18(RequestDenied, FDC, BHQ, Observer, E18), 
+A19(AssignTarget, BHQ, Battery1, Battery2, Battery3, E19), A20(FiringCompleted, Battery1,  BHQ, 
+E20), A21(FiringCompleted, Battery2,  BHQ, E21), A22(FiringCompleted, Battery3,  BHQ, E22), 
+A23(TargetDestroyed, 
+Observer, 
+BHQ, 
+E23), 
+A24(TargetMissed, 
+Observer, 
+BHQ, 
+E24), 
+A25(UpdateField,  BHQ, Field, E25) } 
+ 
+S={S10(BHQReady, A15, S11),  S11(CallFDC, A16, S12), S12(WaitingForApproval, {A17, S13}, {A18, S10}), 
+S13(AssigningTarget, 
+A19, 
+S14), 
+S14(WaitingForFire, 
+{A20, 
+S15}, 
+{A21, 
+S15}, 
+{A22, 
+S15}), 
+S15(WaitingForDamageReport,  {A23, S16}, {A24, S16}), S16(UpdatingField, A25, S10)} 
+ 
+Table 36: BHQ Basic-BOM 
+ 
+FDC = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT =FDC{C11(Id), C12(FD) , C13(CurTarget), C14(Result)} 
+ 
+EvT = {E26(ProcessRequest, BHQ, FDC, target), E27(RequestApproved, FDC, BHQ, Observer, 
+null), E28(RequestDenied, FDC, BHQ, Observer, null)} 
+ 
+Act = {A26(ProcessRequest, BHQ, FDC, E26), A27(RequestApproved, FDC, BHQ, Observer, E27), 
+A28(RequestDenied, FDC, BHQ, Observer, E28)} 
+ 
+S = {S17(FDCReady,  A26, S18),  S18(Processing, {A27, S17}, {A28, S17})} 
+Table 37: FDC Basic-BOM 
+ 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 147 
+ 
+Battery1,2,336 = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Battery1,2,3 {C15(Id), C16(CurTarget) } 
+ 
+EvT = {E29(AssignTarget, BHQ, Battery1, Battery2, Battery3, assign_target), E30(Fire, Battery123, 
+Field, fire),  E31(FiringCompleted, Battery123,  BHQ, null)} 
+ 
+Act = {A29(AssignTarget, BHQ, Battery1, Battery2, Battery3, E29), A30(Fire, Battery1/2/3, Field, E30),  
+A31(FiringCompleted, Battery1/2/3,  BHQ, E31)} 
+ 
+S = {S19(ReadyToFire,  A29, S20),  S20(PreparingCannon,  A30, S21),  S21(Firing,  A31, S19)} 
+Table 38: Battery (1,2,3) Basic-BOM 
+ 
+FA = 〈 AcTIN,  AcTOUT ,  POI 〉 where: 
+AcTIN = AcTOUT = ∅  
+ 
+POI = { POI-0(!A0, ?A8), POI-1(!A9, ?A1), POI-2(!A2, ?A15), POI-3(!A16, ?A26), POI-4(!A27, 
+{?A17, ?A3}), POI-5(!A28, {?A18, ?A4}), POI-6(!A19, ?A29), POI-7(!A30, {?A10, ?A11, ?A12}), 
+POI-8(!A31, {?A20, ?A21, ?A22}), POI-9(!A13, ?A5), POI-10(!A6, ?A23), POI-11(!A7, ?A24), POI-
+12(!A25, ?A14)} 
+Table 39: Field Artillery Composed BOM 
+ 
+Figure 62 shows the formal representation of Field artillery composed model.  
+                                                 
+36 This component has three instances. 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 148 
+ 
+ 
+Figure 62: Field Artillery Composed BOM 
+ 
+8.2.3 Requirement Specification 
+We define Requirement speciation of the field artillery model as: 
+RS0 = 〈O, S〉 where: 
+Objectives O = {o1} and System Constraints S = {s1, s2 s3, s4} 
+o1: All the enemy units must be destroyed 
+s1, 2, and 3: The model should be composable at syntactic and static-semantic level. The 
+state-machines should match and the executable mode should correctly represent the 
+conceptual model.  
+s4: There should never be a friendly fire. 
+
+Observer
+Characteristics:
+CO=ld: Integer, C1=Loc: Integer;
+C2=CurrentTarget: Target; C3=Result : Bool
+BHQ
+FDC
+Target = (Id:Integer, Grid :Integer.
+Characteristics:
+Characteristics:
+C7=ld Integer C8=Loc: Integer;
+C13=CurTarget:TARGET; C14=Result:Bool;
+C11=ld: Integer; C12=FD:FIELD_DATA:
+C9-CurTargetTARGET:
+Action
+C10=TargetStatus:Bool;
+A0=ObserveField, A1=TargetSpotted
+Actions:
+A4=RequestDenied,A5Detonation
+A2=CallForFireSupport, A3=RequestApproved
+A15=CallForFireSupport
+A6=TargetDestroyed, A7=TargetMissed
+A16=ProcessReques
+A28=RequestDenied:
+A17=RequestApproved
+States:
+A18=RequestDenied
+States:
+A19=AssignTarget
+S17=FDCReady: S18=Processing
+S2=RequestingFireSupport;
+S3=WaitingForReponse;
+FDCReady
+A22=FiringCompleted
+A23=TargetDestroyed
+Observing
+A24=TargetMissed
+A25=UpdateField
+essing
+SO
+A3
+S10=BHQReady; S11=CallFDC;
+42
+S3
+uppor
+(A3)
+S15=WaitingForDamageReport;
+A6
+S16=UpdatingField:
+WaitingFor
+BHQReady
+CallFDC
+S10
+11
+A16
+(A18)
+S12
+西
+WaitingFor
+Approv
+A18
+Battery1
+自电
+ingTarge
+Characteristics:
+C15=ld: Integer; C16=CurTarget: TARGET;
+Actions:
+WaitingForFire
+A29=AssignTarget; A30=Fire:
+A31=FiringCompleted:
+A25
+815
+States:
+S19=ReadyToFire; S20=PreparingCannon;
+UpdatingFielc
+S21=Firing:
+ReadyToFire
+PreparingCannor
+S19
+$20
+Field
+C4=ld: Integer; C5=FD: FIELD_DATA;
+Characteristics:
+C6=Impacts: IMPACTS;
+FIELD_DATA = list (ld:Integer, Grid:Integer.
+IMPACTS = list (Grid:Integer)
+Description:S
+Battery1
+Characteristics:
+Actions:
+A10=Fire; A11=Fire; A12=Fire
+A8=ObserveField; A9=TargetSpotted;
+C15=ld: Integer; C16=CurTarget: TARGET;
+A13=Detonation; A14=UpdateField
+Actions:
+A29=AssignTarget; A30=Fire:
+A31=FiringCompleted:
+States:
+S6=FieldReady; S7=BeingObserved;
+S8=TakingFire; S9=WaitingForUpdate
+States
+S19=ReadyToFire; S20=PreparingCannon
+自自自自
+FieldReady
+ReadyToFire
+PreparingCannc
+A29
+S19
+S20
+A30
+TakingFire
+Being
+Obse
+Firing
+WaitingFo
+Update
+Battery1
+Characteristics:
+C15=ld: Integer; C16=CurTarget: TARGET:
+Actions:
+A29=AssignTarget; A30=Fire:
+A31=FiringCompleted; 
+States
+S19=ReadyToFire; S20=PreparingCannon;
+S21=Firing
+ReadyToFire
+S19Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 149 
+ 
+8.3 Verification of the FA model using CPN State-Space 
+Analysis  
+After the BOMs are discovered, selected and composed according to Figure 62, the 
+conceptual model is ready for verification. We select CPN state-space analysis 
+technique for its verification. 
+8.3.1 Static and Dynamic Analysis 
+We assume that the model qualifies syntactic and static-semantic analysis. Also when 
+it undergoes state-machine matching process it is able to make progress until the 
+goal-states are reached. Figure 63 shows how the components interact with each other 
+through the exchange of events (horizontal arrows) due to which their state-
+machines make progress (vertical dotted arrows). Based on the fact that the 
+constraint S1, S2 and S3a are satisfied we proceed to BOM-to-E-BOM extension 
+which is a pre-requisite step for the transformation of conceptual model into 
+executable model.  
+ 
+Figure 63: State-machine Matching of Field Artillery Model 
+8.3.2 BOM to E-BOM extension 
+At this stage all the BOM components are extended to our proposed E-BOM 
+extension with the help of the modeler’s input. The following tables present E-BOM 
+extensions of BOMs in the FA model.  
+
+Observer
+Field
+BHQ
+FDC
+BATTERY
+(1,2,3)
+Observer
+Field
+Ready
+ObserveField
+Ready
+Observing
+Being
+Field
+Observed
+TargetSpotted
+Requesting
+BHQ
+FireSupport
+Ready
++
+CallForFireSupport
+WaitingFor
+Call
+FDC Ready
+Reponse
+FDC
+ProcessRequest
+WaitingFor
+Processing
+Approval
+RequestApproved
+RequestDenied
+Assigning
+Ready
+WaitingFor
+Target
+ToFire
+AssignTarget
+Impact
+Waiting
+Preparing
+ForFire
+Fire (1,2,3)
+Cannon
+-
+Taking
+Fire
+Firing
+Detonation
+FiringCompleted
+(1,2,3)
+-
+WaitingFor
+Evaluate
+DamageReport
+Damage
+TargetDestroyed
+TargetMissed
+WaitingFor
+Updating
+Update
+Field
+UpdateFieldChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 150 
+ 
+Observer E-BOM 
+SV and types {C0(Id:Integer), C1(Loc:Integer), C2(CurrentTarget:TARGET), C3(Result:Bool)} 
+where TARGET = (Id:Integer, Grid37:Integer, Description:String) 
+ 
+Initial States {S0:ObserverReady} 
+Transitions 
+State 
+Event 
+Guard 
+{SVIN} {SVOUT} 
+Action 
+Next State 
+Observer 
+Ready 
+Observe 
+Field 
+ 
+ 
+ 
+ 
+Observing 
+Field 
+Observing 
+Field 
+TargetSpotted 
+ 
+ 
+C2 
+ 
+Requesting 
+FireSupport 
+Requesting 
+FireSupport CallForFireSupport 
+ 
+C2 
+ 
+ 
+WaitingFor 
+Reponse 
+WaitingFor 
+Reponse 
+RequestApproved 
+ 
+ 
+C2 
+ 
+WaitingFor 
+Impact 
+WaitingFor 
+Reponse 
+RequestDenied 
+ 
+ 
+ 
+ 
+Observer 
+Ready 
+WaitingFor 
+Impact 
+Detonation 
+ 
+C2 
+C3 
+Action1 
+Evaluate 
+Damage 
+Evaluate 
+Damage 
+Target 
+Destroyed 
+[Result=true] 
+C3 
+ 
+ 
+Observer 
+Ready 
+Evaluate 
+Damage 
+Target 
+Missed 
+[Result=false] 
+C3 
+ 
+ 
+Observer 
+Ready 
+ 
+Action1 { 
+input (target, detonation); 
+output (result); 
+action 
+let 
+ val grid= #2 target; 
+in 
+ (Destroyed(grid, detonation)) 
+End 
+} 
+fun Destroyed (x, []) = false 
+  | Destroyed (x, h::t) = IsDestroyed(x, h)  orelse Destroyed (x, t); 
+fun IsDestroyed(grid, impact) = 
+let 
+val gridst = Int.toString(grid); 
+val impactst= Int.toString(impact); 
+val gridX = valOf(Int.fromString(substring (gridst, 0, 3))); 
+val impactX = valOf(Int.fromString(substring (impactst, 0, 3))); 
+val gridY = valOf(Int.fromString(substring (gridst, 3, 3))); 
+val impactY = valOf(Int.fromString(substring (impactst, 3, 3))); 
+val X = abs(gridX - impactX); 
+val Y = abs(gridY - impactY); 
+in 
+if (abs(X)<4 andalso  abs(Y)<4)  
+then true 
+ else false 
+end; 
+ 
+Table 40: Observer E-BOM 
+ 
+ 
+Action1 is a CPN-ML script that evaluates if the target is destroyed or not. We 
+assume that the destruction radius of the rounds fired by artillery guns is 4x4 grids 
+i.e. if the round hits the target within this radius it will be destroyed otherwise missed. 
+Note that Action1 calls other functions which are also specified in Table 40. 
+ 
+ 
+ 
+                                                 
+37  In military map, Grid reference system is used to identify a position of an object. We assume that 
+the grid in this scenario is of 6 figures, first three integers define Easting and the other three define 
+Northings. For details see [131] 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 151 
+ 
+Field E-BOM 
+SV:Types 
+{C4(Id:Integer), C5(FD:FIELD_DATA), C6(Impacts:IMPACTS)}          
+FIELD_DATA = list (Id:Integer, Grid:Integer, Description:String, 
+Type:String); IMPACTS = list (Grid:Integer) 
+Initial States 
+{S6: FieldReady} 
+Transitions 
+State 
+Event 
+Guard 
+{SVIN} {SVOUT} 
+Action 
+Next State 
+FieldReady 
+ObserveField 
+ 
+ 
+ 
+ 
+BeingObserved 
+Being Observed 
+TargetSpotted 
+[length 
+FD>0] 
+C5 
+C5 
+Action2 
+FieldReady 
+FieldReady 
+Fire 
+ 
+C6 
+ 
+ 
+TakingFire 
+TakingFire 
+Detonation 
+ 
+ 
+C6 
+ 
+WaitingFor 
+Update 
+WaitingFor 
+Update 
+UpdateField 
+ 
+C5 
+C5 
+Action3 
+FieldReady 
+ 
+Action2 { 
+input (fd); 
+output (target); 
+action 
+let 
+val indx = discrete (0,(length fd)); 
+val F = List.nth(fd, indx); 
+in 
+((#1 F, #2 F, #3 F)) 
+end } 
+Action3 { 
+input(update,fd); 
+output (ufd); 
+action 
+let 
+ val status = #2 update; 
+ val target = #1 update; 
+ val U = (#1 target, #2 target, #3 target, "Enemy"); 
+in 
+ if (status=true) then 
+   rm U fd 
+ else 
+  fd 
+end } 
+ 
+Table 41: Field E-BOM 
+Action2 randomly picks a target from a list of targets (Field Data) and sends it as 
+parameters to the observer, simulating that the observer has spotted a target in the 
+enemy area. Action3 is executed when Update-Field event is received from BHQ. This 
+action removes an object if the target destroyed. 
+ 
+BHQ E-BOM 
+SV:Types 
+{C7(Id:Integer),C8(Loc:Integer),C9(CurrentTarget:TARGET), C10(TargetStatus:Bool)} 
+Initial States 
+{S10: BHQReady } 
+Transitions 
+State 
+Event 
+Guard 
+{SVIN} 
+{SVOUT} 
+Action 
+Next State 
+BHQReady 
+CallForFireSupport 
+ 
+C9 
+ 
+ 
+CallFDC 
+CallFDC 
+ProcessRequest 
+ 
+ 
+C9 
+ 
+WaitingFor 
+Approval 
+WaitingFor 
+Approval 
+RequestApproved 
+ 
+C9 
+ 
+ 
+Assigning 
+Target 
+WaitingFor 
+Reponse 
+RequestDenied 
+ 
+ 
+ 
+ 
+BHQReady 
+Assigning 
+Target 
+AssignTarget 
+ 
+ 
+C9 
+Action4 
+WaitingForFire 
+WaitingForFire 
+FiringCompleted 
+ 
+ 
+ 
+ 
+WaitingFor 
+DamageReport 
+WaitingFor 
+TargetDestroyed 
+ 
+C10 
+ 
+ 
+UpdatingField 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 152 
+ 
+DamageReport 
+WaitingFor 
+DamageReport 
+TargetMissed 
+ 
+C10 
+ 
+ 
+UpdatingField 
+UpdatingField 
+UpdateField 
+ 
+ 
+C10 
+ 
+BHQReady 
+ 
+Action4 { 
+input (target); 
+output (assign_target); 
+action 
+let 
+in 
+if ((#3 target) = "Artillery") then 
+  ((#1 target, #2 target, #3 target, [true, true, true])) 
+else if ((#3 target)  = "Tank") then 
+   ((#1 target, #2 target, #3 target, [true, true, false])) 
+else 
+   ((#1 target, #2 target, #3 target, [true, false, false])) 
+end} 
+ 
+Table 42: BHQ E-BOM 
+Action 4 is used to assign light, medium or heavy targets. If a target is heavy then all 
+three batteries are assigned to hit the target. If the target is medium then battery 1 
+and 2 are assigned otherwise only battery 1 is assigned.  
+ 
+FDC E-BOM 
+SV:Types 
+{C11(Id:Integer), C12(FD:FIELD_DATA) , C13(CurrentTarget:TARGET), 
+C14(Result:Bool)} 
+Initial States 
+{S17: FDCReady } 
+Transitions 
+State 
+Event 
+Guard 
+{SVIN} 
+{SVOUT} 
+Action 
+Next State 
+FDCReady 
+ProcessRequest 
+ 
+C12, C13, 
+C14 
+ 
+Action5 
+Processing 
+Processing 
+RequestApproved 
+[Result=true] 
+ 
+C13, C14 
+ 
+FDCReady 
+Processing 
+RequestDenied 
+[Result=false] 
+ 
+C13, C14 
+ 
+FDCReady 
+ 
+Action5 { 
+input (target, fdcd); 
+output (fdc_result); 
+action 
+let 
+val tid = #1 target; 
+val r = List.nth((listsub fdcd (GetFieldByID tid fdcd)),0); 
+in 
+(if (#4 r = "Enemy") then true else false) 
+end 
+} 
+ 
+Table 43: FDC E-BOM 
+ 
+Action 5 is used to process targets. Here we only check from the internal FDC data 
+that the target is an enemy unit. This method can be expanded to compute target 
+priorities and process other tactical and technical fire direction rules.  
+ 
+ 
+ 
+ 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 153 
+ 
+Battery E-BOM 
+SV:Types 
+{C15(Id:Integer), C16(CurrentTarget:TARGET) } 
+Initial States 
+{S19: ReadyToFire } 
+Transitions 
+State 
+Event 
+Guard {SVIN} {SVOUT} Action 
+Next State 
+ReadyToFire 
+AssignTarget 
+ 
+C16 
+ 
+ 
+PreparingCannon 
+PreparingCannon 
+Fire 
+ 
+ 
+C16 
+Action6 
+Firing 
+Firing 
+FiringCompleted 
+ 
+ 
+ 
+ 
+ReadyToFire 
+Action6 { 
+input (assign_target); 
+output (fire); 
+action 
+let 
+val bid = inst(); 
+val tid = #1 assign_target; 
+val grid = #2 assign_target; val impact=grid; 
+in 
+(bid, tid, grid, impact) 
+end 
+   } 
+ 
+Table 44: Battery E-BOM 
+ 
+Action 6 is used to initiate the fire. It creates a token of type “Fire” to the Field 
+component containing the information of the firing battery id, target id, grid location 
+of the target and the location of the impact.  
+ 
+8.3.3 E-BOM to CPN Transformation 
+In this step, all the extended BOM models (E-BOM) are automatically transformed 
+into our proposed CPN components in such a way that all variables from the 
+corresponding E-BOMs are added in the Structural Layer (shown in Red color in the 
+following figures) and the State-machine is transformed into the Behavioral Layer 
+(shown in Green color). In communication layer (shown in blue color), receive-
+events are transformed into input ports and send-events are converted into output 
+ports. Figure 64Figure 65Figure 66Figure 67Figure 68 represent the CPN component 
+models of each component: 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 154 
+ 
+ 
+Figure 64: Observer CPN Component 
+
+ID
+Observer
+Ready
+LNI
+INT
+Grid
+205405
+ObserveField
+OF
+INI
+Out
+H
+Observing
+Field
+INT
+TargetSpotted
+TS
+In
+TARGET
+Requesting
+FireSupport
+INT
+Current
+Target
+CallForFireSupport
+CFS
+Out
+TARGET
+TARCE
+WaitingFor
+Reponse
+INT
+RequestApproved
+RequestDenied
+RA
+In
+ARCET
+RD
+In
+TARGET
+WaitingFor
+Impact
+LNI
+Detonation
+Occured
+D
+Input (target, detonaony:
+In
+output(result)
+DETONATIOI
+Evaluate
+valgrld=#2target:
+Damage
+result
+(Cestrayed(grld, detonaton3)
+INT
+end
+[target_shabus=true]
+[target_skatus=false]
+Target
+Target
+Destroyed
+Missed
+TD
+Out
+NUL
+TM
+Out
+HChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 155 
+ 
+ 
+Figure 65: Field CPN Component 
+
+ID
+INT
+[length fd > 0]
+Data
+TargetSpotted
+target
+TS
+Out
+FIELD
+DTA
+TARGET
+165505.
+"Bllding
+Neutral
+Input (fd3:
+iEnemy"
+output (target
+1794B1,
+aatlon
+Being
+165465.
+Tank"
+Enemy
+let
+198468.
+Tank"
+"Enemy'
+Observed
+alnox=
+disorete (o.tlength fJ):
+val F
+Ost.ntfan
+INT
+n
+#1F，#2F，#3F
+end
+ObserveField
+OF
+null
+In
+ULI
+Field
+Ready
+INT
+Fire
+E
+Hre
+In
+LFIRE
+Ist.map (ftbkdrtid.
+grld
+mpact)=>
+mpadAre
+Taking
+Impacts
+Fire
+mpacts
+LNI
+Detonation
+D
+Imoaats
+Out
+DETONATION
+In put update, fdy:
+output (ufd):
+let
+Waiting
+val status = #2 updater
+For
+1update
+Update
+val y =[#i target.
+#2 argeat
++3arget.
+"Enemy")
+INT
+If (shatus=true) then
+myfd
+else
+fd
+end
+ufa
+fd
+UpdateField
+UF
+update
+In
+PDATEChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 156 
+ 
+ 
+Figure 66: BHQ CPN Component 
+ 
+ 
+
+BHQ
+ID
+Ready
+LNI
+INT
+CallFor
+FireSupport
+CFS 
+trge
+205405
+In
+Grid
+TARGET
+INT
+arget
+Call
+FDC
+LNI
+Current
+target
+Process
+arge
+Target
+Request
+PR
+Out
+TARGEK
+TARGET
+trger
+WaitingFor
+Approval
+INT
+Request
+Request
+Approved
+Denied
+TARCE
+RA
+In
+Assigning
+trget
+target
+RD
+Target
+In
+TARGET
+INT
+Input (target))
+Assign
+output (asslgn_target):
+Target
+AT
+asslgn_arget
+Out
+ASSIGN TARGE
+Hit := discrete(o,2)F
+If ((3 target)
+"Artillery then
+Waiting
+#1trget
+#2 barget,
+#3target,[huertue,true
+ForFire
+else If ((*3 target)
+ITank") then
+(#1 target,#2 target,#3 target,[true,true,false]3
+else
+((#1 target, #2 target, #3 target, [true, false, false]))
+end
+Firing
+Completed
+In
+FC
+nu
+NLLI
+WaitingFor
+DamageReport
+INT
+Target
+Target
+Destroyed
+Missed
+NULI
+Tue
+In
+TD
+Target
+UpdatingField
+TM
+Status
+In
+1000
+H
+target status
+INT
+target
+UpdateField
+UF
+(target, target shabus)
+Out
+JPDATEChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 157 
+ 
+ 
+Figure 67: Battery CPN Component 
+ 
+
+ReadyToFire
+Assign
+Target
+A.
+Current
+Preparing
+Target
+Cannon
+Fire
+Out
+Firing
+Firing
+Completed
+FC
+OutChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 158 
+ 
+ 
+Figure 68: FDC CPN Component 
+ 
+We assume that each transformed CPN component has passed structural evaluation 
+which is conducted using inspection method and behavioral evaluation conducted 
+using Functional Testing method therefore S3b is also partially satisfied.  
+ 
+8.3.4 Composition of CPN Components  
+In this step all CPN modules are combined together through socket places in a CPN 
+Composed Model as shown in Figure 69. In this composed model some general 
+purpose auxiliary components are also introduced such as Join and Fork to facilitate 
+the composition. 
+
+1B3508.
+"camp
+Enemy'
+Ready
+179481
+EnGmy
+165465
+TEO
+EnemY
+LNI
+Tank
+Field
+Bpy
+Process
+Data
+Request
+PR
+arge
+Input (target, fdcd):
+In
+FIELD_DATA
+target
+output (fdc_
+result
+TARGET
+atn
+Current
+Processing
+let
+Target
+fdyresult
+vald
+LNI
+val r = List.ntht(llstsub faked
+TARGET
+(GetFleldByID tid fdcd)),0)
+Result
+arge
+end
+lt
+1008
+TARGET
+Request
+Request
+Approved
+Denied
+RD
+Out
+fck_result#true]
+[fdk_result=false]
+RA
+target
+Out
+TARGETChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 159 
+ 
+ 
+Figure 69: Field Artillery CPN Composed Model 
+ 
+When the model is composed it is executed in the CPN execution environment. The 
+successful execution of the model (according to Figure 63) satisfies S3b completely.  
+ 
+8.3.5 State space Analysis  
+In the next step the state-space of the entire Field Artillery Model is generated using 
+CPN state-space calculation tool, and is used to perform verification. The generated 
+state-space graph consists of 1960 nodes and 6469 arcs as shown in Figure 70. 
+
+Battery3]
+Battery2
+Battery1
+F3
+F2
+F1
+AT
+AT
+Bty3
+Bty2
+Btv1
+JoinFire
+JoinFC
+ForkAT
+FC
+Field
+OF
+CFS
+Observer
+TD
+BHQ
+TM
+ORD
+ORA
+BR
+BRD
+FDC
+PRChapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 160 
+ 
+ 
+Figure 70: State space of Field Artillery CPN Model (1960 nodes, 6469 edges) 
+ 
+After the state-space is constructed in CPN tools, it is exported into a GraphML file 
+format. It is to be noted that the layout of the state-space graph in Figure 70 is 
+rendered using Gephi Tool. In that “node-1” represents initial marking of the 
+composed model whereas “node-1956” represents the goal state (explained later in 
+this section). Shades of green color (from dark to light) represent proximity from 
+node 1. All the nodes are connected with edges (some of which may not be visible 
+due to light colors). 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Goal State:
+(Main:TS:1' (0,0,")
+1956
+Initial Marking
+:
+8
+O
+O
+C
+5.Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 161 
+ 
+Translation of Requirements specification into CPN Properties 
+To proceed with the verification we first translate Objectives and Constraints defined 
+in the requirement specification to CPN properties. We assume that the default 
+constraints S1, S2 and S3 are already verified. 
+Objective 
+O1: All the enemy units must be destroyed 
+CPN Translation 
+As the Observer detects enemy units, therefore we say that if no more 
+enemy units can be detected (because field-data is empty) then all the 
+enemies should be destroyed. Therefore a marking where TS (Target-
+spotted) place has a null token should exist. If such marking is found then 
+the objective is said to be reached. The following CPN function can be 
+used to verify this property. 
+CPN Function 
+fun AllTargetDestroyed() = 
+let 
+  val token = 1`(0,0,""); /*Create a search criteria */ 
+  val predicate = fn n => (Mark.Main'TS 1 n) =  token; /*Create a predicate 
+function*/ 
+  val TS = PredAllNodes (predicate);  /* Built-in Node search function 
+in 
+ if (length TS > 0)  
+     then true 
+ else false 
+end; 
+Result 
+When the function AllTargetDestroyed() is executed it returns True. This is 
+also evident from Figure 70 where the marking 1956 represents the goal-
+state and is reachable form the initial state 1. 
+ 
+Constraint 
+S4: There should never be a friendly fire. 
+CPN Translation 
+When “UpdateField” (UF) place gets a token from BHQ component 
+(which will be taken as input by the Field component), it shows which 
+field unit is destroyed. We can collect all such nodes in the state-space 
+(where UF field has tokens) and compare that all field units that have 
+been destroyed are of type “enemy”. If this condition holds in the entire 
+state-space then S4 holds. Following CPN-ML function can be used to 
+check if friendly fire has ever happened or not. The result should be false 
+to satisfy S4 
+CPN Function 
+fun CheckFriendlyFire() = 
+let 
+  val predicate = fn n => IsNotEnemy(Mark.Main'UF 1 n) = true;  
+  val ListOfFrieldlyUnits = PredAllNodes (predicate);   
+in 
+ if (length ListOfFrieldlyUnits > 0) /* Means there is a friendly fire */ 
+     then true 
+ else false 
+end; 
+fun IsNotEnemy (update) =  
+let 
+  val upd:UPDATE = List.nth(update, 0); 
+  val target = #1 upd; /*Extract information from the token at the place: UF */ 
+in 
+   if GetType(#1 target) <> "Enemy" then  true /* Checks field unit type*/ 
+else  
+   false 
+end; 
+Result 
+The function CheckFriendlyFire() results false because no such incidence 
+occurred with the data (initial state) provided to the model.  
+To check that this function works correctly we created a counter example 
+in which FDC component is assumed to be erroneous (i.e. it wrongly 
+accepts fire support requests of the friendly units), we ran the routine and 
+found traces of the occurrence of friendly fire. 
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 162 
+ 
+As all the constraints and Objectives are satisfied we say that the field artillery model 
+is composable at all levels and is verified with respect to its given specifications.  
+Therefore the BOM based composed model is qualified for further implementation. 
+ 
+8.4 State Space Reduction 
+In this section the application of our proposed state-space reduction technique is 
+presented. In order to proceed with the state-space reduction of the FA model 
+generated by CPN tools we perform following steps. 
+ 
+1. Trimming Node Description 
+Each node in the CPN state-space graph has a description. This description 
+essentially tells about the presence of tokens (or multiple tokens) in all places of the 
+model, which is called marking. This description is very lengthy if the model has 
+many places or sub-models. In this step we remove all the descriptions and only keep 
+the information related to the places of the main model. To perform this step we use 
+the library function NodeDescriptorOptions() (see manual [73]). When the 
+descriptions are trimmed we will only get information of a node pertinent to the 
+main places otherwise it will be a “Null” string. (This is an important difference for 
+further steps). 
+Conceptually we hypothesize that trimming the node description does not cause loss 
+of information because all the information other than the one in the main places is 
+produced by the internal logic of the composed components. Since the composed 
+components are considered as black boxes and they will eventually output important 
+information (in form of tokens) in any of the main places. This information would be 
+sufficient to answer any verification query related to the model under consideration. 
+ 
+ 
+2. Export to GraphML 
+In the next step we export the state-space graph to an external file. Since CPN state-
+space graph cannot be manipulated internally within the CPN environment therefore 
+we export the graph to a standard GraphML format [128] along with the trimmed 
+node descriptions and the information of the edges. To perform this step we develop 
+a GraphML writer function in CPN-ML.  
+ 
+3. Reduction Algorithm 
+In the next step, we apply our reduction algorithm specified in Table 16. This 
+algorithm is implemented in a Java application which uses JUNG library for graph 
+manipulation functions. In brief, all the nodes which have “Null” descriptions are 
+removed (because they are irrelevant). When a node is removed all its incoming and 
+outgoing edges are removed. So we connect each predecessor of the node with each 
+successor to preserve the structure of the graph. When all the nodes are checked the 
+reduction is completed. The output of the reduce graph of Field Artillery mode is 
+shown in Figure 71.  
+
+Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 163 
+ 
+ 
+Figure 71: Reduced State-Space graph of Field Artillery Model 
+ 
+In Figure 71 node-2 represents initial marking (note that node-1 was trimmed in the 
+reduction process). Also Node-1956 still represents goal state. (Note that the nodes 
+IDs remain the same in the reduction process).  
+ 
+ 
+Reduced Graph Original Graph Percentage 
+Nodes 428 
+1960 
+21% 
+Edges 2503 
+6469 
+38 % 
+Table 45: Reduction Statisitics 
+Table 45 shows that the nodes are reduced to 21% and the edges are reduced to 38% 
+of the original graph.  
+ 
+ 
+
+Initial Marking
+2
+GoalState:
+(Main:TS:1 (0,0,""))
+1956Chapter 8  
+ 
+Model Verification using State-space Analysis techniques 
+   
+Page 164 
+ 
+8.5 Summary 
+In this chapter the verification of BOM based composed models is discussed using 
+CPN based state-space analysis technique. An example model of Field Artillery is 
+introduced and the entire verification process is applied on this model. It is shown 
+how requirement specifications are translated into CPN properties and how they are 
+verified using state-space analysis and Query functions.  
+ 
+State-space analysis is advantageous as it is exhaustive and leads the modeler to all of 
+the possibilities that can occur during the abstract level execution of a composed 
+model. A state-space graph helps to study all of these possibilities and to understand 
+the dynamic behavior of the components in detail. Also a state-space query functions 
+proposed along with the approach help in answering different verification questions 
+and evaluate the correctness of model with respect to the requirements. This 
+approach however is vulnerable to the state-space explosion as for a simple model of 
+Field Artillery 2503 nodes and 6469 edges were formed. To deal with this situation 
+we proposed an effective state-space reduction technique which not only reduces 
+large state-spaces into reasonable size but also preserves important information for 
+correct verification. We demonstrated how our proposed state-space reduction 
+technique is applied to the Field Artillery model for proof of concept. 
+ 
+ 
+
+   
+Page 165 
+ 
+Chapter 9 
+Model Verification using CSP based 
+Model Checking Technique 
+ 
+Model Checking is becoming a standard approach for the software verification due to its numerous 
+advantages over traditional formal methods. Communicating Sequential Processes (CSP) is an event 
+based formal language for describing patterns of interactions in concurrent systems and very useful for 
+concurrent behavioral specification and verification due to its theoretical foundations of process 
+algebra (also called process calculi). The application of CSP based Model Checking technique in the 
+Composability verification also proves to be very useful, especially with a focus on the dynamic 
+semantic composability level. In this chapter the Field Artillery Model presented in Chapter 8 is 
+reused and extended with information to capture the behavior of a real-time probabilistic system. It is 
+shown how the Probabilistic-Timed Field Artillery Model is transformed into a composed CSP 
+model and verified using PAT. 
+ 
+In this chapter, the modified version of Field Artillery Model presented in chapter 8 
+is discussed as an example. The objective of this example is to represent a model of a 
+system with time constraints and probabilistic behavior. To the best of our 
+knowledge the PN algebraic approach does not support verification of the timed 
+models or probabilistic systems at all. Also the CPN based approach has a limited 
+support for the verification of timed system but it does not cover probabilistic 
+systems. We therefore propose to apply Modeling Checking for real-time 
+probabilistic systems and show how a composed model of one such system can be 
+verified using a CSP based Model checker called PAT (see 3.2.7). 
+9.1 
+Field Artillery Scenario 
+The scenario of the Field Artillery Model is slightly different.  It is assumed that a 
+soldier observes the field and detects enemy units. When a target is spotted, he calls 
+BHQ for fire support and provides the target details. In military practice, Time-On- 
+Target (TOT) is a Field Artillery coordination protocol observed by multiple firing 
+units. This technique was developed by the U.S. Army during World War II. It uses a 
+precise pre-determination of the estimated preparation time and the time of flight of 
+the munitions from each firing battery to the target area. When a Time on Target 
+(TOT) is designated each battery that will join in firing on that target subtracts the 
+time of flight from the TOT to determine the time to fire. The firing units fire their 
+rounds so that all the munitions arrive at the target at precisely the same time. This is 
+done in order to achieve maximum target destruction. If there is a gap between the 
+multiple impacts the enemy soldiers get time to prone or takeover in the hideouts 
+and mobile vehicles can escape [129].  
+BHQ assigns target to the batteries, and also schedules a certain “TOT” for the 
+batteries to comply. Each battery needs some time to prepare for loading appropriate 
+ammunition and setting up the correct alignment and orientation of the barrel 
+according to the computed firing solution using range (distance) and bearing (angle) 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 166 
+ 
+of the assigned target. It is assumed that each battery needs a random preparation 
+delay. When each battery is ready, it will fire in its own time such that all the rounds 
+hit the target at the given TOT. We also assume that the probability of hitting on the 
+exact target location for each battery is ‘0.9’. In contrast to the previous scenario in 
+chapter 8, we assume that there is only one target in the field component and all 
+three batteries are taking part in the firing operation. To construct a conceptual 
+model for this scenario, the following BOM components are composed: 
+Field:   
+Target location (We assume there is only one target). 
+Observer:  
+A soldier who request for the fire support from BHQ.  
+BHQ:  
+Supervises the entire operation of fire support, responds to the calls 
+for fire support and assigns targets to the batteries.  
+Battery:  
+Three units of artillery batteries (cannons and crew) responsible to hit 
+the target exactly at a given time.  
+(Note that FDC component is removed from the composition. Also some entity 
+characteristics and event parameters are reduced for simplification). The modified 
+BOM components of the Field Artillery conceptual model are formally defined as 
+follows: 
+ 
+Observer = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Observer { C0(target)} 
+ 
+EvT = {E0(CallForFireSupport, Observer, BHQ, target),  E1(Detonation, Field, Observer, 
+detonation)} 
+ 
+Act = {A0(CallForFireSupport, Observer, BHQ, E0), A1(Detonation, Field, Observer, E1)} 
+ 
+S = {S0(ObserverReady, A0, S1),  S1(WaitingForImpact, A1, S0) } 
+Table 46: Observer Basic-BOM 
+ 
+Field = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Field {C1(destruction[3])} 
+ 
+EvT = {E2(Fire, Battery1, Field, BID), E3(Fire, Battery2, Field, BID), E4(Fire, Battery3, Field, BID), 
+E5(Detonation, Field, Observer, destruction)} 
+ 
+Act = {A2(Fire, Battery1, Field, E2), A3(Fire, Battery2, Field, E3), A4(Fire, Battery3, Field, E4), 
+A5(Detonation, Field, Observer, E5) } 
+ 
+S = {S2(FieldReady, {A2, S3}+{A3, S3}+{A4, S3}) , S3(TakingFire, A5, S2)} 
+Table 47: Field Basic-BOM 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 167 
+ 
+BHQ = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT =BHQ {C2(TOT) } 
+ 
+EvT = {E6(CallForFireSupport, Observer, Field, target), E7(AssignTarget, BHQ, Battery1, Battery2, 
+Battery3, TOT), E8(FiringCompleted, Battery1,  BHQ, null), E9(FiringCompleted, Battery2,  BHQ, 
+null), E10(FiringCompleted, Battery3,  BHQ, null} 
+ 
+Act = {A6(CallForFireSupport, Observer, Field, E6), A7(AssignTarget, BHQ, Battery1, Battery2, 
+Battery3, E7), A8(FiringCompleted, Battery1,  BHQ, E8), A9(FiringCompleted, Battery2,  BHQ, E9), 
+A10(FiringCompleted, Battery3,  BHQ, E10)} 
+ 
+S={S4(BHQReady, A6, S5),  S5(AssigningTarget, A7, S6), S6(WaitingForFire, {A8, S4} + {A9, S4} + 
+{A10, S4})} 
+Table 48: BHQ Basic-BOM 
+ 
+Battery1,2,3 = 〈 EnT, EvT, S, AcT 〉 where: 
+EnT = Battery1,2,3 { C3(BID), C4(Destroyed) } 
+ 
+EvT = {E11(AssignTarget, BHQ, Battery1, Battery2, Battery3, TOT), E12(ReadyToFire, Battery1/2/3, 
+Battery1/2/3, null), E13(Fire, Battery123, Field, BID, Destroyed),  E14(FiringCompleted, Battery123,  
+BHQ, null)} 
+Act = {A11(AssignTarget, BHQ, Battery1, Battery2, Battery3, E11), A12(ReadyToFire, Battery1/2/3, 
+Battery1/2/3, E12),  A13(Fire, Battery1/2/3, Field, E13),  A14(FiringCompleted, Battery1/2/3,  BHQ, 
+E14)} 
+ 
+S = {S7(BatteryIdle,  A11, S7),  S8(Preparing,  A12, S9),  S9(ReadyToFire,  A13, S10), S10(Firing,  A14, S7)} 
+Table 49: Battery (1,2,3) Basic-BOM 
+ 
+FA = 〈 AcTIN,  AcTOUT ,  POI 〉 where: 
+AcTIN = AcTOUT = ∅  
+POI = { POI-0(!A0, ?A6), POI-1(!A7, ?A11), POI-2(!A12), POI-3(!A13, {?A2, ?A3, ?A4}), POI-
+4(!A14, {?A8, ?A9, ?A10}), POI-5(!A5,?A1} 
+Table 50: Field Artillery Composed BOM 
+ 
+The composed field artillery model is shown in Figure 72 using our proposed 
+graphical notation. 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 168 
+ 
+Observer
+S1
+Observer
+Ready
+Waiting
+forImpact
+S0
+Characteristics:
+C0 = target : Integer
+Actions:
+A0 = CallForFireSupport
+A1 = Detonation
+States:
+S0 = ObserverReady
+S1 = WaitingForImpact
+C0
+z
+Field
+A2
+S2
+FieldReady
+TakingFire
+A1
+S3
+Characteristics:
+C1 = destruction : Boolean
+Actions:
+A2 = Fire
+A3 = Fire
+A4 = Fire
+A5 = Detonation
+States:
+S2 = FieldReady
+S3 = TakingFire
+C1
+A3
+A4
+BHQ
+A8
+S6
+WaitingForFire
+BHQReady
+A6
+S4
+Characteristics:
+C2 = TOT : Integer
+Actions:
+A6=CallForFireSupport
+A7=AssignTarget
+A8=FiringCompleted
+A9=FiringCompleted
+A10=FiringCompleted
+States:
+S4=BHQReady
+S5=AssigningTarget
+S6=WaitingForFire
+C2
+A9
+A10
+A7
+S5
+AssigningTarget
+Battery1
+A14
+S9
+ReadyToFire
+BatteryIdle
+A11
+S7
+Characteristics:
+C3=BID:Integer=1
+C4=Destroyed:Boolean
+Actions:
+A11=AssignTarget
+A12=ReadyToFire
+A13=Fire
+A14=FiringCompleted
+States:
+S5=BatteryIdle
+S6=Preparing
+S7=ReadyToFire
+S8=Firing
+C3, C4
+A13
+A12
+S8
+Preparing
+S10
+Firing
+Battery2
+A14
+S9
+ReadyToFire
+BatteryIdle
+A11
+S7
+Characteristics:
+C3=BID:Integer=1
+C4=Destroyed:Boolean
+Actions:
+A11=AssignTarget
+A12=ReadyToFire
+A13=Fire
+A14=FiringCompleted
+States:
+S5=BatteryIdle
+S6=Preparing
+S7=ReadyToFire
+S8=Firing
+C3, C4
+A13
+A12
+S8
+Preparing
+S10
+Firing
+Battery3
+A14
+S9
+ReadyToFire
+BatteryIdle
+A11
+S7
+Characteristics:
+C3=BID:Integer=1
+C4=Destroyed:Boolean
+Actions:
+A11=AssignTarget
+A12=ReadyToFire
+A13=Fire
+A14=FiringCompleted
+States:
+S5=BatteryIdle
+S6=Preparing
+S7=ReadyToFire
+S8=Firing
+C3, C4
+A13
+A12
+S8
+Preparing
+S10
+Firing
+A0
+A1
+ 
+Figure 72: Field Artillery Composed Model 
+9.2 Requirement Specification 
+We define Requirement speciation of the modified field artillery model as: 
+ 
+RS0 = 〈O, S〉 where: 
+Objectives O = {o1, o2} and System Constraints S = {s1, s2 s3, s4} 
+o1: All the firing units should fire precisely at the target location 
+o2: All the firing units should fire at the target exactly at the given time (i.e., the Time 
+on Target property should be satisfied) 
+ 
+s1, 2, and 3: The model should be composable at syntactic and static-semantic level. The 
+state-machines should match and the executable mode should correctly represent the 
+conceptual model.  
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 169 
+ 
+9.3 Verification using Model Checking  
+After the BOM are discovered, selected they are composed to form a conceptual 
+model according to the simuland. This composed model is now ready for 
+verification. At this stage we select model checking technique for its verification. 
+9.3.1 Static and Dynamic Analysis 
+We assume that the model qualifies syntactic and static-semantic analysis. Also when 
+it undergoes state-machine matching process it is able to make progress until the 
+goal-states are reached. Figure 73 shows the interaction of the state-machine of each 
+component during the state-machine matching process. 
+Based on the fact that the constraint S1, S2 and S3a are satisfied we proceed to 
+BOM-to-E-BOM extension. 
+Observer
+BHQ
+Field
+Observer
+Ready
+WaitingFor
+Impact
+Field
+Ready
+Taking
+Fire
+BHQ
+Ready
+Assigning
+Target
+Waiting
+ForFire
+BATTERY
+(1,2,3)
+Idle
+Preparing
+Firing
+CallForFireSupport
+AssignTarget
+Fire (1,2,3)
+FiringCompleted 
+(1,2,3)
+Detonation
+ReadytoFire
+ 
+Figure 73: State-machine Matching of Field Artillery Model 
+9.3.2 BOM to E-BOM extension 
+At this stage all the BOM components are extended to our proposed E-BOM 
+extension with the help of the modeler’s input. Here additional information such as 
+timing constraints and probabilistic factors are proposed to be included. Following 
+tables present E-BOM extensions of BOMs in the FA model.  
+Observer E-BOM 
+SV and types 
+{C0(Target:Integer} 
+Initial States 
+{S0:ObserverReady} 
+Transitions 
+State 
+Event 
+Time 
+Guard 
+Action 
+Next State 
+Observer 
+Ready 
+CallForFireSupport 
+ 
+ 
+ 
+WaitingFor 
+Impact 
+WaitingFor 
+Impact 
+Detonation 
+ 
+  
+ 
+Observer 
+Ready 
+ 
+Table 51: Observer E-BOM 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 170 
+ 
+Field E-BOM 
+SV and types {C1(Firing_Result_Of_Battery1:Boolean},  
+{C1(Firing_Result_Of_Battery2:Boolean}, 
+{C1(Firing_Result_Of_Battery3:Boolean} 
+Initial States 
+{S2:FieldReady} 
+Transitions 
+State 
+Event 
+Time 
+Guard 
+Action 
+Next State 
+FieldReady 
+Fire(1) 
+ 
+ 
+Action1 
+TakingFire 
+Fire(2) 
+ 
+ 
+Action2 
+Fire(3) 
+ 
+ 
+Action3 
+TakingFire 
+Detonation 
+ 
+  
+ 
+FieldReady 
+ 
+   /* A CSP script for defining a probabilistic action, with 95% chance that the 
+target will be destroyed when an event fire is received from battery1 and 5% chance 
+that the target will be missed */ 
+Action1{      
+pcase{        
+               [0.05] : fire?1 → atomic{tau{ Destruction [0]=False;} → Skip} 
+               default : fire?1→ atomic{tau{ Destruction [0]=True;}→ Skip} 
+           } 
+ 
+} 
+/* From battery 2 */ 
+Action2{      
+pcase{        
+               [0.05] : fire?2 → atomic{tau{ Destruction [1]=False;} → Skip} 
+               default : fire?2→ atomic{tau{ Destruction [1]=True;}→ Skip} 
+           } 
+ 
+} 
+/* From battery 3 */ 
+Action3{      
+pcase{        
+               [0.05] : fire?3 → atomic{tau{ Destruction [2]=False;} → Skip} 
+               default : fire?3→ atomic{tau{ Destruction [2]=True;}→ Skip} 
+           } 
+ 
+} 
+ 
+Table 52: Field E-BOM 
+ 
+ 
+BHQ E-BOM 
+SV and types 
+{C2(TOT:Integer} 
+Initial States 
+{S4: BHQReady } 
+Transitions 
+State 
+Event 
+Time 
+Guard 
+Action 
+Next State 
+BHQReady CallForFireSupport 
+ 
+  
+ 
+AssigningTarget 
+AssigningTarget 
+AssignTarget 
+ 
+ 
+ 
+WaitingForFire 
+WaitingForFire 
+FiringCompleted(1) 
+ 
+ 
+ 
+BHQReady 
+FiringCompleted(2) 
+ 
+ 
+ 
+FiringCompleted(3) 
+ 
+ 
+ 
+ 
+Table 53: BHQ E-BOM 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 171 
+ 
+Battery(1,2,3) E-BOM 
+SV and types 
+{C3(BID:Integer} 
+Initial States 
+{S7: BatteryIdle } 
+Transitions 
+State 
+Event 
+Time 
+Guard 
+Action 
+Next State 
+BatteryIdle 
+AssignTarget 
+ 
+  
+ 
+Preparing 
+Preparing 
+readytofire 
+Wait[Prob] 
+ 
+Action1 ReadyToFire 
+ReadyToFire 
+Fire 
+ 
+ 
+ 
+Firing 
+Firing 
+FiringCompleted 
+ 
+ 
+ 
+BatteryIdle 
+ 
+/* A CSP script for defining a probabilistic wait action, with 94% chance that each 
+battery will launch the fire exactly at given time on target, and 6% chance that  it 
+will fire earlier or later */ 
+Action1{      
+pcase{ 
+ 
+[0.01] :  Wait[TOT-3]; readytofire → ReadyToFire(i) 
+ 
+[0.01] :  Wait[TOT-2]; readytofire → ReadyToFire(i) 
+ 
+[0.01] :  Wait[TOT-1]; readytofire → ReadyToFire(i) 
+ 
+[0.94] :  Wait[TOT];     readytofire → ReadyToFire(i) 
+ 
+[0.01] :  Wait[TOT+1]; readytofire → ReadyToFire(i) 
+ 
+[0.01] :  Wait[TOT+2]; readytofire → ReadyToFire(i) 
+ 
+[0.01] :  Wait[TOT+3]; readytofire → ReadyToFire(i) 
+ 
+}; 
+ 
+} 
+ 
+Table 54: BHQ E-BOM  
+9.3.3 E-BOM to CSP# Transformation 
+In this step, all the probabilistic timed extended BOM models are automatically 
+transformed into CSP# using our automatic BOM-to-CSP transformation tool. 
+Figure 74 shows the global code block which is used to define global variables and the 
+communication channels for each BOM send-receive event pair.  
+ 
+//------------Global Block ---------------------------------- 
+#define TOT 30; //Constant pre-defined Time on Target 
+enum {Hit, Miss}; //Hit or Miss flag 
+//Each battery has a hit/miss ratio = 95:5 % 
+var Firing_Result_Of_Battery1=Miss; 
+var Firing_Result_Of_Battery2=Miss; 
+var Firing_Result_Of_Battery3=Miss; 
+ 
+//For each event a channel is defined 
+channel callforfire 0; 
+channel detonate 0; 
+channel assigntarget 0; 
+channel firingcomplete 0; 
+channel fire 0;  
+ 
+Figure 74: Global code Block of Field Artillery Model 
+ 
+ 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 172 
+ 
+Figure 75 shows the CSP code of Observer BOM component. The send-events are 
+transformed into channels with send operator ‘!’ and receive-events are transformed 
+into channels with receive operator ‘?’. 
+ 
+ 
+//=========================================================== 
+//OBSERVER Component 
+//=========================================================== 
+ 
+ObserverSM = ObserverReady(); //Initial State 
+ 
+ObserverReady()= 
+ (callforfire!0 ->WaitingForImpact()); 
+ 
+WaitingForImpact()= 
+ (detonate?0 -> ObserverReady()); 
+ 
+//=========================================================== 
+Figure 75: CSP representation of Observer Component 
+ 
+ 
+ 
+//=========================================================== 
+//BHQ 
+//=========================================================== 
+BHQSM = BHQReady();//Initial State 
+ 
+BHQReady()= 
+(callforfire?0 ->AssigningTarget()); 
+ 
+AssigningTarget()= 
+(assigntarget!> assigntarget!2 -> assigntarget!3 ->   
+ 
+Waitingforfire()); 
+//Sending assigntarget to multiple recievers 
+ 
+Waitingforfire()= 
+ 
+firingcomplete?>firingcomplete?2->firingcomplete?3-> 
+ 
+ 
+BHQReady(); 
+//Recieving firingcomplete from multiple senders 
+ 
+ 
+//=========================================================== 
+Figure 76: CSP representation of BHQ Component 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 173 
+ 
+//=========================================================== 
+//BATTERY i={1,2,3} 
+//=========================================================== 
+BatterySM(i) = BatteryIdle(i); //Initial State 
+BatteryIdle(i)= 
+(assigntarget?i->Preparing(i)); 
+Preparing(i)= pcase{ 
+ 
+ 
+[0.01] :  Wait[TOT-3]; readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.01] :  Wait[TOT-2]; readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.01] :  Wait[TOT-1]; readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.94] :  Wait[TOT];   readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.01] :  Wait[TOT+1]; readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.01] :  Wait[TOT+2]; readytofire-> ReadyToFire(i) 
+ 
+ 
+[0.01] :  Wait[TOT+3]; readytofire-> ReadyToFire(i) 
+ 
+ 
+}; 
+// TOT is a global constant 
+//readytofire is an internal event 
+ 
+ 
+ReadyToFire(i)= fire!i->Firing(i);  
+Firing(i)= firingcomplete!i ->BatteryIdle(i); 
+//=========================================================== 
+Figure 77: CSP representation of Battery Component 
+ 
+//=========================================================== 
+//FIELD Component 
+//=========================================================== 
+FieldSM = FieldReady(); //Initial State 
+ 
+FieldReady()=  
+ 
+ 
+  
+ 
+pcase{        
+  
+[0.05]: fire?1 ->  
+    
+ 
+atomic{tau{Firing_Result_Of_Battery1=Miss;} -> Skip} 
+ default : fire?1 ->  
+         
+atomic{tau{Firing_Result_Of_Battery1=Hit;} -> Skip} 
+ } 
+|||  
+/* ||| is the interleaving operator between the synchronizing events 
+fire(1), fire(2) and fire(3) */ 
+ 
+pcase{        
+  
+ 
+[0.05]: fire?2 ->  
+    
+ 
+ 
+atomic{tau{Firing_Result_Of_Battery2=Miss;} -> Skip} 
+  
+ 
+default : fire?2 ->  
+         
+ 
+atomic{tau{Firing_Result_Of_Battery2=Hit;} -> Skip} 
+ } 
+||| 
+ 
+pcase{        
+  
+ 
+[0.05]: fire?3 ->  
+    
+ 
+ 
+atomic{tau{Firing_Result_Of_Battery3=Miss;} -> Skip} 
+  
+ 
+default : fire?3 ->  
+        
+ 
+ atomic{tau{Firing_Result_Of_Battery3=Hit;} -> Skip} 
+ }; 
+ 
+ 
+/* This code randomly sets hit or miss effect for the firing of each 
+battery */ 
+ 
+Detonation(); 
+Detonation()= detonate!0 -> FieldReady(); 
+//=========================================================== 
+ 
+Figure 78: CSP representation of Field Component 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 174 
+ 
+// FIELD ARTILLERY COMPOSED MODEL 
+//====================================================================== 
+ 
+FieldArtillery = ObserverSM || BHQSM || FieldSM || BatterySM(1)|| 
+BatterySM(2)|| BatterySM(3) 
+ 
+// || is the parallel operator between all the components 
+// BatterySM has three instances initialized with battery id parameter. 
+ 
+Figure 79: Field Artillery Composed Model 
+ 
+Figure 79 shows the CSP representation of how the transformed components are 
+composed using the parallelism operator ‘||’. This means that all the components 
+execute in parallel, however they perform barrier synchronization while exchanging 
+events in their respective communication channels.  
+ 
+9.3.4 Model Checking of Field Artillery Model 
+The CSP based Field Artillery Model can be opened and executed in PAT tool. A 
+successful compilation of this model shows that it has no errors. When this model is 
+executed, and if each component reaches its final states then we say that the 
+constraint S3b of requirement specification is satisfied i.e., the transformed 
+executable model correctly represents the behavior of its conceptual model.  
+In the verification process, we define the following assertions to be verified by PAT 
+built-in model checker. Since the nature of the input model is probabilistic and real-
+time, we use Probabilistic-Real-Time module of the PAT tool.   
+Figure 80 shows how we define goal reachability assertions using PAT’s Probabilistic 
+CSP LTL specification.  
+ 
+//=========================================================== 
+// FIELD ARTILLERY COMPOSABILITY VERIFICATION 
+//=========================================================== 
+ 
+// ASSERT1: Goal state Reachability 
+#assert FieldArtillery |= [](callforfire.0 -> <>detonate.0); 
+ 
+// Goal Definition 
+#define goal (Firing_Result_Of_Battery1==Hit  
+&& Firing_Result_Of_Battery2==Hit  
+&& Firing_Result_Of_Battery3==Hit); 
+ 
+//ASSERT2: //Goal Reachability 
+#assert FieldArtillery |= <>goal with prob; 
+ 
+Figure 80: Field Artillery Verificataion Assertions 
+ 
+Assertion1 uses LTL construct to verify that if there is a “callforfire” then detonation 
+at the target location will eventually occur. If assertion1 is satisfied, it shows that 
+there exists a valid execution path, which leads to the goal state.   
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 175 
+ 
+The result of PAT model checker is shown in Figure 81 which shows that the goal 
+state is reachable.  
+********Verification Result******** 
+The Assertion (FieldArtillery() |= []( callforfire.0-><> detonate.0)) is VALID. 
+ 
+********Verification Statistics******** 
+Visited States:160477 
+Total Transitions:475422 
+Time Used:8.8263759s 
+Estimated Memory Used:111668.696KB 
+Figure 81: Verification Result of assertion 1 
+ 
+Assertion2 uses an LTL construct to verify that the “goal” is eventually reachable 
+where “goal” is defined as a condition that all the batteries successfully hit at the 
+exact location of the target. Note that assertion2 uses “with prob” construct, which 
+makes it a PLTL statement. Figure 82 shows the verification result which means that 
+the probability of reaching the goal is between 77% and 94%. 
+********Verification Result******** 
+The Assertion (FieldArtillery() |= <> goal with prob) is Valid with Probability [0.77378, 0.94526]; 
+ 
+********Verification Statistics******** 
+Visited States:84019 
+Total Transitions:245561 
+MDP Iterations:63123 
+Time Used:5.5017483s 
+Estimated Memory Used:97028.192KB 
+Figure 82: Verification result of assertion 2 
+ 
+Now we check whether the goal is reachable within the time constraints defined by 
+Time-On-Target property. To perform this evaluation we use the PAT’s deadline 
+operator as shown in Figure 83. We define three assertions: Early, Exactly and Late. 
+//=========================================================== 
+// FIELD ARTILLERY COMPOSABILITY VERIFICATION 
+//=========================================================== 
+ 
+//Goal Reachability with TOT constraint 
+Early = FieldArtillery deadline[TOT-3]; 
+Exactly = FieldArtillery deadline[TOT]; 
+Late = FieldArtillery deadline[TOT+3]; 
+ 
+ 
+//ASSERT3: //Goal Reachability at TOT 
+#assert Early reaches goal with prob; 
+#assert Exactly reaches goal with prob;  
+#assert Late reaches goal with prob; 
+ 
+Figure 83: Field Artillery Verificataion Assertions with TOT 
+ 
+ 
+ 
+
+Chapter 9  
+ 
+Model Verification using CSP based Model Checking Technique 
+   
+Page 176 
+ 
+The verification result of assertion 3 is shown in Figure 84 according to which the 
+early reachability of the goal is impossible. Whereas the maximum probability of 
+reaching the goal exactly on TOT is 86% which satisfies objectives O1 and O2 
+However the maximum probability of reaching goal at a later time is 94% which 
+satisfies O1 with a higher probability but does not satisfy O2.  
+ 
+********Verification Result******** 
+The Assertion (Early() reaches goal with prob) is NOT valid. 
+ 
+********Verification Statistics******** 
+Visited States:58414 
+Total Transitions:146548 
+MDP Iterations:2557 
+Time Used:4.46633s 
+Estimated Memory Used:69901.84KB 
+ 
+********Verification Result******** 
+The Assertion (Exactly() reaches goal with prob) is Valid with Probability [0, 0.86271]; 
+ 
+********Verification Statistics******** 
+Visited States:169998 
+Total Transitions:342091 
+MDP Iterations:28115 
+Time Used:9.6064619s 
+Estimated Memory Used:148703.552KB 
+ 
+********Verification Result******** 
+The Assertion (Late() reaches goal with prob) is Valid with Probability [0, 0.94526]; 
+ 
+********Verification Statistics******** 
+Visited States:274934 
+Total Transitions:584498 
+MDP Iterations:112797 
+Time Used:15.4390606s 
+Estimated Memory Used:232989.792KB 
+ 
+Figure 84: Verification result of assertion 3 
+ 
+Based on the verification results, we can say that the field artillery model satisfies it’s 
+given requirements with a certain probability factor. Since it is a non-deterministic 
+model, the reliability of the success depends on the threshold between how tight the 
+Time-On-Target deadline is that BHQ can assign and how efficiently the batteries 
+can prepare and how accurately they can fire on the target. 
+9.4 Summary 
+In this chapter the model checking approach is presented with an example and 
+verified using Process Analysis Toolkit (PAT). The example of Field Artillery Model 
+(from chapter 8) is modified to represent a Probabilistic-Timed model in order to 
+explain how the CSP based model checking approach using PAT can be effective in 
+the composability verification. Using the example of Field Artillery model, it is 
+explained how the verification of time constraints is performed and how different 
+property assertions are verified with probability using PLTL. A successful verification 
+of this approach is a result of satisfaction of all the assertions defined in the 
+requirement specification, with an acceptable probability factor, and hence shows 
+that the components are composable.  
+ 
+
+   
+Page 177 
+ 
+Chapter 10 
+Summary and Conclusion 
+ 
+This chapter makes a comparison between the composability verification approaches presented in this 
+thesis and provides some guidelines for choosing the appropriate approach according to the nature of 
+the composed model. This discussion is followed by a summary of the major contributions of the thesis 
+and some suggestions for future research in this area are suggested.    
+ 
+In this thesis we propose a verification framework that follows the fundamental 
+principles of M&S domain in terms of the notions of model correctness. It integrates 
+several methods, techniques and tools to support different tasks in the multi-tier 
+composability verification process of a composed model. It also inherits useful 
+technological characteristics related to model verification from other communities 
+such as Petri Nets, Model Checking and Process-Algebra community. And utilizes 
+the existing knowledge shared by these communities for the verification of 
+component based simulation models.  These simulation models are called 
+component based models because they are designed in form of components and can 
+further be composed to construct sophisticated models (called composed models or 
+compositions). To ensure correctness, a composed simulation model is required to 
+be verified at its different composability levels, where each level poses certain degree 
+of difficulty in verification. The initial levels of composability require that all the 
+components in a composition can be syntactically connected to each other through 
+valid interfaces. And they can correctly communicate with valid semantics. Whereas a 
+deeper level of composability is the dynamic-semantic composability which requires 
+that all the composed components should possess suitable behavior in order to 
+correctly interact with each other for pursuing mutual objectives. The validity of 
+behavior in a component composition relies on two factors: (i) each component 
+should always be at the right state while interacting with the others and (ii) the 
+composition should satisfy required behavioral properties, as prescribed in the 
+requirement specifications.  
+The proposed verification framework not only provides complete support for the 
+verification of initial composability levels, but also the most important characteristic 
+of this framework is its ability to verify the composed model components at the 
+deeper level of dynamic-semantic composability. Composability Verification at this level is 
+a daunting task and requires a dynamic analysis approach. The behavior of 
+components can be studied when they are set to interplay with each other in an 
+execution environment, where they communicate through the exchange of events 
+and make progress by the change of their internal states. Therefore an appropriate 
+dynamic analysis approach is required which not only provides suitable execution 
+environment but also support built-in verification techniques to evaluate the 
+composability behavior at the runtime.  
+According to our findings not a single approach completely covers all the intricacies 
+required for proving correctness at this the dynamic-semantic composability level 
+due to its complex nature. The effectiveness of a certain approach also varies due to 
+the varied nature of the composed model and the modeling formalism used. Since 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 178 
+ 
+some models have complicated structure and demand rich expressiveness in terms of 
+data-centric details for the abstraction of a system; Whereas others have behavior of 
+complex nature including notions of concurrency and temporal constraints. Besides 
+the system behavior can be deterministic or stochastic. Therefore it is difficult to 
+depend on a single approach for the challenging task of dynamic-semantic 
+composability verification. 
+For this reason we investigated three different dynamic analysis approaches in our 
+framework namely: (i) PN based Algebraic Analysis, (ii) CPN based State-Space 
+Analysis and (iii) CSP based Model Checking Technique. These approaches inherit 
+theories, methods, tools and techniques from their corresponding ancestry 
+communities such as PN, CSP and Model checking. We adapt these inherited 
+resources and integrate them in our framework. We also propose several extensions 
+in each approach to suit the needs of dynamic-semantic composability verification. 
+Some of these extensions are listed as follows: 
+ A component-based description format is proposed. This description format is 
+used to represent the BOM based composed model in the required form in order 
+to apply the selected approach. For instance a CPN based component model is 
+proposed which represents the structural and behavioral aspects of a BOM 
+component in form of a CPN model. Similarly for CSP, a Component oriented 
+CSP process model is introduced which represents a BOM component using CSP 
+notation.  
+ For each approach a rule based transformation technique is proposed which 
+converts BOM components into the description format of the corresponding 
+approach while keeping the structure and behavior of the model preserved. To 
+ensure this fact methods are proposed to compare the original model (BOM) and 
+the transformed model to assert that the latter correctly represents the former.  
+ For PN algebraic approach algorithms are proposed to automate the process. 
+Also a function library is developed for the ease of conducting repeated 
+verification tasks. 
+ In case of state-space analysis, a reduction technique is proposed which helps in 
+reducing a large state-space and ease the process of verification. 
+The advantages and disadvantages of these three approaches are categorized as 
+follows: 
+ 
+Category: 
+Kinds of properties that can be verified 
+PN Algebraic 
+Analysis 
+This approach only verifies a limited number of properties because it depends 
+on the applicability of underlying mathematical theorems which are limited in 
+number and may not cover all types of properties 
+CPN State-
+Space Analysis 
+It constructs state-space of all possibilities that a system could be in. Therefore 
+it allows to specify and verify different kinds of general system properties as 
+well as scenario specific properties.  
+CSP based 
+Model Checking 
+In this approach the verification depends on the specification of properties 
+using LTL or CTL assertions, which along with their variety of extensions 
+provide rich expressiveness to define different kinds of properties. Therefore it 
+covers a bigger pool of verification questions both in terms of generic as well 
+as scenario specific properties 
+Table 55: Kinds of properties that can be verified 
+ 
+Advantage 
+Disadvantage 
+Neutral 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 179 
+ 
+Category: 
+Type of the models that can be verified 
+PN Algebraic 
+Analysis 
+This approach supports simple event-driven PN models. It does not support 
+models with rich data, or models of real-time or probabilistic systems. 
+CPN State-
+Space Analysis 
+This approach support models with rich data-centric structure and behavior 
+since it offers flow of the tokens of complex data-types and their manipulations 
+during the transitions. It also offers limited support for Timed systems. 
+However it does not support model verification of probabilistic nature.  
+CSP based 
+Model Checking 
+This approach limits size of information in the model and does not entertain 
+models with rich data-centric expressiveness. However it offers a variety of 
+types of systems that can be verified such as reactive systems, real-time 
+systems, probabilistic and stochastic systems. Therefore this approach is much 
+stronger in verifying different kinds of systems. 
+Table 56: Type of the models that can be verified 
+ 
+ 
+Category: 
+Scalability 
+PN Algebraic 
+Analysis 
+Verification is dependent on the structure of the PN model (i.e., number of 
+places and transitions). This factor is much less than the number of reachable 
+markings produced by other approaches. Therefore for larger models this 
+approach proves to be scalable 
+CPN State-
+Space Analysis 
+Verification is dependent on the state-space, which tends to grow large for 
+even ordinary models and hence can easily subject to state-space explosion. 
+Some reduction techniques (including one of our own) may minimize this risk 
+but cannot completely omit it. 
+CSP based 
+Model Checking 
+Model checking is also exposed to state-space explosion however it has gone 
+through a continuous evolution of improved algorithms and compact data-
+structures to minimize this risk. Therefore it promises a better resolution of 
+scalability as compared to the State-space analysis. 
+Table 57: Scalability 
+ 
+ 
+Category: 
+Infinite Model Verification 
+PN Algebraic 
+Analysis 
+It is not affected in its reasoning if the model is finite or infinite, because in 
+most of the cases it uses invariants for reasoning which are derived from the 
+algebraic computations and do not depend upon the number of  reachable 
+system states 
+CPN State-
+Space Analysis 
+If the model is infinite it will require a construction of infinite state-space 
+which is infeasible.  
+CSP based 
+Model Checking 
+Infinite model verification using this approach is possible by applying bounded 
+model checking or by abstracting an infinite system into a finite one however 
+this may lead to results with partial correctness. 
+Table 58: Infinite Model Verification 
+ 
+ 
+ 
+ 
+ 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 180 
+ 
+ 
+Category: 
+Usability 
+PN Algebraic 
+Analysis 
+This approach is difficult to use due to complex mathematics and requirement 
+to underlying applicable theorems for correct reasoning.  
+CPN State-
+Space Analysis 
+This approach is easy to use. Most of the operations are automatic.  
+CSP based 
+Model Checking 
+This approach requires some effort to understand the formalisms used for 
+model input and property specifications. However its operations are easy and 
+all run in a black box i.e., the model checker. 
+Table 59: Usability 
+ 
+Category: 
+Automation 
+PN Algebraic 
+Analysis 
+This approach is not automatic because the definition of a property and its 
+theorem applicability requires manual effort. When a property is defined, and a 
+theorem is selected, the modeler has to perform mathematical computations 
+and manually infer whether a condition is satisfied or not. 
+CPN State-
+Space Analysis 
+This approach is semi-automatic because defining a verification task and a 
+suitable verification function requires modeler’s input. However the execution 
+of the function is automatic and it searches all state-space to return a result. 
+CSP based 
+Model Checking 
+This approach is totally automatic. Once a temporal logic assertion is defined, it 
+is executed automatically by and model checker to find out whether it is 
+satisfied or otherwise a counter example is generated. 
+Table 60: Automation 
+ 
+Table 55 compares the proposed approaches in terms of the different types of 
+properties that can be verified. It highlights that the PN algebraic technique is limited 
+to verify only general properties (such as deadlock, liveness, fairness) since it depends 
+on the underlying theorems for the proof of their satisfiability. Whereas the other 
+two approaches are relatively more flexible to the specification and verification of 
+properties of varied types, including general and scenario specific properties.  Table 56 
+presents a comparison of the proposed approaches in terms of the type of models. 
+PN Algebraic approach only supports PN models with simple events without any 
+parameters, guards, actions or input/output state-variables. These features are rather 
+supported by CPN based state-space analysis approach which also provides limited 
+support for Time based CPN models. But for models of complex real-time systems 
+or probabilistic systems Model Checking approach is the suitable choice.  
+ 
+Table 57 compares these approaches in terms of scalability of the models. In case of 
+Algebraic technique most of the operations in the property verification require 
+matrix computations such as Incidence matrix, P-Invariants, T-invariants. Therefore, 
+the scalability factor is dependent on the size of the matrix i.e., the number of places 
+× number of transitions of the composed model. Thus, the algebraic technique is 
+relatively salable. With regard to scalability the CPN based state-space approach has 
+serious limitations due to its rich data expressiveness and enumeration features. It is 
+reported [130] that if the model is very large it generates state-space around 105 -106 
+nodes. Consequently ordinary PCs cannot easily handle such a large state-space. 
+However there are different approaches to make it more scalable. We also believe 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 181 
+ 
+that if our proposed state-space reduction technique is directly implemented in the 
+CPN tools environment, this limitation can further be relaxed. Model Checking 
+technique is relatively more scalable. Since it relies on the usage of PAT tool which 
+can handle about 107 states in a reasonable amount of time [98]. This should be 
+sufficient for the verification of most industrial scale system models. 
+ 
+According to the Table 58 the algebraic approach is indifferent whether the model is 
+finite or infinite in nature. An infinite model is a non-terminating model which keeps 
+on evolving indefinitely. Such models are difficult to be verified using State-space 
+approach because its state-space construction is impossible. Although some 
+techniques have been developed such as coverability graphs, to resolve this problem 
+however they fail in some cases, such as in case of timed models. To verify infinite 
+models using Model Checking is somewhat possible using bounded model checking 
+or by abstracting an infinite system into a finite one. However this may lead to results 
+with partial correctness because only a portion of the system state-space can be 
+considered for the reasoning of property correctness. Table 59: UsabilityTable 59 and 
+Table 60 compare the ease of use of these approaches in terms of their application in 
+a verification task and the extent of automation they provide.  
+In short, there is no ultimate winner and making the right choice of an approach 
+entirely depends on the kind of model under investigation and the types of 
+verification properties in question. There are also no exact rules however some 
+fundamental guidelines can be used to help the modeler select a suitable approach: 
+10.1 Guidelines for choosing an approach 
+In this section some basic guidelines are presented for the modelers in making a 
+suitable choice 
+10.1.1 PN Algebraic Technique 
+This approach is most suitable when the analysis of a BOM composition is in 
+question which with simple state/transitions and does not require any extension (i.e., 
+it does not have state-variables, or complex notions of transitions with parameters, 
+guards, actions, inputs and outputs etc.). Also its requirement specification includes 
+properties which can be translated in form of PN properties (for which the solution 
+of PN algebraic verification exist). Therefore it should be used when the requirement 
+specifications can be defined in terms of PN properties. For instance, in chapter 7 
+the objectives are translated into “Fairness” which means that they can be satisfied if 
+the model is fair so the objective of verification is to prove this assumption and can 
+be done using PN algebraic approach. Also it is not effected by the model size, 
+because it performs computations on matrices of the order of (No. of places × No. 
+of Transitions) which remain static, therefore it can also be used for somewhat larger 
+models.  
+It should not be used if the requirement specification contains reachability 
+properties. Though it is possible to verify them using the PN state equation however 
+it is rather difficult and inefficient as compared to State-Space Analysis approach. 
+This approach cannot be used if the composed model has notions of time, colored-
+tokens (i.e., the BOM events have parameters) or non-determinism.  
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 182 
+ 
+10.1.2 CPN based State-Space analysis Technique 
+This approach is best suitable when the given model has (or requires) rich data-
+centric structure and behavior such as state-variables, events parameters, guards and 
+actions. In this case the BOM components are required to be extended to capture 
+more details. If the modeler has the necessary information to extend the BOM 
+components then he should use this approach otherwise he should choose the 
+Algebraic technique. This approach is also suitable if the modeler wants to execute 
+the composed model at an abstract level to study the behavior of the components 
+before actually implementing them. Although other proposed approaches also have 
+execution/simulation environments, but the CPN based execution is more detailed 
+and comprehensive to study the interaction between composed components, as it 
+provides a hierarchical interconnection between the CPN components and their 
+execution is shown by the flow of data carrying colored tokens among inputs and 
+outputs of each component in an interactive, step-by-step or an automatic fashion. 
+This allows the modeler to closely inspect the composition and its dynamics in a run-
+time environment. Using this approach has many benefits from a component-based 
+development point of view and the chances of its success are further elevated with 
+our proposed state-space reduction technique called “Compositional State-Space”, 
+which reduces the risk of state-space explosion.  
+This approach can also be used for timed systems since CPN environment supports 
+modeling and verifying timed systems. However few limitations exist since the state-
+space of timed systems is much more expensive and memory intensive, due to the 
+fact that each state carries an overhead of timed-stamps so even for a simpler model, 
+its state-space will be much heavier than a similar model with no time.  Moreover, if 
+the model has even one non-terminating loop, its state-space cannot be constructed 
+as it keeps growing to infinity by incrementing the time-stamps.  (i.e., the system may 
+return back to previous states in loops and no new state is being added in the state-
+space but the time increases so the time-stamps keep on increasing. Therefore with 
+different time-stamps the same states keep on adding infinitely).  
+This approach however completely fails when certain non-determinism is involved in 
+the model. Even though CPN specification allows using different probability 
+distribution functions, but when they are used, the resultant state-spaces are 
+generated with variations, which cannot be used for verification reasoning. Therefore 
+we do not recommend this approach if the model is stochastic in nature.  
+ 
+10.1.3 CSP based Model Checking Technique 
+This approach is usually favored by majority of the software verification community. 
+It also has a greater flexibility of adopting a new technique or algorithm with specific 
+requirements at hand, and thus can be useful in a variety of contexts. This approach 
+allows the modeler to execute the composed model using PAT simulator (see 3.2.7) 
+therefore it also contends with CPN based State-space analysis in terms of studying 
+system behavior at runtime. However its main strength is revealed when it offers 
+answers to a variety of verification questions, and to a variety of types of systems 
+(real-time, probabilistic etc.) using model checking.  
+This approach however restricts model expressiveness since it limits the use of data 
+types such as strings, products, records (unlike CPN). This requires an extra effort 
+from the modelers to represent a model in reduced or compact forms using smaller 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 183 
+ 
+data-types. For instance Boolean flags may be used instead of strings in the 
+parameters such as a pair of string parameters: “Target_Destroyed”, “Target_Missed” can 
+be represented as True/False. Similarly a set of string parameters: {“Red”, “Blue”, 
+“Green”} can be represented by corresponding integer values {0, 1, 2}. This kind of 
+reduction is required for this approach to work correctly.  
+For example, we presented a detailed data-centric model of field artillery in chapter 8 
+to be verified with CPN state-space analysis approach. But when it was required to 
+verify a specific timed property (with non-determinism) we reduced unnecessary 
+details and presented a simpler prototype of the Field Artillery model in chapter 9, 
+focusing only on its behavior relevant to the desired property. By doing this 
+modification the model was useable with this approach which successfully verified 
+the required properties that could not be verified using CPN based approach.  
+As a final note, each approach has its own benefits and drawback and the choice 
+depends on the modeler’s objectives, nature of the task and available information. 
+However we also encourage using multiple approaches for a single task and 
+comparing the results. It gives different perspectives and can better help in 
+confirming correctness. 
+ 
+10.2 Thesis Contributions 
+Component based modeling and simulation is a promising approach to develop and 
+simulate system models. It incorporates numerous benefits such as modular design, 
+logical separation, flexible change management, reusability of existing components, 
+cross-domain model integration and thus consequently helps in reducing cost, time 
+and system complexity. A key characteristic in this expedient paradigm is composability 
+that is the ability to add or select and assemble reusable components in order to 
+satisfy user’s requirements. In this thesis we mainly endeavored to investigate 
+different aspects of this quality characteristic of component based model design and 
+proposed a composability verification framework for the assessment of its 
+correctness. Our proposed framework uses Base Object Model (BOM), a SISO 
+standard for component based modeling, and performs composability verification of 
+BOM based model compositions with respect to given requirement specifications. In 
+order to prove the correctness of composability of a set of BOM components, our 
+framework undergoes a prescribed verification process, which has different phases 
+starting from system abstraction, requirement gathering, selection of BOM 
+components, their composition to form a conceptual model and then verifying its 
+different levels of composability, in an iterative top-down refinement fashion. When 
+the entire process is completed successfully the composed model is said to be 
+verified with respect to its specifications and can be used for implementation using 
+an implementation architecture (such as HLA) and simulated to serve its purpose.  
+ 
+Following are the key contributions of this thesis: 
+ We developed a composability verification framework, which stands on 
+fundamental verification principles and backed by the theoretical underpinnings 
+of M&S, the details of which are mainly covered in Part-I. It integrates different 
+methods, techniques, paradigms, algorithms, formalisms, templates, tools and 3rd 
+party libraries (or APIs) to support different tasks in the multi-tier composability 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 184 
+ 
+verification process of a composed model with respect to its requirement 
+specification. 
+ We outlined a component based modeling and simulation (CBM&S) life-cycle by 
+categorizing its different phases, and activities under each phase. A pictorial 
+representation has been used to explain different tasks conducted under each 
+phase. This life-cycle provides guidelines for using various features of our 
+framework, and allows the user to conduct verification operations in a systematic 
+fashion. 
+ A template to define and express requirements in a formal way is proposed. Our 
+requirement specification template can be used to specify a set of objectives and 
+system constraints. Objectives can be seen as ultimate goals while the constraints 
+are necessary quality requirements that must be satisfied for achieving the 
+objectives. 
+ Inspired from the Discovery, matching and composition (DMC) paradigm of 
+model development [19], we propose method for rapid development of BOM 
+based conceptual models. 
+ We propose a formal description of BOM components and their compositions 
+for documentation purpose. We also propose a graphical notation38  to describe 
+the structure of the BOM component and to show how they are connected to 
+each other in a compact form. This notation can be used as blue prints of 
+different model compositions and can be shared among different teams or 
+archived in the repository for reference.  
+ We propose methods for evaluating the structural consistency of the composed 
+BOMs using rule based static analysis technique. The structural analysis involves 
+checking that the components are correctly connected and they can communicate 
+with each other with correct semantics. For semantic analysis, we propose an 
+OWL based differencing approach which checks that the communication of the 
+components is semantically consistent, meaningful and is understood as intended.  
+ We suggest a behavioral evaluation technique which implicates that the 
+components can correctly interact with each other in a right causal order to reach 
+final states or pass through the goal states. For this purpose we propose state-
+machine matching process, which transforms BOM state-machines of each 
+component into an executable SCXML format and execute them to analyze their 
+interaction. If there is no deadlock and all the state-machines make required 
+progress then the behavior of the components is reported to be consistent.  
+ For the evaluation of dynamic-semantic composability level, our framework 
+incorporates three main approaches: (a) PN Algebraic technique (b) CPN-based 
+State-space analysis technique and (c) CSP based model checking. These three 
+approaches are offered to be used as alternatives to each other and their selection 
+is dependent on the nature of the model being investigated and decision of the 
+modeler. We also present basic guidelines to help the modeler choose an 
+appropriate approach.  
+ For each approach we develop automatic transformation tool that transforms a 
+BOM based composed model into its respective executable model description 
+formalism. This method is inspired from Model Driven Architecture, in which a 
+platform independent model is transformed into platform specific model using 
+                                                 
+38 It should be noted that different UML diagrams such as State charts and sequence diagrams are 
+used to describe BOMs informally. Our graphical notation follows the pattern of CBSE. 
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 185 
+ 
+some transformation rules. We also propose BOM extensions based on certain 
+additional details that are required for correct transformation. For this purpose we 
+develop a BOM extension editor that takes modeler’s input for extending BOM 
+components.  
+ We have applied our proposed approaches in three different case studies 
+discussed in chapter 7, 8 and 9 respectively. Each case study provides a proof of 
+concept and validates specific characteristics of our framework. For PN based 
+algebraic technique we presented a manufacturing system, in which fairness 
+property is verified.  For CPN-based state-space analysis approach a field artillery 
+model is presented in which a set of scenario specific properties are verified. For 
+model checking, the same field artillery scenario is modified into a timed non-
+deterministic model and a particular time property is verified with some 
+probabilistic assumptions.  
+ We introduce a CPN based component model in order to describe a BOM 
+component (or any other simulation component) in form of an executable model 
+that can be executed using CPN execution environment. This CPN component 
+model can also represent any other simulation component using its three layers 
+namely (i) structural layer: which is used to define component attributes and 
+variables; (ii) behavioral layer: which is used to describe the state-machine of a 
+component and (iii) communication layer: which is used to describe components 
+interfaces and how it can connect with other components and communicate. We 
+transform all BOM components into the proposed CPN based component model 
+and compose them to form a composed model which can be executed in CPN 
+environment and verified using CPN based state-space analysis technique. 
+ We introduce a State-space reduction technique called Compositional state-space. 
+This technique assumes that all the composed components are black-boxes and 
+their inputs and outputs are exchanged in the main model. Therefore we can 
+select all the nodes from the state-space which are relevant to any activity 
+happening in the main model and filter all the other nodes, by replacing them with 
+edges. The resultant graph will be a reduced state-space representing only those 
+nodes which describe the interactions of components in the main model and 
+provide sufficient information for composability verification.  
+ 
+10.3 Conclusions 
+The verification framework proposed in this thesis expedites the process of 
+composability verification of BOM based composed models with respect to the 
+requirement specifications. A verified composed model ensures consistent structure 
+and behavior and guarantees the satisfaction of its objectives and required 
+constraints. A rapid development of the conceptual model using Discovery, 
+Matching and Composition paradigm, its automatic transformation into an 
+executable form and its composability verification helps in studying its structural and 
+behavioral correctness with respect to the given requirement specifications.  This 
+helps in rectifying any possible defects in the model design before it is actually 
+implemented and simulated to serve its purpose, and thus saves a significant amount 
+of time, cost and achieve robustness. Moreover this process strongly supports 
+reusability as the entire process can easily be repeated to compose same components 
+for different scenarios with varied configurations or with different requirement 
+specifications (as in chapter 8 and 9).  
+
+Chapter 10  
+ 
+Summary and Conclusion 
+   
+Page 186 
+ 
+The entire composability verification framework is acclimated by a systematic 
+Component Based M&S life-cycle which gives an outline of different phases of 
+component based M&S development process, where each phase has different 
+activities. This life-cycle inherits important features and characteristic of some 
+existing M&S development life-cycles and the Model Driven Architecture with an 
+expansion of component based model development and guides the modelers with 
+necessary directions to perform different tasks at different phases.   
+An important feature of this life-cycle is the software engineering principle of top-
+down refinement. According to this principle a conceptual model is refined into an 
+executable form through a number of intermediary steps. Each step generates a 
+relatively detailed version of the abstract model and is easier to reason about its 
+correctness based on assumptions of its previously verified version. For instance, 
+when the state-machine matching process is successful we can proceed to a more 
+detailed dynamic level execution/verification with an assumption that the behavior 
+of the composed components is consistent. 
+Our experience with the three different dynamic analysis approaches proves to be 
+very constructive for composability verification. Each approach in its own way 
+provides significant improvement on efficient and accurate reasoning regarding 
+model correctness. We profess that the cross domain sharing of existing knowledge 
+and valuable contributions from other communities (such as PN, CSP, model 
+checking in our case) bridges cooperation in problem solving and helps in 
+accomplishing quality research. 
+10.4 Future Directions 
+Some of the key future directions of this work include: 
+ We intend to deploy the composability verification framework in different 
+application areas to evaluate its potential and to make use of its valuable features 
+in verification. One area is the component based design for robotics applications. 
+Many software architectures for robotic applications support component oriented 
+design and thus can be explored for the utilization of our composability 
+verification process, such as in studying various aspects of behavioral 
+composability in different robotic applications.  
+ 
+ In the context of improvements in the verification framework following are some 
+key future directions: 
+o We intend to include verification of requirement specifications. Correctness 
+of requirements is a necessary aspect for successful verification. 
+ 
+o We also intend to produce viable solution for the validation of the composed 
+model with respect to the real system. 
+ 
+o We defined Pragmatic composability level in chapter 2 however the 
+composability verification at this level is still under investigation. We intend 
+to explore this direction in future.  
+ 
+ In general we are interested to explore the area of component based design 
+optimization and to study the composability of component design for 
+optimization with multiple objectives.  
+
+   
+Page 187 
+ 
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+"Fairness Verification of BOM-Based Composed Models Using Petri Nets," 
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+Nice, France, June 2011. 
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+and Theory of Petri Nets, Lars Kristensen and Laure Petrucci, Eds.: Springer 
+Berlin / Heidelberg, 2011. 
+[131] Military grid reference system 
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+Stein, Introduction to Algorithms, 3rd ed.: MIT Press, 2009. 
+ 
+                                                                                                                                                                                                                                                                                                                                                    
+
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+page_content='A Verification Framework for Component Based Modeling and Simulation “Putting the pieces together”  Imran Mahmood  Doctoral thesis in Electronics and Computer Systems Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sweden 2013  KTH  VETENSKAP  OCHKONST  6  KTH Informations och  kommunikationsteknik  Page  2  ISBN 978 91 7501 628 3  TRITA ICT/ECS AVH 13:01  ISSN 1653 6363  ISRN KTH/ICT/ECS/AVH 13/01 SE  KTH School of Information and Communication Technology SE-164 40 Kista Sweden  Akademisk avhandling som med tillstand av Kungliga Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen tisdagen den 26 feb 2013 kl 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='00 i Sal E, Forum, Isafjordsgatan 39, Kista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf8e9 Imran Mahmood, February 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Tryck: Universitetservice US AB  Page  3  Acknowledgement I would like to dedicate this manuscript to my loved ones, including some dignitaries, my parents who recently passed away, my guardians Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  & Mrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sajid Latif who raised me well to make me see this day and most important of all: my beloved wife and my little daughter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Their sacrifice for being apart and for my long absence cannot be compensated for anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  I offer my deepest gratitude to my supervisor Professor Rassul Ayani for this devotion and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Instead of just giving me the directions he actually grabbed my hand and took me to the destination like a true guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I am honored to work under his supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I am thankful to Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Professor Vladimir Vlassov who gave sound advice and provided valuable contributions in my research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I offer my affectionate tribute to the esteemed palace of knowledge, the Royal Institute of Technology, and specially the school of Information and Communication Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  I am thankful for continuous support and encouragement from Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Farshad Moradi from Swedish Defense Research agency (FOI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I am grateful for the constructive critics I received from my opponent Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Gary Tan and the member of the evaluation committee Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Oliver Dale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  I am very grateful for the Higher Education Commission of Pakistan to provide entire financial support for my studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I thank Mrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mumtaz Begum for her support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I would like to offer special thanks to Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Awais Ali Sohrawardi and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Muhammad for their moral support during my study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I thank all my colleagues, friends and especially the cricket team for wonderful time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Finally I thank Sweden for its hospitality, care and warm memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Imran Mahmood  January 2013, Stockholm  Page  4  Abstract  The discipline of component-based modeling and simulation offers promising gains including reduction in development cost, time, and system complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This paradigm is very profitable as it promotes the use and reuse of modular components and is auspicious for effective development of complex simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It however is confronted by a series of research challenges when it comes to actually practise this methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One of such important issue is Composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In modeling and simulation (M&S), composability is the capability to select and assemble components in various combinations to satisfy specific user requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore to ensure the correctness of a composed model, it is verified with respect to its requirements specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are different approaches and existing component modeling frameworks that support composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Though in our observation most of the component modeling frameworks possess none or weak built-in support for the composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One such framework is Base Object Model (BOM) which fundamentally poses a satisfactory potential for effective model composability and reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it falls short of required semantics, necessary modeling characteristics and built-in evaluation techniques, which are essential for modeling complex system behavior and reasoning about the validity of the composability at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In this thesis a comprehensive verification framework is proposed to contend with some important issues in composability verification and a verification process is suggested to verify composability of different kinds of systems models, such as reactive, real-time and probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With an assumption that all these systems are concurrent in nature in which different composed components interact with each other simultaneously, the requirements for the extensive techniques for the structural and behavioral analysis becomes increasingly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed verification framework provides methods, techniques and tool support for verifying composability at its different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These levels are defined as foundations of consistent model composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each level is discussed in detail and an approach is presented to verify composability at that level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In particular we focus on the Dynamic-Semantic Composability level due to its significance in the overall composability correctness and also due to the level of difficulty it poses in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to verify composability at this level we investigate the application of three different approaches namely (i) Petri Nets based Algebraic Analysis (ii) Colored Petri Nets (CPN) based State-space Analysis and (iii) Communicating Sequential Processes based Model Checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  All the three approaches attack the problem of verifying dynamic-semantic composability in different ways however they all share the same aim i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', to confirm the correctness of a composed model with respect to its requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Beside the operative integration of these approaches in our framework, we also contributed in the improvement of each approach for effective applicability in the composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Such as applying algorithms for automating Petri Net algebraic computations, introducing a state-space reduction technique in CPN based state-space analysis, and introducing function libraries to perform verification tasks and help the modeler with ease of use during the composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also provide detailed examples of using each approach with different models to explain the verification process and their functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Lastly we provide a comparison of these approaches and suggest guidelines for  Page  5  choosing the right one based on the nature of the model and the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With a right choice of an approach and following the guidelines of our component-based M&S life-cycle a modeler can easily construct BOM based composed models and can verify them with respect to the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Keywords: Modeling and Simulation, Component-based development, Composability, Semantic Composability, Dynamic-Semantic Composability, Verification, Correctness, Petri Nets Analysis, Algebraic Techniques, Colored Petri Nets, State-space Analysis, Communicating Sequential Processes,  Model Checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page  6  Table of  Contents  Acknowledgement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 132 Figure 55: Manufacturing System BOM based Composed Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='. 136 Figure 57: PN model of the manufacturing System .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 81 Table 11: Semantic Matching Algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 89 Table 12: State-machine Matching algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 91 Table 13: Incidence Matrix Calculation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 102 Table 16: Compositional State-space generation algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='. 111 Table 18: Probability Distribution Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='. 111 Table 19: E-BOM to CSP# transformation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' 139 Table 32: Initial Marking and Incidence Matrix (Scenaro II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='. 140 Table 33: P-Invariants and T-Invariants (Scenaro II) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content="  “If you can't explain it simply, you don't understand it well enough”." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' - Albert Einstein  Page  16  Chapter 1  Introduction  This chapter provides the opening statement and general information about the research presented in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It outlines background, history, the formal definition and the basic philosophy of the problem under question and covers the motivation, goals and scope of the research and the contributions of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the end, a section on the thesis organization is rendered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Background and the opening perspective Over the last fifty years, there has been a revolutionary development influencing almost all of the sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This progress is mainly instigated by the astonishing growth of the use of the digital computers and the subsequent rise of the age of computer simulations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is the emergence and widespread availability of computing power and resources that have made possible the new dimension of experimentation with complex models and their simulations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Computer simulations are now widely used in various scientific disciplines and application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are used for studying complex systems and gaining insight into the operation of an existing system without disturbing the actual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Furthermore they are used for testing new concepts of the systems before implementation, visualizing and predicting behavior of a future system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Besides, they are used for analyzing and solving problems, drawing conclusions and aiding the process of crucial decision making [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore computer simulation is regarded as third branch of science [4] and stands alongside of the first two branches namely theory and experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Modeling and Simulation (M&S) is a discipline with its own body of knowledge, theory, and research methodology [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The goals of M&S are aligned with the systems theory, and include modeling & analysis, design & synthesis, control, performance evaluation and optimization of a real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The M&S community has demonstrated a longstanding focus on providing support for these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With the advent of the net-centric era of methods and technologies in designing complex simulation systems, the focus of M&S industry has been driven by the most recognized potential benefits of reduced development cost, time  and system complexity [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is because M&S development process is costly, time consuming, resource intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Models can be large, complex and require a great deal of time, resource and domain specific expertise to develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Beside this, an enormous effort is required to evaluate that the model is correct and meets its requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore M&S community has taken a deep interest in the quality design principles and their underlying supportive theories to alleviate these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It has been realized that constructing a model from scratch each time it is needed is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Instead, the practice of model reuse has been increasingly appreciated and is inspired from the vision of software reuse, which was originally introduced in 1968 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Apparently this approach looks very appealing however it poses many obstacles in implementing, such as lack of flexibility and adaptability in design, difficulty of integration, mismatched interface, incomplete specification etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These obstacles are  Chapter 1  Introduction  Page  17  considered elusive research challenges and are now the primary research interests of the software engineering and M&S communities [8] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Component based Software Engineering Component-based software engineering (CBSE) has been identified as a key enabler in the construction of complex systems by combining software components that are already developed and prepared for integration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Software Component A software component is defined as a unit of composition which is independently developed and can be combined with other components to build larger units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It must have clearly specified interfaces to communicate with its environment while the implementation must be encapsulated in the component and is not directly reachable from the environment [9], and therefore can be easily used by the third party without having to know implementation details [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Building software from components contributes to a major paradigm shift in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The basic philosophy behind the idea of component-based development is to carry out the software development process by (quickly) producing software applications through assembling prefabricated software components and to archive these interoperable software components in form of an increasingly large repository for further reuse [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CBSE promotes the principle of modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' That essentially helps to master the complexity of the reality by decomposing it into parts [12] and enables the designer to use and reuse appropriate parts for different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These parts are the sub-systems built in a component- based fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These subsystem components may have been separately developed by different teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They may also have been developed for different purposes unrelated to the current context of the usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CBSE has many advantages, such as effective management of complexity, logical separation, reduced time and cost, increased productivity, improved quality, a greater degree of consistency, increased dependability, and a wider range of usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In addition, the growing connectivity of real world problems is reflected in the requirement to compose cross domain solutions [13], and therefore support knowledge sharing to a wider user community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  CBSE is therefore a discipline of software engineering that deals with the composition of components to construct software systems which are capable of performing functions according to the user’s requirements [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In CBSE, component integration and component composition are two distinguished terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Component integration is merely the task of connecting components together whereas composition also includes reasoning about the semantic behavior of the resulting assembly [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With the advent of component technology the integration problems are becoming a difficulty of the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Instead more crucial problems of predicting the emergent behavior of assemblies and the problem of reasoning about how well components will play together are now in debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Component composition supports this type of reasoning and provides a foundation for fundamental reasoning to justifying validity of the resulting assemblies, their run-time compatibility and emergent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main reason for the difference between integration and composition is due to the fact that component interfaces do not provide enough information to determine how well the composed components will play together [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An interface can only help to determine if the component can be connected to  Chapter 1  Introduction  Page  18  some other component but cannot supports reasoning about emergent properties of the assemblies [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Component composition promises such rationale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' however is still a subject of open research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Component based Modeling & Simulation Inspired by the discipline of component based software engineering, M&S community has also started to develop simulation models by reusing previously developed and validated “simulation components”, and composing them in a new simulation model according to the desired user objectives [16], [17], [18], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The basic and effective strategy for tackling any large and complex simulation problem is “divide and conquer.” One major idea in component-based simulation development is to create models that are themselves self-contained and independently deployable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus different simulationist will be able to work on different components independently, without needing much communication among each other, (and particularly without the need to share the classified domain knowledge) and yet the components will work together seamlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In addition, during the maintenance phase, it is possible to modify some of the components without affecting all of the others [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In simulation community the research on component based development falls under the rubric of composability [22], where simulation models are considered to be the building blocks and are referred as “model-components”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Model Component1 A model component is an independent element of a simulation model that conforms to certain component standard, has well-defined functionalities (inputs/outputs) and behaviors, presented through its interface describing its communication with other components and a formalized description of its internal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A model component is not a stand-alone component, but can be independently deployed, and it is subject to third-party composition with or without modification [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In component based development, some basic reusable model components are composed together to create complex and sophisticated simulations, without building them from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The model components can be composed if their inputs and outputs physically match each other however it is difficult to say whether this combination is meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Besides it cannot be said for sure if it will perform according to the desired requirements unless the correctness of the composability is checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Composability is the property of the models, as it essentially contends with the alignment of issues on the modeling level [13], where it is viewed as creation of complex models by selection and integration of basic reusable model-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A set of components can be integrated if their inputs and outputs are compatible, but in order to guarantee that their combination is valid in the required executable scenarios, we study the degree of composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With a slightly greater number of components, which are somewhat complex in nature, the composability becomes an increasingly challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the  1The term Model component should be differentiated from the term Component Model, which in text refers to the underlying technology being used by the component based software engineering platforms such as CORBA, EJB etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1  Introduction  Page  19  presence of functional and non-functional application requirements it poses severe implications on the effort involved in verifying the requirements, and increasing dynamism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Even though, the individual components are pre-verified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' their verification is usually done in a limited context, with assumptions that may not hold after composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As a result, the complexity of system verification grows exponentially with the number of applications [23]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Modeling and Analysis using Petri Nets Petri nets (PN) is a mechanism of modeling complex systems, in which states and events can be manipulated according to certain rules and explicit conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN formalism was introduced by Carl Adam Petri in 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It provides an elegant and useful graphical and mathematical formalism for modeling concurrent systems and their behaviors [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN graphs are quite suitable for representing Discrete Event Systems (DES) in which operations depend on potentially complex control schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN graphs are intuitive and capture a lot of structural and behavioral information about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Another motivation for considering PN for the modeling of DES is the body of analysis techniques that have been developed for over three decades and are used for reasoning about structural and behavioral properties of PN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These techniques include reachability analysis, state-space analysis, and model-checking as well as linear-algebraic techniques [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The PN research has been developed in two directions for the past three decades: (i) PN theory that focused on the development of basic tools, techniques and concepts needed for the PN application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (ii) Applied PN theory which is mainly concerned with the PN application for the modeling of systems and their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Successful work in this direction requires good knowledge of the application area in which PN are applied and PN theories and techniques [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Modeling and Analysis using Process Algebra  Process Algebra is an algebraic approach for the modeling and analytical study of concurrent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It has a diverse family of algebraic formalisms for modeling concurrent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These formalisms comprise of algebraic language for the specification of processes and provide calculi in form of algebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning about their correctness and equivalence [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=" The main Process algebraic formalisms are: \uf0a7 CCS, Milner's Calculus of Communicating Systems \uf0a7 CSP, Hoare's Communicating Sequential Processes \uf0a7 ACP, Algebra of Communicating Processes \uf0a7 LOTOS, Language Of Temporal Ordering Specification 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Model Verification In M&S, verification is concerned with building the model right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is typically defined as a process of determining whether the model has been implemented correctly [28] and whether it is consistent with its specifications [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In principle,  2 Even though the referred text corresponds to the electronic components which are physically composable, however the problem of composability complexity is the same and is mutually understood by different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1  Introduction  Page  20  verification is concerned with the accuracy of transforming the model’s requirements into a conceptual model and the conceptual model into an executable model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The distinction of a conceptual model and executable model is of great importance and is a fundamental principle in M&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A conceptual model is abstract description of a real system [30], captured based on given requirements and modeling objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is later refined and implemented into a more concrete executable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In these terms, conceptual modeling is a subset of model design [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conceptual modeling is about moving from a problem situation, through model requirements to a definition of what is going to be modeled, and is independent of its implementation details [30], which are later addressed in form of an executable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Summary of the opening perspective In essence, component-based approach is highly favored in M&S community for building large and complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' But to ensure that the model is correct and meets its requirement specifications, a substantial effort is required to evaluate its degree of composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In M&S community, the discipline of Model Verification provides basic concepts and fundamental principles for the compressive study of the degree of composability and reasoning its correctness with respect to the given specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However the existing component-based simulation frameworks offer limited built-in extensive verification techniques or none at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Therefore third party approaches such as PN analysis techniques and process algebra are considered for the thorough examination of composed models at various levels of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The sub-topics: (i) Component-Based Modeling & Simulation, (ii) PN /CSP Analysis and (iii) Model-Verification are the elementary pillars and theoretical foundations of this thesis and are expanded in details in chapter 2, 3 & 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Preliminaries Based on the previous discussion, the formal definition of the problem of this thesis is furnished in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to define the problem statement, following definitions are used: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Definition 1: Set of Components Let C = {c1, c2, c3 …, cn} be a given set of components discovered and selected from a component repository R, as per the abstraction of the real-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Definition 2: Requirement Specification The Requirement specification of the system model is defined as a tuple: RS = 〈O, S〉 where: O = {o1, o2, o3 …, on} is a set of objectives (or goals) and S = {s1, s2, s3 …, sn} is a set of system constraints (or system properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Objective: An objective oi ∈ O can be defined as a reachable “final-state” of the composed model or an aggregated desirable output (a data value or event) produced by the composed model which cannot be produced by individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1  Introduction  Page  21  System Constraint: In modeling terms, a system constraint si ∈ S is defined as a system property that must be satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' for instance a good state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' which must be reached or a bad state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' which must be avoided (never be reached) during the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The notions of constraints are different from Objectives, because they can be necessary requirements but not the ultimate goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a manufacturing system should not only produce the desired products (objective) but also fulfill safety requirements (constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Definition 3: Composition & Composability Pattern Let CM〈c1, c2, c3 …, cn〉 be a composition of a set of given components C, composed using a particular composability pattern P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A pattern describes how the components are attached to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the topology of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And provide important information for composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A pattern of composability can be sequential, parallel, fork, join, iterative, or composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Definition 4: Satisfiability Operator For each element in the requirement specification RS, a Satisfiability operator╞ maps a given composed model CM to a Boolean (True or False) formally described as follows: • CM〈c1, c2, c3 …, cn〉    ╞ i  oi∈O → true | false • CM〈c1, c2, c3 …, cn〉    ╞ j  sj∈S → true | false For each relation ╞ i we define a verification function (algorithm or theorem) based on which the satisfiability operator maps the resultant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  This verification function determines whether a given composed model satisfies a required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Problem Statement Based on the above definitions the problem statement is defined as follows: “Given a composed model CM, composed from a set of components C using a pattern P, and the requirement specification RS, can we verify that CM fulfills all the objectives and satisfy all the constraints given in the requirement specification”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Formally:  This problem statement is considered as an initial point and basis of the research presented in this thesis proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this work it will be shown how a modeler can correctly compose component models and verify the composition at different levels through utilization of our proposed verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CM=Compose ([c1, c2, c3 …, cn], P) ∧ RS=〈O, S〉  → {CM ╞𝒊 ∀oi ∈ O ∧  CM ╞𝒋 ∀sj ∈ S} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1)  Chapter 1  Introduction  Page  22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Approach In this section an overview of the approach and methodology is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on the software engineering principle, this section is divided into two main parts (i) Problem Domain and (ii) Solution Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Problem Domain Problem domain (or problem space) is an engineering term referring to all information that defines the problem and its constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It includes the goals that the problem-owner 3 wishes to achieve, the context within which the problem exists, and all rules that define required essential functions or other aspects of any solution product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It represents the environment in which a solution will have to operate [Wikipedia].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  All the information provided in this thesis related to modeling & Simulation, component-based model development, conceptual modeling, model components, composability, model-verification and the problem of composability correctness correspond to the problem-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In particular, Chapter 2 covers the main aspect of the problem domain where the component based modeling and simulation is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Following sub-sections briefly describe the selected method of specification of the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Base Object Model (BOM) as Composability Framework BOM is selected in this thesis as a component specification standard which can be used as a foundation for developing model components at conceptual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are composed and are subjected to the composability verification process to evaluate that they satisfy given requirements, hence represent component framework of the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Requirement Specification Template A “Requirement Specification Template” is defined which is used to formulate requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It essentially contains a set of objectives and constraints (of standard or scenario-specific properties), which are required to be satisfied for the proof of correctness of the composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Solution Domain Solution domain (or solution space) is a term referring to all information that defines the proposed solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It includes the concepts, principles, methods, techniques, algorithms, programs, software architects, frameworks, processes and recommended practices, which help in solving the problem under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Following sub-section gives a brief overview of the approach used in this thesis:  3  A problem owner can be the customer, solution buyer, organization or a prospective target community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A problem owner sees the problem as an opportunity, whereas the solution engineer sees the problem for which he/she has to provide a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1  Introduction  Page  23  Multi-tier Composability Verification The composed model undergoes multiple iterations for composability verification at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each level corresponds to a tier in the verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the composability at a particular level is successfully verified, next level is iterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When all the levels are completed, the components are said to be fully composable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These levels are discussed in detail in chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The verification of these levels is discussed in chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN Formalism PN formalism (and specially the Colored Petri Nets extension) is chosen for creating executable models of the BOM based conceptual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed framework automatically transforms BOM components in form of an executable PNML 4 or CPN-based component which can be executed or undergo a verification process using the corresponding PN execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP Formalism CSP5 formalism with an extension of Timed-CSP is picked as another executable modeling language for BOM based conceptual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed framework transforms BOM components into executable CSP process components and composed for execution and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Automatic Transformation Tools  An automatic transformation tool is proposed, which transforms a BOM component model into the selected executable modeling formalism such as PNML, CPN based or CSP based executable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It may also be required to provide additional details, which cannot be modeled or represented by BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Dynamic Analysis Approach Three main dynamic analysis approaches are selected for composability verification of BOM base composed models at dynamic-semantic composability level: Algebraic analysis approach This approach is used to transform a BOM composition into a classical PN model using PNML format and verifies the properties using PN algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State-space analysis approach This approach is based on using Colored Petri Nets and State-Space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN tool is a strong simulation and verification tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State-space analysis is a very accurate correctness reasoning technique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' however it is costly in terms of computational power and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore a reduction technique is also proposed to reduce a state-space graph of a composed model, in order to avoid state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model Checking approach CSP based model checking is used for the formal verification of BOM based composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In formal logic, model checking designates the problem of determining whether a formula or a correctness property ϕ defined using LTL, CTL6 or similar property specification formalism, evaluates to true or false in an  4 Petri Net Markup Language 5 Communicating Sequential Process 6 Linear Temporal Logic, Computational Tree Logic  Chapter 1  Introduction  Page  24  interpretation of a system K, written as K ╞ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Efficient algorithms are selected to determine whether K ╞ ϕ holds [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In summary, these three approaches are extensively being used in formal verification for over a couple of decades and therefore equipped with rich theoretical foundations and practical tools and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We however believe that they are being considered in this thesis for the composability verification of BOM based models (or for that matter any M&S composition framework) for the first time and will prove to be very promising and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A basic foundation is built using these approaches in this thesis, and their usage are shown though appropriate examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also necessary guidelines are provided for developing new verification methods using these approaches and tools, in order to address various verification issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This research aims to propose a multi-tier verification life cycle for defining, development, archiving, discovering, matching, selection and composing, transforming, executing, verifying and finally reasoning about the correctness of the composed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This life-cycle extensively relies on the integrated component development, composition and verification framework that is being proposed in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This life-cycle follows our proposed process to perform verification of a composed model at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  This life-cycle can be adapted by M&S practitioners for rapid model construction, analysis, refinements and reuse and thus it will boost the process of modeling and simulation of complex dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Solution Statement Based on the proposed approach the solution statement is described as follows: “A verified composed model guarantees that the selected components are composable at all composability levels, and they meet the requirement specification by satisfying given objectives and fulfilling the required constraints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A correctly composed model, promotes reuse of base components thus support rapid model development and can be reused as yet another component later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6 Scope of the Thesis In this section, the scope and the boundaries of the thesis are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Correctness In this thesis, “Correctness” is the main focus of the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The approach, methods, process and framework mainly deal with the correctness issue of the composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The other issues such as performance, efficiency and cost estimation of the solution are currently beyond the scope of this thesis and considered as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Validation Validation is a vital part of model evaluation and always goes hand in hand with verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it is beyond the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Although we believe that, our framework is flexible and open-ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it can accommodate necessary extensions to support validation with a minor effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1  Introduction  Page  25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Emergence Emergent behavior due to composition of sub-systems is an important and open research topic in the composability domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We however do not address this issue in this thesis and consider it a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Generalization Currently, the proposed approach is based on Base Object Model, only as a demonstration of how our approach can be applied on an existing component standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However the framework presented in this thesis is open-ended and can be generalized to accommodate any other component standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Furthermore, heterogenic composability can also be supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We however do not address generalization issues in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 Summary of the Contributions The existing work in the area of component based modeling and simulation is fragmentary in nature, especially when the verification of component composability of model at a conceptual level is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Furthermore, even though different composability verification approaches exist, but they have not been studied in depth at different granular levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this research, composability of BOM based model is studied in depth, focusing mainly on the different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A multi-tier component based verification life-cycle is proposed that tackles key issues of such as model development, discovery, selection, matching, composition, requirement specification, transformation, implementation, execution, analysis and most importantly verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In terms of verification, the major contributions of this thesis include development of a composability verification framework, which integrates different methods, and techniques to support different tasks in the composability verification process of a composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These different tasks are categorized in different phase of a proposed component based modeling and simulation (CBM&S) life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We propose methods for evaluating structural and behavioral consistency of the composed BOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For structural evaluation we propose a set of static analysis techniques to verify that the components can be correctly connected and their communication is semantically consistent, meaningful and is understood as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For behavioral consistency of the composition we suggest a state-machine matching technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It verifies that the components can correctly interact with each other in a right causal order to reach final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For the further evaluation of the behavioral composability our framework incorporates three main approaches: (a) PN Algebraic technique (b) CPN-based State-space analysis technique and (c) CSP based model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For each approach we develop automatic transformation tool that transforms a BOM based composed model into the executable model of the corresponding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We present three different case studies for the proof of concept and for the evaluation of our verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We also suggest various extensions in each approach to suit the needs of composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance we propose algorithms for automation of the PN algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also a CPN based component model is proposed for the State-space algebraic approach in order to describe a BOM component (or any other simulation component) in form of an executable model that can be executed using  Chapter 1  Introduction  Page  26  CPN execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also introduce a State-space reduction technique for the CPN based state-space analysis approach to avoid the risk of state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For the CSP based model checking approach we propose an external function library for methods to support various modeling tasks such as definition of probability distribution functions for probabilistic system models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8 Structure of the Thesis This thesis is divided into two main parts:  Part I Episteme: This part mainly covers the theoretical concepts, principles and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It comprises of chapters 1, 2, 3 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 1: Introduction: Chapter 1 gives a bird’s eye view of the research presented in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It addresses the concept, historical background and the basic philosophy of composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The problem is defined and the approach is briefly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A section on the scope of the thesis and main contributions are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 2: Component Based Modeling and Simulation: Chapter 2 introduces and discusses component based modeling and simulation in details, as it is the foundation of the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This chapter mainly covers the theory, issues, different levels, framework and the formalism of model composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also introduces Base Object Model (BOM) in details as a choice of Model composition standard of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Chapter 3: Executable Modeling Formalism This chapter provides introduction, theory, basic definition and classification of PN and CSP as executable modeling formalisms and regarded as solution domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also describes basic concepts of the analysis techniques that are used later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Chapter 4: Verification and Analysis Chapter 4 discusses theory and principles of verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also categorizes some of the important verification techniques that are used in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Part II Techne: This part contains practical aspects including approaches, methods, tools, development frameworks and lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also contains examples related to our proposed solutions for the proof of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It comprises of chapters 5, 6, 7, 8 & 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5: Proposed Approach and Framework Chapter 5 is the center of the thesis as it provides the most important details of our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It describes the proposed verification framework and verification life- cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It covers our proposed methods, techniques, algorithms, procedures as our  Chapter 1  Introduction  Page  27  contributions at different phases of composability verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These phases and their concerning activities are outlined as composability verification life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 6: Composability Verification Process This chapter presents the proposed composability verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It provides essential guidelines of how to use our proposed composability verification framework (discussed in chapter 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It uses work flow diagrams to describe the overall process and gives necessary guidelines to the modeler at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 7: Fairness verification using PN Algebraic Technique Chapter 7 describes a case study of a manufacturing system as an example to explain how the proposed framework helps to verify fairness property in a composed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The purpose of this chapter is to practically demonstrate algebraic verification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8: Model verification using State-space analysis technique Chapter 8 covers an example of the verification of a Field Artillery Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It practically demonstrates how state-space analysis is used to verify a composed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The field artillery model is introduced in detail along with requirement specifications and it is shown how the proposed approach can help to verify its composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 9: Model Checking This chapter demonstrates an example of verification using CSP based Model Checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The field artillery model discussed in chapter 8 is modified into a real-time probabilistic system and is verified using CSP based model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 10: Conclusion and Future work This chapter provides summary and conclusion, discussion and future work of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page  28  Chapter 2  Component Based Modeling and  Simulation  Composability is an important quality characteristic and an effective means to achieve several benefits in M&S discipline, but in reality, it is a challenging and daunting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The community has conducted active research on its theoretical and practical intricacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In theory, composability is studied under various facets and views primarily distinguished, by its different “layers” or “levels” as identified by different research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas in practice, various practical challenges associated with it are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Most important of these issues are component specification, development, integration, composability verification and validation, collectively referred to as phases of a Component based life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter both theoretical and practical aspects of composability are discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Composability in M&S In M&S applications, composability has been defined in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Much of these definitions have been collected by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tolk in his article [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Harkrider and Lunceford defined composability as: The ability to create, configure, initialize, test, and validate an exercise by logically assembling a unique simulation execution from a pool of reusable system components in order to meet a specific set of objectives [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Kasputis and Ng defined composability as: The ability to compose models across a variety of application domains, levels of resolution, and time scales [16] Petty and Weisel recommended the following definition in their article on theory of composability, which later was appended by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Davis: Composability is the capability to select and assemble simulation components in various combinations into valid simulation systems to satisfy specific user requirements, meaningfully [17] [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It has been realized that composing models is more difficult than composing general software components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This argument is predicated on the assumptions that models are more complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' they are developed for particular purposes, and they depend on context-sensitive assumptions [8] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model development is a hard design task, mainly due to the complexity involved in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Nowadays this complexity is increasing to levels in which the utilization of pre-defined models is considered very useful to cut short the development time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus model composition is a paradigm, where existing components are the building blocks for the construction of new larger and more sophisticated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a model is composed, it must be evaluated in  Chapter 2  Component Based Modeling and Simulation  Page  29  terms of correctness with respect to its requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In short the predictability of guaranteeing the correctness of model composition is called Composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 A Brief History of Composability and related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Initiation Composability in M&S has primarily been investigated by the defense research sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The earliest uses of the term composability within the context of defense simulation dates back to the Composable Behavioral Technologies (CBT) project during the mid-1990s [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Later on the Joint Simulation System (JSIMS) project investigated composability as a system objective [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In 1998, a project on model based simulation composition (MBSC) was started in which a prototype composition environment for JSIMS was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In 1999 Page and Opper investigated the composability problem from a computability and complexity theoretic perspective [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Composability became a key system objective for OneSAF project in 1999 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Theoretical evolution Later on a series of numerous articles were published which addressed various issues of and methodologies of composability and became the theoretical foundations for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Important works in this series include: Kasputis and Ng [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Petty & Weisel [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Petty and Weisel extended the work of Page and Opper, provided a broad survey of the uses of the term composability, and examined the composite validation problem within the context of automata theory and computable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Later a comprehensive report was published by Davis and Anderson in 2003 [17] that provides a broad survey of the composability and suggests its applications for the DoD7 in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Standards & Frameworks Later on, the research on composability remained focused on the development of standard composition frameworks and its practical application in various domains of modeling and Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In 2005 the Extensible Modeling and Simulation Framework (XMSF) was initiated by the Naval Postgraduate School to develop a web-based simulation environment [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Advances in M&S technologies, gave rise to different distributed simulation standards and protocols such as Simulation Networking (SIMNET), Distributed Interactive Simulation (DIS), Aggregate Level Simulation Protocol (ALSP) and the High Level Architecture (HLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The details of these standards are well documented by Moradi [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Due to the complex nature of the standards, and distributed simulation itself, different composability frameworks were introduced to co-op with these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' More general-purpose frameworks such as the Discrete Event System Specification (DEVS) [40], the Open Simulation Architecture (OSA) [41], the Base Object Model (BOM) [42], and the Component Oriented Simulation Toolkit (COST) emerged and contributed to various issues of composability in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Technological Advances Due to the technological advances in computer engineering, many approaches emerged with the aim to address issues and high end requirements of modeling and  7 United State Department of Defense  Chapter 2  Component Based Modeling and Simulation  Page  30  simulation such as representation of Complex, Dynamic and Adaptive Systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' integration of large interdependent Systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' multi-resolution and multi-scale modeling [43], and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this period, many tools and techniques were developed using composability paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model Coupling Toolkit (MCT) was developed to support and simplify the construction of parallel coupled models [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' MUSE is another composable simulation environment for astrophysical applications in which different simulation models of star systems are incorporated into a single framework [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some frameworks such as Common Component Architecture (CCA) [46] and Component based Grid Environment (MOCCA) [47], were proposed to be used in high-performance computing, where scientific components are directly connected by their Users and Providers ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A Multi-scale Coupling Library and Environment (MUSCLE) provided a software framework for building composable simulations according to the complex automata theory [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Compo- HLA is an environment proposed for supporting HLA component [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Composability verification and Validation Most of these frameworks lack strong built-in composability evaluation support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Therefore some third-party composition, verification and validation frameworks were developed by individual research teams such as Composable Discrete-Event scalable Simulation (CODES) [20] and Semantic Web-based BOM composition framework [19], where verification and validation of composability are strongly focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Theory of Composability The formal theory of composability was pioneered by Petty and Weisel [34], [38], [50] in an initiative developed at the Virginia Modeling, Analysis & Simulation Center (VMASC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It was also called “semantic composability theory” (SCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The aim of the SCT is to check and prove the semantic composability of components using formal descriptions and reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A model is defined as a computable function: y = ƒ(x), where function is calculable by a finite procedure and relates each input to a unique output, as shown in Figure 1  Figure 1: A model as computable function (acquired from [34])  A simulation is a sequence of executions of a model ƒ(x), where the output from each execution step is the input to the next step of the execution:  Where i = input value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' m=memory value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' o=output value and n=current iteration, as shown in Figure 2 (mn, on) = ƒ(mn-1, in-1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1)  xeX  J(x)e Y  Domain  Codomain  X  YChapter 2  Component Based Modeling and Simulation  Page  31  Figure 2: Sequence of executions (acquired from [50])  The composition is defined as output of one function to be the input of another:  Figure 3 shows the representation of a composed model, which is developed through composing other models (f1, f2 & f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A composed model as a whole has also a set of inputs, outputs, current states and next-states as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 3: Composed Model  (acquired from [50]) The composition of models in SCT is in fact the composition of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since a set of computable functions is closed under composition any set of models can be composed if the composition exists, but there is no guarantee that the resultant will be a useful model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus focus of SCT is semantic composability, the question of whether the model composition is meaningfully valid or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Validity A model is defined as valid, if it is an accurate representation of the real-world with respect to the intended use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For formal validation, the simulation of a composition is represented as Labeled Transition System where nodes are model states, edges are function executions, and labels are model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A composition is valid if and only if its simulation is close to the simulation of a perfect model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Perfect Model A model is perfect with respect to a natural system N 8 if and only if it represents a system of perfect observations of the natural system [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8 A natural system N is a real or imagined system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' h(x) = ƒ(g(x)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2)  io  i,  is  i3  mo  111  1112  1113  m4  01  02  03  04  2  3  4i1  i2  i  X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  m  X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3  Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  → mner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  mz  X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2  Ji,1  X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4  Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2  →> Mno2  Ji  X22X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  m3  X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3  J1,2  X2,3  J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='s Js Y33  → mer3  Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3  fn y2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  tu  X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6  J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4  J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='+Y23  m,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7  Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="s  > mnoxt's  J3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8 J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6  个个  03  So to  % l %Chapter 2  Component Based Modeling and Simulation  Page  32  For details of different classes of models, their equivalence relations, formal theorems and proofs of equivalence, interested readers should refer to [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The basic concepts of a formal theory of semantic composability include formal definitions for model, simulation, validity, and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A theory of composability can facilitate the convenient reuse of simulation components, which holds the potential to the time and cost of simulation development [34] [38] [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Concepts related to Composability In this sub-section, some of the concepts and idea related to composability are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Composability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reusability Composability is differentiated from reusability in many aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Balci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' define Reusability as the degree to which an artifact, method, or strategy is capable of being used again or repeatedly [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' on the other hand suggest that the term simulation model reuse can be taken to mean various things from the reuse of small portions of code, through component reuse, to the reuse of complete models [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Composability offers means to achieve reusability, but reusability might not always be the ultimate objective of model composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance, in a particular situation, a set of modular components are purpose-fully built and composed to construct a large model, but they cannot be reused in a different project, due to their highly specific design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  To be widely reusable, a component must be sufficiently general, scalable, and adaptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A requirement for reusability may lead to another development approach, for example, a design on a more abstract level [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The comparison between usability and reusability of composable components poses a tradeoff between them being very specific in function and behavior so that they can be used in a particular case to satisfy specific user’s requirements or them being very generic and abstract so that they can be reused in different situations again and again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 4: Generic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Specific component design Figure 4 illustrates a component is often more reusable if it has a generic design and less reusable if it has functionally specific design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Both use and reuse of composable components share three levels of transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A component can be seen as a box, which contains the interfaces and internal implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Three levels of composability transparency are defined:  Generic  Functionally  specific  design  More reusable  Less reusableChapter 2  Component Based Modeling and Simulation  Page  33  Black Box Composition In black box composition, the user sees the interface, but not the implementation of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The user documentation is provided that contains the details of the inputs and outputs, requirements and restrictions of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' All the implementation details are hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The clients will get what the contract promises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The changes are not feasible at the deployment end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The advantage of black-box composition is that the testing done at the development side is persevered and there is no need of further testing at the deployment side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Glass Box Composition In glass box composition the inside structure of a component can be viewed, but it is not possible to modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This solution has an advantage when compared to black box reuse, as the modeler can understand the box and its use better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it is not possible to make any changes in the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The advantage of this level remains the same as that of black-box composition however an additional benefit is that the user can gain knowledge of the internal implementation and can understand the mechanics of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' White Box Composition In white box composition it is possible to see and change the inside of the box as well as its interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A white box can share its internal structure and implementation with another box through inheritance or delegation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The advantage of this level is greater flexibility due to the provision of modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  However this level incurs an extra burden of testing at the deployment end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 5: Black Box, Glass Box, White Box Figure 5 illustrates difference between black box, glass box and white box composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Composability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Interoperability Bearing in mind the definition of composability mentioned previously, the IEEE definition of interoperability is: The ability of two or more systems or components to exchange information and to use the information that has been exchanged The concept of interoperability is mainly about inter-connecting systems of various types developed for different purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' for different platforms, and about their syntactically and semantically agreed upon communication [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the context of Internals not known Internals known Internals known No modification No modification Modifiable  Chapter 2  Component Based Modeling and Simulation  Page  34  modeling and simulation, interoperability is the ability of different simulations connected in a distributed system to collaboratively simulate a common scenario [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [52] distinguishes composability and Interoperability as follows: Composability contends with the alignment of issues on the modeling level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The underlying models are purposeful abstractions of reality used for the conceptualization being implemented by the resulting systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' whereas Interoperability contends with the software and implementation details of interoperations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' this includes exchange of data elements via interfaces, the use of middleware, mapping to common information exchange models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Composability Levels Petty and Weisel emphasized on two basic types of composability: syntactic and semantic in their theory of composability [38] [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' According to which the syntactic composability requires that the composable components should be constructed with compatible implementation details such as parameter passing mechanisms, external data accesses, and timing assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The question of syntactic composability is whether the components can be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In contrast, semantic composability is a question of whether the models can be meaningfully composed to form a composed simulation system and whether the combined computation is semantically valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is possible that two components may be syntactically linked, so that one can pass data to the other, but they can be semantically invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 6 represents the difference between syntactic and semantic composability metaphorically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 6 Syntactic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Semantic Composability (acquired from [38]) Composability is studied in more depth under different levels, as identified by different research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Several levels of understanding and agreement are required between the models in order for them to be meaningfully composed—that is, for their composition to produce meaningful results [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Davis recommended five distinctions of levels namely: syntax, semantics, pragmatics, assumptions, and validity to study composability [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' He describes these levels as different consistencies of composability, which all together are examined for the correctness of model composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Petty & Weisel have suggested nine levels of composability in terms of composition units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These levels are: Application, Federate, Package, Parameter, Module, Model, Data, Entity and Behavior [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tolk described a six layered model called Levels of Conceptual Interoperability (LCIM) to study composability and interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This model includes: technical layer, syntactic layer, semantic layer, pragmatic layer, dynamic layer, and the conceptual layer [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly Medjahed & Bouguettaya introduced a composability stack in which the composability of semantic web services is checked at four levels: Syntactic, Static Semantic, Dynamic Semantic and Qualitative level [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' First three levels of  AF  Syntacticcomposability  Semantic composabilityChapter 2  Component Based Modeling and Simulation  Page  35  Medjahed & Bouguettaya’s composability stack were adopted by Moradi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' to study the degree of composability of Base object Model (BOM) components [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, these levels are considered as fundamental benchmarks for the evaluation of model composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The notion of model composability and its correctness strongly depend on the consistency of these levels as explained in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Syntactic level: At this level, the structure of the components is studied to know if they can fit together i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the output of one can be read as an input to the other and that the syntactic information of the connected components, such as message name, mode of action and number of parameters match each other e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', A “passenger airplane” component will be a syntactic misfit in a military training simulation, where a “fighter jet” component is required whose input will be a signal from “ground station” component to engage a target and output will be an airstrike on the “target” component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A passenger plane can neither take a target designation as input, not it can fire on a ground target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So this component is not composable at syntactic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Static-Semantic level: It is concerned with the meaningful interaction of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Static- Semantic level of composability involves in having a concise and mutual understanding of the data exchanged by the components participating in the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  At this level, it is ensured that all the components possess the same understanding of the terms, parameters, data types and units, so basically this level deals with the interpretation of same meaning of concepts for the information exchanged between the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance, if two components being composed interpret units of quantities in a different way, they will incorrectly process data values during the information exchange and thus result in a situation not intended by the user e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', if a integer data value is intended to be the bearing of a target (in degrees) but interpreted as target distance (in Km) by the other component then it is a semantic mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The term “static” is prefixed, because all the information that is required to evaluate this level is static and does not change during the entire component interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Dynamic-Semantic level: Dynamic Semantic Composability implies that the components are dynamically consistent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', they have suitable state-full behavior, necessary to reach the desired goals and subsequently satisfy user requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The dynamic level of composability ensures in having a behavioral consistency and coherency among the participating components in achieving the common goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The dynamic semantic composability can only be achieved if the components are at the right states during their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also they should possess required behavior to make a collective progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', in a composed model of a restaurant, a waiter component may have two different behaviors (i) Classical restaurant where a waiter takes order from customer, serves food and then collects payment or (ii) Fast food restaurant where waiter takes order, collects payment and then serves food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The selection of the correct behavior and the correct customer component (the one who can correctly interact with the classical restaurant waiter or fast food waiter) will affect the overall composability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This example presents how the components should be at right states to make  Chapter 2  Component Based Modeling and Simulation  Page  36  progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A customer (expecting classical treatment) will wait forever for the (fast food waiter) to serve food and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Even if the components are at the right states, but their behavior is not correct, the composition may not reach its goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', in a manufacturing system two machine components produce two different parts that are later combined to make a finished good, and they share a single robot component for input of raw material, it is required that the robot component should be fair so that both machines get more or less equal chance to proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the robot is not fair the proportion of good produced will be unbalanced and therefore the system will fail to meet its objectives even though the components are at right states and continue to progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The term dynamic is prefixed, because the information such as current state of components changes dynamically during component interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Pragmatic level: Consistency of meaning is not always straightforward because the same word means very different things depending on context [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Pragmatic consistency refers to a context based meaningful composition of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In linguistics the study of the relations between linguistic phenomena and aspects of the context of language use is called pragmatics whereas Context is defined as something that consists of the ideas, situations, events, or information that relates to it and makes it possible to fully understand it [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The pragmatic level of composability evaluates the difference of actual effect of the messages with the intended effect of messages during communication [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The research of pragmatic level of composability involves in-depth study of computational linguistics, cognitive technologies and contextual computing [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An important issue at this level is pragmatic ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Pragmatic ambiguity arises when the message is not specific, and the context does not provide the information needed to clarify the statement, and due to which the components do not interact according to the desired objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An example of pragmatic ambiguity is the story of King Croesus and the Oracle of Delphi (derived from [56]): "King Croesus consulted the Oracle of Delphi before warring with Cyrus of Persia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The Oracle replied that, "If Croesus went to war with Cyrus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' he would destroy a mighty kingdom".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Delighted, Croesus attacked Persia, and Croesus’ army and kingdom were crushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Croesus complained bitterly to the Oracle’s priests, who replied that the Oracle had been entirely right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' By going to war with Persia, Croesus had destroyed a mighty kingdom – his own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='"  In essence, a set of components can possibly fit together (syntactically), and their communication is meaningful and understood (semantically), but unless all components preserve essential behavior (dynamically) in order to reach the desired composition goals, and they share the correct contextual knowledge (pragmatically), the composability cannot be qualified as correct with respect to given requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6 Composability frameworks Composability essentially relies on a suitable composition framework that can provide accurate reasoning of its correctness and support means to be able to leverage certain component standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Various component standards and their respective frameworks have been developed for M&S to support composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of these frameworks contribute to conceptual modeling by providing the needed formalism and influence the ability to develop and compose model  Chapter 2  Component Based Modeling and Simulation  Page  37  components at conceptual level, while others support model composition at executable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These frameworks practically support composability, as they usually offer features such as model specification, development, and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A brief description of some of the composability frameworks is provided below: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Discrete Event System Specification (DEVS) DEVS [57] is a component based formalism based on dynamic systems theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It was developed for the purpose of describing the structure and behavior of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It supports the concept of hierarchical and modular model construction through coupling of components [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' DEVS is basically a model specification formalism however it incorporates different implementation frameworks such as DEVS-Java, DEVS-C++ and DEVS-Sharp which are used to implement DEVS models into executable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Two types of DEVS models exist, namely, atomic and coupled [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An atomic DEVS is a tuple M = 〈X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' δint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' δext,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' τ〉 where: X = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈ InPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v∈Xp} is the set of input ports and values Y = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈ OutPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v∈Yp} is the set of output ports and values S is the set of states δint : S →S is the internal transition function δext: Q × X→S is the external transition function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' where Q = {(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' e) | s ∈S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 0 ≤ e ≤ τ(s)} is the total state set e is the time elapsed since last transition λ : S →Y is the output function τ : S →R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='∞ +  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ∞ is the time advance function  A DEVS atomic component has inputs X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' outputs Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and a set of S states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At a given moment, a DEVS model is in a state s∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the absence of external events, it remains in that state for a lifetime defined by τ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When τ(s) expires, the model outputs the valueλ(s) through a port y ∈ Y, and it then changes to a new state given by δint(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A transition that occurs due to the consumption of time indicated by τ(s) is called an internal transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' On the other hand, an external transition occurs due to the occurrence of an external event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this case, the external transition function determines the new state, given by δext (s, e, x), where s is the current state, e is the time elapsed since the last transition, and x∈X is the external event that has been received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The time advance function can take any real value between 0 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A state for which τ(s)=0 is called a transient state (which will trigger an instantaneous internal transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In contrast, if τ(s)=∞, then s is said to be a passive state, in which the system will remain perpetually unless an external event is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 2  Component Based Modeling and Simulation  Page  38  A coupled DEVS is a tuple: M = (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {Md | d ∈ D},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EOC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' IC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Select) where: X = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈ InPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v∈Xp} is the set of input ports and values Y = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈ OutPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v∈Yp} is the set of output ports and values D is the set of component names Md is a DEVS model with Xd = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈ InPortsd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v ∈ Xp} Yd = {(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) | p ∈OutPortsd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v ∈ Yp} EIC is the set of input port couplings EIC  ⊆ {((N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ipN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ipd)) | ipN ∈ InPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' d ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ipd ∈ InPortsd} EOC is the set of output port couplings EOC ⊆ {((d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' opd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' opN)) | opN ∈ OutPorts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' d ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' opd ∈ OutPortsd} IC is the set of internal couplings IC ⊆ {((a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' opa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ipb)) | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' b ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' opa ∈ OutPortsa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ipb ∈ InPortsb} Select is the tie-break function  A system modeled using DEVS can be described as a composition of atomic and coupled components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A coupled model comprises a set of input and output ports, a set of component names D, a set of DEVS components Md, input port EIC and output port EOC couplings, and, a set of internal couplings IC connecting internal components with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The tie-break function decides which component to proceed when two or more components have internal transitions scheduled at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 7 describes a DEVS example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this example two atomic component A & B are coupled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Both components have two states Send τ(s)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 and Wait τ(s)=∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Input port: ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='receive and Output port: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='send are defined and connected to each other in coupled DEVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 7: Ping-Pong DEVS [Wikipedia] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 Base Object Model (BOM) framework The SISO 9 standard BOM is defined as, “a piece part of a conceptual model composed of a group of interrelated elements, which can be used as a building block in the development and extension of simulations and simulation environments” [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BOM provides a simulation standard that allows model developers and simulation engineers to create modular conceptual models in form of composable objects,  9 Simulation Interoperability Standards Organization  Ping Pong  AY  B  Send,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  Send,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  Ised  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='receive  Isend  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='send  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='leceive  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='feceive  Wait, inf  Wait, inf  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='receive  IsendChapter 2  Component Based Modeling and Simulation  Page  39  which can be used as the basis for a simulation or simulation environment [59], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The concept of BOM is based on the assumption that components of models, simulations, and federations can be reused as building blocks in the development of a new simulation or a federation [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BOMs are unique because they provide a means to represent aspects of a conceptual model that captures structural and behavioral descriptions of items abstracted from the real system (simuland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then they allow these conceptual models to be mapped to one or more class definitions, which may be used by a software design, variety of programming languages, or distributed simulation architectures such as HLA or TENA10 [61], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BOM standard also offers a general purpose modeling architecture for defining components to be represented within a live, virtual, or constructive (LVC) simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is well suited for characterizing models including the structural and anticipated behavior of interacting systems, individuals, and other entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Primarily BOMs framework poses a satisfactory potential for effective composability of conceptual models at syntactic and semantic levels, resulting in a framework for the assembly of a system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' simulation) or system of systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' distributed simulation environment) [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In spite of these reasonable qualities, BOM framework still falls short of required behavioral semantics and necessary built-in evaluation techniques, which are essential for modeling complex system behavior and reasoning about the correctness of the composability at each of its different level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it becomes a most suitable candidate and a preferred choice of a composition framework (in this thesis) for studying model composability in depth and applying proposed methods on BOM based compositions to explain the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Structure of BOM A BOM is constituted of elements specifying metadata information, conceptual model and the class structure information defined using HLA OMT constructs [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 8 presents different parts of BOM, explained as follows: Model Identification Model Identification associates the metadata information with the BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Its purpose is to document certain key identifying information within the BOM description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It provides a minimum but sufficient degree of descriptive information about a BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='10 Test and Training Enabling Architecture Figure 8: BOM structure (acquired from [59])  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Identification (Metadata)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual Model Definition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Pattern of Interplay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='State Machine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Entity Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Event Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Entity Type Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Event Type Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Object Model Definition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Object Classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Object Classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Object Class Attributes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Interaction Classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Interaction Classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Interaction Class Parameters  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='HLA Data Types  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Notes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Lexicon (definitions)Chapter 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Component Based Modeling and Simulation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual Model Definition From the composability point of view,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' this is the most important part of BOM and therefore the main focus of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To understand this part, the definition of a conceptual model should first be considered: Conceptual Model A Conceptual model is an abstract description or an appropriate simplification of a real (or proposed) system, which is later, refined and implemented in to a more concrete executable model (or simulation model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In these terms, conceptual modeling is a subset of model design which is formed through an iterative process according to the objectives of system modeling [63], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The term conceptual model is used in different ways in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A conceptual model could be a specific diagram like UML class diagram or it could be documentation of a particular aspect of the simuland11 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To better understand the concept of BOMs, consider the home construction analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a new house is to be built the conceptual understanding of features of the building is captured in architectural drawings, which is analogous to a conceptual model (BOM) [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BOM Conceptual Model definition consists of following parts: Pattern of Interplay (POI) POI models a specific purpose or capability and is represented by one or more pattern actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For each pattern action, one or more senders and receivers are specified to provide a means for understanding and the behavioral relationship among conceptual entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  POI is represented by UML sequence diagram [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State Machine The state machine is used to model the behavior of a BOM’s conceptual entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state machine is specified by a set of states where each state may transit to a subsequent state called next state, upon an exit action, which is identified in a pattern of interplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' UML state-machine diagram is used to represent BOM’s state-machine [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Entity Type A conceptual entity is an abstraction of a real world entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It defines a relationship with other entities within a pattern of interplay and acts as a sender or receiver of the events [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Event Type Conceptual events include information about the source, target, and content (parameters) of a message or trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The difference between a trigger and a message is that a trigger is used to broadcast information whereas the messages are directed exchanges of information where the sender knows about the intended receiver of the message [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Entities and Events represent data about the real world objects and their interaction (physical description), whereas the pattern of interplay and state-machine collectively represents the dynamic behavior of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  11 A simuland is the real world system of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is the object, process, or phenomenon to be simulated [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 2  Component Based Modeling and Simulation  Page  41  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 BOM Assembly The BOM concept provides a mechanism for combining BOMs and creating High- Level BOMs, called BOM Assemblies, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A BOM Assembly representing a composition of BOMs, is built in a hierarchical manner and includes information about composed BOMs, which in turn is used to identify a composite interface, and represent a federate, federation within the simulation space12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Typically a developer of a simulation would search a BOM repository for suitable BOM candidates for use in a simulation and combine those into a BOM Assembly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' a simulation model), which is then used to create the actual simulation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A BOM assembly contains Model Identification, and pattern of the interplay among conceptual entities being represented, which is provided through the association of BOMs to the various Pattern Description actions that the BOM Assembly identifies, within the Conceptual Model view [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 9: BOM Assembly BOM models can be created using XML script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' But for constructing BOM models graphically, a free IDE tool called BOM Works [65] is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 10 represents an example developed using BOM Works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is similar to the DEVS example shown in Figure 7, to compare the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 10: (a) PingPong BOM in BOM Works  (b) POI (c) State-machineA (d) State-machineB (e) EntityA (f) EntityB (g) EventA (h) EventB  12 Although use of HLA is not a mandatory subsequent step, it is likely that BOM assemblies are intended to support an HLA based federation [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Assembly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Repository  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Discovery  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM3PingPong  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OModelIdentification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='旦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Sending  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActionA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Waiting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActionA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='eAuthor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='@PatiternsofInterplay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActionA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='百@PingPong  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='@ActionA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActlonB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActionB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='@ActionB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Waiting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ActionB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Sending  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='State Machines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Sending  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Waiting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Entity Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='日 Entity Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Waiting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Sending  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='semantics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='semantics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='idtag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='id2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='idtag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Entity Types  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='id3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='characteristic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='characteristic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Message  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Message  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='白?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Event Types  ID  ID  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='EventA  (e)  (f)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='EventB  Model Mapping  Event Type  Event Type  Entity Type Mappings  name  EventA  name  EventB  EventType Mappings  triggerCondition  triggerCondition  Object Model Definition  semantics  idtag  semantics  @ objects  id1  idtag  tp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Interactions  日 sourceCharacteristic  name  sourceCharacteristic  name  由DataTypes  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ID  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ID  @ Notes  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='targetCharacteristic  name  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' targetCharacteristic  name  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ID  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ID  日contentCharacteristic  name  contentCharacteristic  name  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Message  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Message  (a)  (g)  (h)Chapter 2  Component Based Modeling and Simulation  Page  42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Model Mapping and Object Model Definition The model mapping provides a mechanism for mapping between the elements of the conceptual model and the class structure elements of the Object Model Definition that are described using HLA OMT 13 specification constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The object model definition defines the structure of an object and interaction class, and their associated attributes and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' HLA Object classes include HLA attributes and HLA interaction classes include HLA parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These parts of BOM are not used in this thesis, however interested readers can find more details in [58], [59], [60], [61], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Formal specification for the Compositon of BOM Unlike DEVS, BOM does not have a graphical and mathematical formalism for specifying how components are composed (even though parts of BOM such as state- machine and POI can be represented in UML and BOM documents can be described using XML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This initiates a need for a graphical and formal representation of BOM composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this section, we introduce a formal and graphical specification of BOM14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We define two types of BOM: (i) Basic BOM and (ii) Composed BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A basic BOM is an undividable atomic BOM component, with an assumption that it represents only one conceptual entity at the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A composed BOM is a hierarchical combination of basic and other composed BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Basic BOM We propose that a basic BOM (BB) can formally be defined as: Where: \uf0a7 EnT is an entity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We assume that a basic BOM has only one entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EnT is defined as: EnT =  Name {Characteristic: Type} Where Name is the name of an entity uniquely defined by an identifier15 and characteristic is a set of attributes of an entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each characteristic is uniquely defined by an identifier and has a type16  \uf0a7 EvT is a set of event types,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' each with sender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' receiver and content  Evt = {(Name,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {Content: Type}) | Name ∈ Identifier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sender & Receiver ∈ EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Content∈ Identifier: Type ∈ type}  \uf0a7 S is a set of states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' each has an exit-condition and a next state: S = {(Name,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ExitCondition{Action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' NextState})} | Name ∈ Identifier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Action ∈ Act,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' NextState ∈ S  13 High Level Architecture Object Model Template 14 These concepts are not new and exist in literature for other component-based approaches [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, their application in BOM is intended for facilitating specification and ease of understanding 15 An identifier is a unique sequence of letters & digits, starting with a letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 16 Type := Integer | String | Double | Complex BB = 〈 EnT, EvT, AcT,  S 〉 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3)  Chapter 2  Component Based Modeling and Simulation  Page  43  \uf0a7 AcT is a set of actions, each has name, sender, receiver and an associated event: AcT= {(Name, Sender, Receiver, Event) | Name ∈ Identifier, Sender & Receiver ∈ EnT, Event ∈ EvT}  Composed BOM A composed BOM (CB) can formally be defined as: Where: \uf0a7 AcTIN is a set of input actions that are received from other BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This set can be empty if the Composed BOM is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcTIN = {(Name, Sender, Receiver, BOM) | Name ∈ Identifier, Sender & Receiver ∈ EnT, BOM ∈ File}  \uf0a7 AcTOUT is a set of input actions that are sent to other BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This set can also be empty if the Composed BOM is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcTOUT = {(Name, Sender, Receiver, BOM) | Name ∈ Identifier, Sender & Receiver ∈ EnT, BOM ∈ File}  \uf0a7 POI is the pattern of interplay that defines how basic or composed BOMs are connected to each other (through actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It maps a list of send actions to a list of receive actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ‘ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ’ symbol means send and ‘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ’ symbol means receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' POI = {({!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='AcTSEND} , {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='AcTRECV})} | AcTSEND & AcTRECV ∈AcT  Example As an example, BOMs from Figure 10 can formally be represented as: BB0 = 〈 EnT, EvT, AcT, S 〉 where: EnT = EntityA {C0(Message:String)} EvT = {E0(EventA, BB0, BB1, BB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C0), { E1(EventB, BB1, BB0, BB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C0)} Act = { A0(ActionA, BB0, BB1, E0), A1(ActionB, BB1, BB0, E1)} S = { S0(Sending, A0, S1), S1(Waiting, A1, S0)}  Table 1: Entity A  CB = 〈 AcTIN,  AcTOUT ,  POI 〉 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3)  Chapter 2  Component Based Modeling and Simulation  Page  44  BB1 = 〈 EnT, EvT, S, AcT 〉 where: EnT = EntityB {C1(Message:String)} EvT = {E2(EventA, BB0, BB1, BB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C0), { E3(EventB, BB1, BB0, BB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C1)} Act = { A2(ActionA, BB0, BB1, E2), A3(ActionB, BB1, BB0, E3)} S = { S2(Waiting, A2, S3), S3(Sending, A3, S2)}  Table 2: Entity B  Similarly a composed BOM CB0 can be formally described as: CB0 = 〈 AcTIN,  AcTOUT ,  POI 〉where: AcTIN = ∅ (since there is no incoming actions from any other BOM) AcTOUT = ∅ POI = {I/O0(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2), I/O1(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1)}  Table 3: Composed BOM  We propose a graphical notation for representing basic BOM and their composition shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this figure two basic BOM EntityA and EntityB are composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 11: Composed BOM The general information of a component such as entity name, characteristics, actions and states are defined in the main block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the lower block the states and their transitions (with blue arrow) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each transition is mapped with actions (in red arrow) with parameter labels (the IDs of characteristics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The direction of the arrow shows the type of the associated action (send or receive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The composition of BOMs is shown through connectors (in green color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A  Action Connector  EntityA  EntityB  S  State Connector  Characteristics:  Characteristics:  C0 Message1 String  C1 Message2 : String  Initial State  Actions:  Actions:  Exit condition  A ActionA  A2 ActionA  A1 ActionB  A3 ActionB  State Transistion  States:  States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  SO=Sending  S2= Waiting  Input/Output  S1 Waiting  S3 Sending  connection  A0  1  Waiting  Sending  Sending  WaitingChapter 2  Component Based Modeling and Simulation  Page  45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Summary In this thesis, we harness the capability of BOM as a conceptual modeling framework, because it provides a component standard using an XML specification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' gives guidelines for the further development of the executable model and helps determine the appropriateness of the model or its parts for model reuse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and most importantly due to its strong support for syntactic and semantic composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It will be shown, how BOM with its existing potential can be facilitated by composability evaluation for accurate and rapid construction and modification of its corresponding federates in HLA based simulations and hence brings forth an improvement in the distributed simulation community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page  46  Chapter 3  Executable Modeling Formalisms  In this chapter two popular model description formalisms are discussed namely Petri Nets and Communicating Sequential Processes (CSP)17, which are normally used for modeling, execution (or simulation) and verification of concurrent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This chapter provides an introduction, theory, properties, classification, modeling methods and analysis techniques of PN and CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN and CSP are both considered as a part of solution domain in this thesis, because of their impressive accumulation of knowledge in concurrency modeling and analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These aspects are imported in this thesis and used for composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  PN and CSP formalisms are relatives since they are used to model same class of systems called concurrent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Unlike other systems such as transitions systems or automata, the formalisms of concurrent systems are strongly based on concurrency theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One of the major contributors of concurrency theory are: Carl Adam Petri who initiated concept of interacting sequential processes and introduced Petri Nets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hoare who focused on developing programming language (CSP) for concurrent systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and Robin Milner who introduced Calculus of Communicating System (CCS) and  π-Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These are variants of approaches for formally modeling concurrent systems and are the member of the family of mathematical theories of concurrency known as process algebras, or process calculi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP is also a member of process algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main difference between PN and CSP is that the former are based on graphs, while the latter are based on a textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However both offer strong formal semantics for modeling executable systems and share a broad pool of knowledge of theoretical principles and practical techniques for the analysis and verification of models of complex behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, we propose using these two formalisms to model executable form of components and study their composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Petri Nets PN were introduced by Carl Adam Petri (and named after him) in 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They provide an elegant and useful graphical and mathematical formalism [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  With PN the main idea is to represent states of subsystems separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this way, the distributed activities of a system can be represented very effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN are widely used for modeling and control in a variety of the sorts of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Particularly, in Discrete Event Dynamic Systems (DEDS) 18 in which many properties such as synchronization, sequentiality (producer-consumer problem), concurrency and  17 The "Sequential" word of the CSP name is now something of a misnomer, since modern CSP allows component processes to be defined both as sequential processes, and parallel [Wikipedia].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 18 Examples of DEDS are air traffic control systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' automated manufacturing systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' computer and communication networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' embedded and networked systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and software systems etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The activity in these systems is governed by operational rules designed by humans and their dynamics is often driven by asynchronous occurrences of discrete events [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  47  conflict (mutual exclusion) concurrency, and choices can be well presented and analyzed using PN [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Their structural and behavioral properties have been successfully exploited for solving various problems of complex and dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Significant progress in these directions was made over three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Most essential features of PN are the principles of locality, concurrency, graphical and algebraic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They can be used not only for the specification and analysis of the structural system design but also for design of the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [66], [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN present two interesting characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Firstly, they make it possible to model and visualize systems with complex behaviors including parallelism, concurrency, synchronization and resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Secondly the properties of these nets, their analysis and theorems have been extensively studied [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 PN Definitions and Concept In PN, two basic elements of modeling are places and transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Events are associated with transitions which occur when some conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Information related to these conditions is contained in places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are two types of places namely: Input places and Output places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Input places are associated with the conditions required for this transition to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Output places are associated with conditions that are affected by the occurrence of this transition [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Transitions, places, and certain relationships between them define the basic components of a Petri net graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A PN graph has two types of nodes, places and transitions, and arcs connecting these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is a bipartite graph in the sense that arcs cannot directly connect nodes of the same type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' rather, arcs connect place nodes to transition nodes and transition nodes to place nodes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Petri net graph Mathematically a PN is a 5 tuple: PN = 〈P, T, F, W, M0〉 where: \uf0a7 P is a finite set of places P = {p1, p2… pm} represented as oval shaped node in the PN graph \uf0a7 T is a finite set of transitions T = {t1, t2… tn} represented as a line or a rectangular shaped node in the graph \uf0a7 F is a flow function such that F ⊆ (P ×T)∪(T×P) →N 19 \uf0a7 W: F →N + where N∈{1, 2, 3…} is arc weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 M0: P→N is a function called the initial marking, where each element M0(p) has N number of tokens20 initially in place p where N is a set of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 For each transition t∈T a set of input places denoted as •t are those places which are connected to t through incoming arcs: \uf0a7 Similarly, for each transition t∈T a set of output places denoted as t• are those places to which t is connected through outgoing arcs:  19Such that P∩T= ∅ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' P&T are disjunctive sets) and P ∪ T≠∅ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' neither P nor T are isolated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also an arc can be connected from place to transition (input arc) or from transition to place (output arc) but not to the node of same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 20 In classical PN, tokens are represented as black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are assigned to, and can be thought to reside in, the places of a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' •t = {pi | (pi, t) ∈F} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1) t• = {pi | (t, pi)∈F} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2)  Chapter 3  Executable Modeling Formalisms  Page  48  Definition: Marking A marking is an assignment of tokens to the places of a PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The number and position of tokens defines a system state, and it may change when the tokens move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This movement of tokens due to the firing of transitions causes the execution of a PN [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The marking M can be defined as an n-vector, M = (m0, m1, m2 … mn), where n = |P| (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' of places), and each mn ∈ N, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The vector M gives for each place pi in a PN the number of tokens in that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Definition: PN State-space The state of a PN model is defined by its marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The firing of a transition represents a change in the marking of the net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state space of a PN with n places is the set of all markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State-space will be discussed in detail later in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Definition: Enabling of a Transition A transition t in a given PN is called enabled or fire-able by a marking Mi  iff for each input place p∈•t its marking is equal or greater than the weight of the arc from it to t, (or t has no input place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mathematically, a transition t is fire-able iff Definition: Firing of a Transition If a transition t is enabled, it may fire by removing W(p, t) number of tokens from each input place p and putting W(t, p’) tokens in each output place p’, due to which a new marking Mn+1 is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mn+1 is immediately reachable from Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mn is reachable from M0 if firing a sequence σ = t1, t2 … tk of enabled transitions leads M0 to Mn, written as M0 σ→ Mn  Example21  Consider the PN model PN = 〈P, T, F, W, M0〉 as shown in Figure 12 where: P = {p1, p2, p3, p4, p5} and T = {t1, t2, t3, t4}, Let W = 1 for all arcs Initial marking M0 = [1 0 0 0 0]  21 This example is inspired from [68] ∀p ∈ •t | M(p)≥W(p, t) ∨ •t=∅ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) Mn+1 𝑡→ Mn | M(p’) = M(p) - W(p, t) +  W(t, p’) ∀p∈P (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4)  Chapter 3  Executable Modeling Formalisms  Page  49  Figure 12: Transition firing sequence  (acquired from [68])  σ1: M0 = [1 0 0 0 0] 𝑇1 ��  M1 = [0 1 0 1 0] 𝑇2 ��   M2  [0 0 1 1 0] 𝑇3 ��    M4  [0 0 1 0 1] 𝑇4 �� M4  [1 0 0 0 0]  σ2: M0 = [1 0 0 0 0] 𝑇1 ��  M1 = [0 1 0 1 0] 𝑇3 ��   M3  [0 1 0 0 0] 𝑇2 ��   M4  [0 0 1 0 1] 𝑇4 �� M4  [1 0 0 0 0]  In this example there are two possible transition firing sequences σ1= T1, T2, T3, T4 and σ2 = T1, T3, T2, T4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Properties of PN Just like other models, PN are constructed from informal requirement specifications, which is not a trivial task, and requires a great deal of modeling experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a system being modeled is very complex, a PN model may differ considerably from its original specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A model can only be useful if it is logically correct with respect to its specifications [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Different concepts of correctness exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A system is said to be correct when two aspects, namely the specification and the implementation, are equivalent, or when the system satisfies a set of desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These desirable properties allow the system designer to identify the correctness of the system [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In PN literature a “basic kit of PN properties” is referred to a set of properties that are related to frequently occurring problems or the key issues related to the logical structure and behavior of complex systems, therefore they are classified into two main categories namely (i) Structural Properties and (ii) Behavioral Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is important to note that fulfillment of these properties answer many questions of  m1m2mChapter 3  Executable Modeling Formalisms  Page  50  system correctness, therefore they contribute in the analysis of PN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of the selected behavioral PN properties are listed and briefly discussed informally22 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reachability Reachability is a fundamental property for studying the dynamic behavior of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In PN, reachability property is studied to analyze if a particular system state (in terms of markings) can be reached or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A marking Mn is said to be reachable from an initial marking M0 if there exists a sequence of firings that transforms M0 to Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In reachability analysis, a set of all possible firing sequences from M0 are populated in a reachability graph R(N, M0) and the reachability problem for PN is the problem of finding if a given marking Mn ∈ R(N, M0) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Boundedness In classical systems theory, a state variable that is allowed to grow to infinity is generally an indicator of instability in the system [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it is desirable that a system holds boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A PN is said to be bounded (or k-bounded) if the number of tokens in each place does not exceed a finite number k for any marking reachable from initial marking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', M(p) ≤ k for every place p and every marking Mn ∈ R(N, M0) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Deadlock-free and Liveness A PN is said to be deadlock-free if from any reachable marking at least one transition can always occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A stronger condition than deadlock-freeness is liveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A transition is live if it is potentially fire-able in all reachable markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In other words, a transition is live if it never loses the possibility of firing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A net is live if all transitions are live [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reversibility A PN is said to be reversible if, from each marking Mn, the initial marking M0 is reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus, in a reversible net one can always get back to the initial marking or state [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Fairness Fairness has different meanings and understanding in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In specific terms, fairness means to give some contenders an equal number of chances, such that no one proceeds for more than “k-times” without letting the others to take their turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In PN s, two transitions tl and t2 are said to be in a bounded-fair (or B-fair) relation if the maximum number of times that either one can fire while the other is not firing is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A PN is said to be a B-fair net if every pair of transitions in the net are in a B-fair relation [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mutual Exclusion This property captures constraints such as the impossibility of a simultaneous access of a critical section (resource) by two or more processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In PN, mutual exclusion can be defined in terms of places or transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Two places p and q are mutually  22 In literature these properties are discussed in detail with mathematical definitions and proofs [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter they are only discussed for background concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of these properties are used later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  51  exclusive in a PN if their token counts cannot be both positive in the same marking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', ∀m ∈ RS m(p)·m(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly, two transitions in a PN are mutually exclusive if they cannot be both enabled in any marking [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Some of the important structural properties of PN are defined below: Controllability: A PN is said to be completely controllable if any marking is reachable from any other marking [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conservativeness: A PN N is said to be (partially) conservative if there exists a positive integer y(p) for every place p such that the weighted sum of tokens, is a constant, for every marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Given a PN model, we are often required to ensure conservation with respect to certain weights representing the fact that resources are not lost or gained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Persistence A PN is said to be persistent if, for any two enabled transitions, the firing of one transition will not disable the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A transition in a persistent net, once it is enabled, will stay enabled until it fires [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 PN Analysis The major strength of PN is the modeling of systems that exhibit concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However modeling by itself is of little use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is necessary to be able to analyze the modeled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The analysis leads to important insights into the structure and behavior of the modeled system [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are many techniques available for the analysis of PN models and can be employed for verification depending upon the nature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each technique may also have different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this section two of the most commonly used techniques for the analysis of a PN model are discussed:  Figure 13: Petri Net Analysis Techniques  These techniques provide solutions and mechanism for verifying the properties mentioned in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, these techniques are selected for composability verification and their application is shown in Part II with suitable examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter, they are briefly explained and discussed, with their advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Petri Net Analysis Techniques Algebraic Method State-Space Analysis  Chapter 3  Executable Modeling Formalisms  Page  52  Algebraic Method This technique is also called Linear-Algebraic Technique (or Linear Invariant due to its abundant use of invariants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the framework of using algebraic techniques for reasoning about PN, solving a PN problem is reduced to finding a solution for an algebraic equation associated with the PN [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Due to the nature of this technique, the method is in general efficient (and in most cases, polynomial in the size of the PN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The dynamic behavior of PN models can be described by algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to work with Algebraic method, the following basic concepts are applied: Matrix Definitional Form (MDF) A PN model has a Matrix Definitional Form (MDF) that consists of three n×m2F23 matrices: (i) Output matrix A+  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', if pj is connected to the output of ti then 𝒂𝒊𝒋 + is equal to the weight of output arc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 0 otherwise [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  (ii) Input matrix A- i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', if pj is connected to the input of ti then 𝒂𝒊𝒋 − is equal to the weight of output arc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 0 otherwise [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  (iii) Incidence matrix A In the incidence matrix A, each entry aij represents the change of tokens in place j when transition i fires once [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Firing Count Vector A marking Mk is an m × 1 column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The jth entry of Mk denotes the number of tokens in place j after the kth firing in some firing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An n×1 column vector X of nonnegative integers is called firing count vector, where the ith entry of X denotes the number of times transition t must be fired to transform Mk-1 to Mk [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State Equation State equation for a PN is written as: Where: Mk-1 is the current marking  23 n×m refers n transitions and m places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A+ = [𝒂𝒊𝒋 +] n×m, where 𝒂𝒊𝒋 + = w(ti, pj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' if pj ∈ ti•, and i ∈ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' j ∈ m (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5) A- = [𝒂𝒊𝒋 −] n×m, where 𝒂𝒊𝒋 − = w( pj , ti);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' if pj ∈•ti (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6) A =  A+ - A- , where [𝒂𝒊𝒋] = [𝒂𝒊𝒋 + − 𝒂𝒊𝒋 −] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7) Mk =  Mk-1 + A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7)  Chapter 3  Executable Modeling Formalisms  Page  53  Mk is the new marking A is incidence matrix X is the firing count vector Example An example of a Producer-Consumer PN model is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 14: Producer Consumer Example  Using equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 the incidence matrix A of this model is calculated as follows:  A+ T1  T2  T3  T4  P1  1  0  0  0  P2  0  1  0  0  P3  0  1  0  0  P4  0  0  0  1  P5  0  0  1  0 A T1  T2  T3  T4  P1 0  1  0  0  P2 1  0  0  0  P3 0  0  1  0  P4 0  0  1  0  P5 0  0  0  1  =  A  T1  T2  T3  T4  P1  1 1  0  0  P2 1  1  0  0  P3  0  1 1  0  P4  0  0 1  1  P5  0  0  1 1  Table 4: Incidence Martic A  In this model, the initial marking is [1 0 0 1 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With a firing sequence σ = t2, t1, t2 the firing count vector will be [1 2 0 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using the state equation, the marking Mx can be generated as follows:  M0  P1 1  P2 0  P3 0  P4 1  P5 0  +  A  T1  T2  T3  T4  P1  1 1  0  0  P2 1  1  0  0  P3  0  1 1  0  P4  0  0 1  1  P5  0  0  1 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  X  T1 1  T2 2  T3 0  T4 0  =  Mx  P1 0  P2 1  P3 2  P4 1  P5 0  Table 5: State equation  Figure 15 graphically illustrates, how a firing sequence of σ = t2, t1, t2 can lead M0 to M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Green color highlights the firing of a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be noted that the marking M3 in the lower right corner matches the marking generated by matrix state-equation in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  P4  P3  T  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  T4  P2  P5Chapter 3  Executable Modeling Formalisms  Page  54  Figure 15: M0 to M3 throguh firing sequece σ = t2, t1, t2 State equation alone can only help to algebraically compute a future marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to analyze the model algebraically, some more concepts are used, such as PN Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  PN Invariants Occurrences of transitions transform the token distribution of a net, but they often respect some global properties of markings, regarded as Linear Invariant Laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Invariants are very useful for analyzing structural and behavioral properties of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' From an initial marking, the marking of a PN can evolve by the firing of transitions (and if there is no deadlock) the number of firings is unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However, not just any marking can be reached, all the reachable markings have some properties in common;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' a property which does not vary when the transitions are fired is said to be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly, not just any transition sequence can be fired;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' some invariant properties are common to the possible firing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hence, invariants enable certain properties of the reachable markings and firable transitions to be characterized, irrespective of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 16 illustrates a PN model of different seasons in a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be seen that, regardless of the change of seasons, there will always be one and only one token for all 4 places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus at all times, M(p1) + M(p2)  + M(p3)  + M(p4)  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This invariant property has an obvious meaning that at all time there is one and only one season [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also means that the net is structurally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 16: Seasons in a year (acquired from [68])  Spring  T1  Summer  P  J  T4  T3  Winter  AutumnTChapter 3  Executable Modeling Formalisms  Page  55  There are two important types of invariants of PN: P-Invariant Place Invariants formalize invariant properties regarding places in PN, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', if in a set of places the sum of tokens remains unchanged independently of any firing, then this set can define a place invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are useful to evaluate structural properties of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In simple words, a place belonging to a P-invariant is bounded [24], [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A P-invariant exists in a PN if Where y is an m × 1 column vector of integers such that ∃ y = (y1, y2 … yn) > 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', has at least one positive non-zero entry [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It means the firing of any transition does not change the weighted sum of tokens in the PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' More generally, a vector y is called P-Invariant if A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' y =  0 It is easy to see that if there is a P-invariant, for all p ∈ P, then the PN is guaranteed to be structurally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hence, place invariants can be used for reasoning about structural boundedness [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' P-invariant is a P-semi-flow if every element of it is non-negative [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' T-Invariants Transition Invariants on the other hand formalize properties regarding transition firing sequences applicable to a PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are useful to evaluate behavioral properties such as liveliness and fairness [24], [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A n × 1 firing count vector X, is called a T-Invariant if A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' X =  0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', firing each transition the number of times specified in X, brings the PN back to its initial marking M0 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' T-invariant is a T-semi-flow if every element of J is non- negative [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A T-Invariant X is a minimal T-invariant, if there is no other T-invariant X′ such that x′i ≤ xi for all i∈T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There can be multiple T-invariants for a PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A minimal T- Invariant is called the Reproduction vector of the net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The intrinsic difference between P- and T-invariants are the facts that all places in a PN if covered by P-invariants is a sufficient condition for boundedness, whereas the existence of T- invariants is only a necessary condition for a PN model to be able to return to a starting state, because there is no guarantee that a transition sequence with transition count vector equal to the T- invariants can actually be fired [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Advantages and Disadvantages The advantage of algebraic analysis is that the net structure is much less than the number of reachable markings and therefore there is no risk of state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Various properties of PN consequently can be proven using linear algebraic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However the weakness of this method is that it only entertains limited set of properties and provides only sufficient or necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also this method � 𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 𝑦𝑝 = � 𝑚0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 𝑦𝑝 𝑛 𝑝=1  𝑛  𝑝=1  ∀m ∈ R(N, m0)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8)  Chapter 3  Executable Modeling Formalisms  Page  56  involves complex underlying mathematical theorems, each one different for different property verification and thus cannot be generalized for automated reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State-Space Analysis State space analysis is one of the most prominent approaches for conducting formal analysis and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In contrast to algebraic techniques, it is relatively simpler approach for analyzing the behavior of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The basic idea in this approach is to calculate all possible system states and the events which cause the change of states and represent them in a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the graph is completely constructed, different search techniques can be applied to analyze the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In PN terms, this method is also commonly known as Reachability graph analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-space analysis of a PN model is performed by exhaustively generating all the reachable markings from a given initial marking, and then reasoning about the PN properties of the model by examining the structure of the reachability graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The reachability graph consists of vertices which correspond to reachable markings and of arcs corresponding to firing of transitions resulting in the passing from one marking to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A simple example of reachability graph is shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 17: (a) PN Model (b) Reachability Graph (acquired from [68])  In some cases, the construction of reachability graphs becomes infinite if the PN or some of its parts are repetitive and the net is unbounded, or in other words the PN has infinite number of reachable markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore instead of keep on constructing nodes of the graph infinitely, an alternative technique is used, in which a finite graph is constructed by abstracting out certain details and inserting the symbol ω (the symbol of “infinity”) to representing the marking of an unbounded place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is called cover-ability graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The coverability graph of the Producer-Consumer PN model is shown in Figure 18  Figure 18: Producer Consumer PN Model and its Coverability Graph  It can be seen that the markings in which place P3 is unbounded contain ω symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  P  0  2  []}[]  m2  mo  m1  0  1  m3  (a)  (b)P4  P3  T  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  T4  P2  P5(1,0,0,1,0)  t1  (0,1,0,1,0)  t3  t4  ti  (0,1,1,1,0)  (0,1,0,0,1)  (1,0,0,0,1)  t1  t3  t2  (1,0,0,1,0)  (1,0,0,0,1)  (0,1,0,0,1)  t4  t,  t2  ti  (0,1,0,1,0)Chapter 3  Executable Modeling Formalisms  Page  57  A constructed state space can help in answering a large set of analytical questions concerning the structure and behavior of the model such as verifying deadlock- freedom, absence of live-locks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' presence of liveness, the possibility of being able to reach good states, and impossibility of reaching bad states and the guarantee of fulfilling the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Following are some examples of how state space analysis help in model verification: Boundedness The problem of boundedness is easily solved using a coverability tree with an assumption that a PN is bounded if the symbol ω never appears in its coverability tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since ω represents an infinite number of tokens in some place, therefore its absence can guarantee that the PN is structurally bounded [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Deadlock freedom A deadlock freedom problem is solved, if there is no node in the graph (which is not a final node), and yet it does not have an outgoing arc meaning there is no further enabled transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Existence or one or more such nodes shows that the model has possibility of deadlock and can also help to find out the exact cause of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Live lock freedom Similarly, a live-lock can be detected using state space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For concurrent systems, a process is tasked to perform some particular actions [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These actions are normally intended to make progress and are called progress actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A live lock is detected, if there exists a cycle within the reachability graph, in which no progress action is being executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  State Reachability Reachability of good states (or bad states) can be guaranteed using state space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A state is reachable if there is a valid firing sequence that leads to that state from the initial marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (In graph, there exists a path from the initial node to the corresponding node of the desired state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There could be multiple paths in a graph that reach the desired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A shortest path analysis can be useful to analyze the minimum number of steps required to reach that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For details on how state space analysis are conduced, interested readers are recommended to refer to a very informative step by step tutorial on PN state space analysis [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Advantages and Disadvantages The main advantage of state space method is that it is a way to explore all the possible states of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also it provides counter examples as to why an expected property does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Furthermore, the automatic calculation and generation of state-space provides an ease of use, due to the fact that the computer tool hides a large portion of the underlying complex mathematics from the user, who is only required to formulate the property which is to be investigated and a suitable query function to evaluate it [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  58  The main disadvantage of using state spaces is the state explosion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The construction of the reachability graph is very expensive and intensive from a computational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is because the size of the state space may grow exponentially with respect to the size of the PN model (measured, for example, by the number of places).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Even relatively small systems may have an astronomical or infinite number of reachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This problem escalates severely, when the models includes time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A lot of effort has been invested in the development of reduction methods to alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reduction methods represent the state space in a compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The reduction should not affect the properties of the system and they should be preserved and can still be derived from the reduced state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However, due to the complexity and diversity in verification, there is no single reduction method which works well in all situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the choice of a reduction method completely depends on the nature of the system being verified [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of the important reduction methods are Sweep line method [75], Hash Compaction Method [76], Symmetry Method [77] and Equivalence Method [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, we propose another reduction method which suits our need (Composability verification) and can help to alleviate the state explosion problem, if the model under consideration becomes large and resource intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 PN Classes The computational power of basic or classical PN is weak as it has been shown that PN are not as expressive as Turing machines, making them inadequate for modeling certain real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To overcome this shortcoming, a number of extended PN have been introduced to enhance the expressive capabilities of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are different ways to classify PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=" In structural sense, they can be classified into three main categories [79]:  Level-1 PN: are characterized by 'Boolean tokens', i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' places are marked with at most one unstructured token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="  Level-2 PN:  are characterized by 'Integer tokens', i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' places are marked with several unstructured tokens - they represent counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Level-3 PN:  are characterized by high-level tokens, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' places are marked with structured tokens where information is attached to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  There are many extensions of PN formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this section we only discuss some of the extensions of PN, which are used in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Colored Petri Nets (CPN) CPN is a level-3 extension of PN, in which places are marked with structures token representing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN is a graphical language for constructing models of concurrent systems and analyzing their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN is a general purpose discrete event language which combines the capabilities of PN, as a foundation of the graphical notation and a programming language (CPN ML), which is based on Standard ML [80] functional programming language, that provides the primitives for the definition of data types and for specifying data manipulation routines [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  59  CPN is formally defined by the tuple [81]: CPN = (P, T, A, Σ, V, C, G, E, I) where: P is a finite set of places T is a finite set of transitions such that:  P ∩ T = ∅ A ⊆ P×T ∪T ×P is a set of directed arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Σ is a finite set of non-empty color sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' V is a finite set of typed variables such that: Type[v] ∈ Σ for all variables v ∈ V C: P→Σ is a color set function that assigns a color set to each place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' G: T → Expression is a guard function that assigns a guard to each transition t E: A→ Expression is an arc expression function that assigns an arc expression to each arc a I: P → Expression is an initialization function that assigns an initialization expression to each place p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tokens of an ordinary PN have no types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With CPN it is possible to define token using data types and complex data manipulation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', each token has attached a data value called the token color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The token colors can be investigated and modified by the occurring transitions [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' “CPN Tools” is a software package for the editing, simulation, state space analysis, and performance analysis of CPN models [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The tool acts as an integrated development environment (IDE) for the construction of CPN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It provides a canvas for creating PN graphs, offers features for writing CPN ML code with a facility of incremental syntax checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also comes along with a bundled simulator that efficiently handles the execution of untimed and timed nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The most important feature of CPN tool from our point of view is the generation and analysis of state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The analysis of state space includes various built-in state-space querying functions, and support for creating analysis report which altogether greatly contributes to the verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For further details of CPN formalism and its application [78], [81] are referred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 19: A CPN Model  Figure 19 shows a basic example of a CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The nodes A and B in oval shape represent places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The place is initialized with three tokens of String type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The rectangular shaped node represents transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An input arc connects Place A with the transition with an arc variable v of type String (to carry tokens of the same type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1\'"Token1"++  1\'"Token2"++  1\'"Token3"  [v="Token2"]  V  Trans  A  B  STRING  STRING1\'"Token1"++  "Token3"  [v="Token2"]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1\'"Token2"++  V  Trans  V  A  B  STRING  STRINGChapter 3  Executable Modeling Formalisms  Page  60  Similarly an output arc connects transition to place B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The transition has a guard expression that checks the token value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the expression is true only then the transition can be fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The second part of the Figure 19 shows the result of the firing of transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the token “Token2” being deposited to place B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hierarchical CPN CPN model can be organized as a set of modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' where modules can be seen as black boxes which make it possible to work at different abstraction levels, concentrating on one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Substitute Transitions CPN tools offer facility to construct hierarchical CPN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In hierarchical nets a transition can represent an entire piece of net structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Such a transition is called substitution transition [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sub-page /Super-page A page that contains a substitution transition is called a super-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a CP-net uses a substitution transition the logic that the transition represents is kept on a page called a subpage [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Ports and sockets Super-pages and sub-pages are connected by ports and sockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A socket is a place in the super-page that has at least one arc between a substitution transition and a socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A port on the other hand is a place in a subpage, marked with one of the port-type tags: (i) In-Port (ii) Out-Port or (iii) In/Out-Port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is bound with a socket in the main page using Port & socket assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This relationship is used to define how a subpage should be connected with the surroundings of its super-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Some of the assignment rules are as follows: • A port with an In-tag must be assigned to a socket which is an input arc of the substitution transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • An Out-tag indicates that the port must be related to a socket which is an output arc, • I/O-tag indicates that the socket must be both an input and output arc [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 20: Hierarchical Colored Petri Net  Figure 20 presents an example of hierarchical CP-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the super-page (above), a substitute transition Process is shown which represents a sub-module (below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A  1""Token1"++  1\'"Token2"++  1""Token3"  Process  B  STRING  Process  STRING  Stage1  Stage2  Stage3  In  Out  STRING  STRING  Q  R  STRING  STRINGChapter 3  Executable Modeling Formalisms  Page  61  process has three stages, and input and an output marked with In and Out ports which are connected with A and B socket places in the super-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Timed Petri Nets PN with timing dependencies can be classified according to the way of specifying timing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These constraints can be timing intervals or single numbers, or elements of the net these constraints are associated with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', places, transitions or arcs [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The next criterion is an interpretation of the timing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When associated with a transition, the constraint can be viewed as  (i) Firing time A transition consumes the input tokens when it becomes enabled, but does not create the output tokens until the delay time associated with it has elapsed [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  (ii) Holding time When the transition fires, the actions of removing and creating tokens are performed instantaneously, but the tokens created are not available to enable new transitions until they have been in their output places for the time specified as the duration time of the transition which created them [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  (iii) Enabling time A transition is forced to be enabled for a specified period of time before it can fire, and tokens are removed and created in the same interval [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Timed extensions are known also for high-level PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One of them is timed Colored Petri nets [78], in which the time concept is based on introducing a global clock used to represent the model time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tokens are equipped with time stamps, which describe the earliest model times at which they can be used to fire a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Stamps are modified according to expressions associated either with transitions, or with their output arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Timing intervals can be interpreted as periods of non-activity of tokens, and the transitions are fired according to the strong earliest firing rule [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Formally a time PN is a tuple:  N = (P, T, F,m0,Eft, Lft)  Where: (P, T, F, m0) is a PN, Eft = Earliest firing time for each t∈T Lft = Latest firing time for each t∈T  Chapter 3  Executable Modeling Formalisms  Page  62  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Communicating Sequential Processes CSP is the second formalism that is selected in this thesis for the modeling of executable components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP is a language developed by Sir Charles Antony Richard Hoare [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It aimed to be used for specification and reason about the concurrent interaction of the system processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The idea of CSP was conceived for the study of concurrent processes using formal notation with required expressive power and algebraic laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The formal notation and the associated algebraic laws allow the process models to be controlled and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They also enable formal reasoning about their correctness and prove equivalences between the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They also provide sufficient theoretical foundations for the development of the necessary tools for these purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Basic Concepts and Definitions The main primitives of CSP formalism are (i) Processes (ii) Events and (iii) Algebraic Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Process In CSP terms, a process is an independent, self-contained, modular description of an entity and a basic unit to capture behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A process has particular interface, captured by events that are used to interact with the environment which itself is a process, called the universe of the system (Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The environment can be viewed as a system of concurrently evolving processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In any run a process performs a sequence of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A process has a name, list of parameters and expression which determines its computational logic: Process (parameters) = Expression Expression is behavior of a process which can be described as an occurrence of an event or the sequence of some events, known as a trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A process can only perform a finite number of events in any finite time, and thus all traces have finite length [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Events The ultimate unit in the behavior of a process is an event [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Events characterize communications or interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Events are abstraction of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each event forms an interaction between the process and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the interaction does not occur then the process is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Event can be defined with no data or data with typed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A set of all events of a Process P are called Alphabet of P (αP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following line describes a simple vending machine which takes in a coin and dispatches a coffee every time [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' VM() = insert-coin → coffee → VM();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Where VM() is a process (with no parameters) and its expression contains a sequence of atomic events: insert-coin and coffee and then the process is self-referenced (recursion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Events can be written in compound form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', with parameters as shown in the following line: VM() = insert-coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='10 → coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 → VM();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also there could be data operations using statement blocks inside the event body: VM() = insert-coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='10{Balance= Balance +10} → coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1{coffee--} → VM();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  63  A statement block could be a complete sequential program contains assignment statements, if-then-else clauses, for or while loops and math functions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Input/output Channels Processes may also communicate through channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Channels are special type of events, called communication events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Usually a communication on a channel results from an input and output occurring in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The input channel is represented by ‘?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' symbol whereas the output channel is represented by ‘!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The channel parameters can be send or received using the form: c !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' x or c ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' y Algebraic operators There are many different useful operators that are used to represent different notions of process behavior and their compositions [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some are described as follows: \uf0a7 Prefix a → P The prefix operator combines an event and a process to produce a new process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Sequential composition  P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Q It composes two processes P and Q in a sequential order i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the latter only starts when the former terminates  \uf0a7 Deterministic Choice P � Q The deterministic (or external) choice operator allows choosing between two component processes, and allows the environment to resolve the choice externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Non-deterministic Choice P ⊓ Q The nondeterministic (or internal) choice operator allows a choice between two component processes, but does not allow the environment any control over which one of the component processes will be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Conditional Choice if cond  P  else  Q The choice depends on the evaluation of a condition to choose between P or Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Interleaving P |||  Q The interleaving operator represents completely independent concurrent activity between the processes P and Q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', without barrier synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Parallel Composition P || Q The parallel composition operator represents concurrent activity between P and Q that requires barrier synchronization between the component processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If an event is in the alphabet of both P and Q, then it can only occur when both processes are able to engage in that event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  64  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 CSP Analysis Techniques Many techniques have been developed for the analysis of CSP models however Model Checking has surpassed them all in many aspects and is commonly favored by most of the CSP based modeling environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this section Model Checking technique is briefly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model Checking “Model checking is an automated technique that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' given a finite-state model of a system and a formal property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' systematically checks whether this property holds for that model [86]” The instigation and rapid advancements of model checking methods is one of the towering achievements in the area of model based software verification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' especially with the advent of difficulties faced by the computing communities when the struggle of sequential program verification was followed by even more daunting exertion of verifying concurrent programs [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The growing difficulty in error tracing of such programs is due to the increase of complexity of the system behavior and the arbitrariness of large portion caused by emergent system states which cannot be easily tacked by ordinary testing and debugging methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Starting from late 70’s Model checking and other similar algorithmic and automata theoretic approaches are the result of efforts of notable researchers who pioneered different standards that can be marked as a collective foundation of principles that shaped the modern model checking techniques [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Model checking became successful in different communities due to following reasons: \uf0a7 Unlike traditional testing methods it is an exhaustive approach that provides an in-depth analysis of a system model to certify absence of bugs (instead of just finding few of them through debugging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Model checking returns answers — either successful outcomes or counterexamples showing the exact trace of errors and their causes \uf0a7 Improvements in model checking techniques have effectively alleviated the risk of state-space explosion problem [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Model Checking has a sound and mathematical underpinning and is based on theory of graph algorithms, data-structures, and logic [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Model checking support formalism both for the specification of the input models (such as FSM, PN, CSP or others) and the specification of system properties being verified (which are mostly in the form of LTL or CTL or their extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore any 3rd party community can use a model checker as a black box without knowing the insights and complexity of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Beside its various strengths some of the weaknesses include: \uf0a7 Most model checkers require the models to have reduced details using compact and less expressive states and without specifying enumerations due to the risk of state-explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the reduction in the system expressiveness may cost extra effort and possibly lead to overlooking important features and getting inadequate verification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 Despite the development of several very effective methods and improved data- structures to combat the state-explosion problem, models of realistic systems may still be too large to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  65  Types of Model Checking Model checking approaches are classified into two types: (i) Explicit and (ii) Symbolic based on how they enumerate states [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Explicit model checking techniques store the explored states in a hash table, where each entry corresponds to a single system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For just a few hundred states the nodes in the state space graph becomes as large as ~1011 [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' On the other hand explicit model checkers support state-enumeration that gives detailed expressiveness of the system states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Symbolic model checking techniques store sets of explored states symbolically by using efficient data structures represented by canonical structures such as Binary Decision Diagrams (BDDs) [89], and traverse the state-space symbolically by exploring a set of states in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The use of these BDD-based methods has greatly improved scalability in comparison to explicit state enumeration techniques, yet they have performance degradation because  BDDs constructed in the course of symbolic traversal grow extremely large, and BDD size is critically dependent on variable ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This causes a newer trend of research towards separating Boolean reasoning and representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hence Boolean Satisfiability (SAT) [90] has been studied and explored for Boolean reasoning and efficient semi-canonical representations which results in  the development of SAT-solvers which are efficient and have compact representation compared to BDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  SAT, together with efficient representation, have become a viable alternative to BDDs for model checking applications [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Bounded model Checking is a model checking approach where the number of steps in forward traversal of the state space are bounded and checks whether a property violation can occur in k or fewer steps [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The approach reports either “violation found” or “no violation possible within the bounded depth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', k steps), which can be incremented to look ahead for possible violation of the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This method is promising because it does not cause state-space explosion or at least let the user control its possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis all three model checking approaches are accompanied by the tools selected for composability verification of CSP based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Temporal Logics Logic provides formal languages containing formulas for the representation of the statements and their logical reasoning within some area of application [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Generally, a logical language is given by an alphabet of different symbols and the definition of the set of formulas which are strings over the alphabet [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In logic, the term temporal logic is used for representing and reasoning about propositions qualified in terms of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Temporal logic has found an important application in formal verification, where it is used to specify system requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Linear Temporal Logic (LTL) and Computational Tree Logic (CTL) are its two main variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' LTL formulas are interpreted on computation paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Let A and B be atomic predicates and ¬ , ∧ , ∨ , ↔ and True be the operators of classical logic, whereas , , and U are the operators of linear temporal logic called Next, Always and Eventually and Until.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  66  The intuitive meanings of some LTL statements are: • ¬ A : A does not hold • A ∧ B : Both A & B hold • A: A holds at the next state • A: A holds in all states • A: A will eventually hold • A U B: A will hold until B holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In CTL there are additional path quantifiers ‘∃’ and ‘∀’ denoting ‘there exists a path’ and ‘for all paths’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CTL formulas are interpreted on computation trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With respect to a tree the intuitive meanings of the formulas mentioned above are: • ∃ A: There exists a path in which A holds at the next state • ∀ A: For all paths A always holds in all states  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Time CSP CSP has been in evolution for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One of the major extensions of CSP is devised with timing primitives, denoted as TCSP, to support time sensitive process modeling [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In TCSP, each of the untimed CSP operators is interpreted in a timed context, and two primitive timing operators are added: (i) timeout and (ii) interrupt, with a Newtonian Time assumption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', that all the processes have a single global clock with same progress rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Timeout P ⊳d Q Timeout operator can be used to introduce delay in the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Timed Interrupt P △e Q Interrupt is used if the process is permitted to run for no more than a particular length of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The concept of TCSP is used later in this thesis to model and perform verification of real-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Probabilistic Systems Systems that exhibit probabilistic aspects essential for designing randomized algorithms, modeling unreliable or unpredictable behavior or specifying model-based performance evaluation are called probabilistic systems [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to model random phenomena in such systems, transition systems are enriched with probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Probabilistic systems can be specified in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Two very popular ways are: (i) Markov chains (MC) and (ii) Markov decision processes (MPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis, we considered MPDs as specification formalism for probabilistic systems because they support both nondeterministic and probabilistic choices and unlike MC they can model the interleaving behavior of the concurrent processes in an adequate manner [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 3  Executable Modeling Formalisms  Page  67  A Markov Decision Process is a tuple 〈S, Act, P, linit, AP, L 〉 [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Where: S = Set of states Act = set of actions P: S × Act × S → [0, 1] is the transition probability function such that for all states s∈S and actions α∈ Act: � P(s, α, s′)∈{0,1} s′∈S  linit: S → [0, 1] is the initial distribution such that: � 𝑙𝑖𝑛𝑖𝑡(s) = 1 s′∈S  AP is a set of atomic propositions L: S → 2AP is a labeling function The concept of MDP is used later in this thesis to model and perform verification of probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6 CSP Implementation Tools There are a variety of implementation support tools and languages for developing CSP models such as CTJ (Java), CSP++ (C++), CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='NET, PyCSP (Python), JCSP (Java) and CSP# (C-Sharp) [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Similarly various techniques exist for CSP analysis such as: • FDR2 model checker is developed by Formal Systems Europe Ltd [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • ARC, the Adelaide Refinement Checker, is a CSP verification tool [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • ProB is an animator and model-checker and support refinement checking and LTL model-checking of CSP [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • PAT is a model checker, simulator and refinement checker for CSP [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In this thesis we selected PAT model checker because of its user friendly environment for modeling CSP models, fast simulator and model checker and above all its support for CSP extensions such as Real-Time CSP, Probabilistic CSP and Real-time Probabilistic CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 Process Analysis Toolkit (PAT) PAT is an established tool developed by National University of Singapore in concurrent system verification and has been used in real-world industrial projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PAT is designed to develop, compose, simulate and analyze event-based system models using an extension of CSP formalism called CSP-Sharp (or CSP#24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This extension comprises of some additions such as shared variables and asynchronous message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Moreover it supports using complex data types (such as Set, Queue, and Stacks) and functions from external libraries written in C# therefore allow to  24It uses C# like syntax for the specification of CSP processes  Chapter 3  Executable Modeling Formalisms  Page  68  model complex process behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PAT also supports automated refinement checking and model checking of LTL extended with events [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PAT is an appropriate modeling, composition, simulation, verification and reasoning framework of CSP based process models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These models can be of different nature such as concurrent, real-time and probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main strength of this framework is that it implements various model checking techniques and provide verification support for different properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' That includes general system properties such as deadlock-freeness, divergence-freeness or reachability and user specific properties defined in terms of LTL assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also includes refinement checking, model checking of real-time and probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To achieve good performance, advanced optimization techniques are also implemented in PAT, such as partial order reduction using BDD, symmetry reduction and parallel model checking [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Summary In this chapter we have discussed two executable modeling formalisms namely: (i) Petri Nets and (ii) Communicating Sequential Processes and their associated concepts, tools and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Both formalisms are used in this thesis for describing executable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The conceptual background of both PN and CSP is required to understand the approach presented later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page  69  Chapter 4  Verification and Analysis  Verification and Validation are important aspects of any software engineering expedition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are independent procedures with different characteristics that are used to check that a program, service, model or a system is correct, meets requirements specifications and that it fulfills its intended purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are critical constituents for achieving the necessary levels of quality assurance, and are essential prerequisites for a credible and reliable use of the delivered product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main focus of this chapter is on Verification and its different analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The aim of this chapter is to outline basic concepts, principles, issues and different approaches of software verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This chapter can be viewed as a manual to understand the verification process being proposed later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The correctness of a program is a relative concept, meaning that the program is doing no less than prescribed by its specification [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Verification, Validation and Testing (VVT) in combination is a broader and more complex discipline of system engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In M&S the combination of Verification, Validation and Accreditation (VVA) is generally referred where “Accreditation” is the formal certification that a model or simulation is acceptable to be used for a specific purpose [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Nevertheless the goal is to assure the quality of the product and the impetus behind this assurance is intensified when the systems are highly critical, either because they are very expensive to produce, such as land rovers investigating outer planets, or because human lives depend on them, such as computers controlling airplanes and cars, and life assisting real-time systems in hospitals [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These systems need to be correct, because their failure can lead to loss of human lives or enormous economic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Moreover correct systems can be used in a wrong manner which can also results in a failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is a general problem when systems are designed in a modular fashion, and are implemented with assumptions on a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A similar case caused a drastic failure at the launch of Ariane-5 expendable rocket launch system, because a software module was reused from Ariane-3 with certain assumptions that did not hold for Ariane-5 which self-destructed just because one single variable of 64 bit floating point value was erroneously converted to a 16 bit integer causing the system to crash [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So for critical systems it is worth the effort to have a guarantee that they are correct and have no errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Verification and validation aim to increase the credibility of models and simulation results by providing evidence and indication of correctness and suitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Verification in particular deals with the correctness of the model perceived from a real-system, whereas validation deals with the suitability or fitness of the model with respect to its real-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Testing on the other hand aims to uncover incorrectness in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the following section, definitions and concepts of these inter-related terms are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 4  Verification and Analysis  Page  70  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Some Basic Concepts in Modeling and Simulation The first applied technical discipline that began to struggle with the methodology and terminology of V&V was the operations research (OR) community, also referred to as systems analysis or modeling and simulation (M&S) community [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Verification According to the Department of Defense (DoD) Defense Modeling and Simulation Office verification is defined: as a process of determining that a model implementation accurately represents the developer’s conceptual description and specification [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In general verification refers to an evaluation process that determines whether a product is consistent with its specifications or compliant with applicable regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In M&S, verification is typically defined as the process of determining if a model is consistent with its specification [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Verification deals with the model correctness and is concerned with building the model right [28], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a model which works correctly and has no bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In principle, verification is concerned with the accuracy of transforming the model’s requirements into a conceptual model and the conceptual model into an executable model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For the sake of clarity the notions of correctness are defined as follows: Correct: Free from error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in accordance with the fact, truth, or reason;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conforming to the acknowledged standards of a method, routine or behavior [Oxford Dictionary] Correctness The degree to which a program, model or a system as a whole is free from defects in its specification, design, and implementation [105] The ability of a software product (or a simulation model) to perform the exact task, as defined by its specification [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We define a composed model to be correct if its structure and behavior matches its specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Correctness of a composed model is therefore relative to its specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A software entity can exist in three apparent states of correctness namely: (i) correct when it has been established correct against its specification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (ii) defective when it has been established incorrect against its specification and (iii) unknown when its correctness has not been established against a specification [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=" In SE a software entity's specification is the sum of all its passing unit-tests [107]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We define specification to be a set of goals (or objectives) and property constraints (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2) that must be fulfilled by the composed model to be established as correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Validation According to the Department of Defense (DoD) Defense Modeling and Simulation Office validation is defined: as a process of determining the degree to which a model is an accurate representation of the real world from the perspective of intended uses of the model [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model validation on the contrary, deals with building the right model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the model which is an accurate representation of the real system [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model validation is usually defined to mean “substantiation that a computerized model within its domain of  Chapter 4  Verification and Analysis  Page  71  applicability possesses a satisfactory range of accuracy consistent with the intended application of the model [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Testing Model Testing on the other hand, ascertains whether inaccuracies or errors exist in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The objective of testing is to show that the model (or system) is incorrect (rather than proving that it is correct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Testing can only find errors but cannot guarantee the absence of errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' therefore it is more of an ad-hoc and inexpensive method of necessity, where the correctness is established merely on the fact that all tests have passed, which is insufficient and unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the test fails, it succeeds in revealing an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a test is passed, it fails to detect an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a number of tests fail to detect a bug, they increase a confidence level in the system even if the correctness cannot be guaranteed [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Verification and Validation in a Modeling Process A Modeling Process has been defined by Sargent [108] as shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this process Verification is referred to as an activity which ensures that the computer programming and implementation of the conceptual model is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 21: Modeling Process (acquired from [108])  Whereas validation is defined in three perspectives: Conceptual model validity is defined as determining that the assumptions underlying the conceptual model are correct and that the model representation of the problem entity (simuland) is “reasonable” for its intended purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Operational validity is defined as determining that the model’s output behavior has sufficient accuracy for the model’s intended purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Data validity is defined as ensuring that the data necessary for the model execution and model experiments to solve the problem are adequate and correct [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Mike Petty in his article [29] also clarifies the difference between the two terms at different stages of model evaluation process as illustrated in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Problem  Entity  Conceptual  Operational  Validity  Model  Analysis  Validity  Experimentation  and  Modeling  Data  Validity  Computerized  Computer Programming  Conceptual  Model  Model  and Implementation  Computerized  Model  VerificationChapter 4  Verification and Analysis  Page  72  Figure 22: Modeling Process (acquired from [29])  A simuland is the real system that is to be simulated whereas a model is a representation of the simuland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' developed with its intended application in mind and therefore captures only the necessary abstractions of the simuland and omit others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The requirements are driven by the intended application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conceptual models document those aspects of the simuland including the structural and behavioral aspects such as objects, entities, events, functions, environmental phenomena etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The executable model is the computer program that can be executed and is intended to simulate the simuland as detailed in the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the conceptual model can be viewed as a design specification for the executable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The results are the output produced by a model during a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 22 presents Verification and Validation as activities that compare one thing to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Verification compares the requirements with the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this comparison, verification seeks to determine if the conceptual model satisfies the given requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The second comparison is between the conceptual model and the executable model, where the goal is to determine if the implemented executable model is consistent with respect to the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Validation compares the simuland with the conceptual model to determine if the simuland has been accurately described in the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The second comparison is between the simuland and the results which determine if the output of the simulation is sufficiently accurate with respect to the actual behavior of the simuland [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Another comprehensive VV&T model is presented by Balci [28] in the form of a simulation study life-cycle as shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The phases are shown by oval symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The dashed arrows describe the processes which relate the phases to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The solid arrows refer to the credibility assessment stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Every phase of the life-cycle has an associated VV&T activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Problem Formulation (or problem definition) is the process of formulating a problem which is sufficiently well-defined to enable specific research action and the investigation of suitable solution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of system investigation results in the System and objective definition which further aids in model formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model formulation is the process of defining a conceptual model which abstracts or envisions the real system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The conceptual model is further represented inform of a Communicative Model which is a model representation and can be communicated to other designers and can be compared against the system and the study objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is further  Requirements  analysis  Requirements  Simuland  Accreditation  Modeling  Validation  Talidation  Verification  Conceptual  Results  model  Execution  Implementation  Verification  Transformation  Executable  Comparison  modelChapter 4  Verification and Analysis  Page  73  transformed into an executable model through the process of programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An Experimental Model is the programmed model incorporating an executable description of operations along with the design of experiments, for experimenting with the simulation model with a specific purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The process of experimentation produces the Simulation Results, which are presented for decision makers for their acceptance and implementation or undergo refinements if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 23: Simulation study life-cycle (acquired from [28]) The model-evaluation life-cycles shown in Figure 21, Figure 22 and Figure 23 have been considered as guidelines and they are used as inspiration for the verification life-cycle proposed and presented later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 The Principles of Top-Down Refinement The principle of top-down refinement has been appreciated in the area of model verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Constructing a highly detailed model that satisfies all levels of correctness in one attempt is very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Instead it is easy to construct a less detailed abstract model at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Let S1 be an initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To get from S1 to the final shape of the model, the Top-Down Refinement paradigm advocates the derivation  COMMUNICATED  PROBLEM  Problem  Formulated Problem  Formulation I  VV&T  FORMULATED  PROBLEM  Investigation of  Feasibility Assessment  Solution Techniques I  of Simulation  DECISION MAKERS  PROPOSED SOLUTION  Acceptability of  TECHNIQUE  Simulation Results  (Simulation)  INTEGRATED  DECISION  System  System and Objectives  SUPPORT  Investigation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Definition VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='SYSTEMAND  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OBJECTIVES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='DEFINITION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Formulation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Simulation Results  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Presentation VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Presentation of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Qualification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CONCEPTUAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='MODEL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Communicative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='SIMULATION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Experimental  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='COMMUNICATIVE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RESULTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='MODEL(S)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Programmed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/ Programming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='PROGRAMMED  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='MODEL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Experiment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Design VV&T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='EXPERIMENTAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Design of Experiments  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='MODELChapter 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Verification and Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='74  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='of an (ordered) sequence S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2…Sf of models of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='f, model Si+1 is a refinement of its immediate predecessor model Si if the following conditions are met:  (i) Si+1 is more expressive than Si  (ii) Si+1 is less abstract than Si  (iii) It is relatively easy to evaluate Si+1 on the basis of verified Si  Consequently, the last model in the refinement sequence should be correct by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following are some consequences of the top-down refinement paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' First, Si+1 is harder to understand than Si and therefore harder to prove on its own;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' it is precisely the refinement step that allows the verification of Si+1 under the assumption that Si has already been proved correct [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis the proposed verification process is based on this fundamental principle where the verification is performed iteratively and on a relatively refined shape of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Verification techniques There exist a large variety of verification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The diversity is due to the range of different simulation project types, different subjects (simuland), and different types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Most of the verification methods are inspired from software engineering domain, because the executable models in simulation projects are almost always realized as software [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In literature, Verification techniques are generally classified into four main categories as show in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 24: Verification Techniques  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Informal Techniques These techniques are most commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are called informal because the tools and methods used rely heavily on human reasoning and inspection without any underlying mathematical formalism [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These techniques are well structured and are conducted with proper guidelines by following standard policies and procedures, however these techniques are tedious and not very much effective [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Verification  Techniques  Informal  Techniques  Static Analysis  Dynamic  Analysis  Formal Analysis  Chapter 4  Verification and Analysis  Page  75  Some of the commonly used informal methods are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Audit An audit is undertaken to assess how adequately the system study is conducted with respect to established plans, policies, procedures, standards and guidelines [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Desk checking Desk checking or self-inspection is a thorough examination performed by an individual as a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this method syntax checking, specification comparison, code, control flow graph analysis are performed [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Inspections Inspections are conducted by a team and performed at different phases of developments such as problem definition, conceptual modeling, executions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Inspections are conducted to find and document faults [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Turing Tests Turing test is performed by domain experts (of the system under study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are presented with two sets of output data obtained one from the model and one from the specification (without identifying which one is which) and are asked to differentiate both and based on their feedback model corrections are made [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 6: Informal Verification Techniques 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Static Analysis: These techniques are applied to assess the static model design and the implementation (source code), without executing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They aim at checking the structure of the model, the dataflow and control flow, the syntactical accuracy, and the consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of the commonly used static analysis methods are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Structure Analysis Structure Analysis is used to examine the model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It is conducted by constructing a control flow graph of the model structure [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Data Analysis It involves data dependency tests and data flow analysis to ensure that data used by the model is properly defined and proper operations are applied to data objects [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Cause- Effect Graphing Cause-Effect graphing assists model correctness evaluation by answering “what causes what” questions in the model representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is performed by identifying causes and effects in the model and checking if they are reflected accurately in the specification [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Syntactic Analysis Syntactic analysis is usually performed by the compiler of the simulation language being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Syntactic analysis can also be performed using a set of rules applied on the model representation to verify if it satisfies given specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Semantic Analysis This technique is used to determine the modeler’s intent and verify that the true intent is accurately reflected in the model representation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 7: Static Analysis Techniques  Chapter 4  Verification and Analysis  Page  76  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Dynamic Analysis: Dynamic analysis techniques are based on the execution of the model in order to evaluate its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They do not simply examine the output of an execution but also observe the model as it is being executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The insertion of additional code into the model called instrumentation is needed to collect or monitor the behavior during its execution [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 8 presents some of the important dynamic analysis verification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Assertion Checking An assertion is a statement that should be true during the execution of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Assertions are placed in various parts of the model and monitored during execution [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Bottom up Checking This technique is used in conjunction with the bottom up model development strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The sub models are checked individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then the parents at the higher level are checked [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Fault/Failure insertion This approach is used to insert a fault or a failure in the model and observe whether the expected incorrect behavior is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach is effective to detect unexplained behavior and hence uncover errors [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Functional Testing This technique is used to assess the accuracy of model input- output transformation, to evaluate how accurately a model transforms a given input into a set of output data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sensitivity Analysis Sensitivity analysis is performed by changing the values of model input variables and parameters over some range of interest and observing the effect on model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Unexpected effects may reveal errors [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 8: Dynamic Analysis Techniques  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Formal Analysis Formal analysis refers to mathematical analysis of proving or disproving the correctness of a system with respect to a certain unambiguous specification or property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The methods for analysis are known as formal verification methods, and unambiguous specifications are referred as formal specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Formal verification can provide complete coverage on an abstract model of the system, modeled using finite state machines, PN or any other specification formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it should be noted that formal verification can ensure the correctness of a design only with respect to certain properties that it is able to prove [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are many formal analysis techniques, which we classify in four main groups:  Chapter 4  Verification and Analysis  Page  77  Equivalence Checking It is also called Reference Model Checking, which is widely used verification technique that allows two behavioral models to be compared with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In general, one of the two is taken as the reference model and represents the so-called golden model (or perfect model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It verifies that the behavior of two models is the same for the exercised scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This technique has limitation that it does not actually verify that the design is bug free, and provides proof of relative correctness [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Theorem Proving This method involves verifying the truth of mathematical theorems that are postulated or inferred throughout the design using a formal specification language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The procedure involves two main components: (i) proof checker (which can be completely automated in most cases) and (ii) an inference engine (which may require occasional human guidance) [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Property Verification Formal properties specify the requirements of the correct system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The objective of this method is to check whether an implementation satisfies these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Static Assertion-based Verification (ABV) and dynamic [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model Checking Model checking establishes a solid confidence in a reliable V&V process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model checking is an automated and comprehensive verification technique that can be used to verify whether the properties specified (usually using Temporal Logic) for a given design or its components are satisfied for all legal design inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Model checking also faces a limitation, since it suffers from the well-known state explosion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In a worst-case scenario, the state space of the design may grow exponentially large with the number of state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model checking can be fully automated for design verification and can yields results much more quickly than theorem proving [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 9: Formal Analysis Techniques  Some of these techniques have been adopted in our proposed verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Summary In this chapter, different concepts of verification, validation and testing are discussed as they collectively contribute to proving the correctness and accuracy of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some existing model development processes (devised mainly by M&S community) are also discussed, since they are the bases of the proposed verification life-cycle presented later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed framework essentially focuses on Verification (however its design is also open to adopt validation techniques).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Different verification techniques are classified into four main groups and some of the selected techniques are briefly explained, as they will be used later in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 4  Verification and Analysis  Page  78  Part II  Techne  Technê in Greek is translated as craftsmanship or craft or art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In science it is the practice of knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Techne resembles Epistēmē in the implication of knowledge of principles, although techne differs in that its intent is making or doing, as opposed to "in-depth understanding";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Applied- Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It deals with “How” of the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Part-II covers the technology of the research under discussion, where the theoretical concepts provided in Part I are applied, and technically discussed under an integrated framework of methods, techniques, algorithms and processes and their practical implications are provided in the form of a proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  “Without knowledge the practice is useless, and without practice the knowledge is useless” – Ali bin Usman Hajvery (Kashaf-Almahjoob)  Page  79  Chapter 5  Proposed Methodology and the  Verification Framework  This chapter renders the core of the solution framework proposed in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter, a collection of methods, techniques, algorithms, sub-processes, activities and approaches are presented, as proposed solution to various issues in the composability verification of BOM based model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' All these contributions are integrated into a unified framework which we refer to as: Composability Verification Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The proposed verification Framework consists of different methods, techniques, algorithms, sub-processes, activities and approaches which all together encompass the component based modeling & simulation (CBM&S) life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Component-based Modeling & Simulation life-cycle CBM&S life-cycle is inspired by different modeling architectures proposed by Sargent, Petty and Balci and discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is extended with our proposed contributions at its different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed CBM&S life-cycle is mainly divided into four main quadrants: (i) Inception (ii) Modeling (iii) Execution and (iv) Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each quadrant has different phases and in each phase there are multiple activities (or cycle of activities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each activity consists of methods and techniques pertinent to its respective phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These phases are revisited iteratively during the life-cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' where each iteration represents a tier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' hence the entire CBM&S life-cycle is a multi-tier process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' whilst each tier results into a refinement of the solution of the problem under investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' as it follows the principle of top down refinement, discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' All the above mentioned features of the CBM&S life-cycle are shown in Figure 25 divided into four quadrants:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 25: CBM&S life cycle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ANALYSIS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INCEPTION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Refinement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Simuland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requirements Engineering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase II  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Abstract Level Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase IV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase III  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Executable Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Fransformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Activity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Formal Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='EXECUTION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='MODELINGChapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='The following sub-sections provide microscopic details of each quadrant along with their associated inside activities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' methods and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Inception The first quadrant of the CBM&S life-cycle called “Inception” initiates the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first the abstraction of a real-system is accumulated as simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A simuland can be ingested in the form of UML diagrams (Figure 26) or using any other formal or informal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 26: Simuland using UML Diagrams The basic idea is to gather the body of knowledge so that the modelers can envision the real system under a certain frame of reference i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the context under which the system is being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the simuland is ingested into the framework, it is used (i) to gather requirements, through the process of requirement engineering and (ii) to search and discover suitable components from a BOM repository for the construction of a composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a required component does not exist in the repository then it is built from scratch and added in the repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The outcome of the requirement engineering activity results in formulation of requirements specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The requirement specification formalism (as defined in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2) is used to express formal requirements for this framework: RS = 〈O, S〉 Where  O = {o1, o2, o3 …, on} is a set of objectives or goals that must ultimately be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These goals are usually defined in the context of the scenario of the modeling domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the properties expressed as goals or objectives may be scenario- specific and not the standard system properties e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in a restaurant model the objectives could be that the customers are served food and payments are collected, and not that the model should be deadlock free (which however might be a necessary condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S = {s1, s2, s3 …, sn} is a set of system constraints (system properties or scenario- specific safety/liveness properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Deadlock freedom (or other similar system properties) could be the required constraints necessary to fulfill the above objectives and therefore must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We propose to define the following mandatory (or default) constraints in the requirements specification of the composability verification framework:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='81  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S1 = All the interacting components25should be composable at Syntactic level S2 = All the interacting components should be composable at static-semantic level S3a = State-machines of the interacting components should match each other such that they can continue to progress until they reach the final or goal states26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3b = If the conceptual model is transformed into an executable model, the latter should correctly represent the structure and behavior of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 10: Mandatory constraints in composability verification We assert that [S1 ∧ S2 ∧ (S3a ∧ S3b)] is a necessary condition for the overall composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S1 and S2 ensure that the composed model is structurally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas S3a confirms that the behavior of the composed model is coherent for reaching given objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The satisfaction of S3b obeys the definition of Model verification (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1) in the sense that it confirms the second part of the definition that is: “the accuracy of transforming the conceptual model into an executable model” and therefore the overall success of the verification process depends on the satisfaction of S3b constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The conjunction of these default constraints impose the three C’s of requirements namely (i) Consistency, (ii) Completeness, and (iii) Correctness [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Consistency is required for the evenness in the input and output connections of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Completeness is required for the totality of the information of the components being composed to check that the composition does not lack required inputs for making progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Correctness is needed to confirm that the composed components interact in a correct way as they are supposed to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If all the objectives are fulfilled and all the constraints are satisfied and then we say that the model is composable at all levels and is verified with respect to its specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The overall objective of our proposed framework is to provide environment and tool support to assess this postulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The outcome of discovery results in a set of candidate BOMs and their matching with the simuland and the requirements results in a selection of BOMs suitable for the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This selection is composed to form a conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Modeling In the Modeling quadrant, a BOM based composed model is taken as an input and the conceptual model is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also a formal model and its graphical notation (as proposed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4) are produced for the purpose of documentation of the conceptual model26F27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Considering that BOM itself is a conceptual framework and is used to model passive components which cannot undergo any form of execution therefore the conceptual model is subjected to a series of extensions and refinements  25  In a composed model it is not necessary that every component interacts with every other component for instance A, B and C are composed such that A interacts with B and B interacts with C but A does not interact with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 26 If there are no final-states defined in a model and the model is non-terminating then we assume that certain important states called goal-states are present in the model, reachability of which confirms that the goals are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 27 This step is optional but beneficial if different teams are working on different phases of the development life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This documentation makes it easy to understand the structure and behavior of basic components and their composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page  82  using external input and our proposed model transformation algorithms so that it can be implemented into executable forms and sent to the “Execution” quadrant (Figure 25) for abstract level execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our proposed extensions and refinements are listed as follows: • BOM State-machines to State Chart XML (Transformation) • Composed-BOM to Petri Net –PNML (Transformation) • Basic-BOM to Extended-BOM (Extension) • Extended-BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(E-BOM) component to Colored Petri Net (CPN) Component Model (Transformation) • Basic-BOM to Extended-BOM with Time (Extension) • Basic-BOM to Extended-BOM with probabilistic factors (Extension) • BOM to CSP based Process Model (Extension & Transformation) In the later section these extensions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' refinements and transformations will be explained in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is important to note that each time the conceptual model is extended or refined the Modeling quadrant is revisited in iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Execution As previously discussed this quadrant is mainly for the abstract-level execution activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It takes following implemented and executable forms of the conceptual model from the Modeling quadrant as input: • State Chart XML (SCXML) • Petri Net –PNML • Colored Petri Net (CPN) Composed Component Model • Communicating Sequential Process (CSP) based Component Processes In the later section these executable forms and their abstract level execution processes will be discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Analysis The outcome of an execution process yields some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  These results are analyzed in the Analysis quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our verification framework supports different analysis techniques listed as follows: • State-machine matching Analysis • Petri Nets based Algebraic Analysis • Colored Petri Net based State-Space Analysis • Model Checking Analysis  These analysis techniques will be discussed in later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When all the necessary steps in the composability verification are complete and the composed model under investigation is said to be verified with respect to the given requirement specification then the CBM&S life-cycle proceeds to the further steps for implementation and simulation as shown in Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The details of these steps are out of the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page  83  Figure 27: Implemenation and Simulation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6 Composability Verification Framework In this section different method, techniques, procedures, algorithms and modules of our proposed composability verification framework are discussed in detail and considered as building blocks in the CBM&S life-cycle and will be connected to its different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These details are necessary to understand the composability verification process being presented in chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Discovery Matching and Composition (DMC) In component based development, it is a normal practice to construct reusable components and store them in a library or repository so that they can be reused later as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To reuse an existing component, a Discovery, Matching, Composition (DMC) paradigm [19] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that a library of BOM components is maintained in a repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using the information given in the simuland a modeler attempts to search and discover BOM components from the repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a collection of candidate components is retrieved, they are filtered through matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A matching process matches the candidate components from the simuland and requirement specifications and results in a selection of components suitable for the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The aspects of syntactic and semantic matching during the discovery and selection of BOM components are proposed and discussed in detail in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In this article a set of discovery rules are presented which must be fulfilled while matching a candidate selection from the simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We apply these rules for the syntactic and semantic matching of the candidate selection with the simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We further suggest matching the candidate selection with given requirements, because a selection may match with its respective simuland but if it does not match with its requirements then the composability verification will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We implement the concept of DMC process in our framework as shown in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is also assumed that if a required component does not exist in the repository, then it is constructed from scratch and is added in the repository for reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The result of DMC process is a BOM-based composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This composed model is taken as input in the Modeling quadrant and considered as a conceptual model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is recommended that the modelers also use our proposed formal specification and graphical notation presented in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 to construct a formal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This formal model can be used for documentation and shows how the components are composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is however an optional step and is not considered as a phase in our  IMPLEMENTATION  Composed Model  Code Generation  Simulation Model  Successful Completion of  the Composability  Decision Support  Design of Experiment  Verification Process  Experimental  Model  Simulation  Simulation ResultsChapter 5  Proposed Methodology and the Verification Framework  Page  84  CBM&S life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In chapter 7 & 8 the formal models of the examples are also described for reader’s understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 28: Discovery, Matching, Composition (DMC) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Structural and Behavioral Evaluation The conceptual model ingested in the Modeling quadrant requires structural and behavioral evaluation so that we can confirm that the model is consistent, complete and correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And it is suitable for thorough verification at different levels of composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Checking the structure and behavior of the conceptual model before subjecting it to the deeper levels of composability verifications is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the model is structurally and behaviorally consistent then the confidence level is increased based on which different useful assumptions can be made later during the in-depth verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there are discrepancies in the structure or behavior of the model then we can skip further steps, save time and computational resources and perform necessary design refinements before the entire process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This setup obeys the principle of top-down refinement as discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The structure of the model is analyzed using static analysis techniques (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2), whereas the behavior of the model is evaluated using dynamic analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Static Analysis We propose two types of Static analysis procedures (i) Syntactic Matching and (ii) Static-Semantic Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These procedures are used to evaluate the structure and verify composability at syntactic and static-semantic levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are called static analysis because they are evaluated based on pre-defined rules and do not require any form of execution and the information on which these rules are applied is static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Phase I  Simuland  Reguirements  Component Search  Engineering  Phase II  Reguirements  BOM  Repository  Matching  Discovery  Candidate  Matching  BOMs  Selection  Modeling  12N  Phase I1  Composition  Conceptual  Model  Formal  Model  TransformationChapter 5  Proposed Methodology and the Verification Framework  Page  85  Syntactic Matching (SM) This module is responsible for evaluating BOM composability at syntactic level based on the following rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The outcome of this module verifies that the components can be correctly connected to each other syntactically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These rules were introduced in a BOM matching technique presented in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' SM-Rule 1: The name of each event28 exchanged between the two components should be same i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the send-event should have the same name as the receive-event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A send-event is defined in the BOM’s event types where the sender is the BOM itself and the receiver is some other BOM (in the composition) whereas a receive-event is the definition of an event in the BOM event types, where the sender is some other BOM (in the composition) and the receiver is the BOM itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' SM-Rule 2: Each send-event should have at least one corresponding receive-event and vice-versa i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the send/receive pair should be complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  SM-Rule 3: The number of parameters (content characteristics of event types) of the send-events should be the same as the number of parameters of the receive-events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The satisfaction of Syntactic Matching rule1, rule2 and rule3 fulfills the default constraint S1 (see Table 10) which is a necessary condition for the overall composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 29  shows different steps in the syntactic matching activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 29: Syntactic Matching  28 It is assumed that in the BOM construction the events and their corresponding actions are given the same name  Phase I  Refinement  Simuland  Reguirements  Engineering  Phase V  Phase II  Analysis  Constraint Sl satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Requirements  Technique  Static Analysis Technique  Rule evaluation  Analysis  Satisfy  Rule3  Rule2  Rulel  Modeling  Abstract Level  /Violate  BOM  Execution  Components  Phase I1I  Phase IV  Conceptual  Executable  Model  ModelChapter 5  Proposed Methodology and the Verification Framework  Page  86  Static-Semantic Matching (SSM) This module is responsible for evaluating BOM composability at static-semantic level based on certain rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The outcome of this module verifies that the composition of the components is meaningful and the communication between the components is understood as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to certify these facts we propose static-semantic matching at two levels: (i) Operational Level matching and (ii) Message level [53]: (i) Operational Level matching In BOM-based composed models Operations are described by Pattern-of-Interplay (POI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' POI is formed by a collection of actions from the basic BOMs being composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In operational-level semantic matching, it is ensured that the composed components share the same “domain of interest” and they are composed for the same purpose (or aim) so that we can guarantee that the composition is (static) semantically meaningful and without any pragmatic ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Even with the same domain of interest, the component may serve for varied purposes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', in Military domain a Battalion Head Quarter (BHQ) component may have many purposes and can take part in many different operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it is also important that the purpose of the selected components should be clear for a meaningful outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In order to ensure semantic consistency at operational-level we propose to specify following semantic-attributes 29 in the definition of actions at the time of the construction of Basic BOMs and in the POI when the basic BOMs are being composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the static-semantic matching these attributes are used to compare that the correct actions are involved in the BOM composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  o Area-of-Interest: It describes the area or the domain of interest of the system that is being modeled using the components and the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We propose to define “Area-of-Interest” as a semantic-attribute in each action of Basic BOM and also in the POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This attribute will confirm that all the components share the same domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If of some general purpose components that may belong to multiple- domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', Queues etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=') we propose to construct a specialization of the component and make it a member of the selected area-of-interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', In a restaurant composed model a generic queue component can be specialized into a restaurant-queue with actions JoinRestaurantQueue() and ServeCustomer() instead of Put() and Get() actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  o Purpose: Purpose describes the aim or goal of the entire operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In BOM composition, POI represents a single operation being performed by the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it is also possible that one or more composed components may be designed to serve multiple purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and in a given scenario only some part of the multi-purpose components is involved in the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a Customer component could be generic and can have multiple purposes whereas a Restaurant waiter component is specific to a restaurant scenario, so it is important that if a Customer component is selected in a Restaurant scenario then its purpose should be aligned with the other components in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Hence we define “purpose” as a semantic-attribute of actions in the basic BOM (with multiplicity  ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  29 In BOM the conceptual modeling elements (Entities, Events States and Actions) support semantic fields [65]  Chapter 5  Proposed Methodology and the Verification Framework  Page  87  (ii) Message Level matching BOM represent event driven components and function by sending or receiving events (messages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At the message level it is required that the communication between composed components is meaningful and semantically understood by the receivers as intended by the senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At this level we propose to match Data-Types and Units of measurements of the parameters of send-events and receive-events [53] [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It is assumed that the BOM components have corresponding OWL attachments as proposed in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The BOM-OWL attachments are used to define semantic classes of the domain ontology, their properties, data-types and the individuals and stored in the BOM repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to evaluate static-semantic matching at both Operational and Message levels, we apply following rules:  SSM-Rule 1 The intersection of the “Area-of-Interest” attribute of all the actions (involved in an operation) should be exactly the same as that of POI or should belong to an equivalent class30 in the respective ontology: � Acti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AOI n i=1 ≅ POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AOI  SSM-Rule 2 The intersection of the “Purpose” attribute of all the actions should be exactly the same as that of POI or should belong to an equivalent class in the respective ontology: � Acti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' purpose n i=1 ≅ POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' purpose  SSM-Rule 3 Data types of each element in the event parameters of the send-event and receive-events should be of the same class, equivalent class or should be in direct hierarchical relationship i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the sender’s parameter data-type should belong to the direct child class of the receiver’s parameter data-type (but not the inverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a send-event contains a parameter of type ‘second’, whereas the receive-event expects a parameter of type ‘time’ which according to the rule it is a semantic match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 30 presents primitive data-types as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In real situations BOM components will have more domain specific complex data-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  30In OWL two classes can be marked equivalent if they have same semantic meanings and both classes have the same individuals (instances) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', Healthcare and Medical are synonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We denote it as  ≅  Chapter 5  Proposed Methodology and the Verification Framework  Page  88  Figure 30: Some of the sub-classes of Data Type ontololgy SSM-Rule 4 The units of the measurements expressed in the event parameters should be same or equivalent or should belong to a direct class hierarchy such that they are convertible without (or with acceptable) loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that if two measurement units are in either of the direct relationship i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', parent or child then their conversion loss will be acceptable e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a send-event has a parameter with unit m/s (meter per second) to express speed whereas the receive- event expects Km/hr (Kilometer per hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is a valid semantic match because the quantities are convertible without loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Semantic Matching Technique In order to match two elements we propose a semantic matching technique as shown in Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This technique uses OWL-API [112], a semantic reasoning engine (FaCT++, Pellet, or HermiT) and an OWL ontology document to process a query of any two elements A & B and outputs their semantic relationship as one of the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Exact (A = B) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Equivalent (A ≅ B i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', A and B belong to equivalent classes) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Direct-Parent (A is a direct parent of B) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Direct-Child (A is a direct child of B) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Indirect (A and B are not in direct contact but belong to same hierarchy) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No relationship (A and B are not related)  Figure 31: Semantic Matching Technique This technique is used to evaluate Static-Semantic Matching Rules 1, 2, 3 & 4 using the algorithm31 given in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  31 The Pseudo-code conventions and format of the algorithms provided in this thesis, for most parts, follows the guidelines set by [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' OWL Doc A Reasoner OWL-API B Query Relation Result  ODay  OYear  Binary  ODate  OMonth  OInteger  Number  ODataType  ODouble  Time  Minute  Text  Language  OHour  Second  OCharacter  O StringChapter 5  Proposed Methodology and the Verification Framework  Page  89  Algorithm: Semantic Matching Input: {Actions}, POI, BOM-OWL Output: TRUE, FALSE 1 Owl ←  Load Ontology(BOM-OWL) 2 {CommonAOI} ← ⋂ 𝑎𝑖 𝑛 𝑖=0 ∈ Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='AOI  ⊳ Gives a set of common area of interest of all actions 3 for caoi ∈ {CommonAOI} do 4 SR1 ← Get-Semantic-Relation(caoi, POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='AOI, Owl) ⊳ It is assumed that Get-Semantic-Relation() 5 function is implemented using semantic matching technique shown in Figure 31 ⊲ 6 if SR1 = “Exact” or “Equivalent” then  ⊳ Rule1 satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='continue 7 next 8 else 9 Return FALSE 10 end if 11 end for 12 13 {CommonP} ← ⋂ 𝑎𝑖 𝑛 𝑖=0 ∈ Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='purpose  ⊳ Gives a set of common purpose of all actions 14 for cp ∈ { CommonP } do 15 SR2 ← Get-Semantic-Relation(cp, POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='purpose, Owl) 16 if SR2 = “Exact” or “Equivalent” then  ⊳ Rule2 satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='continue 17 next 18 else 19 Return FALSE 20 end if 21 end for 22 23 {Events} ←  Get-Events(Actions) ⊳ gets corresponding Events of Actions 24 for  e ∈ Events  do 25 if e=Send-Event then 26 f ← Get-Receive-Event(e, Events) ⊳ gets corresponding Receive Event of e 27 {PE} ← e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Parameters ⊳ Set of parameters of send-event e 28 {PF} ← f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Parameters  ⊳ Set of parameters of receive-event f 29 ⊳ No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  of parameters of e and f must be same because of SM-Rule3 30 for pe∈PE & pf ∈PF do 31 SR3 ← Get-Semantic-Relation(pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Type, pf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Type, Owl) ⊳ Compare Parameter types 32 if SR3 = “Exact” or “Equivalent” or “Direct-Child” then 33 ⊳ Rule3 satisfy…continue to rule4 34 SR4 ← Get-Semantic-Relation(pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Unit, pf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Owl) ⊳ Compare Units 35 if SR3 = “Exact” or “Equivalent” or “Direct-Parent” or “Direct-Child” then 36 Return TRUE ⊳ Static-Semantic Matching Successful 37 else 38 Return FALSE 39 end if 40 else 41 Return FALSE 42 end if 43 end for 44 else 45 next 46 ⊳ Goes to the next send-event and need not to check receive-events (because SM-Rule2) 47 end if 48 end for Table 11: Semantic Matching Algorithm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='The semantic matching algorithm takes a set of actions (parsed from Basic BOMs which are being composed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the pattern of interplay (POI) which specifies how the actions are connected to each other and the corresponding OWL ontology document as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of this algorithm is TRUE if the static-semantic matching is successful otherwise FALSE if any of the rule is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 32 shows steps in the verification of BOM composability at Static-Semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 32: Static-Semantic Matching If the semantic matching is successful, it will fulfill the default constraint (S2) of the requirement specification (see Table 10) which is a necessary condition for the overall composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Dynamic Analysis We use Dynamic Analysis technique (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) to evaluate the behavior of the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first the components undergo a state-machine matching process for the evaluation of the behavior consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When this evaluation is successful, we proceed with the in-depth verification at the dynamic-semantic composability level, choosing one of the different proposed set of dynamic analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These analyses are called dynamic analysis because they require execution at different abstract levels as mentioned in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 State-Machine Matching (SMM) State-machines represent behavior of the components and are the essential dynamic part of BOM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the verification of BOM composability at dynamic- semantic level, it is important that the behavior of the composed components should be coherent with each other i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', their interactions are consistent in order to make progress towards composition goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To ensure this fact we assert (as a necessary condition) that the state-machines of the composed components should match each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BOM state-machines are event driven in nature and make progress by exchanging events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In order to ensure that the state-machines of the composed BOM components match each other they are required to be executed at an abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore we proposed a technique in [113] which transforms each BOM state- machine to SC-XML (State-Chart XML) [114] format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A sample of SCXML is shown in Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Simuland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Refinement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reguirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Engineering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase II  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Constraint S2 satisfied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Static Semantic Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Technigue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OWL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OWL API  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Doc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ouery  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rulel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rule2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reasoner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rule3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Abstract Level  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rule4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase III  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase IV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Violate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Satisfy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Executable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='XChapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='91  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 33: SCXML format We develop a runtime environment using SCXML API for the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This environment parses SCXML files (transformed BOM state-machines) and creates instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then it initializes all the state-machines to their initial states and simulates sending and receiving of the events to observe state-machine transitions until they reach their final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The state-machine matching process is based on the following algorithm: Algorithm: State-Machine Matching Input: {SM} ∈ BOM State-Machines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {Actions} Output: TRUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FALSE 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{SCXML} ←  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TransformSMtoScXML(SM) 2 ⊳ Transform all BOM-Statemachines in SCxml format ⊲ 3 4 Create and Initialize EventController: EC 5 ⊳ Event Controller controls sending and receiving of events ⊲ 6 7 for scxml ∈ { SCXML } do 8 SC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='← Parse(scxml)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='⊳ Parse scxml document 9 Create and Initialize  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='SCXMLExecutor(SC) 10 ⊳Instantiate SCXMLExecutor thread for each state-machine ⊲ 11 12 Done ← FALSE 13 while  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(Done =FALSE) do 14 CurrentState ← GetCurrentState() ⊳ SCXMLExecutor returns current state 15 if CurrentState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='IsFinal = TRUE then 16 Done ← TRUE 17 end if 18 ⊳Get Next Action to send or receive ⊲ 19 {NextActions}← CurrentState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='GetActions() 20 for next ∈ NextActions  do 21 if next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Type = “Send” then 22 EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Put(next) ⊳ Simulate sending of next action 23 SCXMLExecutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Trigger(next) ⊳Transit from the current state to next state 24 else 25 EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Get(next) ⊳ Simulate recieving of next action 26 SCXMLExecutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Trigger(next) ⊳Transit from the current state to next state 27 end if 28 end for 29 end while ⊳Due to either of the send or receive actions the state-machine will 30 transit to the next state and therefore the current state will be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 31 If the final state is reached then the state-machine matching will be 32 terminated successfully⊲ 33 end for Table 12: State-machine Matching algorithm  <scxml initialstate= STATE1> <state id = STATE1 final = True> <transition event=EVENT NAME target=STATE2/> </state> <state id = STATE2 final = false> <transition event=EVENT NAME target=STATE1/> </state> </scxml>Chapter 5  Proposed Methodology and the Verification Framework  Page  92  Figure 34 shows the state-machine matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It takes BOM state-machines as modeling objects, automatically transforms it into a SCXML executable format and perform state-machine matching using abstract level execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A successful run of this routine implies that all the state-machines match each other, which satisfies a necessary (but not sufficient) condition of BOM composability i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' constraint S3a of the requirement specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The fulfillment of S3a certifies consistency and completeness of the behavioral design of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Consistency is due to the fact that the components are in correct causal order and Completeness, because their inputs and outputs (send and receive-events) are complete to reach their final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However we still cannot guarantee correctness the 3rd C of requirements, unless the composition satisfies its requirement specification i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', all the assigned objectives and required constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the state- machine matching approach may result in reaching final-states but it does not explore all possibilities of the behavioral interaction of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So it is required to analyze the model at a greater depth using an appropriate dynamic analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 34: State-machine Matching Process Therefore for deeper evaluation we propose to utilize the modeling and analytical strength of Petri Net and CSP formalism and incorporate three analysis approaches in our verification framework as introduced and discussed chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The selection of a suitable approach for the composability verification at dynamic-semantic level depends on the nature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the following subsections, each of these approaches is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Phase-I Refinement Simuland Reguirements Engineering Phase-V Phase-II Analysis X Necessary condition of Constraint S3 satisfied Reguirements Technique Fail Success 4 Put( Analvsis Is Final state 1 8 SM-1 SMIM reached?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 0155 BOM Components Event Putl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2 BOM-SM to Contro ler SCXML SM-2 Get 04 Transformation Abstract Level Modeling Action Execution Lookup SM-N Table Puto Phase-I1I Phase-IV sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='XvL Executors BOM Conceptual Executable Execution Model Model Transformation TransformationChapter 5  Proposed Methodology and the Verification Framework  Page  93  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 PN Algebraic Technique The basic idea of this technique is to transform BOM into Petri Net format and verify the properties given in the requirement specifications using algebraic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the verification framework, following steps are proposed to conduct algebraic analysis: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 BOM to PNML Transformation In the first step, BOM components are transformed into Petri Net Markup Language PNML format [115] which is an XML based form to specify Place/Transition Nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first BOM state-machines of all components are parsed and each state is transformed as a Place in the PN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly each event (send or receive event) is transformed into a Transition in PN with no duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An outgoing arc is connected from a place-P to a transition-t if the corresponding state-S (of the sender) has a corresponding event-t as its exit condition and next state S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An incoming arc is connected from transition-t to another place-P′ which represents the next state S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly state-R (of the receiver) is transformed into place-Q and the next state R′ into Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The incoming and outgoing arcs are connected to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The sender and receiver entities (of BOM) are represented as tokens in the places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 35 shows how part of a sender and receiver state-machine is transformed into a PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The place P and Q have tokens showing the current state (or marking) of the composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  When transition t is fired (meaning event t is sent by P and received by Q) the tokens are transported to P′ and Q′ showing the next marking of composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 35: BOM to PN transformation The transformation process is complete, when all the states and events of every state- machine in BOM are plotted in the PN model such that no element is duplicated, and each place or transition is connected so that there are no broken links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 PN Algebraic computations In this step the PN incidence matrix and Place/Transition invariants are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To perform this step we use Platform Independent Petri Net Editor (PIPE) API [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PIPE is a java based open source API for performing different Petri Net related operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It offers API functions to automatically compute algebraic resources of a PN model such as Incidence matrix and Place/Transition invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Incidence Matrix An incidence matrix of a PN model is calculated by subtracting A- from A+ incidence matrices:  R  PChapter 5  Proposed Methodology and the Verification Framework  Page  94  Algorithm: Incidence Matrix Calculation Input: PN Model (P-places × T-transitions) Output: m × n Matrix A 1 Initialize a Matrix Aminus of size m × n such that m=|P| and n=|T| 2 for i=0 to m do 3 for j=0 to n do 4 if pi ∈ P is connected to tj ∈ T  then   ⊳ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', p is the input place of t 5 A[i][j] ← arc weight  ⊳ arc weight is always ≥ 1 6 else 7 A[i][j] ← 0 8 end if 9 end for 10 end for 11 12 Initialize a Matrix Aplus of size m × n such that m=|P| and n=|T| 13 for i=0 to m do 14 for j=0 to n do 15 if tj ∈ T is connected to pi ∈ P  then ⊳ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', p is the output place of t 16 A[i][j] ← arc weight  ⊳ arc weight is always ≥ 1 17 Else 18 A[i][j] ← 0 19 end if 20 end for 21 end for 22 23 Initialize a Matrix A of size m × n 24 for i=0 to m do 25 for j=0 to n do 26 A[i][j] ←  Aplus[i][j] - Aminus[i][j] 27 end for 28 end for 29 Return A Table 13: Incidence Matrix Calculation Lines 10 calculate the A- matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Lines 12-21 calculate A+ matrix and lines 23-28 calculate the final incidence matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Place and Transition Invariants The methods for calculating P-Invariants and T-Invariants of a PN model have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The basic principle to compute the fundamental set of P- invariants and T-Invariants is based on Farkas Method [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The algorithm for finding P-Invariant is presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The input of the procedure is the Incidence Matrix A and an Identity matrix B, both of size m × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output is a matrix C whose rows are the fundamental set of P-Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page  95  Algorithm: P-Invariant Calculation Input: Incidence Matrix A, Identity Matrix B Output: Matrix C (rows of C = P-Invariants) 1 C ← A | B                 ⊳ Augmentation of A with m × n identity matrix B 2 for i=1 to n  do      ⊳ n = |T| 3 for each pair of rows c1, c2 in C[i-1] where c1[i]  and c2[i]  have the opposite signs  do 4 c ← |c2[i]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' c1 + |c1[i]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' c2 5 c´ ← c/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='d of each element of row c   ⊳ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='d =Greatest common divisor 6 augment matrix C[i-1]with row c´ 7 end for 8 Delete all rows of C[i-1] whose ith component is non-zero, the result is C 9 end for 10 Return C Table 14: Place-Invariants  The same procedure is used to find T-invariants by taking the transpose of the Incidence Matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Details and a discussion about the improvement of this algorithm are presented in [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These algorithms are implemented in PIPE API and can be used in form of function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Property Verification Method The outcome of algebraic analysis technique is the satisfaction or violating of a property with respect to a PN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are different methods to perform property verification however there is usually certain theorems behind the reasoning of necessary and sufficient conditions for the fulfillment of a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In Petri Net literature many solutions (proofs) for the property proving theorems are contributed and can be applied to prove different properties when required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using these theorems and the available algebraic resources a property verification method (algorithm) is developed which evaluates the conditions given in the theorem on the PN model and results in satisfaction or violation of the required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 36 presents the mechanism of algebraic verification technique in the verification framework:  Chapter 5  Proposed Methodology and the Verification Framework  Page  96  Figure 36: PN Algebraic Technique  To explain our approach we present the theorems and an example property verification method for the analysis of fairness property in a PN model in chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PNML Execution and State-space Graph It should be noted that PIPE library also offers an execution environment which can be used to run the transformed PNML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the tokens (each representing a BOM entity) eventually reaches its final state (place) then the execution is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This asserts that the model is correctly transformed and it correctly represents the behavior of its source i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PIPE library also offers a function to generate and visualize state-space graph of the PNML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This can be useful to find deadlocks and verify other system properties through graph reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8 CPN based State-Space Analysis Technique The second approach proposed for the dynamic semantic composability verification is based on Colored Petri Nets and State-space analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach effectively utilizes the potential of Colored Petri Net formalism, CPN modeling and programming language, its execution environment and supporting tools in order to verify a composed model at dynamic-semantic level with respect to the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The unique feature of this approach is its data-centric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 CPN supports level-3 PN modeling where tokens are structured and can represent data objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Also the transitions cover greater details of the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the structure and the behavior of the system can be modeled with greater details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to exploit the data-centric nature of our approach we proposed the following stages:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Refinement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Simuland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reguirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Engineering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='All constraints satisfied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase II  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reguirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Fail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Success  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Constraints as System Properties  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Property  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analvsis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Properties  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proving  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='to prove  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='SM 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Algebraic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Theorems  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Verification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Property  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='PN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Verification Method  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM to PNMIL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S1I 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Abstract Level  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='PIPE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='SM N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase Ill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase IV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='State machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Execution of the  Conceptual  Executable  Werification  Model  Model  method  Transformation  TransformationChapter 5  Proposed Methodology and the Verification Framework  Page  97  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 BOM Extension The current BOM standard lacks certain structural and behavioral semantics which are essential for modeling complex system behavior therefore we require specification of additional modalities that can help in capturing the structure and behavior of a system at a greater detail [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We therefore propose to extend the BOM conceptual model specification by applying the concept of Extended Finite State- Machines (EFSM), which is introduced and discussed with detail in [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An Extended Finite State Machine (EFSM) is defined by the tuple: M = (Q, I, Σ1, Σ2, V, Λ) where: Q (≠∅) is a finite set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' I ⊂ Q is the set of initial states Σ1 is a finite set of (send or receive) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Σ2 is a finite set of actions (Actions are the instructions to be executed and should not be confused with the BOM actions, which are used in pattern of interplay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' V is the set of state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Λ is a set of transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' each transition λ ∈ Λ  Where q and q′ ∈ Q e ∈ Σ1 is an event g is a condition (or guard) a ∈ Σ2 is an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It means if the system is at a state q, an event e occurs, and the guard g is satisfied, then action a will be executed and the system will transit to the next state q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During the firing of transition λ ∈ Λ the variables {vin} are used as input and the variables {vout} are used as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Example: This example is a modified version of an extended finite state-machine of a queue discussed in [120] and is intended to explain the notions of EFSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A queue component is either empty or nonempty, and in which insertions are done at the rear of the queue and deletions are done at the front of the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the queue has a maximum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Two events put and get are used to update the states of the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 37: Buffer Extended finite state-machine [120]  [g] / a λ = q q′ {vin} | {vout}  1  empty  nonEmpty  4  3Chapter 5  Proposed Methodology and the Verification Framework  Page  98  The EFSM model of the buffer is: M = (Q, I, Σ1, Σ2, V, Λ) where Q= {empty, nonempty} Σ = {put(string obj), get} q0: empty V = {front, rear, M, Data} Λ: Transition Specifications:  Transition 1 allows Queue to transit from empty state to non-empty when put event is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During this transition the variable rear is incremented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the parameter “Obj” of Put event is stored in Data at the rear location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Transition 2 lets Queue to revisit non-empty state when put event is received if rear is less than the maximum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During this transition the variable rear is incremented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the parameter “Obj” of Put event is stored in Data at the rear location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Transition 3 lets Queue to revisit non-empty state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is fired if rear variable is greater or equal to front+1 and less than the maximum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It will send Get event with data at the front location is sent as parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During this transition the variable front is incremented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Transition 4 allows Queue to return back to empty state when if front+1 reaches the maximum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It will send Get event with Data at front location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During this transition both front and rear variables are reset to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We apply the concept of EFSM to the BOM conceptual model, so that we can introduce state-variables and extended representation for transitions (events, guards, actions), to a form, which we name: Extended BOM or E-BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are several advantages in the BOM extension: The usage of variables (or state-variables) in BOM state-machines allows to model the attributes of a component (structure) and their effects caused due to the change of states and occurrence of transitions (behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And values of these attributes can Put(obj) [ ] / action{ rear++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Data[rear]=obj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} 1: empty nonempty {rear} | {rear, Data} Put(obj) [rear<M]/ action{ rear++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Data[rear]=obj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} 2: nonempty  nonempty {M, rear} | {rear, Data} Get(Data[front]) [rear≥front+1∧rear <M]/ action{ front++} 3: nonempty  nonempty {M, front, rear, Data } | { front } Get(Data[front])  [front+1=M]/ action{rear=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' front=0} 4: nonempty  empty {M, front, Data} | { rear, front }  Chapter 5  Proposed Methodology and the Verification Framework  Page  99  be used in the arithmetic or logical evaluation to assess trigger conditions of the transitions and to transfer variable values from one model to another model, thus in a composition, a value output by one component can be consumed by the other in the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly by introducing actions we can manipulate state-variables of a component during an occurrence of a transition and thus we can model realistic and complex behavior of a system which cannot be obtained through simple labeled transitions (such as events/actions in standard BOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the process of BOM to E-BOM extension some of the features are imported from the existing BOM while others are provided by the modelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 38 provides a comparison of which parts can be imported from BOM and how they are used in E-BOM and which parts are externally provided by the modelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A user input utility is designed for the modeler’s input where the modeler can choose initial states, select state-variables from the existing BOM entity characteristic or add new variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Assign data-types to these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Design transitions using events from existing BOM and adding guards (conditional statements) and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Currently we allow the modelers to use SML language for scripting guards and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is because the extended BOM will later be used in CPN environment which uses SML for code scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When each BOM component is extended to the respective E-BOM we proceed to the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 38: BOM and E-BOM comparison 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 E-BOM to CPN Component Transformation We propose a three layered Component Model based on CPN constructs 32 and transform previously developed E-BOM into this component model using Meta- Model transformation rules (defined later in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our proposed CPN component Model is defined as: CPN-CM = 〈SL, BL, CL〉 where: SL = Structural layer BL = Behavioral layer CL = Communication layer  32  To understand this mechanism background knowledge of CPN formalism and CPN tools environment is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We recommend to consult the online documentation [82]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM E-BOM Conceptual Model States Entity Types Initial States M Characteristics - State Variables Pattern of Interplay SV Data types M Actions Transitions State-machine · Current State States · Event Exit Action Guard Next State Action Events Input Variables Source Output Variables + Target Next State Contents -Events ·Parameters Parametertypes - Can be imported from BOM + Modeler will provide this in formationChapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Structural layer The structural layer represents physical attributes or properties of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-variables defined in E-BOM are transformed into the structural layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In CPN-CM these State-Variables are defined in form of CPN-places with distinct data- types represented by colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They are manipulated by the firing of connected transitions during the execution of the CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-variables are connected to the transitions if they are defined in their input or output sets (as discussed in EFSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The values of the state-variables are specified using CPN-tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The data- types of the tokens are the same as that of their respective state-variable places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The formal definition of the structural layer is: SL = 〈SV, ΣSV, CSV, VSV, ISV〉 where • SV is a set of places (that represent state-variables of the Conceptual Model) • ΣSV represents a set of color sets that represent state-variable data types • CSV: SV→ΣSV is a function that assigns a color set to each state-variable • VSV is a finite set of typed variables33, assigned to each member of SV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • ISV: SV → EXPR is an initialization function that assigns an initial value expression of type Type [SV] to each state-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Behavioral Layer The Behavioral Layer represents state-machine of an E-BOM component in form of CP-Net, where each state from E-BOM component becomes a CPN place of integer type, each E-BOM transition becomes a CPN transition, and the flow integer type token(s), represent the change of component’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-variables of the structural layer are connected to the transitions of this layer, if they are specified as inputs or outputs in the E-BOM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The guards and actions from E-BOM are transformed into CPN transition inscriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They should satisfy CPN-ML scripting syntax otherwise the CPN-ML complier will give errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Formally the Behavioral layer is: BL = 〈S, T, AT, ASV, AC, G, Act〉 where • S is a set of places, called states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each place is assigned a type INT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This set represents the states of E-BOM state-machine • T is a set of transitions (representing E-BOM transitions) • AT is the set of arcs, called transiting-arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each transiting-arc connects a state s ∈ S to a transition t ∈ T, and t to another state s′ ∈ S, if s → t → s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each arc in AT is assigned a variable v of type INT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a transition t is fired a token is transferred from state-s to state-s′ to reflect a change of state in the state- machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' • ASV is another set of arcs, called “sv-arcs” that connects state-variables (of the structural layer) to the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  If a state-variable SV is assigned as an input variable in the E-BOM transition, then it is connected to the transition with an incoming arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a state-variable is assigned as an output variable in E-BOM transition, then the transition is connected to the state-variable with an outgoing arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A variable v is assigned on each sv-arc whose type is same as that of the attached state-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  33 Note that these variables are built-in CPN variables that are used to define arc bindings for transporting tokens and are different from our notions of state-variables  Chapter 5  Proposed Methodology and the Verification Framework  Page 101  AC is another set of arcs, called “communication  arcs”, that connects communication ports (CP) of the communication layer to the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A cp∈CP is connected to a transition t through a communication  arc if it corresponds to a receiving event (as specified in E  BOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas a transition t is connected to a cp∈CP through an arc if it corresponds to a sending event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A variable v is assigned to each communication  arc whose type is same as that of the attached CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  G: T→EXPR is a guard function that assigns a guard to a transition t (as specified in E  BOM) such that Type[G(t)] = Bool  Act: T→EXPR is an action and assigned to a transition t (as specified in E  BOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The action can be a single executable statement or a complete procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Usually the actions are used to read data from attached input state  variables or communication ports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' to process the data and to produce an output to the attached state  variables or communication  ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Actions can also be used to type  cast the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Communication Layer The Communication Layer is responsible for communication with the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It provides interface for connecting the inputs and outputs of the components through “port places” and also provides information about the type of data exchange i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the tokens of complex data  types that carry message parameters contents, as described in E  BOM event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The transitions of behavioral layer are connected to the port places of this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The arc direction depends on the type of the transition (send or receive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a message (E  BOM event) is received at the In  port of the communication layer it causes the transition in the behavioral layer to fire as a result of which the state  machine (in the behavioral layer) progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This may also happen when a transition causes an outgoing event at the Out port which is expected by another component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Formally the Communication layer CL is: 〈CP, ΣCP, CCP, VCP, PT〉  CP is a set of port  places called “communication  ports”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A communication  port connects the inputs/outputs of the component to other components (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each CP corresponds to an event (send or receive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ΣCP represents a set of color sets that represents communication ports data types (as specified by the corresponding event parameters of the E  BOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each color set is constructed by combining the data  types of all the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there are no parameters then NULL type is assigned, which carries a blank token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  CCP: CP→ΣCP is a function that assigns a color set to each port  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  VCP is a finite set of typed variables, assigned to each CP with Type[v]∈ Type[CP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  PT: CP→ {In, Out, I/O} is a port type function that assigns a port type to each port place (as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 102  E-BOM to CPN-CM Transformation Rules The rules of transformation from E-BOM to CPN-ML are summarized as follows:  E BOM  →  CPN Component Model  Structural Layer  Behavioral Layer  Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Layer  1  States  →  State Places of Type[int]  2 Initial-States → Add token type[int], Value = {0, 1…n} for each initial- state  3 State-Variables (SV) → SV-Places of (type color-sets defined in 4), Add tokens if SV need initialization  4  SV Data Types  → Define Color Sets of SV Places  5  Events  →  Add Communicating-Place for each Event of type color- set defined in 6 6 Event Parameters and their types →  Add a color-set of type same as event parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there are multiple parameters then add color-set of type product, where each event parameter type is added in the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (If there is no parameter, use Null type) 7 BOM-Action → Transitions  8 Exit-Condition (Action, Next-State) → Connect a transiting arc from a state-place to transition and from transition to next state- place, Declare v variable of type[int] on each arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  9  Guards of Transitions  →  Add guard on each  transition  10 Actions of Transitions → Add action script on each transition  11 {Input-Variables} of Transitions → Connect SV-arc from each SV to the transition, if it is added as input variable in that transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Declare sv of Type SV on the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  12 {Output-Variables} of Transitions → Connect SV-arc to each SV from a transition if it is added as output variable in that transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Declare sv of Type[SV] on the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  13  Receive Events ∈  Events  →  Connect an arc from CP to the transition if it has a receive event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Declare cp variable of type[CP] on the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Mark the CP as in-port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 14 Send-Events ∈ Events →  Connect an arc from transition to the CP if it has a send event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Declare cp variable of type[CP] on the arc, Mark the CP as Out port Table 15: Transformation Rules  Chapter 5  Proposed Methodology and the Verification Framework  Page 103  Example Let us consider the E-BOM of the Queue example discussed in the previous section:  Based on the above E-BOM the transformation will result in the CPN-CM model shown in Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This model is completely executable in CPN environment and can be composed with other components through communication layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 39: CPN-CM represention of Queue component  The red area represents structural layer and consists of state-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The green area represents behavioral layer and shows the state-machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Blue area represents communication layer and shows the Communication Ports (CPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When some data is received at CP-Put, the value of rear and Max variables are read as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Guard rear<M is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A token is taken from the empty state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Action is executed that E-BOM: Queue Component States {empty, nonempty} Initial States {empty} State Variables & their Data types {front : integer, rear : integer, Max : integer, data: string[ ]} Events {Put (obj:String), Get(obj:String)} Transitions  Put Put Get Get Current State empty nonempty nonempty empty Event Put(obj) Put(obj) Get(null) Get(null) Guard [ ] [rear<M ] [ rear≥front+1∧rear <M ] [ front+1=M  ] Action rear++ rear++ front++ rear=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' front=0 Input Variables rear rear, M rear, front, M, data[front] rear,  front, data[front] Output Variables rear, data(obj) rear, data(obj) front rear, front Next State nonempty nonempty nonempty empty  Structural Layer  Behavioral Layer  Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Empty  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[rearAM]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='rea  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='put  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Put  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Put  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Input (rear):  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='output (R):  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='STRING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='rtar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='atn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='vet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1H+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='al R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(regpput)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='rear+1s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='end  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Nonempty  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M Item  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FonE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ffront+1=rear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='then 1 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='elseempty  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[rear>=front+]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='andalsorearM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Front  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Get  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Get  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Input(front):  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='output (Fy:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='STRING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ket  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='val F=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='font+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='endChapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 104  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='increments the rear variable then the transition Put is finally fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' After which a token is produced at the nonempty state showing the state-transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also rear and data variables are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Max variable retains the token (due to bi-directional arc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If Put is fired again it will repeat the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If Get is fired provided the guard is satisfied, then front, Max and data variables are read as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The data (picked from the front of the queue) will be sent to the out-CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the data is emptied the token will be sent to the empty state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Automated Transformation Tool In order to automate the E-BOM to CPN transformation process, we develop a transformation utility, which takes an E-BOM component as input and produces CPN- code for all three layers of CPN component model automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The code follows CPN-XML specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For each E-BOM component, a separate CPN sub-page is generated (programmatically) and the necessary CPN elements (places, transitions, arcs, color sets, variable declarations, initial markings multi-sets, guards, actions, code segments, CPN ML functions, ports, ports-tag) are generated in one CPN output file, which can be loaded in CPN tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Once all the CPN models of the BOM composition are generated, the modeler creates a main model and “manually” combines the generated CPN-CM modules (using CPN hierarchical features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of this step is a composed CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The modeler is also required to initialize each component with data (in form of token assignments i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the initial values of the tokens of state-variables and initial states of the state-machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3b Evaluation The S3b constraint in the requirement specification requires that “If the conceptual model is transformed into an executable model, the latter should correctly represent the structure and behavior of the former” (see Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore we have to compare each CPN component (executable model) with its respective BOM (Conceptual Model) to check that its structure and behavior is preserved after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  To show that S3b holds after the transformation we rely on the following assertions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As BOM is extended to E-BOM hence BOM ⊂ E-BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Any information added by the modeler in E-BOM cannot cause loss of structural information of BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore E-BOM structurally preserves BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To check that the generated CPN component contains all the Events and their parameters, States and their exit-conditions, Actions and their senders/receivers we need at least one transformation rule that is responsible to transform these elements: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Rules 6 & 7 (see Table 15) are responsible for transforming Events and their parameters into CPN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Rules 1 & 8 are responsible for transforming states and their exit- conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Rule 7 is responsible for specification of BOM-actions as transitions in CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also rule 5 defines port-places which are used to connect senders or receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Existence of Rule 1, 6, 7 & 8 confirm that the structure of corresponding BOM is preserved in the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 105  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN Tools provide a built-in compiler for the compilation of CPN models and report if there is any syntax error in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The absence of error confirms that the transformed model is structurally consistent and behaviorally functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For the behavioral bi-similarity we propose an inspection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first we evaluate that all the generated components possess the same behavior as defined in the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So we test the functional output of each CPN model by giving the needed inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If by giving correct inputs, the model produces desired output then its functional behavior is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To perform functional testing, the modeler initializes all the IN-type communicating-ports (CPs) with tokens of required parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (See Figure 39 for an example where IN-CP “Put” is initialized with tokens of type String).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then the model is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the model produces desired output on the corresponding Out-CPs (In Figure 39 the desired output should be a token of type string retrieved at Get Out-CP), then the functional test is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The modeler performs functional test on all generated CPN components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the second step, when all CPN components are composed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the socket-places of the main model are connected to the Communicating-Port places of the CPN components then the modeler is required to inspect that CPN components are connect exactly according to the Pattern of Interplay of the BOM composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Also when the composed model is executed the sequence of sending and receiving events from one component to another (which can be observed at the main model by seeing the movement of tokens) follows the pattern of interplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the execution is according to the pattern of interplay and the components make progress until they reach their final states, then we say that the behavior of the transformed model is bi- similar to the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This confirms the satisfaction of S3b constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The execution can be automated or interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In automated mode the choices between multiple progressive paths are randomly picked whereas in interactive model the modelers can pick a path of his choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using this option the modeler can probe paths that can lead to a successful execution scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' During the execution CPN tool also offers Data Collection Monitors for recording the data values, which are very valuable for collecting statistics and results of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Verification of the composed CPN model In the next step, the state-space analysis is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first the state-space of the composed CPN model is generated using CPN state space calculation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 a state-space is a graph of nodes (of system-states or markings) and arcs (transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the state-space is generated, different query functions can be used to explore the state space graph for various verification questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A query function is like an algorithm that explores the state-space graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These algorithms are based on theoretical concepts of Petri Nets state-space analysis and are used to verify PN properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore we translate a system property given in the requirement specification into a suitable PN property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There have been a lot of contributions in the PN literature in specifying PN properties and methods of reasoning of their satisfiability or violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In CPN state-space analysis, the existing methods can be utilized in developing query functions for their respective PN properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN tools provide some built-in-functions for the common query tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also propose a library of additional functions to perform queries specific to our  Chapter 5  Proposed Methodology and the Verification Framework  Page 106  composability verification framework and the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 40 illustrates the state-space analysis of a composed CPN model using a query function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We divide these query functions into two categories:  (i) General System Properties This category includes commonly known system properties such as freedom of deadlock, live lock, starvation, or existence of boundedness, mutual exclusion, fairness, sequentiality, time- synchronization etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' if any of these or similar system properties are included as a constraint in the requirement specification then it is translated in CPN terms and a suitable query function is selected from the Function library to perform verification using state-space of the composed CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 40: CPN State-space analysis For instance a deadlock freedom property can be translated into CPN terms as: “An absence of a marking with no outgoing arcs in the entire state-space graph” So essentially we need to find such a node in the state-space graph that violates above condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If no such node is found then the model is said to be deadlock free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A library function ListDeadMarking() returns a set of all those markings (if any) which have no outgoing arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the result of this query is an empty list, then we assert that the model is deadlock free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly there are other library functions that deal with the evaluation of other system properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (ii) Scenario Specific Properties These properties are specific to the scenario (of the real system) under which the model is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The objectives or goals from the requirement specification are usually translated in form of scenario specific properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In CPN terms a typical goal or objective can be translated as a certain desirable marking, where the values of state- variables in structural layer evaluate to a particular criteria or reaching of particular state(s) in behavioral layer is desired or certain data at the output port(s) of the communication layer is looked-for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A goal or objective can be expressed in a combination of all these possibilities too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Scenario specific properties may also include certain safety or liveness assumptions, which represent certain desirable (or un-desirable) situations that must (or must not) occur in order to satisfy (or violate) the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These properties are mostly the CPN translations of the constraints defined in the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conceptual Model Requirement Specification  System Property  Satisfie  Violate  State Space  System property  CPN Translation  Composed CPN  Model  Query function  (Algorithm)  Function  Library  Chapter 5  Proposed Methodology and the Verification Framework  Page 107  Unlike general system properties, verifying scenario specific properties is not a standard operation, and depends on the way they are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Most commonly, we make use of our proposed library functions: IsEqual(), IsNotequal(), IsBetween(), IsUpperBound() or IsLowerBound() to construct a “predicate”, that serves as a condition evaluation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then we use SearchNode(predicate)function to find those nodes, which satisfies the predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If one or more nodes are found, then it is verified that the goal is reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In cases, where it is important to know how a sequence of the occurrence of transitions, leads to a particular situation when a property is satisfied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', how an objective or goal is reached) we use SearchArc()function with the predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This tells us the path in the graph that leads to fulfillment of a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also develop an export function, that creates a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='DOT file of the entire state-space and can be viewed in graph tools such as GraphViz or Gephi, for visualization and performing further tests on the graph such as finding certain paths/shortest paths/longest paths between two particular nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When, a CPN composed model satisfies all the properties in the requirement specification, we say that it is verified at dynamic semantic composability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In chapter 8 we discuss a Field Artillery Scenario as an example of CPN state-space analysis to explain our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  An example of translating a scenario-specific property in CPN terms is a restaurant model where we assume that customers may leave the restaurant without paying the bill because they have been waiting for a long time for the waiter to bring bill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This act of the customers is known as “Balking” and is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Its translation in CPN can be as follows:  “There should be no arc with the name “balk” that leads to any marking in the graph”  Arcs are generated due to firing of the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Existence of balk arc means somewhere in the model an incidence occurred when a customer balked (by firing balk transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So essentially we need to find that such arc is absent in the state- space graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This can be done by using SearchArc()function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Note that this is a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There could be cases in which a sequence of transitions (called traces) or cycles are searched to verify a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reference source not found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' describes the overall process of state-space analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 108  Figure 41: State space Analysis Technique  State-Space Reduction Technique In order to alleviate the well-known problem of state-space explosion we propose a reduction technique called “compositional state space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main idea of this technique is that in a hierarchical composition of CPN model, we propose to only consider the places in the main model and treat all the composed components as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The inputs and outputs of each component can be observed using the flow of Tokens and the data they carry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore in the state-space graph we only keep the markings in which any token is present in the Main model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', any of the place in the main model has at least one token) and delete all other nodes in the state-space graph using the algorithm presented in Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The resultant graph will be a reduced form of the actual graph and only considers those markings that reflect a compositional state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is called compositional state-space because it only represents a part of the actual state-space which is the result of interactions due to the composition of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In our experience this subset state-space of the whole state- space is sufficient to evaluate whether the objectives, goals and the constraints are satisfied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Refinement  Phase I  Function  Simuland  Library  Reguirements  Select  Engineering  Phase V  Phase II  Analvsis  Query function  (Algorithm)  Technique  RS completely satisfied  Reguirements  State Space Analysi:  Analvsis  CPN propert:  Translate  Property  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN State Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="Modeler's Input  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM to E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Extension  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modehng  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Abstract Level  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN CM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Automatbe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transfomation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Editor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase IV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase Im  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CRepeat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Executable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='I to N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN Model Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Trans formation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Trans formationChapter 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Proposed Methodology and the Verification Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Algorithm: Compositional State-Space Generation Input: Original State-Space Graph G |              ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Output: Reduced State-Space Graph G 1 {Vertices}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='← Get-Vertices(G) ⊳ Retrieve all the nodes of the graph in a collection 2 for v∈to {Vertices} do 3 If  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='False ← Is-Filtered(v) then 4 G ← Remove-Vertex(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) 5       Else 6 next 7 end if 10 end for 11 Return G ⊳ Reduced state-space 12 13 Procedure Remove-Vertex(Graph G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Vertex v) 14 {Predecessors} ← Get-Predecessors(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) ⊳ Retrieve all the predecessor vertices of v in G 15 {Successors} ← Get- Successors (G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' v) ⊳ Retrieve all the successors vertices of v in G 16 for p∈to { Predecessors } do 17 for s∈to { Successors } do 18 G ← Add-Edge(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' “DIRECTED”) ⊳ Add a directed arc from each predecessor 19 to each successor ⊲ 20 G ← Delete-Vertex(v) 21 end for 22 end for 23 Return G 24 25 Procedure Is-Filtered (Vertex v) 26 If {places}← GetData(v) then ⊳ Each vertex is a marking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' which contains data of all 27              ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='⊳ the places of the model with their names and their token values ⊲ 29            ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='for p∈to { places } do 30 if p is a main place and it is not empty then 31 Return  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TRUE ⊳ A valid marking with a non-empty Main place is found 32 else 33 next 34 end if 35 end for 36 ⊳ if the loop is complete then there is no main place which is not empty 37 Return FALSE Table 16: Compositional State-space generation algorithm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Using the Compositional state-space generation algorithm we can filter unnecessary nodes and reduce the size of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The current limitation of this approach is that we first need to construct the actual state-space which is a bottleneck if the model is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' But this limitation is due to the fact that the process of CPN state-space graph generation cannot be externally modified otherwise if the principle of our reduction technique is applied to the state-space generation algorithm it will directly generate the reduced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In chapter 8 we will present the results of our reduction technique by applying it to the Field Artillery example model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 110  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9 CSP based Model Checking Technique The third approach proposed for the dynamic semantic composability verification is based on Model Checking, which is widely accepted as formal technique for software verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this approach we propose to use Communicating Sequential Processes formalism as a model description language and Process Analysis Toolkit (PAT) as its execution and verification environment in order to verify a composed model at dynamic-semantic level with respect to the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The strength of this approach is in its ability to answer a large variety of verification questions due to the fact that the verification criteria can be specified using LTL, CTL or any of the temporal logic extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The Model Checking technique is becoming more promising and acceptable by many software verification users since there is an abundance of improved algorithms, efficient data-structures and faster techniques which are constantly being contributed by the model checking community in order to manage large models with complex modeling requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We propose to integrate CSP based model checking verification approach in our composability verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following stages are proposed in order to perform composability verification using model checking approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 BOM Extension The E-BOM extension for CSP based Model Checking approach is also inspired from the concept of Extended Finite State-Machine as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The extended BOMs for CSP can also have state-variables but since CSP# specification does not allow declaration of strings or higher data-types, the state- variable definitions are restricted to integer and Boolean34, which in our experience are sufficient to model the behavior of BOM components using CSP (or otherwise it is required to narrow it down to a less detailed version of the component, where only the necessary behavioral details are specified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The transitions of E-BOM contain current-state, event (with parameters), guard, actions and next states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However in this case the action scripts are written in CSP# specification language instead of CPN- ML language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And instead of input and output variables, we have local variables which are accessible only to the component and global variables which are accessible to all the components of the composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some additional information such as time constraints and probability factors are further proposed to be included in the BOM extension so that the behavior of complex systems such as real-time systems and probabilistic systems can be modeled and verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since Timed-CSPs support a number of timed behavioral patterns to capture quantitative timing requirements, such as delay, timeout, deadline, therefore we suggest using these patterns as time functions in the BOM extension, which helps in the automatic transforming of E-BOM into Timed-CSP components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  These time functions are essentially assigned to the E-BOM transitions as explained in the following table:  34 Generally the high level or user defined data types are not permissible in most of the model checking description languages due to the economy of state size, and to avoid risk to state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However if the use of such type is inevitable, the PAT tool do provide mechanisms of importing classes from external libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If this is the case then the modeler is required to program the components in PAT manually, instead of relying on our automatic transformation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 111  Time Function Usage and Explanation Wait[duration] Wait is assigned to model the delay in an activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An enabled transition waits for the given duration before it is fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' TimeOut[duration, next] When a timeout function is assigned to a transition, it waits for an event to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the event occurs before timeout, it transits to the next state described in the transition definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' otherwise it transits to the next state described in the timeout parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Deadline[duration] A transition is constrained to fire when the deadline is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The difference between Wait[] and Deadline[] function is that the former makes the system inactive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', it cannot do anything but wait, whereas when the latter is used the system is active and can respond to events etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', until its deadline is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 17: Time functions in E-BOM Using any of these time functions during the BOM extension is useful to capture the behavior of the real-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to further capture the behavior of the complex reactive systems, we also propose to introduce probabilistic factors in the BOM extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These probabilistic factors can either be used to model the system behavior in form of Markov Decision Processes (MDP) as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Or the probability factors can be used to model random time delays, timeouts or deadline, using a particular probability distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  For modeling the MDP behavior probability factors can be assigned to multiple transitions of a component’s state using the following notation provided by the PAT tool: Pcase { [P1]: Transition 1 [P2]: Transition 2 … [Pn]: Transition n } ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Where  ∑ 𝑃𝑖 𝑛 𝑖=1 = 1 For randomizing time functions, we propose to assign the commonly used probability distribution functions as parameters in the E-BOM: Probability Distribution Functions Usage and Explanation Normal[mean, variance] Returns a random value from a normal distribution with a given mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (Since PAT does not support higher types so we have confined these functions to use integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Discrete[a, b] Returns a random value from a discrete uniform distribution between a and b (a and b included), such that a < b Exponential [1/lambda] Returns a random value from a an exponential distribution with parameter 1/lambda Table 18: Probability Distribution Functions  In order to implement these assignments we develop an external function library in C# which can be imported and used in PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A call to these functions generates a random number according to the specified probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Beside the time functions, these functions can also be used to generate random values for global or local variables, which can help in modeling different probabilistic system behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When each BOM component is extended to the respective E-BOM we proceed to the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 112  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 E-BOM to CSP# Transformation At this stage, each E-BOM component is transformed into a CSP# process component and composed into an executable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main idea of this transformation is based on [121], which discusses the transformation of UML state machines to CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We however extend this transformation with Communication channels, Time-functions and probability factors to be able to use it for E-BOM transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 19 shows the rules used in the transformation process: E-BOM  CSP# Statement and Description States → State-Name() Each state in E-BOM is defined as a CSP process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Final-State() = Skip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This statement defines a final state in CSP where Skip is a reserved word means the process terminates successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If no such statement exists in any component of the composed model then it is said to be a non-terminating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Component → Component-Name = Initial-State() or Component-Name(i) = Initial-State(i) An initial state is defined and assigned to the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a component has multiple instances it is passed a parameter ‘i’ which represents the instance number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Transitions → Simple Transitions:  State() = [guard] event !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parameters {action} → NextState();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The transitions are defined using the above format, where State() is the current state of a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [guard] is a conditional statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If it is true only then the transition will be enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Event is sent using ‘!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' symbol or received using ‘?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' symbol through an event channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For each event in an E-BOM component, we define a channel as follows: channel event-name 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In CSP# “0” means the buffer size of the communication channel is zero, which further means that it is a synchronous channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Parameters are the values that are passed during an event exchange and a separated using ‘.’ Actions are scripts that should be executed when the transition is fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Usually these actions are used to update local or global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' NextState() is the new state which will be reached when the transition is fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It must be defined within the CSP component body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Transitions with Time functions:  Following statement represents a transition with timed-functions:  State() = [guard] Wait[d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  event !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parameters {action} → NextState();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  State()=[guard] event!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='parameters {action} → NextState() deadline[d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  State()=[guard]event?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='parameters{action}→NextState() timeout[d] NextState2 ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Note that in case of timeout, the transitions should only be receiving an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Markov Decision Process style Transitions: State() = pcase{ [Prob1]: [guard] event !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parameters {actionA} → NextStateA()  [Prob2]: [guard] event!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parameters {actionB} → NextStateB() };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Note the postfixes A and B in action or next states of the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using this CSP# code style multiple transitions can be modeled with different probabilities for either creating a variation of the action which is fired when one of these transitions is selected in a simulation run, or the next states (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Transitions with Probability distribution functions: For using probability functions, at first it is required to import our external probability function library using:  #import "PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ProbabilityDistributionFunctions";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Following are some examples of how the function calls can be made: var x = call(Normal, 10, 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An integer variable is defined which will randomly be assigned a value using normal distribution with mean=10 and variance=4  Chapter 5  Proposed Methodology and the Verification Framework  Page 113  Wait[call(Exponential, 1/4)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Delay function with an exponential distribution, where the inter-arrival rate is ¼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' State- variables → var Variable-Name=Initial-Value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' or #define Constant initial-value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In CSP# weakly typed variables are used which means that while declaring a variable, the type is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The global variables can be accessed by all components whereas the local variables can only be accessed by the component they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Component → Component-Name = Initial-State() or Component-Name(i) = Initial-State(i) An initial state is defined and assigned to the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a component has multiple instances it is passed a parameter ‘i’ which represents the instance number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Composed Model → Composed-Model = Component1 ||| Component2 ||| … ComponentN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The composed model (name) is defined as a composition of CSP process components with an interleaving operator ‘|||’ between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However if there are broadcast events (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', one event is sent to all components);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' or one to many;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' or many to one synchronization events are used then a parallel operator ‘||’ is used to compose CSP process components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 19: E-BOM to CSP# transformation rules We develop a transform tool that takes all the E-BOMs as input, and outputs a single composed model using CSP# description language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The generated CSP# composed model can be opened in PAT tool and compiled for checking errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If no errors are found then the transformed model is said to be structurally consistent and behaviorally functional and it is ready for simulation and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can also be directly compiled, executed and verified using command line operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S3b Evaluation The S3b constraint in the requirement specification requires that “If the conceptual model is transformed into an executable model, the later should correctly represent the structure and behavior of the former” (see Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to evaluate S3b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', to check that the structure and the behavior of the generated executable model (CSP composed model) correctly represents its conceptual model (BOM composition), we propose following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For each CSP component, manually inspect that it contains all the states that exist in its corresponding BOM component 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Inspect that the exit condition(s) of each State in BOM correspond to a transition(s) and a next state(s) in CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Execute the generated CSP model in PAT simulator and observer that all the components reach their final states (or in case of a non-terminating model each component re-visit its initial state iteratively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Step 1 & 2 confirms by inspection that the structure of the generated model correctly represents its conceptual mode whereas step 3 confirms that it behavior is bi-similar to the conceptual model and therefore satisfies S3b constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Verification of the composed CPN model At this stage, the CSP composed model undergoes composability verification using PAT model checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first the requirement specification is translated into CSP# property (or assertion) description language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  This language is based on a mix of classical Linear Temporal Logic (LTL) and its different extensions such as Real-Time LTL and Probabilistic LTL and is used to construct assertions (verification questions) of various types, such as reachability properties, safety properties, liveness properties, deadlock freeness etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We use the syntax of assertion specification language of PAT to translate the objectives and constraints of given requirement  Chapter 5  Proposed Methodology and the Verification Framework  Page 114  specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Following are some generic examples of how to specify PAT assertions:  1 #assert System deadlockfree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This assertion checks deadlock freedom in the ‘System’ 2 #assert System reaches goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This assertion checks that whether the ‘System’ can reach its goal (by receiving a goal event with ‘0’ parameters) 3 #assert System |= <>goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is an equivalent LTL assertion It checks if the goal is eventually reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 4 #assert System |= []<>goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This LTL assertion checks if the goal is always eventually reachable by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Note that it is different from assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 5 #assert System |= <>goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 deadline[50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This assertions verifies goal reachability with time constraint i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', its checks if the goal is reachable within 50 time units or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 6 #assert System |= <>goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 with prob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This assertion checks the min and max probability of the goal reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 7 #define goal (Some-Variable == True);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #assert System reaches goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is another way to verify goal reachability, where the goal definition is based on some value of a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 20: Some examples of PAT Assertions  When an assertion is defined and its syntax is correct, we can verify it by running the PAT model checker and select the desired assertion from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The model checker will present the verification results with success, showing that the assertion is verified or it will provide a counter example if the assertion is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In chapter 9, an example of field artillery is presented to show how a CSP composed model is verified with requirement specifications defined as PAT assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 42 describes the overall process of state-space analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 5  Proposed Methodology and the Verification Framework  Page 115  Figure 42: CSP based Model Checking Technique  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='10 Summary In this chapter the proposed composability verification framework is discussed in details with its structural and functional specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each activity, algorithm, technique and the process is explained in the perceptive of Component based M&S life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The composability verification is performed at three levels of composability called static, semantic and dynamic-semantic composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The main objective of the proposed framework is to verify composability at these levels with respect to requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The first two levels are suggested to be evaluated using static-analysis techniques whereas the third level is proposed to be verified using dynamic analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first the behavior of the composed components is evaluated using State-machine matching technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If they pass this step, they are subjected to one of the three proposed approaches called (i) Algebraic Analysis Technique, (ii) State-space analysis technique or (iii) Model checking for dynamic-semantic composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The choice of these approaches depends on the nature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In chapter 10 we will present some guidelines on how to choose an appropriate approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  When the entire composability verification process is successful, it implies that the BOM based composed model is structurally and behaviorally consistent, it is composable at syntactic, semantic and dynamic-semantic level and is correct with respect to the given requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Refinement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Simuland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reuirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Engineering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase II  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rs completely satisfied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reguirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CSP Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Checking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analvsis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='GlobalVariables  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="Modeler's Input  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='and Modal Checker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Translate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM to E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Extension  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Abstract Level  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CSP#  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Automztis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transfomztion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Edlitor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Phase I1I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Assertions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Executable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='04  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CSP Model Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 116  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composability Verification Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 5 mainly presented the specification of our proposed composability verification framework including details of different modules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' their mechanics and the procedures they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter we present how to use our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be used as a manual of our composability verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At the end of this chapter we also provide necessary recommendations for the selection of appropriate approach based on the given inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The description of shapes used in the following flow diagrams is as follows: Object, Data, Model, Component etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Any shape of this color express a 3rd party tool  List, Collection or Set of objects, Data, Model  Stop  means  that  the  process has failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Process or action  Go means it is successful, therefore process with the implementation phase  Iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Compare two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Extension or Transformation  of object  Compare multiple objects  Extension or transformation  of many objects  Comments  Data, Process, Information  flow  Page connector  Repeat  process  (Go  to  previous step)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Composability Verification Process Figure 43 to Figure 53 illustrate the composability verification process in the form of a flow chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (The illustrated steps are explained later in this section): Stop Go X  Chapter 6  Composability Verification Process  Page 117  Figure 43: Formulation of Simuland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Requirements and Conceptual Model Abstraction Real System Requirement Engineering Simuland UML diagrams (or others) representing structure and behavior of the system Formal or Informal BOM Repository Construct components from scratch and store them in the repository Search components that match the requirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Candidate BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Select suitable BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Filter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='and select the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='most  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='suitable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='set of BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='that match the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='requirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Compose selected BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Matching and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Selection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requirements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Objectives  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Constraints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Selected BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composability Verification Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 118  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='If all candidate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='are  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='tried  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='without  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='success  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='repeat step 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Syntactic Matching Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Start  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Rule  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Syntactic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Matching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='process to check that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='the components can  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='correctly fit to gather  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='and their inputs and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='outputs match each  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Sender BOM i  Events  Entities  States  Actions  Receiver BOM j  Events  Entities  States  Actions  Check SM Rule 1:  Name of the send event and  receive event  should  be  same  Send Event  Rule 1  Passed  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  No  Stop  and  repeat from  step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8  Yes  Proceed  to Rule 2  Check SM Rule 2:  Each send event should have at  least one corresponding receive event and vice versa  Sender BOM i  Events  Entities  States  Actions  Receiver BOM j  Events Entities States Actions Equal Names Rule-2 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes Proceed to Rule-3 Check SM-Rule 3: The number of parameters (content characteristics of event types) of the send-events should be the same as the number of parameters of the receive-events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Sender BOM-i Events Entities States Actions Receiver BOM j  Events Entities States Actions Equal no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' of parameters Receive-Event Rule-3 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes 9 10 11 2 1 Figure 44: Syntactic Matching Process  Chapter 6  Composability Verification Process  Page 119  Static-Semantic Matching Process Check SSM-Rule 1: “Area-of-Interest” attribute of all the actions should be exactly same as that of POI or should belong to an equivalent class in the respective ontology Rule-1 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 12 Yes Proceed to Rule-2 13 3 2 Start Rule based Static- Semantic Matching process to check that the composition is meaningful and the components can correctly understand each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Pattern of Interplay (POI) BOM-1 Events Entities States Actions BOM N  Events  Entities  States  Actions  BOM OWL  AOI Purpose Data Type Units Check SSM-Rule 2: “Purpose” attribute of all the actions should be exactly same as that of POI or should belong to an equivalent class in the respective ontology Rule-2 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes Proceed to Rule-3 Check SSM-Rule 3: Data types of each element in the event parameters of the send-event and receive-events should be of same class, equivalent class or should be in direct hierarchical relationship Rule-3 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes Proceed to Rule-4 14 15 Figure 45: Static-Semantic Matching Process  Chapter 6  Composability Verification Process  Page 120  Check SSM-Rule 4: The units of the quantities expressed in each element in the event parameters of the send-event and receive-events should be of same class, equivalent class or should be in direct hierarchical relationship No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 3 Yes If Static-Semantic Matching Rule 1, 2, 3 & 4 are satisfied then we can confirm that the communication among the components is meaningful as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Perform State-machine Matching Process  Rule-4 Passed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 15 16 Start Dynamic-Semantic Composability evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the first step, the state- machines of all the composed BOM components are matched to check their behavior compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Candidate BOMs Transform BOM State-machines to SCXML Abstract level execution SCXML 1 SCXML 2 SCXML N Is Final?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Stop and repeat from step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes The composed components are behaviorally compatible!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the SCXML instances of composed BOM components are executed at an abstract level, and they all reach their final states then the state-machines are said to be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 4 17 18 Figure 46: State-machine Matching Process  Chapter 6  Composability Verification Process  Page 121  Figure 47: Approach Selection | PN Algebraic Technique  4 Select an appropriate approach for dynamic- semantic composability analysis PN Algebraic Technique 19 CPN State-Space Analysis Model Checking If standard BOM components are composed (with no information available for their extension) and only general structure and behavior is to be analyzed then this approach is suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Conceptual Model Transform BOM to PNML PNML Model 23 21 20 22 If details of the BOM components are available which are required to extend them into to E- BOMs and functional specification of the components is to be evaluated then this approach is suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  5 Transform BOM- state machines into a single PN model If the model requirements contain time constraints and (or) the model possess non- deterministic behavior and (or) the model has a large number of state (but does not require detailed enumerations) then we propose to use this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 6 PNML Execution 24  If  the  execution  is  successful such that it  leads to final states,  then the model satisfies  S3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Meaning  the  behavior  of  transformed  model  correctly represents its  conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  PIPE execution  Environment  No  Stop  and  repeat from  step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Yes  Continue  Verification  7  Success  Chapter 6  Composability Verification Process  Page 122  Figure 49: Implementation  Requirements  Can  did  Can  did  Objectives  Can  dida  Can  dida  Constraints  Property Verification  Translate objectives  and Constraints into  PN properties  26  25  Next  property  Algebraic Computation  Resources  (Incidence Matrix, P Invariants,  T Invariants etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=')  \\  Property Proving  Theorem  Property Verification Method PIPE Function library Calculate algebraic computation resources of the PNML model using PIPE library functions Construct a property verification method using an appropriate PN property proving theorem RS Satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' No Composability verification failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Modifications in the conceptual model are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Yes Composability verification is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='The conceptual model is qualified for the implementation phase Go Stop PNML Model 7 Composed Model Simulation Model Experimental Model Code Design of Experiment Simulation Results Simulation Go Figure 48: PN Algebraic Technique (continued)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composability Verification Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 123  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 50: State Space Analysis Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='\\  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM to E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Extension  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeler’s input is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='required here to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='extend BOM into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transform E BOM to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN Component Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Each  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='automatically  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='transform into CPN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='component Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN Component models 28 5 E-BOM Extension Utility CPN-CM is our proposed component specification based on CPN language Structural Comparison with the Conceptual Model Does the transformed model contain all Events,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Actions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' States,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' exit-conditions as specified in the conceptual Model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Is the behavior of the transformed model bi- similar to the conceptual model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Perform functional test  on each component to  check that the inputs  produce desired output  according  to  the  conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Perform CPN execution to compare that the progress is made by all the components according to the pattern of interplay defined in the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Behavioral Comparison (Functional Testing) CPN Execution Environment Perform Model inspection and check that all the elements of Conceptual model are present in the transform model Perform functional testing: Initialize inputs of each transformed component separately and execute the component in the CPN execution environment to check if it produces desired output according to the conceptual l model  Is  Successful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  No  Stop  and  repeat from  step 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8  Yes  S3b  partially  satisfied  Chapter 6  Composability Verification Process  Page 124  Figure 51: State Space Analysis Technique (continued)  Requirements  Can  did  Can  did  Objectives  Can  dida  Can  dida  Constraints Translate objectives and Constraints into CPN properties Next property Modeler is required to manually compose all the generated CPN components into a main CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Compose CPN Components CPN Composed Model Initialize and Execute CPN Model Successful No There are exceptions in the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Check and Repeat step 29, or 28, 26  29  30  Generate State Space  31  Yes  S3b completely  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Perform CPN Property  Verification  32  RS  Satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  No  Composability  verification failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Modifications  in  the  model  are  required  Yes  Composability  verification  is  successful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' go to implementation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Go  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Stop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Hierarchical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeling Tool  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN Execution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN State Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Analysis Tool  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Property Verification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Query Function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='(written in CPN ML)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN ML  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Programming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CPN Query  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Function library  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Next  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Composability Verification Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 52: Model Checking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Continue from step 22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='\\  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Conceptual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BOM to E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Extension  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Modeler’s input is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='required here to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='extend BOM into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transform E BOM to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CSP Process  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Each  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='E BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='automatically  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='transform into CSP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  CSP Process Components 34 6 E-BOM Extension Utility CSP process components are represented using PAT’s CSP# specification Structural Comparison Does the transformed model contain all states, and events as specified in the conceptual Model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Behavioral Comparison PAT Simulation Environment Perform Model inspection and check that all the elements of Conceptual model are present in the transform model Perform behavioral similarity evaluation by simulating the CSP composed model in the PAT simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If it reaches final states then it correctly represents the behavior of its corresponding conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Pass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  No  Stop  and  repeat from  step 33, 19  or 5  9  Yes  S3b satisfied  E BOM with Time  constraints  and  probabilistic factors  Compose CSP  Components  CSP Composed Model  35  Compose all the generated  CSP components using  parallel operator  Chapter 6  Composability Verification Process  Page 126  Figure 53: Model Checking (continued)  The Composability Verification Process is explained as follows:  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Formulation of Simuland, Requirements and Conceptual Model In step 1, the Real system is studied and a suitable simuland is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be described formally or informally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that UML diagrams are used to describe the simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The system is also studied to gather requirements and formulate requirements using our proposed formal requirement specification method (step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With this information at hand, suitable components are searched in the BOM repository, with an assumption that a composition of these components will form a conceptual model that represents the simuland (step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a desired component is Requirements Can did  Can  did  Objectives  Can  dida  Can  dida  Constraints  Translate objectives  and Constraints into  CSP assertions  Start PAT  Model Checking Tool  36  Verify Assertion  37  RS  Satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  No  Composability  verification failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Modifications  in  the  model  are  required  Yes  Composability  verification  is  successful, go to implementation  phase  Go  Stop  9  PAT Model  Checker  Assertions  LTL, CTL, RT LTL, PLTL  Next  Assertion  Chapter 6  Composability Verification Process  Page 127  not found it is constructed form the scratch, added to the repository, and then used in the current context (step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The discovered components are called candidate components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Among these candidates, most suitable ones i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', those that best match the simuland and the given requirements, are selected (step 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These BOM components are composed and a conceptual model is constructed (step 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We recommend that the modeler also creates a formal model of the conceptual model using our proposed BOM formalism and graphical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This will help in documentation and understanding details of conceptual model and its composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Syntactic Matching Process When the verification process starts the Composed BOM model (conceptual model) is passed through a rule-based static analyzer to verify the composability at syntactic level (step 8-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If this level is passed then the constraint S1 (as defined in Table 10) is satisfied and only then the model is cleared for the next step (otherwise the verification process is stopped and another candidate selection is picked, composed and this step is revised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Static-Semantic Matching Process In the next step the components are analyzed at static-semantic level using the semantic analyzer (step 12-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When this step is passed then the constraint S2 (as defined in Table 10) in is satisfied and the BOM composition is ready to be verified at dynamic-semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 State-machine Matching Process At this level, the first step is to perform state-machine matching of BOM components using State-machine checker (step 15 – 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A successful state-machine matching satisfies the constraint S3a (as defined in Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Approach Selection for Dynamic-Semantic Composability Verification In the next stage the verification framework offers three choices of verification technique for the analysis of dynamic-semantic composability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The modeler can choose algebraic technique if there is no information available to extend the BOM components into E-BOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the conceptual model will be transformed into PNML without requiring any extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the modeler has details and data available to transform BOM into E-BOM and the model does not represent a real-time system, then it is highly recommended that the second proposed approach (CPN state-space analysis) should be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the model represents a real-time system and it is stochastic in nature then the modeler should choose the third approach (Model checking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These are general guidelines and are not concrete rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The ultimate choice of the approach depends on the nature of the model, nature of the requirement specification properties and the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 6  Composability Verification Process  Page 128  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6 PN Algebraic Technique When Algebraic technique is selected, at first the conceptual model is transformed into PNML model (step 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This PNML model is executed in PIPE execution environment to evaluate S3b (step 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If successful then the requirement specification properties are taken one by one and translated into a PN property (step 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thereafter a property proving theorem is selected that proves this PN property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on this theorem a property verification method is constructed inform of an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Running this algorithm proves or falsifies the requirements specification property (step 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If all the properties in the requirement specification are satisfied then the model is successfully verified otherwise the process is stopped and model refinements are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7 State-Space Analysis Technique When the CPN state-space analysis technique is selected, at first each BOM component is extended to E-BOM (step 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This step requires modeler’s input and can be delivered using the BOM-to-E-BOM extension utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the extension is complete, each E-BOM is transformed into our proposed CPN component model using our automatic transformation tool (step 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of this step is a set of CPN components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At this step it is required to conduct structural and behavioral comparison between the generated components and the respective BOM using inspection and functional testing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the comparison is successful then S3b constraint of the requirement specification is partially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The modeler is then required to compose these generated components in a main model using CPN hierarchical tool (step 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (Binding IN-ports and OUT-ports of each component using sockets in the main model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the model is composed, it is executed (step 30) using CPN execution environment to test that all components correctly interact with each other and make necessary progress to reach the final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the execution is successful then the constraint S3b is fully satisfied, conforming that the structure and behavior of the executable model correctly represents its respective conceptual model and therefore any verification operation performed on the executable model will imply correctness of its conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In the next step the CPN model is subjected to the state-space analysis (step 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At first a state-space graph of the model is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Then for each objective and constraint in the requirement specification a verification query function is either created or selected from the function library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The execution of this function is done using CPN-ML program execution environment and the result of this function tells if the property is satisfied or violated (step 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If all the properties are satisfied we say that the composability verification process is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8 Model Checking When the Model Checking technique is selected, at first each BOM component is extended to E-BOM (step 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This step requires modeler’s input and can be delivered using the BOM-to-E-BOM extension utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is possible to assign Time constraints and the probabilistic factors with the states or the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the extension is complete, each E-BOM is automatically transformed into CSP# process specification (step 34) and composed (step 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The composition of each CSP  Chapter 6  Composability Verification Process  Page 129  process representing a BOM component be done using sequential operator ‘;’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' parallel operator ‘||’ interleaving operator ‘|||’ or (non-deterministic or user’s) choice operator ‘[ ]’ depending upon the nature of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We suggest composing each CSP in parallel so that each process executes in parallel and synchronizes with each other by sending or receiving events at their respective communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The composed model can then be simulated using the PAT simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A successful simulation run with at least one path leading to the final state(s) shows that the behavior of the composed model correctly represents its conceptual model and thus satisfies constraint S3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the next step, the assertions defined in CSP format (using LTL, Real-Time LTL or PLTL) are verified using PAT model checker which results in its satisfaction or violation (step 36, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If all the assertions are satisfied we say that the composability verification process is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Summary In this chapter, a flow diagram of composability verification process is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It indicates different steps and forms of inputs and outputs of each step in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This flow diagram can be used as a guideline to perform composability verification using three different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some recommendations are also presented in making a suitable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Once the verification process is completed successfully, the composed model can undergo implementation phase where it is programmed and simulated using a suitable simulation platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the experimental model can be constructed to perform different experiments on the implemented model and simulation results are generated for study and decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The implementation phase is out of the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page 130  Chapter 7  Fairness verification using PN  Algebraic Techniques  This chapter explains how algebraic techniques can help in verifying system properties of a Composed Model, using an example of a manufacturing system in which fairness is selected to be the required system characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Fairness Fairness has been defined in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 in terms of a Petri Nets property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter the concept of fairness is covered in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Intuitively, fairness is a liveness property that means no component of the system which becomes possible (or becomes enabled) sufficiently often should be delayed indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' On the basis of the extent of sufficiency, fairness is generally categorized in the following three types in literature: Unconditional Fairness Also called Impartial implies that every component in a system proceeds infinitely often without any condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The term “proceed” means to make progress, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', firing of a transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Unconditional fairness is also known as non-deterministic choice and is usually present among the components that are independent of each other [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Weak Fairness Also called Just, implies that every component in a system that is enabled continuously from some point onwards eventually proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Strong fairness Every component in a system that is enabled infinitely often proceeds infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A noticeable difference in weak and strong fairness is that weak fairness involves persistent enabling of a component that wants to proceed, whereas strong fairness is not persistently enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Some important generalizations of fairness exist in literature [122]: Equi fairness: means to give each component an equal chance to proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be regarded as Justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This type of fairness does not always apply in real world scenarios because of priority policies or some other reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Bounded fairness: means to give each component an equal number of chances such that no component proceeds for more than “k-times” without letting the others to take their turn For instance there is a check-in service at the airport that serves two types of queues: (i) Business class and (ii) Economy class at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It will be called fair, if it mostly serves business class passengers but not more than (say) 10 times, without serving a passenger from the economy class queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 131  In Petri Nets, fairness can be viewed in two perspectives namely: Transition fairness and Marking fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The former corresponds to fairness of choice of transitions, and the latter deals with the fair reachability of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Fairness Verification There are different ways to verify fairness of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The focus of this chapter is to discuss the technique for the verification of fairness property using PN Algebraic analysis and provide the necessary and sufficient conditions for a PN model to be fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The evaluation of these conditions in a PN model involves theorems and linear algebraic computations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' therefore it is classified as an Algebraic technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on the theorems below, we propose an algorithm for automatic fairness verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In Petri Nets, fairness is mainly perceived in terms of occurrences (or firing) of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Two transitions t1 and t2 are said to be in a fair relation if there exists a positive integer k such that for any reachable marking M and any firing sequence σ: (The symbol #(t/σ) denotes the number of times a transition t occurs in a firing sequence σ) In words, neither of the transitions should occur more than a finite number of times (k) without letting the other to occur at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is known as bounded fairness (or B-Fairness) with upper bound = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If every pair of transition is in a bounded fair relation, then the entire net is said to be fair [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  For the algebraic verification of fairness property in a PN model the following theorems are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Details and proofs of these theorems are discussed in [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Theorem I Given a PN with an incidence matrix A, if there exists a firing-count vector X, such that: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='X ≥ 0 and X≠0 Then a necessary condition for the PN to be fair is that each entry of X is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Theorem II If a Petri Net N is bounded for any initial marking M 0 then the condition in Theorem I is necessary and sufficient for N to be fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Corollary: If there exists a P-Invariant Y of positive integers such that: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Y=0 then the PN is guaranteed to be structurally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Theorem III A fair Petri Net PN has only one reproduction vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', a minimal T-Invariant) at the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Based on the above definition of the bounded fairness and theorems I, II and III a PN is said to be fair if it satisfies two conditions: (i) There must exist a single T- Invariant X of a given PN model whose each entry is non-zero and the product AX = 0 and (ii) There must be at least one P-Invariant, which means that the net is structurally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' # (t1/σ) = 0 ⇒ #(t2/σ) ≤ k  ∧ #(t2/σ) = 0 ⇒ #(t1/σ) ≤ k (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1)  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 132  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Manufacturing system In this section, a component based composed model of a manufacturing system is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using this composed model, it is shown how the proposed verification framework is used to verify the required specification, which in this example consists of fairness as an important quality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Two different scenarios of this example are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the first scenario the model is shown to be unfair as verified by our verification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the second scenario the model is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is then verified and it satisfies the fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Scenario I It is assumed that the manufacturing system model is composed of two machines M1 & M2 and a shared Robot R as shown in Figure 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The robot loads raw material on the machines and operate on them for producing goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The Robot is assigned (“loaded”) to either of the machines at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the Robot is loaded, it deposits raw material on the machine and process it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the good is produced the robot is unloaded and is available for the other machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 54: Manufacturing System (acquired from [124]) The process of composability verification is initiated as follows:  Simuland and Requirement Specification In the first step, the entities, events and the states of the simuland are perceived according to Figure 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The simuland and the requirement specifications are used to construct an appropriate conceptual model according to the steps given in the composability verification process described in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We define Requirement speciation of the manufacturing system as: RS0 = 〈O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S〉 where:  Objectives O = {o1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' o2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' o3} o1: Machine1 should continuously produce product1 without any infinite delay o2: Machine2 should continuously produce product2 without any infinite delay o3: Both machines should produce products with a ratio of 1:1  Loading  Processing  A  Unloading  Finished  Raw Material  Product A  Machine A  Robot  Loading  Processing  B  Unloading  Finished  Product B  Machine AChapter 7  Fairness verification using PN Algebraic Techniques  Page 133  System Constraints S = {s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s4} s1: Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2 and the Robot components should be composable at syntactic level s2: Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2 and the Robot components should be composable at static- semantic level s3a: State-machine matching of the composed model should be successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since the models are non-terminating so there are no final states, instead the goal-states: “Machine1 completes production” & “Machine2 completes production” will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s3b: The transformed executable model correctly represents the structure and behavior of the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s4: The shared robot should treat both machines with fairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', k-fairness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' k=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Conceptual Model The formal specification and graphical representation of each BOM model participating in the manufacturing composed model are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For the ease of readability following color codes are used for different BOM elements in the formal definition:  BB0 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Machine1 {C0(Id:Integer)}  EvT = {E0(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1(UnloadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E2(ResetM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C0)}  Act = { A0(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1(UnloadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A2(ResetM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E2)}  S = {S0(M1Waiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S1(M1Processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2(M1Completed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S0)} Table 21: Formal definition of Machine1 Base-BOM  BB1 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S 〉 where: EnT = Machine2 {C1(Id:Integer)}  EvT = {E3(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4(UnloadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5(ResetM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C1)}  Act = { A3(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A4(UnloadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A5(ResetM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5)}  S = {S3(M2Waiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S4(M2Processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S5(M2Completed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3)} Table 22: Formal definition of Machine2 Base-BOM  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 134  BB2 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S 〉 where: EnT = Robot {}  EvT = {E6(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7(UnloadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E8(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9(UnloadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)}  Act = {A6(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A7(UnloadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A8(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A9(UnloadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BB2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9)}  S = {S6(Idle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{A8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7} ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S7(Busy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6})}  Table 23: Formal definition of Robot Base-BOM  CB0 = 〈 AcTIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  AcTOUT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  POI 〉 where: AcTIN = AcTOUT = ∅  POI = {POI0(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A6 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0), POI1(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A7 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1),  POI2(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2),  POI3(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A8 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3), POI4(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A9 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A4), POI5(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5) }  Table 24: Formal definition of Manufacturing System composed BOM  Figure 55: Manufacturing System BOM based Composed Model  Figure 55 represents the BOM based Conceptual Model of the manufacturing system which includes three BOMs, formally defined using our proposed graphical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The figure shows how the characteristics, Events, Actions and states are mapped to each other (using dotted red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In machine 1 characteristic C0 is mapped to Event E2 which means event uses characteristic C0 as parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly Event E0 is mapped to A0, E1 to A1 and E2 to A2 respectively which means the Actions uses their mapped events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The mapping of actions to the states in the figure shows which action will cause which state to transit to the new state (shown by blue arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Characteristics: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Characteristics: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Characteristics: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Co ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Actions: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Actions: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A6=LoadingM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Actions: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0=LoadingM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A7=UnloadingM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3=LoadingM2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1=UnloadingM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A8=LoadingM2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='M2CompletedChapter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Fairness verification using PN Algebraic Techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 135  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='basic BOM components are connected to each other using the formal definition shown in Table 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' which describes the source (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=') and destination (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=') of an action from one component to other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In Figure 55 this is shown using black arrow lines with their input/output (I/O) label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is called Pattern of Interplay (in BOM specification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Static Analysis Rules Machine1 Machine2 Robot • Name of the send-event and receive-event should be same • Each send-event should have at least one corresponding receive- event and vice-versa • The number of parameters (content characteristics of event types) of the send-events should be the same as the number of parameters of the receive- events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='LoadingM1(null)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='UnloadingM1(null)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='LoadingM2(null)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UnloadingM2(null)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UnloadingM2(null)  Table 25: Syntactic Matching  It can be seen in Table 25 that the name of the send-event and receive-events are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='=Send, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='=Receive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And they are in one-to-one relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' of parameters of each event is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on these facts the components are said to be syntactically composable (S1 satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We assume that Machine1, Machine2 and Robot components have the semantic- attributes as shown in Table 26 which satisfy all the static-semantic matching rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The attributes highlighted in red color are semantically equivalent (Exact match) therefore S2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Machine1  Machine2  Robot  AOI = {Production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Production line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Lathing}  AOI = {Production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Production line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Polishing}  AOI = {Production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conveyer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Automation} Purpose = {Manufacture Product1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Manufacture Product3} Purpose = {Manufacture Product2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Manufacture Product3} Purpose = {Manufacture Product3} Data Types of parameters= {null} Data Types of parameters = {null} Data Types of parameters = {null} Units of Measurement = {} Units of Measurement = {} Units of Measurement = {} Table 26: Static-Semantic Matching  Dynamic Analysis The state-machine matching process is successfully conducted as both Machine1 and Machine2 reach their goal-states namely: Mcompleted and M2-completed and satisfy S3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 136  Figure 56: State-machine matching of manufacturing system  BOM to PNML Transformation In the next step the components are subjected to PNML transformation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of the transformation process is a PN model shown in Figure 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It can be seen from the inspection that the States and their exit conditions, Events and Actions all are present in the transformed model (as specified in the original conceptual model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also this PN model is executed in the PIPE runtime environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The execution is successful because the places P3 and P6 acquired tokens (showing that these goal states were reached during the execution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This satisfies S3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 57: PN model of the manufacturing System  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Robot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='P2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='P5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='T6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M1Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='P8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M2Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Busy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UnloadingM1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Robot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UnloadingM2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='T2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='T5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='P3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='P6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M1Completed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M2Completed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine2Chapter 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Fairness verification using PN Algebraic Techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 137  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Algebraic Resource Computation At this step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the initial marking M0 and the Incidence Matrix A of the PN composed model shown in Figure 57 are calculated using PIPE library functions as follows:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='M 0 P1 P2 P3 P4 P5 P6 P7 P8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A P1 P2 P3 P4 P5 P6 P7 P8 T1 -1 1 0 0 0 0 -1 1 T2 0 -1 1 0 0 0 1 -1 T3 1 0 -1 0 0 0 0 0 T4 0 0 0 -1 1 0 -1 1 T5 0 0 0 0 -1 1 1 -1 T6 0 0 0 1 0 -1 0 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Table 27: Initial Marking and Incidence Matrix (Scenaro I) Note that the labels of rows and columns in A and elements in M0 correspond to places and transitions in Figure 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The matrix A is given as input to the Invariant calculation module that calculates the following P-Invariants and T-Invariants in the PN model of the Manufacturing System:  P1 P2 P3 P4 P5 P6 P7 P8 1 1 1 0 0 0 0 0  T1 1  T2 1  T3 1  T4 0  T5 0  T6 0  T1 0  T2 0  T3 0  T4 1  T5 1  T6 1  Table 28: P Invariants and T Invariants (Scenaro I)  Property Verification Function In order to proceed with the verification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' we have to translate the objectives and constraints of the requirement specification into PN properties: o1: Machine1 should continuously produce product1 without any infinite delay o2: Machine2 should continuously produce product2 without any infinite delay o3: Both machines should produce products with a ratio of 1:1 s4: The shared robot should treat both machines with fairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', k-fairness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' k=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  It is clear from the {o1, o2, o3 and s4} that if the robot serves both machines with fairness (S4) only then both of them will be able to produce their respective products continuously without indefinite delay (O1 & O2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And if the fairness is bounded such that k=1, then both machines will produce products with equal ration 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the translation of the requirement specification in PN form is as follows:  PN Property P1 = “The model should be Bounded-Fair (with K=1) such that the robot serves both machines alternatively”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In order to verify bounded-fairness we consider the property proving theorems I, II & III defined in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on these theorems we construct a property verification function using the following algorithm:  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 138  Algorithm: B-Fairness Verification Input: {P-Invariants}, {T-Invariants}, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Output: TRUE 1 If |{T-Invariants}| = 1 then ⊳ List of T-Invariants has exactly 1 invariant, meaning it is a 2 Reproduction vector , Theorem III⊲ 3 XT ← T-Invariants[0]        ⊳ Get the only T-Invariant from the list 4 if A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='XT ≥ 0 and each element in XT >0 then   ⊳ Multiply XT with Incidence matrix and 5 Check that each element of T-invariant is 6 positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Theorem I⊲ 7 if |{P-Invariants}|>0  then            ⊳ Check if there is any P-Invariant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' meaning PN 8 Model is bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Theorem II⊲ 9 Return TRUE 10 else 11 Return FALSE ⊳ Theorem II violated 12 end if 13 else 14 Return FALSE  ⊳ Theorem I violated 15 end if 15 else 17 Return FALSE ⊳ Theorem III violated 18 end if  Table 29: B Fairness Verification  Based on this algorithm we perform property verification of the given PN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is evident that the T-invariants (see Table 28) contain zero entries which violate Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also there is more than one T-invariant which violates Theorem III therefore the net is said to be unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As the PN is unfair, it is impossible to guarantee that objectives o1, o2, o3 will be satisfied because either of the machines may over perform by acquiring robot multiple times without letting the other to get the robot for at least once (failure of o1 & o2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore either of the machines may face a situation in which it is unable to produce enough number of products to meet the required objectives i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the ratio 1:1 for producing products cannot be fulfilled (failure of o3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' consequently the composed model may fail to satisfy given specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Scenario II In order to understand the fairness verification process, a counterexample is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this example another component is added to the composition called Controller that can supervise the robot assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The job of the Controller is to enforce fairness in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The BOM model of the controller is defined as follows:  Chapter 7  Fairness verification using PN Algebraic Techniques  Page 139  BB3 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S 〉 where: EnT = Controller {}  EvT = {E10(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Controller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E11(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Controller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)}  Act = {A10(LoadingM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Controller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  E0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A11(LoadingM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Controller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Machine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1)}  S = {S8(AssignM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S9}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S9(AssignM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8})}  Table 30: Formal definition of Controller Base-BOM  CB0 = 〈 AcTIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  AcTOUT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  POI 〉 where: AcTIN = AcTOUT = ∅ POI = {POI0(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A10 , {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A6}), POI1(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A7, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1),  POI2(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2), POI3(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A11 , {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A8}),  POI4(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A9 , ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A4), POI5(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5) }  Table 31: Formal definition of Modified Manufacturing System composed BOM The other components have the same definition except that the sender of Event E0 and E1 is BB3 (controller) and the receivers of event E0 are BB0 (Machine1) and BB2 (Robot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' whereas the receivers of event E1 are BB1 (Machine2) and BB2 (Robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 58 shows the composed BOM of modified manufacturing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 58: Modified manufacturing system composed BOM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Machine1 Robot Machine2 Characteristics: Characteristics: Characteristics: CO = Id : Integer C1 = Id : Integer Actions: Actions: A6=LoadingM1 Actions: A0=LoadingM1 A7=UnloadingM1 A3=LoadingM2 A1=UnloadingM1 A8=LoadingM2 A4=UnloadingM2 A2=ResetM1 A9=UnloadingM2 A5=ResetM2 States: States: States: S0=M1Waiting S6=ldle S3=M2Waiting S1=M1Processing S7=Busy S4=M2Processing S2=M1Completed S5=M2Completed M1Waiting M1Processing Idle S6 M2Waiting M2Processing SO S1 A6 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 S3 S4 A4 S2 A2 A7 S7 A9 S5 A5 M1Completed Busy M2Completed Controller Characteristics: A Action Connector Actions: A10=LoadingM1 S Initial State A11=LoadingM2 States: State S8=AssignM1 S9=AssignM2 Exit condition State Transistion AssignM1 Input/Output A10 S8 connection S9 A11 AssignM2Chapter 7  Fairness verification using PN Algebraic Techniques  Page 140  When the verification process is started,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the BOM components are transformed into PN model as shown in Figure 59 where the controller component is attached to both machines and the robot and controls the machine assignment to enforce Kfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is evident from the figure that when the robot is assigned to Machine1 once, it cannot be reassigned (because of the lack of token in P9), and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the number of tokens are increased to ‘n’, the same model can work for k=n fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the initialization phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the initial marking M0 and Incidences Matrix A were calculated as follows: M0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10  1 0 0 1 0 0 1 0 1 0  A P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 T1 -1 1 0 0 0 0 -1 1 -1 1 T2 0 -1 1 0 0 0 1 -1 0 0 T3 1 0 -1 0 0 0 0 0 0 0 T4 0 0 0 -1 1 0 -1 1 1 -1 T5 0 0 0 0 -1 1 1 -1 0 0 T6 0 0 0 1 0 -1 0 0 0 0  Table 32: Initial Marking and Incidence Matrix (Scenaro II)  Figure 59: Modified PN model of the manufacturing System  When the Invariant calculation module is executed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' the following T-Invariant and P- Invariant were discovered for the model shown in Figure 59: T1 1 T2 1 T3 1 T4 1 T5 1 T6 1  P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 1 1 1 0 0 0 0 0 0 0  Table 33: P Invariants and T Invariants (Scenaro II)  Having only one T-Invariant (and the only one) with non-zero entries and having a P-Invariant (with some non-zero entries),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' satisfies the conditions (of Theorem I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' II & III) required for the model to be bounded fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Controller  P1  P4  P9 AssignM1  M2Waiting  [个  LoadingM1  P10AssignM2  LoadingM2  T4  T3  Idle  T6  ResetM1  P2  P7  P5  ResetM2  M1Processing  M2Processing  P8  Busy  UnloadingM1  UnloadingM2  T2  Robot  T5  P3  P6  M1Completed  M2Completed  Machine1  Machine2Chapter 7  Fairness verification using PN Algebraic Techniques  Page 141  Based on these result from PN algebraic analysis technique, we can confirm that the composed model satisfies given requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Due to the supervised controller, the Robot is bound to operate fairly between the two machines, which results in fulfillment of the objectives O1, O2 and O3 and also satisfied required constraint S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Summary Fairness property becomes significant in the composability verification of a composed model because it does not allow any component to dominate and excessively proceed, while other components do not proceed even for once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As illustrated by the example of the manufacturing system, fairness of Robot allocation can ensure that both machines will perform to produce a required number of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there is no fairness we cannot guarantee that this objective will be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using the example of Fairness verification in the manufacturing system, we explain how our Algebraic Verification Technique works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is a notable fact that this technique does not face state-space explosion because it does not involve reachability graph construction and can work only by calculating incidence matrix and P/T invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are a lot of PN properties which can be verified using these PN algebraic computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' On the other hand this approach can only be used to verify a limited set of PN properties (for which suitable theorems exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page 142  Chapter 8  Model Verification using State space  Analysis techniques  Colored Petri Nets and its analysis techniques are very useful for accurate and efficient verification as it is one of the competitive formalisms in the specification of the concurrent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Its application in the Composability verification proves to be very constructive, especially with a focus on the dynamic semantic composability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The analysis techniques contributed by the CPN community over a couple of decades provide a significant improvement on efficient and accurate reasoning regarding the model correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter a Field Artillery Model is presented as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is shown how the BOM based Field Artillery Model is transformed into our proposed Colored Petri Net components and verified using state-space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Combat Modeling is about the models that describe or represent weapon systems and combat situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There are numerous types of combat models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These types are distinguished by their modeling objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of the fundamental objectives of combat modeling are training, war-games, weapon testing etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Combat Modeling Combat modeling purposefully abstracts and simplifies combat entities, their behaviors, activities, and interrelations to answer defense-related research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There cannot be a general model that answers all questions however there is a concept of a generic situated environment and four core activities that can be found on every battlefield [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Situated Environment Combat Modeling starts with analyzing the challenges to model the Situated Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' All modeled combat entities are situated in the environment, the virtual battlefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They perceive the environment including other entities, and map their perception to an internal representation based on the knowledge and goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' They communicate and act with other combat entities within the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The environment contains all objects, passive ones like obstacles, as well as active ones like enemy or friendly units [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Moving Moving is the core activity of combat modeling that deals with the movement of individual entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These entities could be weapon, people etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' or aggregate models that are used to model the movements of groups of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The models use patches and grids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' they use physical models for weapon systems and reference schemas for unit movement [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8  Model Verification using State space Analysis techniques  Page 143  thermal, and optical sensors, can contribute to perceiving the environment and the other entities as close to reality as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Intelligence, surveillance, and reconnaissance operations contribute to similar requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to sense special properties of an entity, each of these special properties needs to be modeled explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If it is modeled explicitly, it needs to make a difference in the reconnaissance process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Furthermore, if a detail is important for the military decision process, it needs to be part of the perception, and hence needs to be observed by sensors, which requires that the respective things are modeled as properties of the entities [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Shooting Modeling the outcomes of duels between weapon systems and battles between units is still a topic of major interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' On the weapon system level, direct and indirect fires are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Direct fire means that the target is in the line of sight of the shooter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In case of indirect fire systems such as Artillery and other ballistic weapons, they do not need to see the target and shoot at it straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Their weapons follow a ballistic curve being described by the term indirect fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Many models have been developed to keep up with the score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance a game based point systems that count how often and where a target is hit and use “hit-and-kill” probabilities (which are based on real- world data) to simulate hitting or missing a target  [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Communication: This core activity deals with the modeling of Communications, Command and Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It ties all the earlier activities together as command and control is situated in the environment and commands the entities to shoot, move, observe, and communicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Several models of command and control in military headquarters are discussed, as more and more simulation models have to come up with decisions based on available information where until recently human decision makers had to be involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The better command and control is modeled the less military experts are needed to provide a realistic training environment [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on these principles of combat modeling an example model of Field Artillery is presented to explain the approach of composability verification using state-space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 60 highlights the activities of combat modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 60: Activities of Combat Modeling  Situated  Environment  Moving  Looking  or Sensing  Shooting  Communication  (Command &  Control)  Chapter 8  Model Verification using State space Analysis techniques  Page 144  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Field Artillery Field Artillery (FA) is one of the indirect fire systems35 that engage the opponent without requiring line of sight between the shooting system and the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Infantry uses small, medium or heavy howitzers (artillery guns) that provide fire support for combat units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly Navy artillery provides fire power, where missiles can be fired on land based or sea based direct or indirect targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The general mission of FA is to destroy, neutralize or suppress the enemy by cannon, rocket, and missile fires and to help integrate all fire support assets into combined arms operations [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The field artillery system provides close support to maneuver friendly forces, counter fire and interdiction as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These fires neutralize, canalize, or destroy enemy attack formations or defenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' obscure the enemy’s vision or otherwise inhibit his ability to acquire and attack friendly targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and destroy targets deep in the enemy rear with long-range rocket or missile fires [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FA weapons are usually located in defiladed areas in order to protect them from enemy detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This nature of FA gunnery makes it an indirect fire problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observed fire (the technique that solves the indirect FA gunnery problem) is carried out by the coordinated efforts of the Forward Observers, Head Quarter (HQ), the Fire Direction Center (FDC), and firing sections of the firing unit (Batteries) [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 61 gives an overview of the essential elements of a field artillery and the situation of an indirect fire, where a forward observer spots an enemy unit and requests fire support from a nearby friendly unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It should be noted that this scenario is only assumed and simplified for the sake of an example, whereas the today’s state of the art of field artillery systems is much more modernized and technologically advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 61: Elements of Field Artliiery & Indirect Fire 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Simuland Based on Figure 61 an Indirect Fire Support scenario is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this scenario the enemy units are not in the line of sight of the firing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A soldier (forward observer) from the observation post observes the enemy field and detects potential targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a target is spotted, he calls BHQ for fire support and provides the target details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ requests FDC to process the target tactically & technically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In tactical terms, the target should be of high importance to gain tactical advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In technical terms the target should be in the firing range of the supporting artillery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If  35 Although there are some exceptions, in which Field Artillery engages in direct fire mode  ForwardObserver  Spotted Target  FiringUnits  BHQ&FDCChapter 8  Model Verification using State space Analysis techniques  Page 145  the target is valid FDC approves the request otherwise the request is denied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the request is approved BHQ assigns the target to the firing units (batteries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We suppose that the target can be one of three types: light (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', camps, troops, and trucks), medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', tanks, light guns) or heavy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', artillery units, missile launchers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The target is assigned to one, two or three batteries respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is because medium and heavy targets require the fire power of more than one battery for complete destruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on this assumption, BHQ assigns target to the batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery components align themselves for correct orientation and elevation by computing the target’s range and bearing (angle), load appropriate ammunition and fire the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a Field component receives fire, and if the detonation is within a destruction radius then the target is said to be destroyed otherwise it is missed, as will be observed by the observer, who provides this information to the BHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This process is restarted for other potential targets, until all the enemy-units are suppressed, which is the ultimate goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Field Artillery Model Based on the above informal description of the simuland a Field Artillery Model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' There could be multiple objectives of modeling field artillery including training, exercises, weapon testing or operational optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following BOM based models were discovered, selected and composed with respect to the simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Field Artillery Conceptual Model The BOM based conceptual model of Field Artillery is formally defined as follows:  Observer = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Observer {C0(Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C1(Loc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C2(CurrentTarget),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C3(Result)}  EvT = {E0(ObserveField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1(TargetSpotted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E2(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' currtgt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E3(RequestApproved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4(RequestDenied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' detonation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E6(TargetDestroyed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7(TargetMissed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)}  Act = {A0(ObserveField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1(TargetSpotted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A2(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A3(RequestApproved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A4(RequestDenied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A5(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A6(TargetDestroyed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A7(TargetMissed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7)}  S = {S0(ObserverReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S1(ObservingField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2(RequestingFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3(WaitingForReponse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S0}) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4(WaitingForImpact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S5(EvaluateDamage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S0}} Table 34: Observer Basic-BOM  Chapter 8  Model Verification using State space Analysis techniques  Page 146  Field = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Field {C4(Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C5(FD),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C6(Impacts)}  EvT = {E8(ObserveField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9(TargetSpotted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E10(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fire),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E11(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fire),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E12(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fire),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E13(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Impacts),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E14(UpdateField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' update) }  Act = {A8(ObserveField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A9(TargetSpotted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A10(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A11(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A12(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A13(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A14(UpdateField,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E14) }  S = {S6(FieldReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S7(BeingObserved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8(TakingFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S9) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S9(WaitingForUpdate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6)} Table 35: Field Basic-BOM  BHQ = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT =BHQ {C7(Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C8(Loc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C9(CurTarget),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C10(TargetStatus) }  EvT = {E15(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E16(ProcessRequest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E17(RequestApproved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E18(RequestDenied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E19(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' assign_target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E20(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E21(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E22(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E23(TargetDestroyed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E24(TargetMissed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT =FDC{C11(Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C12(FD) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C13(CurTarget),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C14(Result)}  EvT = {E26(ProcessRequest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E27(RequestApproved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E28(RequestDenied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)}  Act = {A26(ProcessRequest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A27(RequestApproved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A28(RequestDenied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E28)}  S = {S17(FDCReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S18(Processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S17},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S17})} Table 37: FDC Basic-BOM  Chapter 8  Model Verification using State space Analysis techniques  Page 147  Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='336 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 {C15(Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C16(CurTarget) }  EvT = {E29(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' assign_target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E30(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fire),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  E31(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)}  Act = {A29(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A30(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A31(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E31)}  S = {S19(ReadyToFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S20(PreparingCannon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S21(Firing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S19)} Table 38: Battery (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) Basic-BOM  FA = 〈 AcTIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  AcTOUT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  POI 〉 where: AcTIN = AcTOUT = ∅  POI = { POI-0(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A8), POI-1(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A9, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1), POI-2(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A15), POI-3(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A16, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A26), POI-4(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A27, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A17, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3}), POI-5(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A28, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A18, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A4}), POI-6(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A19, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A29), POI-7(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A30, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A10, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A11, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A12}), POI-8(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A31, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A20, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A21, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A22}), POI-9(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A13, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5), POI-10(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A6, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A23), POI-11(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A7, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A24), POI- 12(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A25, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A14)} Table 39: Field Artillery Composed BOM  Figure 62 shows the formal representation of Field artillery composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  36 This component has three instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8  Model Verification using State space Analysis techniques  Page 148  Figure 62: Field Artillery Composed BOM  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Requirement Specification We define Requirement speciation of the field artillery model as: RS0 = 〈O, S〉 where: Objectives O = {o1} and System Constraints S = {s1, s2 s3, s4} o1: All the enemy units must be destroyed s1, 2, and 3: The model should be composable at syntactic and static-semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-machines should match and the executable mode should correctly represent the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' s4: There should never be a friendly fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Observer Characteristics: CO=ld: Integer, C1=Loc: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C2=CurrentTarget: Target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C3=Result : Bool BHQ FDC Target = (Id:Integer, Grid :Integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Characteristics: Characteristics: C7=ld Integer C8=Loc: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C13=CurTarget:TARGET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C14=Result:Bool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C11=ld: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C12=FD:FIELD_DATA: C9-CurTargetTARGET: Action C10=TargetStatus:Bool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A0=ObserveField, A1=TargetSpotted Actions: A4=RequestDenied,A5Detonation A2=CallForFireSupport, A3=RequestApproved A15=CallForFireSupport A6=TargetDestroyed, A7=TargetMissed A16=ProcessReques A28=RequestDenied: A17=RequestApproved States: A18=RequestDenied States: A19=AssignTarget S17=FDCReady: S18=Processing S2=RequestingFireSupport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3=WaitingForReponse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FDCReady A22=FiringCompleted A23=TargetDestroyed Observing A24=TargetMissed A25=UpdateField essing SO A3 S10=BHQReady;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S11=CallFDC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 42 S3 uppor (A3) S15=WaitingForDamageReport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A6 S16=UpdatingField: WaitingFor BHQReady CallFDC S10 11 A16 (A18) S12 西 WaitingFor Approv A18 Battery1 自电 ingTarge Characteristics: C15=ld: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C16=CurTarget: TARGET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Actions: WaitingForFire A29=AssignTarget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A30=Fire: A31=FiringCompleted: A25 815 States: S19=ReadyToFire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S20=PreparingCannon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' UpdatingFielc S21=Firing: ReadyToFire PreparingCannor S19 $20 Field C4=ld: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C5=FD: FIELD_DATA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Characteristics: C6=Impacts: IMPACTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' FIELD_DATA = list (ld:Integer, Grid:Integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  IMPACTS = list (Grid:Integer) Description:S Battery1 Characteristics: Actions: A10=Fire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A11=Fire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A12=Fire A8=ObserveField;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A9=TargetSpotted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C15=ld: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C16=CurTarget: TARGET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A13=Detonation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A14=UpdateField Actions: A29=AssignTarget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A30=Fire: A31=FiringCompleted: States: S6=FieldReady;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7=BeingObserved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S8=TakingFire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S9=WaitingForUpdate States S19=ReadyToFire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S20=PreparingCannon 自自自自 FieldReady ReadyToFire PreparingCannc A29 S19 S20 A30 TakingFire Being Obse Firing WaitingFo Update Battery1 Characteristics: C15=ld: Integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C16=CurTarget: TARGET: Actions: A29=AssignTarget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A30=Fire: A31=FiringCompleted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' States S19=ReadyToFire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S20=PreparingCannon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S21=Firing ReadyToFire S19Chapter 8  Model Verification using State space Analysis techniques  Page 149  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Verification of the FA model using CPN State-Space Analysis After the BOMs are discovered, selected and composed according to Figure 62, the conceptual model is ready for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We select CPN state-space analysis technique for its verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Static and Dynamic Analysis We assume that the model qualifies syntactic and static-semantic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also when it undergoes state-machine matching process it is able to make progress until the goal-states are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 63 shows how the components interact with each other through the exchange of events (horizontal arrows) due to which their state- machines make progress (vertical dotted arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on the fact that the constraint S1, S2 and S3a are satisfied we proceed to BOM-to-E-BOM extension which is a pre-requisite step for the transformation of conceptual model into executable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 63: State-machine Matching of Field Artillery Model 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 BOM to E-BOM extension At this stage all the BOM components are extended to our proposed E-BOM extension with the help of the modeler’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following tables present E-BOM extensions of BOMs in the FA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Observer  Field  BHQ  FDC  BATTERY  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3)  Observer  Field  Ready  ObserveField  Ready  Observing  Being  Field  Observed  TargetSpotted  Requesting  BHQ  FireSupport  Ready  +  CallForFireSupport  WaitingFor  Call  FDC Ready  Reponse  FDC  ProcessRequest  WaitingFor  Processing  Approval  RequestApproved  RequestDenied  Assigning  Ready  WaitingFor  Target  ToFire  AssignTarget  Impact  Waiting  Preparing  ForFire  Fire (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3)  Cannon Taking  Fire  Firing  Detonation  FiringCompleted  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) WaitingFor  Evaluate  DamageReport  Damage  TargetDestroyed  TargetMissed  WaitingFor  Updating  Update  Field  UpdateFieldChapter 8  Model Verification using State space Analysis techniques  Page 150  Observer E-BOM SV and types {C0(Id:Integer),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C1(Loc:Integer),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C2(CurrentTarget:TARGET),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C3(Result:Bool)} where TARGET = (Id:Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Grid37:Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Description:String)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Initial States {S0:ObserverReady}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transitions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='State  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Guard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{SVIN} {SVOUT}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Next State  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TargetSpotted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requesting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FireSupport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requesting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FireSupport CallForFireSupport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestApproved  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Impact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestDenied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Impact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Detonation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Evaluate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Damage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Evaluate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Damage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Destroyed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[Result=true]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Evaluate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Damage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Missed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[Result=false]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action1 { input (target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' detonation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (result);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let val grid= #2 target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in (Destroyed(grid, detonation)) End } fun Destroyed (x, []) = false | Destroyed (x, h::t) = IsDestroyed(x, h)  orelse Destroyed (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fun IsDestroyed(grid, impact) = let val gridst = Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='toString(grid);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val impactst= Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='toString(impact);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val gridX = valOf(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='fromString(substring (gridst, 0, 3)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val impactX = valOf(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='fromString(substring (impactst, 0, 3)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val gridY = valOf(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='fromString(substring (gridst, 3, 3)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val impactY = valOf(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='fromString(substring (impactst, 3, 3)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val X = abs(gridX - impactX);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val Y = abs(gridY - impactY);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in if (abs(X)<4 andalso  abs(Y)<4) then true else false end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Table 40: Observer E BOM  Action1 is a CPN-ML script that evaluates if the target is destroyed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that the destruction radius of the rounds fired by artillery guns is 4x4 grids i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' if the round hits the target within this radius it will be destroyed otherwise missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Note that Action1 calls other functions which are also specified in Table 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  37  In military map, Grid reference system is used to identify a position of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that the grid in this scenario is of 6 figures, first three integers define Easting and the other three define Northings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For details see [131]  Chapter 8  Model Verification using State space Analysis techniques  Page 151  Field E-BOM SV:Types {C4(Id:Integer), C5(FD:FIELD_DATA), C6(Impacts:IMPACTS)} FIELD_DATA = list (Id:Integer, Grid:Integer, Description:String, Type:String);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' IMPACTS = list (Grid:Integer) Initial States {S6: FieldReady} Transitions State Event Guard {SVIN} {SVOUT} Action Next State FieldReady ObserveField  BeingObserved  Being Observed  TargetSpotted  [length  FD>0]  C5  C5  Action2  FieldReady  FieldReady  Fire  C6  TakingFire  TakingFire  Detonation  C6  WaitingFor  Update  WaitingFor  Update  UpdateField  C5  C5  Action3  FieldReady  Action2 { input (fd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (target);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let val indx = discrete (0,(length fd));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val F = List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='nth(fd, indx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in ((#1 F, #2 F, #3 F)) end } Action3 { input(update,fd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (ufd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let val status = #2 update;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val target = #1 update;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val U = (#1 target, #2 target, #3 target, "Enemy");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in if (status=true) then rm U fd else fd end }  Table 41: Field E-BOM Action2 randomly picks a target from a list of targets (Field Data) and sends it as parameters to the observer, simulating that the observer has spotted a target in the enemy area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Action3 is executed when Update-Field event is received from BHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This action removes an object if the target destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ E BOM  SV:Types  {C7(Id:Integer),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C8(Loc:Integer),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C9(CurrentTarget:TARGET),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C10(TargetStatus:Bool)}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Initial States  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{S10: BHQReady }  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Transitions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='State  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Guard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{SVIN}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='{SVOUT}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Next State  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BHQReady  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CallForFireSupport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CallFDC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CallFDC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ProcessRequest  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Approval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Approval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestApproved  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Assigning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestDenied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BHQReady  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Assigning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='AssignTarget  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingForFire  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingForFire  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FiringCompleted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='DamageReport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TargetDestroyed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UpdatingField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Verification using State space Analysis techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 152  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='DamageReport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='DamageReport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TargetMissed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UpdatingField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UpdatingField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='UpdateField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BHQReady  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Action4 { input (target);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (assign_target);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let in if ((#3 target) = "Artillery") then ((#1 target, #2 target, #3 target, [true, true, true])) else if ((#3 target)  = "Tank") then ((#1 target, #2 target, #3 target, [true, true, false])) else ((#1 target, #2 target, #3 target, [true, false, false])) end}  Table 42: BHQ E-BOM Action 4 is used to assign light, medium or heavy targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If a target is heavy then all three batteries are assigned to hit the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the target is medium then battery 1 and 2 are assigned otherwise only battery 1 is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  FDC E BOM  SV:Types  {C11(Id:Integer), C12(FD:FIELD_DATA) , C13(CurrentTarget:TARGET),  C14(Result:Bool)}  Initial States  {S17: FDCReady }  Transitions  State  Event  Guard  {SVIN}  {SVOUT}  Action  Next State  FDCReady  ProcessRequest  C12, C13,  C14  Action5  Processing  Processing  RequestApproved  [Result=true]  C13, C14  FDCReady  Processing  RequestDenied  [Result=false]  C13, C14  FDCReady  Action5 { input (target, fdcd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (fdc_result);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let val tid = #1 target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val r = List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='nth((listsub fdcd (GetFieldByID tid fdcd)),0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in (if (#4 r = "Enemy") then true else false) end }  Table 43: FDC E BOM  Action 5 is used to process targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Here we only check from the internal FDC data that the target is an enemy unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This method can be expanded to compute target priorities and process other tactical and technical fire direction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8  Model Verification using State space Analysis techniques  Page 153  Battery E BOM  SV:Types  {C15(Id:Integer), C16(CurrentTarget:TARGET) }  Initial States  {S19: ReadyToFire }  Transitions  State  Event  Guard {SVIN} {SVOUT} Action  Next State  ReadyToFire  AssignTarget  C16  PreparingCannon  PreparingCannon  Fire  C16  Action6  Firing  Firing  FiringCompleted  ReadyToFire Action6 { input (assign_target);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' output (fire);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' action let val bid = inst();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val tid = #1 assign_target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val grid = #2 assign_target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val impact=grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in (bid, tid, grid, impact) end }  Table 44: Battery E BOM  Action 6 is used to initiate the fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It creates a token of type “Fire” to the Field component containing the information of the firing battery id, target id, grid location of the target and the location of the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 E-BOM to CPN Transformation In this step, all the extended BOM models (E-BOM) are automatically transformed into our proposed CPN components in such a way that all variables from the corresponding E-BOMs are added in the Structural Layer (shown in Red color in the following figures) and the State-machine is transformed into the Behavioral Layer (shown in Green color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In communication layer (shown in blue color), receive- events are transformed into input ports and send-events are converted into output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 64Figure 65Figure 66Figure 67Figure 68 represent the CPN component models of each component:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Verification using State space Analysis techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 154  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 64: Observer CPN Component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ID  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='LNI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Grid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='205405  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ObserveField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='OF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='H  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TargetSpotted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TARGET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Requesting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FireSupport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Current  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CallForFireSupport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CFS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TARGET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TARCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Reponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='INT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestApproved  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RequestDenied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ARCET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='RD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='TARGET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='WaitingFor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Impact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='LNI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Detonation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Occured  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Input (target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' detonaony:  In  output(result)  DETONATIOI  Evaluate  valgrld=#2target:  Damage  result  (Cestrayed(grld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' detonaton3)  INT  end  [target_shabus=true]  [target_skatus=false]  Target  Target  Destroyed  Missed  TD  Out  NUL  TM  Out  HChapter 8  Model Verification using State space Analysis techniques  Page 155  Figure 65: Field CPN Component  ID INT [length fd > 0] Data TargetSpotted target TS Out FIELD DTA TARGET 165505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' "Bllding Neutral Input (fd3: iEnemy" output (target 1794B1, aatlon Being 165465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tank" Enemy let 198468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Tank" "Enemy\' Observed alnox= disorete (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='tlength fJ): val F Ost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ntfan INT n #1F，#2F，#3F end ObserveField OF null In ULI Field Ready INT Fire E Hre In LFIRE Ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='map (ftbkdrtid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' grld mpact)=> mpadAre Taking Impacts Fire mpacts LNI Detonation D Imoaats Out DETONATION In put update, fdy: output (ufd): let Waiting val status = #2 updater For 1update Update val y =[#i target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #2 argeat +3arget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' "Enemy") INT If (shatus=true) then myfd else fd end ufa fd UpdateField UF update In PDATEChapter 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Verification using State space Analysis techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 156  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 66: BHQ CPN Component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='BHQ ID Ready LNI INT CallFor FireSupport CFS trge 205405 In Grid TARGET INT arget Call FDC LNI Current target Process arge Target Request PR Out TARGEK TARGET trger WaitingFor Approval INT Request Request Approved Denied TARCE RA In Assigning trget target RD Target In TARGET INT Input (target)) Assign output (asslgn_target): Target AT asslgn_arget Out ASSIGN TARGE Hit := discrete(o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2)F If ((3 target) "Artillery then Waiting #1trget #2 barget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #3target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[huertue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='true ForFire else If ((*3 target) ITank") then (#1 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='#2 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='#3 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='[true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='false]3 else ((#1 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #2 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #3 target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' [true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' false])) end Firing Completed In FC nu NLLI WaitingFor DamageReport INT Target Target Destroyed Missed NULI Tue In TD Target UpdatingField TM Status In 1000 H target status INT target UpdateField UF (target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target shabus) Out JPDATEChapter 8  Model Verification using State space Analysis techniques  Page 157  Figure 67: Battery CPN Component  ReadyToFire  Assign  Target  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Current  Preparing  Target  Cannon  Fire  Out  Firing  Firing  Completed  FC  OutChapter 8  Model Verification using State space Analysis techniques  Page 158  Figure 68: FDC CPN Component  We assume that each transformed CPN component has passed structural evaluation which is conducted using inspection method and behavioral evaluation conducted using Functional Testing method therefore S3b is also partially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Composition of CPN Components In this step all CPN modules are combined together through socket places in a CPN Composed Model as shown in Figure 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this composed model some general purpose auxiliary components are also introduced such as Join and Fork to facilitate the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1B3508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' "camp Enemy\' Ready 179481 EnGmy 165465 TEO EnemY LNI Tank Field Bpy Process Data Request PR arge Input (target, fdcd): In FIELD_DATA target output (fdc_ result TARGET atn Current Processing let Target fdyresult vald LNI val r = List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ntht(llstsub faked TARGET (GetFleldByID tid fdcd)),0) Result arge end lt 1008 TARGET Request Request Approved Denied RD Out fck_result#true] [fdk_result=false] RA target Out TARGETChapter 8  Model Verification using State space Analysis techniques  Page 159  Figure 69: Field Artillery CPN Composed Model  When the model is composed it is executed in the CPN execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The successful execution of the model (according to Figure 63) satisfies S3b completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 State space Analysis In the next step the state-space of the entire Field Artillery Model is generated using CPN state-space calculation tool, and is used to perform verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The generated state-space graph consists of 1960 nodes and 6469 arcs as shown in Figure 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Battery3]  Battery2  Battery1  F3  F2  F1  AT  AT  Bty3  Bty2  Btv1  JoinFire  JoinFC  ForkAT  FC  Field  OF  CFS  Observer  TD  BHQ  TM  ORD  ORA  BR  BRD  FDC  PRChapter 8  Model Verification using State space Analysis techniques  Page 160  Figure 70: State space of Field Artillery CPN Model (1960 nodes, 6469 edges)  After the state-space is constructed in CPN tools, it is exported into a GraphML file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is to be noted that the layout of the state-space graph in Figure 70 is rendered using Gephi Tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In that “node-1” represents initial marking of the composed model whereas “node-1956” represents the goal state (explained later in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Shades of green color (from dark to light) represent proximity from node 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' All the nodes are connected with edges (some of which may not be visible due to light colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Goal State:  (Main:TS:1\' (0,0,")  1956  Initial Marking  :  8  O  O  C  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 8  Model Verification using State space Analysis techniques  Page 161  Translation of Requirements specification into CPN Properties To proceed with the verification we first translate Objectives and Constraints defined in the requirement specification to CPN properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We assume that the default constraints S1, S2 and S3 are already verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Objective O1: All the enemy units must be destroyed CPN Translation As the Observer detects enemy units, therefore we say that if no more enemy units can be detected (because field-data is empty) then all the enemies should be destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore a marking where TS (Target- spotted) place has a null token should exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If such marking is found then the objective is said to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The following CPN function can be used to verify this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN Function fun AllTargetDestroyed() = let val token = 1`(0,0,"");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' /*Create a search criteria */ val predicate = fn n => (Mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="Main'TS 1 n) =  token;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' /*Create a predicate function*/ val TS = PredAllNodes (predicate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  /* Built-in Node search function in if (length TS > 0) then true else false end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Result When the function AllTargetDestroyed() is executed it returns True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is also evident from Figure 70 where the marking 1956 represents the goal- state and is reachable form the initial state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Constraint S4: There should never be a friendly fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN Translation When “UpdateField” (UF) place gets a token from BHQ component (which will be taken as input by the Field component), it shows which field unit is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We can collect all such nodes in the state-space (where UF field has tokens) and compare that all field units that have been destroyed are of type “enemy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If this condition holds in the entire state-space then S4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Following CPN-ML function can be used to check if friendly fire has ever happened or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The result should be false to satisfy S4 CPN Function fun CheckFriendlyFire() = let val predicate = fn n => IsNotEnemy(Mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content="Main'UF 1 n) = true;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val ListOfFrieldlyUnits = PredAllNodes (predicate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' in if (length ListOfFrieldlyUnits > 0) /* Means there is a friendly fire */ then true else false end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' fun IsNotEnemy (update) = let val upd:UPDATE = List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='nth(update, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' val target = #1 upd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' /*Extract information from the token at the place: UF */ in if GetType(#1 target) <> "Enemy" then  true /* Checks field unit type*/ else false end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Result The function CheckFriendlyFire() results false because no such incidence occurred with the data (initial state) provided to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To check that this function works correctly we created a counter example in which FDC component is assumed to be erroneous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' it wrongly accepts fire support requests of the friendly units), we ran the routine and found traces of the occurrence of friendly fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8  Model Verification using State space Analysis techniques  Page 162  As all the constraints and Objectives are satisfied we say that the field artillery model is composable at all levels and is verified with respect to its given specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore the BOM based composed model is qualified for further implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 State Space Reduction In this section the application of our proposed state-space reduction technique is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to proceed with the state-space reduction of the FA model generated by CPN tools we perform following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Trimming Node Description Each node in the CPN state-space graph has a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This description essentially tells about the presence of tokens (or multiple tokens) in all places of the model, which is called marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This description is very lengthy if the model has many places or sub-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this step we remove all the descriptions and only keep the information related to the places of the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To perform this step we use the library function NodeDescriptorOptions() (see manual [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When the descriptions are trimmed we will only get information of a node pertinent to the main places otherwise it will be a “Null” string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (This is an important difference for further steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Conceptually we hypothesize that trimming the node description does not cause loss of information because all the information other than the one in the main places is produced by the internal logic of the composed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since the composed components are considered as black boxes and they will eventually output important information (in form of tokens) in any of the main places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This information would be sufficient to answer any verification query related to the model under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Export to GraphML In the next step we export the state-space graph to an external file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since CPN state- space graph cannot be manipulated internally within the CPN environment therefore we export the graph to a standard GraphML format [128] along with the trimmed node descriptions and the information of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To perform this step we develop a GraphML writer function in CPN-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Reduction Algorithm In the next step, we apply our reduction algorithm specified in Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This algorithm is implemented in a Java application which uses JUNG library for graph manipulation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In brief, all the nodes which have “Null” descriptions are removed (because they are irrelevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a node is removed all its incoming and outgoing edges are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' So we connect each predecessor of the node with each successor to preserve the structure of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When all the nodes are checked the reduction is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The output of the reduce graph of Field Artillery mode is shown in Figure 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 8  Model Verification using State space Analysis techniques  Page 163  Figure 71: Reduced State-Space graph of Field Artillery Model  In Figure 71 node-2 represents initial marking (note that node-1 was trimmed in the reduction process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also Node-1956 still represents goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (Note that the nodes IDs remain the same in the reduction process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Reduced Graph Original Graph Percentage Nodes 428 1960 21% Edges 2503 6469 38 % Table 45: Reduction Statisitics Table 45 shows that the nodes are reduced to 21% and the edges are reduced to 38% of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Initial Marking  2  GoalState:  (Main:TS:1 (0,0,""))  1956Chapter 8  Model Verification using State space Analysis techniques  Page 164  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5 Summary In this chapter the verification of BOM based composed models is discussed using CPN based state-space analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An example model of Field Artillery is introduced and the entire verification process is applied on this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is shown how requirement specifications are translated into CPN properties and how they are verified using state-space analysis and Query functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  State-space analysis is advantageous as it is exhaustive and leads the modeler to all of the possibilities that can occur during the abstract level execution of a composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A state-space graph helps to study all of these possibilities and to understand the dynamic behavior of the components in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also a state-space query functions proposed along with the approach help in answering different verification questions and evaluate the correctness of model with respect to the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach however is vulnerable to the state-space explosion as for a simple model of Field Artillery 2503 nodes and 6469 edges were formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To deal with this situation we proposed an effective state-space reduction technique which not only reduces large state-spaces into reasonable size but also preserves important information for correct verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We demonstrated how our proposed state-space reduction technique is applied to the Field Artillery model for proof of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page 165  Chapter 9 Model Verification using CSP based Model Checking Technique  Model Checking is becoming a standard approach for the software verification due to its numerous advantages over traditional formal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Communicating Sequential Processes (CSP) is an event based formal language for describing patterns of interactions in concurrent systems and very useful for concurrent behavioral specification and verification due to its theoretical foundations of process algebra (also called process calculi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The application of CSP based Model Checking technique in the Composability verification also proves to be very useful, especially with a focus on the dynamic semantic composability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this chapter the Field Artillery Model presented in Chapter 8 is reused and extended with information to capture the behavior of a real-time probabilistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is shown how the Probabilistic-Timed Field Artillery Model is transformed into a composed CSP model and verified using PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In this chapter, the modified version of Field Artillery Model presented in chapter 8 is discussed as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The objective of this example is to represent a model of a system with time constraints and probabilistic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To the best of our knowledge the PN algebraic approach does not support verification of the timed models or probabilistic systems at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also the CPN based approach has a limited support for the verification of timed system but it does not cover probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We therefore propose to apply Modeling Checking for real-time probabilistic systems and show how a composed model of one such system can be verified using a CSP based Model checker called PAT (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Field Artillery Scenario The scenario of the Field Artillery Model is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is assumed that a soldier observes the field and detects enemy units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a target is spotted, he calls BHQ for fire support and provides the target details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In military practice, Time-On- Target (TOT) is a Field Artillery coordination protocol observed by multiple firing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This technique was developed by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Army during World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It uses a precise pre-determination of the estimated preparation time and the time of flight of the munitions from each firing battery to the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a Time on Target (TOT) is designated each battery that will join in firing on that target subtracts the time of flight from the TOT to determine the time to fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The firing units fire their rounds so that all the munitions arrive at the target at precisely the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This is done in order to achieve maximum target destruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there is a gap between the multiple impacts the enemy soldiers get time to prone or takeover in the hideouts and mobile vehicles can escape [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ assigns target to the batteries, and also schedules a certain “TOT” for the batteries to comply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each battery needs some time to prepare for loading appropriate ammunition and setting up the correct alignment and orientation of the barrel according to the computed firing solution using range (distance) and bearing (angle)  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 166  of the assigned target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is assumed that each battery needs a random preparation delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When each battery is ready, it will fire in its own time such that all the rounds hit the target at the given TOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also assume that the probability of hitting on the exact target location for each battery is ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='9’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In contrast to the previous scenario in chapter 8, we assume that there is only one target in the field component and all three batteries are taking part in the firing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To construct a conceptual model for this scenario, the following BOM components are composed: Field: Target location (We assume there is only one target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer: A soldier who request for the fire support from BHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ: Supervises the entire operation of fire support, responds to the calls for fire support and assigns targets to the batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery: Three units of artillery batteries (cannons and crew) responsible to hit the target exactly at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (Note that FDC component is removed from the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also some entity characteristics and event parameters are reduced for simplification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The modified BOM components of the Field Artillery conceptual model are formally defined as follows:  Observer = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Observer { C0(target)}  EvT = {E0(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  E1(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' detonation)}  Act = {A0(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E1)}  S = {S0(ObserverReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S1(WaitingForImpact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S0) } Table 46: Observer Basic-BOM  Field = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Field {C1(destruction[3])}  EvT = {E2(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BID),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E3(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BID),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BID),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' destruction)}  Act = {A2(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A3(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A4(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A5(Detonation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E5) }  S = {S2(FieldReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3}+{A3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3}+{A4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3}) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S3(TakingFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S2)} Table 47: Field Basic-BOM  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 167  BHQ = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT =BHQ {C2(TOT) }  EvT = {E6(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' TOT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E8(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E10(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null}  Act = {A6(CallForFireSupport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A7(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A8(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A9(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A10(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E10)}  S={S4(BHQReady,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S5(AssigningTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S6(WaitingForFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' {A8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4} + {A9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4} + {A10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S4})} Table 48: BHQ Basic-BOM  Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 = 〈 EnT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' EvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' AcT 〉 where: EnT = Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 { C3(BID),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C4(Destroyed) }  EvT = {E11(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' TOT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E12(ReadyToFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E13(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Destroyed),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  E14(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' null)} Act = {A11(AssignTarget,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A12(ReadyToFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A13(Fire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A14(FiringCompleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Battery1/2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  BHQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' E14)}  S = {S7(BatteryIdle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S8(Preparing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  S9(ReadyToFire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S10(Firing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  A14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' S7)} Table 49: Battery (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) Basic-BOM  FA = 〈 AcTIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  AcTOUT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  POI 〉 where: AcTIN = AcTOUT = ∅ POI = { POI-0(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A6), POI-1(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A7, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A11), POI-2(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A12), POI-3(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A13, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A3, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A4}), POI- 4(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A14, {?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A8, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A9, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A10}), POI-5(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A5,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1} Table 50: Field Artillery Composed BOM  The composed field artillery model is shown in Figure 72 using our proposed graphical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Verification using CSP based Model Checking Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 168  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Observer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Ready ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Waiting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='forImpact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Characteristics: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='C0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Actions: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='CallForFireSupport ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Detonation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='States: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='ObserverReady ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content='C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C4 A13 A12 S8 Preparing S10 Firing Battery2 A14 S9 ReadyToFire BatteryIdle A11 S7 Characteristics: C3=BID:Integer=1 C4=Destroyed:Boolean Actions: A11=AssignTarget A12=ReadyToFire A13=Fire A14=FiringCompleted States: S5=BatteryIdle S6=Preparing S7=ReadyToFire S8=Firing C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C4 A13 A12 S8 Preparing S10 Firing Battery3 A14 S9 ReadyToFire BatteryIdle A11 S7 Characteristics: C3=BID:Integer=1 C4=Destroyed:Boolean Actions: A11=AssignTarget A12=ReadyToFire A13=Fire A14=FiringCompleted States: S5=BatteryIdle S6=Preparing S7=ReadyToFire S8=Firing C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' C4 A13 A12 S8 Preparing S10 Firing A0 A1  Figure 72: Field Artillery Composed Model 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Requirement Specification We define Requirement speciation of the modified field artillery model as:  RS0 = 〈O, S〉 where: Objectives O = {o1, o2} and System Constraints S = {s1, s2 s3, s4} o1: All the firing units should fire precisely at the target location o2: All the firing units should fire at the target exactly at the given time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the Time on Target property should be satisfied)  s1, 2, and 3: The model should be composable at syntactic and static-semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The state-machines should match and the executable mode should correctly represent the conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 169  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Verification using Model Checking After the BOM are discovered, selected they are composed to form a conceptual model according to the simuland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This composed model is now ready for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' At this stage we select model checking technique for its verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Static and Dynamic Analysis We assume that the model qualifies syntactic and static-semantic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also when it undergoes state-machine matching process it is able to make progress until the goal-states are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 73 shows the interaction of the state-machine of each component during the state-machine matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Based on the fact that the constraint S1, S2 and S3a are satisfied we proceed to BOM-to-E-BOM extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer BHQ Field Observer Ready WaitingFor Impact Field Ready Taking Fire BHQ Ready Assigning Target Waiting ForFire BATTERY (1,2,3) Idle Preparing Firing CallForFireSupport AssignTarget Fire (1,2,3) FiringCompleted (1,2,3) Detonation ReadytoFire  Figure 73: State-machine Matching of Field Artillery Model 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 BOM to E-BOM extension At this stage all the BOM components are extended to our proposed E-BOM extension with the help of the modeler’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Here additional information such as timing constraints and probabilistic factors are proposed to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Following tables present E-BOM extensions of BOMs in the FA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Observer E-BOM SV and types {C0(Target:Integer} Initial States {S0:ObserverReady} Transitions State Event Time Guard Action Next State Observer Ready CallForFireSupport  WaitingFor  Impact  WaitingFor  Impact  Detonation  Observer  Ready  Table 51: Observer E BOM  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 170  Field E BOM  SV and types {C1(Firing_Result_Of_Battery1:Boolean},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  {C1(Firing_Result_Of_Battery2:Boolean},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  {C1(Firing_Result_Of_Battery3:Boolean}  Initial States  {S2:FieldReady}  Transitions  State  Event  Time  Guard  Action  Next State  FieldReady  Fire(1)  Action1  TakingFire  Fire(2)  Action2  Fire(3)  Action3  TakingFire  Detonation  FieldReady  /* A CSP script for defining a probabilistic action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' with 95% chance that the target will be destroyed when an event fire is received from battery1 and 5% chance that the target will be missed */ Action1{ pcase{ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05] : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 → atomic{tau{ Destruction [0]=False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} → Skip} default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1→ atomic{tau{ Destruction [0]=True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='}→ Skip} }  } /* From battery 2 */ Action2{ pcase{ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05] : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 → atomic{tau{ Destruction [1]=False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} → Skip} default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2→ atomic{tau{ Destruction [1]=True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='}→ Skip} }  } /* From battery 3 */ Action3{ pcase{ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05] : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 → atomic{tau{ Destruction [2]=False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} → Skip} default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3→ atomic{tau{ Destruction [2]=True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='}→ Skip} }  }  Table 52: Field E BOM  BHQ E BOM  SV and types  {C2(TOT:Integer}  Initial States  {S4: BHQReady }  Transitions  State  Event  Time  Guard  Action  Next State  BHQReady CallForFireSupport  AssigningTarget  AssigningTarget  AssignTarget  WaitingForFire  WaitingForFire  FiringCompleted(1)  BHQReady  FiringCompleted(2)  FiringCompleted(3)  Table 53: BHQ E BOM  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 171  Battery(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3) E BOM  SV and types  {C3(BID:Integer}  Initial States  {S7: BatteryIdle }  Transitions  State  Event  Time  Guard  Action  Next State  BatteryIdle  AssignTarget  Preparing  Preparing  readytofire  Wait[Prob]  Action1 ReadyToFire  ReadyToFire  Fire  Firing  Firing  FiringCompleted  BatteryIdle  /* A CSP script for defining a probabilistic wait action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' with 94% chance that each battery will launch the fire exactly at given time on target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and 6% chance that  it will fire earlier or later */ Action1{ pcase{  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='94] :  Wait[TOT];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='     readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire → ReadyToFire(i)  };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  }  Table 54: BHQ E-BOM 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 E-BOM to CSP# Transformation In this step, all the probabilistic timed extended BOM models are automatically transformed into CSP# using our automatic BOM-to-CSP transformation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 74 shows the global code block which is used to define global variables and the communication channels for each BOM send-receive event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //------------Global Block ---------------------------------- #define TOT 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Constant pre-defined Time on Target enum {Hit, Miss};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Hit or Miss flag //Each battery has a hit/miss ratio = 95:5 % var Firing_Result_Of_Battery1=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' var Firing_Result_Of_Battery2=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' var Firing_Result_Of_Battery3=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //For each event a channel is defined channel callforfire 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' channel detonate 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' channel assigntarget 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' channel firingcomplete 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' channel fire 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 74: Global code Block of Field Artillery Model  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 172  Figure 75 shows the CSP code of Observer BOM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The send-events are transformed into channels with send operator ‘!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' and receive-events are transformed into channels with receive operator ‘?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //===========================================================  //OBSERVER Component  //===========================================================  ObserverSM = ObserverReady();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Initial State  ObserverReady()=  (callforfire!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 >WaitingForImpact());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  WaitingForImpact()=  (detonate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 > ObserverReady());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //=========================================================== Figure 75: CSP representation of Observer Component  //===========================================================  //BHQ  //===========================================================  BHQSM = BHQReady();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='//Initial State  BHQReady()=  (callforfire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 >AssigningTarget());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  AssigningTarget()=  (assigntarget!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='> assigntarget!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 > assigntarget!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 >  Waitingforfire());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Sending assigntarget to multiple recievers  Waitingforfire()=  firingcomplete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='>firingcomplete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 >firingcomplete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 >  BHQReady();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Recieving firingcomplete from multiple senders  //=========================================================== Figure 76: CSP representation of BHQ Component  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 173  //=========================================================== //BATTERY i={1,2,3} //=========================================================== BatterySM(i) = BatteryIdle(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Initial State BatteryIdle(i)= (assigntarget?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='i->Preparing(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Preparing(i)= pcase{  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire > ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire > ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire > ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='94] :  Wait[TOT];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='   readytofire-> ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire-> ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire-> ReadyToFire(i)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='01] :  Wait[TOT+3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' readytofire-> ReadyToFire(i)  };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' // TOT is a global constant //readytofire is an internal event  ReadyToFire(i)= fire!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='i->Firing(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Firing(i)= firingcomplete!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='i ->BatteryIdle(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //=========================================================== Figure 77: CSP representation of Battery Component  //=========================================================== //FIELD Component //=========================================================== FieldSM = FieldReady();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //Initial State  FieldReady()=  pcase{  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05]: fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 >  atomic{tau{Firing_Result_Of_Battery1=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} > Skip}  default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 >  atomic{tau{Firing_Result_Of_Battery1=Hit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} -> Skip} } ||| /* ||| is the interleaving operator between the synchronizing events fire(1), fire(2) and fire(3) */  pcase{  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05]: fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 >  atomic{tau{Firing_Result_Of_Battery2=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} > Skip}  default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 >  atomic{tau{Firing_Result_Of_Battery2=Hit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} > Skip}  }  |||  pcase{  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='05]: fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 >  atomic{tau{Firing_Result_Of_Battery3=Miss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} > Skip}  default : fire?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 >  atomic{tau{Firing_Result_Of_Battery3=Hit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='} > Skip}  };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  /* This code randomly sets hit or miss effect for the firing of each battery */  Detonation();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Detonation()= detonate!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 > FieldReady();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='//===========================================================  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Figure 78: CSP representation of Field Component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Chapter 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Model Verification using CSP based Model Checking Technique  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='Page 174  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='// FIELD ARTILLERY COMPOSED MODEL //======================================================================  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='FieldArtillery = ObserverSM || BHQSM || FieldSM || BatterySM(1)|| BatterySM(2)|| BatterySM(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='// || is the parallel operator between all the components // BatterySM has three instances initialized with battery id parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 79: Field Artillery Composed Model  Figure 79 shows the CSP representation of how the transformed components are composed using the parallelism operator ‘||’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This means that all the components execute in parallel, however they perform barrier synchronization while exchanging events in their respective communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Model Checking of Field Artillery Model The CSP based Field Artillery Model can be opened and executed in PAT tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A successful compilation of this model shows that it has no errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When this model is executed, and if each component reaches its final states then we say that the constraint S3b of requirement specification is satisfied i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the transformed executable model correctly represents the behavior of its conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In the verification process, we define the following assertions to be verified by PAT built-in model checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since the nature of the input model is probabilistic and real- time, we use Probabilistic-Real-Time module of the PAT tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 80 shows how we define goal reachability assertions using PAT’s Probabilistic CSP LTL specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //=========================================================== // FIELD ARTILLERY COMPOSABILITY VERIFICATION //===========================================================  // ASSERT1: Goal state Reachability #assert FieldArtillery |= [](callforfire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0 -> <>detonate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  // Goal Definition  #define goal (Firing_Result_Of_Battery1==Hit  && Firing_Result_Of_Battery2==Hit  && Firing_Result_Of_Battery3==Hit);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //ASSERT2: //Goal Reachability #assert FieldArtillery |= <>goal with prob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 80: Field Artillery Verificataion Assertions  Assertion1 uses LTL construct to verify that if there is a “callforfire” then detonation at the target location will eventually occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If assertion1 is satisfied, it shows that there exists a valid execution path, which leads to the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 175  The result of PAT model checker is shown in Figure 81 which shows that the goal state is reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ********Verification Result******** The Assertion (FieldArtillery() |= []( callforfire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0-><> detonate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='0)) is VALID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ********Verification Statistics******** Visited States:160477 Total Transitions:475422 Time Used:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='8263759s Estimated Memory Used:111668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='696KB Figure 81: Verification Result of assertion 1  Assertion2 uses an LTL construct to verify that the “goal” is eventually reachable where “goal” is defined as a condition that all the batteries successfully hit at the exact location of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Note that assertion2 uses “with prob” construct, which makes it a PLTL statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Figure 82 shows the verification result which means that the probability of reaching the goal is between 77% and 94%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' ********Verification Result******** The Assertion (FieldArtillery() |= <> goal with prob) is Valid with Probability [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='77378, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='94526];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ********Verification Statistics******** Visited States:84019 Total Transitions:245561 MDP Iterations:63123 Time Used:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='5017483s Estimated Memory Used:97028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='192KB Figure 82: Verification result of assertion 2  Now we check whether the goal is reachable within the time constraints defined by Time-On-Target property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To perform this evaluation we use the PAT’s deadline operator as shown in Figure 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We define three assertions: Early, Exactly and Late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' //=========================================================== // FIELD ARTILLERY COMPOSABILITY VERIFICATION //===========================================================  //Goal Reachability with TOT constraint Early = FieldArtillery deadline[TOT-3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Exactly = FieldArtillery deadline[TOT];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Late = FieldArtillery deadline[TOT+3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  //ASSERT3: //Goal Reachability at TOT #assert Early reaches goal with prob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #assert Exactly reaches goal with prob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' #assert Late reaches goal with prob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Figure 83: Field Artillery Verificataion Assertions with TOT  Chapter 9  Model Verification using CSP based Model Checking Technique  Page 176  The verification result of assertion 3 is shown in Figure 84 according to which the early reachability of the goal is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas the maximum probability of reaching the goal exactly on TOT is 86% which satisfies objectives O1 and O2 However the maximum probability of reaching goal at a later time is 94% which satisfies O1 with a higher probability but does not satisfy O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ********Verification Result******** The Assertion (Early() reaches goal with prob) is NOT valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ******* Verification Statistics******* Visited States:58414  Total Transitions:146548  MDP Iterations:2557  Time Used:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='46633s  Estimated Memory Used:69901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='84KB  ********Verification Result******** The Assertion (Exactly() reaches goal with prob) is Valid with Probability [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='86271];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ******* Verification Statistics******* Visited States:169998  Total Transitions:342091  MDP Iterations:28115  Time Used:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='6064619s  Estimated Memory Used:148703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='552KB  ********Verification Result******** The Assertion (Late() reaches goal with prob) is Valid with Probability [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='94526];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  ******* Verification Statistics******* Visited States:274934  Total Transitions:584498  MDP Iterations:112797  Time Used:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4390606s  Estimated Memory Used:232989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='792KB  Figure 84: Verification result of assertion 3  Based on the verification results, we can say that the field artillery model satisfies it’s given requirements with a certain probability factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since it is a non-deterministic model, the reliability of the success depends on the threshold between how tight the Time-On-Target deadline is that BHQ can assign and how efficiently the batteries can prepare and how accurately they can fire on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Summary In this chapter the model checking approach is presented with an example and verified using Process Analysis Toolkit (PAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The example of Field Artillery Model (from chapter 8) is modified to represent a Probabilistic-Timed model in order to explain how the CSP based model checking approach using PAT can be effective in the composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using the example of Field Artillery model, it is explained how the verification of time constraints is performed and how different property assertions are verified with probability using PLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A successful verification of this approach is a result of satisfaction of all the assertions defined in the requirement specification, with an acceptable probability factor, and hence shows that the components are composable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Page 177  Chapter 10  Summary and Conclusion  This chapter makes a comparison between the composability verification approaches presented in this thesis and provides some guidelines for choosing the appropriate approach according to the nature of the composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This discussion is followed by a summary of the major contributions of the thesis and some suggestions for future research in this area are suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  In this thesis we propose a verification framework that follows the fundamental principles of M&S domain in terms of the notions of model correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It integrates several methods, techniques and tools to support different tasks in the multi-tier composability verification process of a composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also inherits useful technological characteristics related to model verification from other communities such as Petri Nets, Model Checking and Process-Algebra community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And utilizes the existing knowledge shared by these communities for the verification of component based simulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These simulation models are called component based models because they are designed in form of components and can further be composed to construct sophisticated models (called composed models or compositions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To ensure correctness, a composed simulation model is required to be verified at its different composability levels, where each level poses certain degree of difficulty in verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The initial levels of composability require that all the components in a composition can be syntactically connected to each other through valid interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' And they can correctly communicate with valid semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas a deeper level of composability is the dynamic-semantic composability which requires that all the composed components should possess suitable behavior in order to correctly interact with each other for pursuing mutual objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  The validity of behavior in a component composition relies on two factors: (i) each component should always be at the right state while interacting with the others and (ii) the composition should satisfy required behavioral properties, as prescribed in the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The proposed verification framework not only provides complete support for the verification of initial composability levels, but also the most important characteristic of this framework is its ability to verify the composed model components at the deeper level of dynamic-semantic composability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Composability Verification at this level is a daunting task and requires a dynamic analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The behavior of components can be studied when they are set to interplay with each other in an execution environment, where they communicate through the exchange of events and make progress by the change of their internal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore an appropriate dynamic analysis approach is required which not only provides suitable execution environment but also support built-in verification techniques to evaluate the composability behavior at the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' According to our findings not a single approach completely covers all the intricacies required for proving correctness at this the dynamic-semantic composability level due to its complex nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The effectiveness of a certain approach also varies due to the varied nature of the composed model and the modeling formalism used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since  Chapter 10  Summary and Conclusion  Page 178  some models have complicated structure and demand rich expressiveness in terms of data-centric details for the abstraction of a system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas others have behavior of complex nature including notions of concurrency and temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Besides the system behavior can be deterministic or stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it is difficult to depend on a single approach for the challenging task of dynamic-semantic composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For this reason we investigated three different dynamic analysis approaches in our framework namely: (i) PN based Algebraic Analysis, (ii) CPN based State-Space Analysis and (iii) CSP based Model Checking Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These approaches inherit theories, methods, tools and techniques from their corresponding ancestry communities such as PN, CSP and Model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We adapt these inherited resources and integrate them in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also propose several extensions in each approach to suit the needs of dynamic-semantic composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some of these extensions are listed as follows: \uf0a7 A component-based description format is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This description format is used to represent the BOM based composed model in the required form in order to apply the selected approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance a CPN based component model is proposed which represents the structural and behavioral aspects of a BOM component in form of a CPN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly for CSP, a Component oriented CSP process model is introduced which represents a BOM component using CSP notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 For each approach a rule based transformation technique is proposed which converts BOM components into the description format of the corresponding approach while keeping the structure and behavior of the model preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To ensure this fact methods are proposed to compare the original model (BOM) and the transformed model to assert that the latter correctly represents the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 For PN algebraic approach algorithms are proposed to automate the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also a function library is developed for the ease of conducting repeated verification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 In case of state-space analysis, a reduction technique is proposed which helps in reducing a large state-space and ease the process of verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The advantages and disadvantages of these three approaches are categorized as follows:  Category: Kinds of properties that can be verified PN Algebraic Analysis This approach only verifies a limited number of properties because it depends on the applicability of underlying mathematical theorems which are limited in number and may not cover all types of properties CPN State- Space Analysis It constructs state-space of all possibilities that a system could be in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it allows to specify and verify different kinds of general system properties as well as scenario specific properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking In this approach the verification depends on the specification of properties using LTL or CTL assertions, which along with their variety of extensions provide rich expressiveness to define different kinds of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it covers a bigger pool of verification questions both in terms of generic as well as scenario specific properties Table 55: Kinds of properties that can be verified  Advantage  Disadvantage  Neutral  Chapter 10  Summary and Conclusion  Page 179  Category: Type of the models that can be verified PN Algebraic Analysis This approach supports simple event-driven PN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It does not support models with rich data, or models of real-time or probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN State- Space Analysis This approach support models with rich data-centric structure and behavior since it offers flow of the tokens of complex data-types and their manipulations during the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also offers limited support for Timed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it does not support model verification of probabilistic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking This approach limits size of information in the model and does not entertain models with rich data-centric expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However it offers a variety of types of systems that can be verified such as reactive systems, real-time systems, probabilistic and stochastic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore this approach is much stronger in verifying different kinds of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 56: Type of the models that can be verified  Category: Scalability PN Algebraic Analysis Verification is dependent on the structure of the PN model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', number of places and transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This factor is much less than the number of reachable markings produced by other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore for larger models this approach proves to be scalable CPN State- Space Analysis Verification is dependent on the state-space, which tends to grow large for even ordinary models and hence can easily subject to state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Some reduction techniques (including one of our own) may minimize this risk but cannot completely omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking Model checking is also exposed to state-space explosion however it has gone through a continuous evolution of improved algorithms and compact data- structures to minimize this risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it promises a better resolution of scalability as compared to the State-space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 57: Scalability  Category: Infinite Model Verification PN Algebraic Analysis It is not affected in its reasoning if the model is finite or infinite, because in most of the cases it uses invariants for reasoning which are derived from the algebraic computations and do not depend upon the number of  reachable system states CPN State- Space Analysis If the model is infinite it will require a construction of infinite state-space which is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking Infinite model verification using this approach is possible by applying bounded model checking or by abstracting an infinite system into a finite one however this may lead to results with partial correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 58: Infinite Model Verification  Chapter 10  Summary and Conclusion  Page 180  Category: Usability PN Algebraic Analysis This approach is difficult to use due to complex mathematics and requirement to underlying applicable theorems for correct reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN State- Space Analysis This approach is easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Most of the operations are automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking This approach requires some effort to understand the formalisms used for model input and property specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However its operations are easy and all run in a black box i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the model checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 59: Usability  Category: Automation PN Algebraic Analysis This approach is not automatic because the definition of a property and its theorem applicability requires manual effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' When a property is defined, and a theorem is selected, the modeler has to perform mathematical computations and manually infer whether a condition is satisfied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CPN State- Space Analysis This approach is semi-automatic because defining a verification task and a suitable verification function requires modeler’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However the execution of the function is automatic and it searches all state-space to return a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' CSP based Model Checking This approach is totally automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Once a temporal logic assertion is defined, it is executed automatically by and model checker to find out whether it is satisfied or otherwise a counter example is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 60: Automation  Table 55 compares the proposed approaches in terms of the different types of properties that can be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It highlights that the PN algebraic technique is limited to verify only general properties (such as deadlock, liveness, fairness) since it depends on the underlying theorems for the proof of their satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Whereas the other two approaches are relatively more flexible to the specification and verification of properties of varied types, including general and scenario specific properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 56 presents a comparison of the proposed approaches in terms of the type of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' PN Algebraic approach only supports PN models with simple events without any parameters, guards, actions or input/output state-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These features are rather supported by CPN based state-space analysis approach which also provides limited support for Time based CPN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' But for models of complex real-time systems or probabilistic systems Model Checking approach is the suitable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Table 57 compares these approaches in terms of scalability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In case of Algebraic technique most of the operations in the property verification require matrix computations such as Incidence matrix, P-Invariants, T-invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore, the scalability factor is dependent on the size of the matrix i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the number of places × number of transitions of the composed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Thus, the algebraic technique is relatively salable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' With regard to scalability the CPN based state-space approach has serious limitations due to its rich data expressiveness and enumeration features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It is reported [130] that if the model is very large it generates state-space around 105 -106 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Consequently ordinary PCs cannot easily handle such a large state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However there are different approaches to make it more scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also believe  Chapter 10  Summary and Conclusion  Page 181  that if our proposed state-space reduction technique is directly implemented in the CPN tools environment, this limitation can further be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Model Checking technique is relatively more scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Since it relies on the usage of PAT tool which can handle about 107 states in a reasonable amount of time [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This should be sufficient for the verification of most industrial scale system models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  According to the Table 58 the algebraic approach is indifferent whether the model is finite or infinite in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An infinite model is a non-terminating model which keeps on evolving indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Such models are difficult to be verified using State-space approach because its state-space construction is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Although some techniques have been developed such as coverability graphs, to resolve this problem however they fail in some cases, such as in case of timed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' To verify infinite models using Model Checking is somewhat possible using bounded model checking or by abstracting an infinite system into a finite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However this may lead to results with partial correctness because only a portion of the system state-space can be considered for the reasoning of property correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Table 59: UsabilityTable 59 and Table 60 compare the ease of use of these approaches in terms of their application in a verification task and the extent of automation they provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In short, there is no ultimate winner and making the right choice of an approach entirely depends on the kind of model under investigation and the types of verification properties in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  There are also no exact rules however some fundamental guidelines can be used to help the modeler select a suitable approach: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 Guidelines for choosing an approach In this section some basic guidelines are presented for the modelers in making a suitable choice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1 PN Algebraic Technique This approach is most suitable when the analysis of a BOM composition is in question which with simple state/transitions and does not require any extension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', it does not have state-variables, or complex notions of transitions with parameters, guards, actions, inputs and outputs etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also its requirement specification includes properties which can be translated in form of PN properties (for which the solution of PN algebraic verification exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore it should be used when the requirement specifications can be defined in terms of PN properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance, in chapter 7 the objectives are translated into “Fairness” which means that they can be satisfied if the model is fair so the objective of verification is to prove this assumption and can be done using PN algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Also it is not effected by the model size, because it performs computations on matrices of the order of (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' of places × No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' of Transitions) which remain static, therefore it can also be used for somewhat larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It should not be used if the requirement specification contains reachability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Though it is possible to verify them using the PN state equation however it is rather difficult and inefficient as compared to State-Space Analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach cannot be used if the composed model has notions of time, colored- tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the BOM events have parameters) or non-determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 10  Summary and Conclusion  Page 182  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 CPN based State-Space analysis Technique This approach is best suitable when the given model has (or requires) rich data- centric structure and behavior such as state-variables, events parameters, guards and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this case the BOM components are required to be extended to capture more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If the modeler has the necessary information to extend the BOM components then he should use this approach otherwise he should choose the Algebraic technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach is also suitable if the modeler wants to execute the composed model at an abstract level to study the behavior of the components before actually implementing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Although other proposed approaches also have execution/simulation environments, but the CPN based execution is more detailed and comprehensive to study the interaction between composed components, as it provides a hierarchical interconnection between the CPN components and their execution is shown by the flow of data carrying colored tokens among inputs and outputs of each component in an interactive, step-by-step or an automatic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This allows the modeler to closely inspect the composition and its dynamics in a run- time environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Using this approach has many benefits from a component-based development point of view and the chances of its success are further elevated with our proposed state-space reduction technique called “Compositional State-Space”, which reduces the risk of state-space explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  This approach can also be used for timed systems since CPN environment supports modeling and verifying timed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However few limitations exist since the state- space of timed systems is much more expensive and memory intensive, due to the fact that each state carries an overhead of timed-stamps so even for a simpler model, its state-space will be much heavier than a similar model with no time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Moreover, if the model has even one non-terminating loop, its state-space cannot be constructed as it keeps growing to infinity by incrementing the time-stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=', the system may return back to previous states in loops and no new state is being added in the state- space but the time increases so the time-stamps keep on increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore with different time-stamps the same states keep on adding infinitely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach however completely fails when certain non-determinism is involved in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Even though CPN specification allows using different probability distribution functions, but when they are used, the resultant state-spaces are generated with variations, which cannot be used for verification reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore we do not recommend this approach if the model is stochastic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 CSP based Model Checking Technique This approach is usually favored by majority of the software verification community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It also has a greater flexibility of adopting a new technique or algorithm with specific requirements at hand, and thus can be useful in a variety of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach allows the modeler to execute the composed model using PAT simulator (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='7) therefore it also contends with CPN based State-space analysis in terms of studying system behavior at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However its main strength is revealed when it offers answers to a variety of verification questions, and to a variety of types of systems (real-time, probabilistic etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=') using model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This approach however restricts model expressiveness since it limits the use of data types such as strings, products, records (unlike CPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This requires an extra effort from the modelers to represent a model in reduced or compact forms using smaller  Chapter 10  Summary and Conclusion  Page 183  data-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance Boolean flags may be used instead of strings in the parameters such as a pair of string parameters: “Target_Destroyed”, “Target_Missed” can be represented as True/False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Similarly a set of string parameters: {“Red”, “Blue”, “Green”} can be represented by corresponding integer values {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This kind of reduction is required for this approach to work correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For example, we presented a detailed data-centric model of field artillery in chapter 8 to be verified with CPN state-space analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' But when it was required to verify a specific timed property (with non-determinism) we reduced unnecessary details and presented a simpler prototype of the Field Artillery model in chapter 9, focusing only on its behavior relevant to the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' By doing this modification the model was useable with this approach which successfully verified the required properties that could not be verified using CPN based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' As a final note, each approach has its own benefits and drawback and the choice depends on the modeler’s objectives, nature of the task and available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' However we also encourage using multiple approaches for a single task and comparing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It gives different perspectives and can better help in confirming correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='2 Thesis Contributions Component based modeling and simulation is a promising approach to develop and simulate system models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It incorporates numerous benefits such as modular design, logical separation, flexible change management, reusability of existing components, cross-domain model integration and thus consequently helps in reducing cost, time and system complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A key characteristic in this expedient paradigm is composability that is the ability to add or select and assemble reusable components in order to satisfy user’s requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In this thesis we mainly endeavored to investigate different aspects of this quality characteristic of component based model design and proposed a composability verification framework for the assessment of its correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our proposed framework uses Base Object Model (BOM), a SISO standard for component based modeling, and performs composability verification of BOM based model compositions with respect to given requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' In order to prove the correctness of composability of a set of BOM components, our framework undergoes a prescribed verification process, which has different phases starting from system abstraction, requirement gathering, selection of BOM components, their composition to form a conceptual model and then verifying its different levels of composability, in an iterative top-down refinement fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  When the entire process is completed successfully the composed model is said to be verified with respect to its specifications and can be used for implementation using an implementation architecture (such as HLA) and simulated to serve its purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Following are the key contributions of this thesis: \uf0a7 We developed a composability verification framework, which stands on fundamental verification principles and backed by the theoretical underpinnings of M&S, the details of which are mainly covered in Part-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' It integrates different methods, techniques, paradigms, algorithms, formalisms, templates, tools and 3rd party libraries (or APIs) to support different tasks in the multi-tier composability  Chapter 10  Summary and Conclusion  Page 184  verification process of a composed model with respect to its requirement specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We outlined a component based modeling and simulation (CBM&S) life-cycle by categorizing its different phases, and activities under each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A pictorial representation has been used to explain different tasks conducted under each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This life-cycle provides guidelines for using various features of our framework, and allows the user to conduct verification operations in a systematic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 A template to define and express requirements in a formal way is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our requirement specification template can be used to specify a set of objectives and system constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Objectives can be seen as ultimate goals while the constraints are necessary quality requirements that must be satisfied for achieving the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 Inspired from the Discovery, matching and composition (DMC) paradigm of model development [19], we propose method for rapid development of BOM based conceptual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We propose a formal description of BOM components and their compositions for documentation purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also propose a graphical notation38  to describe the structure of the BOM component and to show how they are connected to each other in a compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This notation can be used as blue prints of different model compositions and can be shared among different teams or archived in the repository for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 We propose methods for evaluating the structural consistency of the composed BOMs using rule based static analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The structural analysis involves checking that the components are correctly connected and they can communicate with each other with correct semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For semantic analysis, we propose an OWL based differencing approach which checks that the communication of the components is semantically consistent, meaningful and is understood as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We suggest a behavioral evaluation technique which implicates that the components can correctly interact with each other in a right causal order to reach final states or pass through the goal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For this purpose we propose state- machine matching process, which transforms BOM state-machines of each component into an executable SCXML format and execute them to analyze their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' If there is no deadlock and all the state-machines make required progress then the behavior of the components is reported to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 For the evaluation of dynamic-semantic composability level, our framework incorporates three main approaches: (a) PN Algebraic technique (b) CPN-based State-space analysis technique and (c) CSP based model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' These three approaches are offered to be used as alternatives to each other and their selection is dependent on the nature of the model being investigated and decision of the modeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also present basic guidelines to help the modeler choose an appropriate approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 For each approach we develop automatic transformation tool that transforms a BOM based composed model into its respective executable model description formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This method is inspired from Model Driven Architecture, in which a platform independent model is transformed into platform specific model using  38 It should be noted that different UML diagrams such as State charts and sequence diagrams are used to describe BOMs informally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our graphical notation follows the pattern of CBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 10  Summary and Conclusion  Page 185  some transformation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We also propose BOM extensions based on certain additional details that are required for correct transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For this purpose we develop a BOM extension editor that takes modeler’s input for extending BOM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We have applied our proposed approaches in three different case studies discussed in chapter 7, 8 and 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each case study provides a proof of concept and validates specific characteristics of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For PN based algebraic technique we presented a manufacturing system, in which fairness property is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For CPN-based state-space analysis approach a field artillery model is presented in which a set of scenario specific properties are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For model checking, the same field artillery scenario is modified into a timed non- deterministic model and a particular time property is verified with some probabilistic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We introduce a CPN based component model in order to describe a BOM component (or any other simulation component) in form of an executable model that can be executed using CPN execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  This CPN component model can also represent any other simulation component using its three layers namely (i) structural layer: which is used to define component attributes and variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' (ii) behavioral layer: which is used to describe the state-machine of a component and (iii) communication layer: which is used to describe components interfaces and how it can connect with other components and communicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We transform all BOM components into the proposed CPN based component model and compose them to form a composed model which can be executed in CPN environment and verified using CPN based state-space analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' \uf0a7 We introduce a State-space reduction technique called Compositional state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This technique assumes that all the composed components are black-boxes and their inputs and outputs are exchanged in the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Therefore we can select all the nodes from the state-space which are relevant to any activity happening in the main model and filter all the other nodes, by replacing them with edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' The resultant graph will be a reduced state-space representing only those nodes which describe the interactions of components in the main model and provide sufficient information for composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='3 Conclusions The verification framework proposed in this thesis expedites the process of composability verification of BOM based composed models with respect to the requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A verified composed model ensures consistent structure and behavior and guarantees the satisfaction of its objectives and required constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' A rapid development of the conceptual model using Discovery, Matching and Composition paradigm, its automatic transformation into an executable form and its composability verification helps in studying its structural and behavioral correctness with respect to the given requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This helps in rectifying any possible defects in the model design before it is actually implemented and simulated to serve its purpose, and thus saves a significant amount of time, cost and achieve robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Moreover this process strongly supports reusability as the entire process can easily be repeated to compose same components for different scenarios with varied configurations or with different requirement specifications (as in chapter 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  Chapter 10  Summary and Conclusion  Page 186  The entire composability verification framework is acclimated by a systematic Component Based M&S life-cycle which gives an outline of different phases of component based M&S development process, where each phase has different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' This life-cycle inherits important features and characteristic of some existing M&S development life-cycles and the Model Driven Architecture with an expansion of component based model development and guides the modelers with necessary directions to perform different tasks at different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' An important feature of this life-cycle is the software engineering principle of top- down refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' According to this principle a conceptual model is refined into an executable form through a number of intermediary steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each step generates a relatively detailed version of the abstract model and is easier to reason about its correctness based on assumptions of its previously verified version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' For instance, when the state-machine matching process is successful we can proceed to a more detailed dynamic level execution/verification with an assumption that the behavior of the composed components is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Our experience with the three different dynamic analysis approaches proves to be very constructive for composability verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Each approach in its own way provides significant improvement on efficient and accurate reasoning regarding model correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  We profess that the cross domain sharing of existing knowledge and valuable contributions from other communities (such as PN, CSP, model checking in our case) bridges cooperation in problem solving and helps in accomplishing quality research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='4 Future Directions Some of the key future directions of this work include: \uf0a7 We intend to deploy the composability verification framework in different application areas to evaluate its potential and to make use of its valuable features in verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' One area is the component based design for robotics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Many software architectures for robotic applications support component oriented design and thus can be explored for the utilization of our composability verification process, such as in studying various aspects of behavioral composability in different robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 In the context of improvements in the verification framework following are some key future directions: o We intend to include verification of requirement specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Correctness of requirements is a necessary aspect for successful verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  o We also intend to produce viable solution for the validation of the composed model with respect to the real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  o We defined Pragmatic composability level in chapter 2 however the composability verification at this level is still under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' We intend to explore this direction in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content='  \uf0a7 In general we are interested to explore the area of component based design optimization and to study the composability of component design for optimization with multiple objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+page_content=' Leiserson, Ronald L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' Rivest, and Clifford Stein, Introduction to Algorithms, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
+page_content=' : MIT Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfUQOY/content/2301.03088v1.pdf'}
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+A Lightweight Blockchain and Fog-enabled Secure Remote Patient
+Monitoring System
+Omar Cheikhrouhoua,e,g,∗, Khaleel Mershadb, Faisal Jamilc, Redowan Mahmudd, Anis
+Koubaae, Sanaz Rahimi Moosavif
+aCES Laboratory, University of Sfax, Tunisia
+bComputer Science and Mathematics Department, Lebanese American University, Beirut, Lebanon
+cDepartment of Computer Engineering, Jeju National University, Korea
+dSchool of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth,
+Australia
+eRobotics and Internet of Things Lab, Prince Sultan University, Riyadh, Saudi Arabia.
+fCalifornia State University, Dominguez Hills (CSUDH)
+gISIMA, Mahdia, University of Monastir, Tunisia
+Abstract
+IoT has enabled the rapid growth of smart remote healthcare applications. These IoT-based
+remote healthcare applications deliver fast and preventive medical services to patients at risk
+or with chronic diseases. However, ensuring data security and patient privacy while exchang-
+ing sensitive medical data among medical IoT devices is still a significant concern in remote
+healthcare applications. Altered or corrupted medical data may cause wrong treatment and
+create grave health issues for patients. Moreover, current remote medical applications’ ef-
+ficiency and response time need to be addressed and improved. Considering the need for
+secure and efficient patient care, this paper proposes a lightweight Blockchain-based and Fog-
+enabled remote patient monitoring system that provides a high level of security and efficient
+response time. Simulation results and security analysis show that the proposed lightweight
+blockchain architecture fits the resource-constrained IoT devices well and is secure against
+attacks. Moreover, the augmentation of Fog computing improved the responsiveness of the
+remote patient monitoring system by 40%.
+Keywords:
+IoT, Healthcare monitoring, Lightweight Blockchain, Fog computing,
+consensus protocol.
+1. Introduction
+Healthcare IoT networks are evolving from centralized to distributed systems to con-
+nect with each other to provide patients with high-quality healthcare. According to pre-
+∗I am corresponding author
+Email addresses: omar.cheikhrouhou@isetsf.rnu.tn (Omar Cheikhrouhou),
+khaleel.mershad@lau.edu.lb (Khaleel Mershad), faisal@jejunu.ac.kr (Faisal Jamil),
+mdredowan.mahmud@curtin.edu.au (Redowan Mahmud), akoubaa@psu.edu.sa (Anis Koubaa),
+srahimimoosavi@csudh.edu (Sanaz Rahimi Moosavi)
+Preprint submitted to Elsevier
+January 10, 2023
+arXiv:2301.03551v1  [cs.CR]  9 Jan 2023
+
+dictions, the current hospital-centered healthcare monitoring systems will develop first to
+hospital–home-balanced in 2025 and then ultimately to home-centered in 2030 [1]. New
+system architectures, technologies, and computing paradigms are needed to realize such
+evolution, specifically in the Healthcare Internet of Things (HIoT) [2].
+Emerging tech-
+nologies like IoT, blockchain, and artificial intelligence have made deploying smart remote
+patient monitoring systems a fact. Indeed, IoT devices permit them to sense and moni-
+tor patients’ physiological parameters, hence exempting them from a long waiting queue at
+a doctor’s visit. All necessary physiological parameters needed by doctors can be sensed
+by the biomedical IoT devices (also known as the Internet of Medical Things devices) and
+sent remotely to the doctor, allowing the latter to decide the appropriate treatment for the
+patient [3].
+The evolution of sophisticated security attacks and the rising need for individualized
+healthcare has made it essential for medical institutions to embrace blockchain technology.
+The arrival of the blockchain provides solutions to several problems that the healthcare sys-
+tem has been facing for a long time. The growing numbers of healthcare data breaches, pa-
+tient privacy violations, counterfeit drugs, and many other issues are major reasons for steer-
+ing the blockchain market’s growth in the healthcare industry. In general, the blockchain
+brings a large number of opportunities to smart healthcare, which can be summarized as
+follows:
+• Secure access to personal health records: the decentralized blockchain system offers the
+power of controlling data access to the owner of the data itself. Smart contracts register
+and authorize users to access the patient’s data according to the patient consent policy.
+• Patient Consent Management: the fundamental features of the blockchain, such as
+transparency and immutability, enables healthcare applications to build trust among
+patients and verify compliance with consent management policies.
+• Traceability of remote treatment: the blockchain permits healthcare applications to
+create immutable and coherent electronic records (EHRs) that can be viewed by all
+stakeholders. The transparency and consistency of blockchain EHRs aid in tracing the
+medical history of patients to offer the appropriate treatment.
+• Traceability of in-home medical kits and devices: the blockchain provides immutable
+and transparent record transactions to the ownership and performance of medical kits.
+Reputation scores of medical devices and kits are saved in the blockchain using smart
+contracts.
+• Reputation-aware specialist referral services: during the treatment of a remote pa-
+tient, medical referrals and expert suggestions are acquired through smart contracts.
+Blockchain enables healthcare providers to store these referral documents on an Inter-
+Planetary File System (IPFS) server, such that an IPFS hash of the document is stored
+securely in the blockchain. The hash prevents the alteration of the stored document
+and maintains its integrity.
+2
+
+• Automated payments: blockchain provides digitally signed automatic payments to
+guarantee non-repudiated secure transactions.
+A complete discussion on the blockchain benefits to smart healthcare applications can
+be found in [4].
+Ensuring the security of the remote patient monitoring (RPM) system is a must. Since a
+vulnerability in such a system could enable attackers to steal/modify sensitive information
+and endanger the patient’s life. The blockchain has emerged as a promising technology that
+can store and secure assets through a transparent and distributed ledger. In healthcare,
+where patient data is a critical asset that needs to be securely managed, the blockchain could
+become the right technology to address this challenge and provide a secure, transparent, and
+tamper-proof management of patient healthcare data. However, the blockchain is a heavy
+system requiring much processing and communication. Lightweight IoT devices would face
+problems if they were to act as full blockchain nodes. Hence, a solution should be adopted to
+enable IoT devices to participate in the blockchain network without affecting their limited
+resources.
+The lightweight blockchain [5, 6] has been proposed to achieve this purpose.
+Here, the blockchain architecture and processes are modified to assign light roles to the IoT
+devices while allowing them to benefit from the blockchain services.
+In traditional RPM systems, patient healthcare data is stored in an Electronic Healthcare
+Record (EHR) and saved in the cloud.
+Cloud computing provides ubiquitous access to
+patients’ data through a user-centric access control model, where the user chooses which
+data and to whom he/she should give access. However, a cloud computing system presents
+the disadvantages of high latency and, therefore, cannot fit critical healthcare application
+requirements where immediate intervention is needed. More precisely, real-time detection
+and notification of abnormal situations must be implemented in the context of a heart disease
+use case. Otherwise, the patient’s life will be at risk.
+To overcome the high latency limits of cloud computing and to fit the real-time require-
+ments of most healthcare applications, we propose leveraging fog computing technology in
+this paper. In our proposed architecture, fog computing will not replace cloud computing
+but will cooperate via the lightweight blockchain to provide real-time and efficient service.
+More precisely, we introduce the fog computing layer that will host a lightweight blockchain
+application with low latency requirements. On the other hand, complex AI algorithms can
+be executed at the cloud computing layer.
+Currently, smart cities are moving towards adopting blockchain technology in many smart
+city applications. In healthcare, and especially in remote patient monitoring, the blockchain
+can change the methods in which the application is executed and managed. Integrating the
+blockchain allows healthcare managers to guarantee the transparency of public healthcare
+data and removes the need to apply trust-based mechanisms and systems to achieve this
+target. In addition, the blockchain guarantees the privacy of patients’ personal data through
+smart contracts. Moreover, the blockchain allows for fast and direct connectivity between
+healthcare officials, providers, staff, and patients. Issuing blockchain transactions allows
+these entities to communicate securely via the blockchain without intermediaries. Finally,
+the blockchain allows healthcare and smart city officials to know the origin and destination
+3
+
+of each medical resource. They can also find out how healthcare services are being used
+without compromising people’s privacy.
+To sum up, we propose a smart and secure remote patient monitoring system based
+on three technology pillars: IoT, fog computing, and blockchain. More precisely, the key
+contributions of this paper are as follows:
+• We propose the architecture of a smart and secure remote patient monitoring system.
+The proposed architecture uses IoT for patient vital signs collection and blockchain to
+guarantee the privacy and security of the patient-collected data.
+• The efficiency of the proposed architecture is achieved through the introduction of the
+fog computing layer to provide real-time response and aggregate the patients’ collected
+data.
+• To reduce the heavy demands of traditional blockchain, we modify the blockchain
+structure to include a local blockchain within the IoT ecosystems and a global chain
+at the cloud layer. Each IoT ecosystem saves the block headers of all blockchain blocks,
+the bodies of the blocks of interest to the local chain, and the smart contract functions
+needed within the local chain. On the other hand, the global chain comprises whole
+blocks and smart contracts.
+• We propose a lightweight consensus model that enables the fog nodes to participate
+in the consensus protocol without consuming a lot of processing and energy resources
+and allows IoT nodes to store only the information they need to verify the legitimacy
+and integrity of the blockchain data that they obtain from fog nodes and cloud servers.
+The remainder of this paper is as follows.
+Section 2 outlines the existing literature
+on the remote patient monitoring system using blockchain and Fog Computing. Section 3
+gives an overview of the proposed remote patient monitoring architecture with its different
+components. Section 4 describes the details of the proposed lightweight blockchain model.
+Section 5 describes the fog computing layer functions and properties.
+The performance
+evaluation of the system is discussed in Section 6. Section 7 analyses the security of the
+proposed system. Finally, we conclude and give future directions in Section 8.
+2. Related Work
+As our work is based on three technologies, namely the IoT, fog computing, and blockchain,
+in this section, we present relevant work that uses one or more of these technologies to deploy
+a healthcare solution. The discussed works are summarized in Table 1.
+4
+
+Table 1: Summary of related work
+Ref Contribution
+Use case
+Used Technologies
+Pros(+)/Cons(-)
+IoT BC
+FC
+[7]
+Cloud based remote health
+monitoring system with sig-
+nal watermarking
+ECG-based
+health monitor-
+ing
+✓
+✓
++Providing sig-
+nal
+authentica-
+tion using water-
+marking
+[8]
+A
+hierarchical
+fog-
+computing-assisted
+ar-
+chitecture for IoT health
+monitoring system
+Arrhythmia de-
+tection
+✓
+✓
++
+Map
+the
+IBM’s
+MAPE-
+K
+computing
+model
+to
+the
+healthcare
+ap-
+plication
+[1]
+They developed a smart e-
+Health gateway localized at
+the edge.
+Heart disease
+✓
+✓
++Full-system
+implementation
+[9]
+Improved the energy con-
+sumption of sensor nodes
+during
+data
+transmission
+and processing.
+Migraine disease
+✓
+✓
++Energy
+con-
+sumption reduc-
+tion
+[10]
+An
+Edge-Based
+Architec-
+ture for IoT-Healthcare ap-
+plication.
+Detect
+high-
+stress conditions
+for workers and
+athletes.
+✓
+✓
+-security
+is-
+sues
+are
+not
+addressed.
+[11]
+Used retraining of SDA in
+the
+testing
+phase
+of
+ar-
+rhythmia
+classification
+to
+add or merge features in
+the
+anomaly
+detector
++
+Blockchain for access con-
+trol
+arrhythmia clas-
+sification
+✓
++High accuracy
+[12] Used blockchain to secure
+remote patient monitoring
+General
+✓
+✓
+-Time issue
+-Key
+manage-
+ment issue
+Continued on next page
+5
+
+Table 1: Summary of related work
+Ref Contribution
+Use case
+Used Technologies
+Pros(+)/Cons(-)
+IoT BC
+FC
+[13]
+Remote health monitoring
+system using Tor to min-
+imize the latency of the
+blockchain network.
+Cardiac
+Pa-
+tients.
+Sleep
+Apnoea
+Pa-
+tients. Epileptic
+Patients
+✓
+✓
+-The accuracy of
+the system is not
+tested.
+[14]
+HealthFog: A heart disease
+analysis system based on
+ensemble deep learning and
+using integrated IoT and
+Fog computing
+Heart disease
+✓
+✓
+-Security
+is-
+sue
+are
+not
+addressed.
+[15]
+They developed an intelli-
+gent e-Health architecture
+integrating
+AI,
+IoT,
+and
+cloud computing.
+ECG-based
+arrhythmia
+detection
+✓
+✓
++Hardware
+im-
+plementation of
+AI algorithms
+[16]
+An IoT and fog comput-
+ing architecture with par-
+allelization and core allo-
+cation capabilities to accel-
+erate healthcare processor-
+intensive services
+ECG-based
+arrhythmia
+detection
+✓
+✓
++Response
+Time
+was
+im-
+proved.
+[17]
+Lightweight identity man-
+agement and access control
+scheme for IoT devices us-
+ing IOTA.
+General
+✓
+✓
+-Does not sup-
+port smart con-
+tracts
+[18]
+Blockchain based architec-
+ture to provide patient cen-
+tric data access
+Healthcare
+✓
+✓
++
+Use
+cluster-
+ing techniques to
+improve system
+scalability
+BC: Blockchain, FC: Fog Computing or any cloud computing related technologies
+Hossain et al. [7] proposed a cloud-based architecture for ECG signal monitoring. To
+authenticate the captured ECG signal, the authors add a watermark that will be checked on
+the cloud side. Moreover, the authors proposed additional services, including ECG signal
+enhancement, classification, and analysis. Azimi et al., [8] proposed a hierarchical computing
+architecture leveraging fog and cloud computing technologies.
+The authors proposed a
+methodology to partition the existing machine learning methods for fog-enabled healthcare
+IoT systems. The authors in [1] developed a smart e-Health gateway localized at the edge to
+provide several functions, including local storage, real-time local data processing, embedded
+data mining, etc. By releasing the small IoT devices from these functions, a considerable
+6
+
+amount of energy can be saved by outsourcing some loads from sensor nodes to these smart
+gateways.
+In [9], the authors also proposed the integration of the IoT and cloud computing tech-
+nologies to predict migraine disease.
+The authors’ main contributions are the design of
+low-power techniques in the radio and data processing for the sensor nodes. Moreover, the
+authors proposed workload-balancing policies for cloud computing servers.
+The authors in [10] proposed BodyEdge: an edge-based architecture for IoT-healthcare
+applications. The system was implemented in two examples of edge gateway: Raspberry Pi3
+and Zotac CI540 NANO Pc, and its performance was compared to cloud systems. Moreover,
+as a validation example, the authors have implemented the system to detect high-stress
+conditions for users in two different scenarios, namely Workers in a factory and Athletes
+training in a fitness center. In [11], authors used blockchain to secure access control to patient
+EHR. The authors proposed to store patient data in an off-chain database to overcome the
+storage constraint of the blockchain.
+The authors in [12] used a consortium blockchain based on the IBM Hyperledger platform
+to secure remote patient monitoring.
+In their proposed system, sensors interact with a
+gateway (such as a mobile phone) that implements smart contracts for data analysis and
+sends essential notifications to patients and healthcare providers. The blockchain was used
+to securely log transactions (such as data reads and doctor’s commands). However, patient
+data was stored on a local database.
+They have proved that blockchain could be used
+to resolve security concerns about the transfer and logging of data transactions in an IoT
+healthcare system. The limitation rests in perfecting the time of the transmission of the
+aggregated data sent by the gateway to the blockchain nodes.
+The authors in [13] proposed a decentralized peer-to-peer remote health monitoring sys-
+tem. The proposed architecture uses Tor hidden services for off-chain data delivery between
+patients and doctors. The authors in [14] proposed HealthFog, a Fog-based healthcare sys-
+tem that integrates Edge computing and IoT. Their work was motivated by latency-sensitive
+healthcare applications, especially deep learning-based algorithms. The proposed system was
+validated for a health disease use case.
+In [15], an AI-driven e-health solution was proposed. The solution integrates IoT with
+cloud computing. The key difference of this solution is the distribution of the AI intelligence
+across the three architecture layers, namely: the Device layer, Fog layer, and Cloud layer.
+Moreover, the authors proposed hardware implementation of the SVM, ANN, and CNN
+algorithms using digital circuits. The authors in [16] proposed a framework for accelerating
+the response to remote patients requiring the execution of smart eHealth services. Their
+proposed framework supports distributed offloading to fog servers and multicore processors’
+capacity to accelerate its execution.
+The authors in [17] proposed a blockchain-based lightweight authentication and autho-
+rization scheme for IoT devices. The proposed scheme uses distributed ledger technology
+IOTA to design a lightweight and scalable mechanism for identity management and access
+control of IoT devices. However, this solution did not support smart contacts. The authors
+in [18] proposed a blockchain-based architecture that enables data owners to define their
+desired access policies over their privacy-sensitive healthcare data. The architecture used
+7
+
+two separate chains; one for storing data transactions and one for storing access policies.
+Several lightweight blockchain architectures have been proposed in the literature [19, 20,
+21, 22, 23, 24]. For example, the ECLB protocol in [19] saves the full blockchain on edge
+nodes, while the IoT nodes store what the authors call the fragmented ledger structure,
+which contains the block headers and some of the transactions in each block that are needed
+by the lightweight node. A multi-layer blockchain model is proposed in [20]. The blockchain
+network is divided into three layers. At the first layer, ordinary IoT nodes are divided into
+clusters. At the second layer, IoT cluster heads (CHs) store the local (i.e., cluster) copy of
+the BC. The cellular base stations (BSs) store the full global BC at the third layer. Nodes
+in Layers 2 and 3 collaborate to create new blocks and execute the consensus algorithm. On
+the other hand, some IoT nodes at layer one can be peers that maintain a copy of the local
+BC and act as transaction endorsers or committers, while CHs and BSs act as Hyperledger
+orderers who order transactions and create blocks.
+From the study of the existing work, we note that several works address only the per-
+formance and energy aspect of their proposed RPMs by adding the fog layer that manages
+the computing and data processing tasks [7, 8, 1, 9, 10, 14, 15, 16]. However, these schemes
+did not address the security aspects, and therefore, they are vulnerable to attacks. Other
+schemes such as [11, 12, 13, 17, 18], addressed the security aspect by adopting the blockchain
+technology, however, they used classical blockchain platforms and architectures that cannot
+fit the resource-constrained IoT devices. In this paper, we leverage the fog computing layer
+not only to improve the performance of the RPM system but also to lighten the load of
+blockchain technology. More precisely, the fog layer permits the proposal of a lightweight
+blockchain architecture that provides security services adaptable to resources constrained
+IoT devices. Moreover, the proposed consensus mechanism frees the architecture from the
+burden of classical consensus algorithms such as PoW or PoS [25, 26].
+3. The Proposed RPM System Overview
+This section gives an overview of the proposed RPM system. It highlights its three-layer
+architecture and the different communication interaction between the components.
+3.1. RPM System Architecture
+The proposed remote patient monitoring system is a three-layer architecture as shown
+in Figure 1, and which are:
+• The IoT devices layer: This layer, composed of biomedical sensor nodes, wearable
+sensor nodes, and IoT medical devices, is responsible for collecting the vital signs of the
+monitored patient. These sensor readouts are collected continuously; however, their
+transmission to the gateway node located at the fog layer can be done periodically. The
+transmission period depends on the nature of the vital sign and generally is determined
+by the patient supervising doctors.
+• The Fog Computing layer: This layer is responsible for the lightweight processing
+of vital signs received from the IoT layer. For example, suppose the monitored vital
+8
+
+Cloud Layer
+DB
+DB
+DB
+Gateway
+Gateway
+Gateway
+Gateway
+Edge/ Fog Layer
+IoT Layer
+Global Blockchain
+Big data Analytics
+Patient
+Doctor
+IoMT devices
+GetVitalSign
+Write/Order
+Local Blockchain
+GiveAccess/
+Audit Data
+GetVitalSign/ 
+GetNotification
+cloud 
+server 
+cloud 
+server 
+cloud 
+server 
+Blockchain 
+update
+Figure 1: The three-layer architecture of the proposed remote patient monitoring system
+sign exceeds a specific threshold. In that case, an alert message will be triggered and
+sent to the patient and the supervising doctor to make the right decision. Moreover,
+the fog layer first decides which data needs to be recorded in the blockchain network
+and then interacts with this latter. The fog layer will also aggregate continuous sensed
+data before sending it to the cloud server for permanent storage and data analytics.
+Additionally, the fog layer contains IoT gateways that include the local blockchain
+network, which is a subset of the global blockchain network (please refer to Section 5 for
+full description). In the proposed system, the fog computing module consists of many
+geographical intelligent gateways, that is, forming the fog. Each gateway supports
+different protocols for communication and serves as a point of contact between the
+sensor network and the cloud. It collects data from different sub-networks, translates
+protocols, and offers other higher-level services, including filters, data aggregation,
+analysis, and so on. The fog computing layer extends cloud computing to the edge of
+the network and its facilities. From cloud to end users/devices, the fog recognizes real-
+time interaction, dense geographical distribution, heterogeneity, accessibility support,
+pre-processing interoperability along with cloud interaction [27]. This enables latency
+to be decreased, particularly for real-time applications such as in-house IoT monitoring
+of patients. The fog reduces contact with the cloud, particularly in the event of a loss
+of cloud connectivity, where the data is stored locally on these gateways, and patients’
+data is sent to the cloud when the connection is restored.
+• The Cloud Computing layer: This layer is responsible for permanent data storage
+and data analytics. Complex AI and deep learning algorithms can be implemented
+at this layer for data classification, disease detection and prediction, and treatment
+9
+
++OAEplan decision.
+Moreover, in the proposed architecture, cloud servers play the role
+of full blockchain nodes that store the full copy of the blockchain and participate in
+transaction validation, block generation, and consensus. The blockchain records pa-
+tient data and actions of patients and caregivers and permits the patients to decide to
+whom they give access to their data. Moreover, blockchain technology contains pieces
+of code called smart contracts that can be automatically triggered when an event is
+achieved. These smart contracts are a powerful tool for a remote patient monitoring
+system as they can trigger an alarm and notify the doctor in an abnormal situation
+(for example, when the vital sign value exceeds a specific threshold). In addition, the
+blockchain is used to ensure patient data privacy and the system’s security. First,
+thanks to blockchain technology, the patient will be given an anonymous identity.
+This permits hiding the real patient’s identity; therefore, doctors can treat his/her
+data privately. Moreover, in our system, we propose to use a private blockchain. This
+type of blockchain has the advantage of restricting access to users’ data to only au-
+thorized persons (such as patients, doctors, and caregivers). Furthermore, blockchain
+architecture permits a patient-centric data management architecture. More precisely,
+the patient will decide to whom he/she shares data access (please refer to Section 4
+for more details).
+3.2. Communication Models
+The fog layer enables us to control access to IoT devices for medical applications. Each
+fog node manages and operates a group of medical IoT devices. This layer also interacts
+with a network of fog nodes allowed by blockchain, which function together on the Internet.
+All the related smart medical devices are connected with the closest blockchain-enabled
+fog node, e.g., in an in-house monitoring scenario. These blockchain-enabled fog nodes are
+communicated by IoT nodes and system users for authentication, authorization, and safe
+communication synchronization. An intelligent contract with a collection of rules can also
+be established on top of the fog nodes allowed by blockchain. Furthermore, the consensus
+algorithm is performed to validate the transactions and blocks for those transactions after
+they are created.
+Transaction blocks can be exchanged between cloud servers and the
+blockchain-enabled fog nodes or between the fog nodes to support robust authentication,
+permission, and distributed secure communication. The proposed solution mainly includes
+four forms of communication:
+(1) Medical caregiver-to-fog communication: Where the end user (e.g., a healthcare
+provider) is willing to use a particular IoT system, he first sends a request for au-
+thentication with a query authentication function specifying the sensor details to the
+blockchain-enabled fog node. The fog node with the blockchain feature will search for
+that medical attendant in the available list of approved sensor equipment. A reject
+message will be given when the user is not allowed to access the requested data. Oth-
+erwise, if the user is approved, the blockchain-enabled fog node issues an access token
+containing Unique Identification (UID) information, length, time of access, blockchain
+address for the data, user blockchain address, and blockchain address of the fog node
+10
+
+that stores the requested data. Notice that every fog node, sensor, and the user has a
+unique blockchain address.
+(2) Medical sensor-to-fog communication: The sensor-to-fog correspondence has two
+principle objectives in our framework.
+IoT medical services system mainly aims to
+validate and authorize the clinical sensors. The following goal is to insert a blockchain-
+enabled fog connected to sensor devices. It helps new sensors to enlist with the mist and
+ensures that all sensors are recognizable by the blockchain network. In our context, each
+IoT medical care system has at least one blockchain-enabled fog node close to the entire
+blockchain network and is used for the enlistment, confirmation, and authorization of
+IoT medical care gadgets with the same framework. Initially, the gadgets will enroll
+with their associated blockchain fog node. As an exchange and blocks are made for
+them, data concerning these gadgets are placed in the blockchain. These blocks would
+then be transported between the wide range of different blockchain fog nodes. Should a
+system with a collecting place need confirmation and consent, the associated blockchain-
+enabled fog node should be given its certifications. The blockchain approves the provided
+accreditation, and if there are significant requirements, the IoT gadgets for medical care
+are effectively checked and authorized. If the certification is not valid, the gadget is
+refused and will not obtain permission to access the blockchain data.
+(3) Fog-to-fog communication: The main goal is to synchronize the information associ-
+ated with IoT medical service confirmation and approval across all blockchain-enabled
+fog nodes [28]. Several biological or physiological parameters are obtained by medical
+sensors transmitted by patients. Medical IoT programs should be reliable and diligent
+in supporting patients moving to a hospital or home. Typically, the mobility support
+of the medical sensors from the upper layer (i.e., fog layer) should be given so that zero
+reconfiguration in the sensor layer is essential. The strategic location and distribution
+of smart gates in the fog layer can be used to provide smooth mobility for medical
+sensors and relieve processing loads. Fog-to-fog contact helps patients wander around
+the hospital wards, ensuring their health monitoring is not disrupted. The patient-free
+movement provides a high level of medical services using a portable patient monitoring
+system. Support of mobility for healthcare IoT systems is one of the most critical prob-
+lems [29]. The improvements to patients’ quality of life in such programs are essential
+[27]. It is important to encourage patients to walk into the hospital/medical facilities
+knowing that monitoring their well-being is not disrupted. It is necessary to establish
+self-configuration or transfer mechanisms to ensure safe and successful data transfer be-
+tween different Medical Sensor Networks (MSNs) [30] to achieve ongoing monitoring of
+patients considering mobility support. For example, when a patient is moving across the
+clinics, a data transfer mechanism is described as the process of modifying or updating
+the registration of mobile sensors on its MSN base. Data handover solutions should
+allow ubiquity when they need to function independently without human interference.
+(4) Medical sensor-to-medical sensor communication: When two clinical devices are
+effectively tested and approved (both have a position with a similar system or another
+one), they may create a safe link to each other and convey information.
+In a case,
+for example, where a patient is released from a clinic but still needs to be constantly
+11
+
+monitored. The doctors bind the patient’s body before he/she leaves the medical cen-
+ter to health tracking devices, including blood pressure monitors, pulse sensors, blood
+glucose monitoring sensors, etc. These devices sense the patient’s blood pressure, heart
+rate, and glucose level and transmit them through a safe channel to the health workers.
+These devices can also interact with the patient’s intelligent home devices. For exam-
+ple, if the patient’s condition becomes severe or a fall is detected, an immediate alarm
+may automatically be activated. The hospital-related devices must interact to check the
+availability of hospital beds in a smart city to ensure a correct count. The proposed
+framework provides medical devices with access control in the IoT healthcare system.
+Under this mechanism, devices can only communicate with recorded and successfully
+authenticated and certified devices with blockchain-enabled fog nodes. A device not reg-
+istered in the blockchain cannot authenticate itself or communicate with other devices
+within the same healthcare ecosystem or external ones. The contact between malicious
+devices and legitimate devices would also be alleviated.
+In what follows, we detail the proposed lightweight blockchain model and the Fog layer
+functions and properties.
+4. The Blockchain Module Description
+Blockchain technology provides a decentralized, transparent, authenticated platform that
+applies a consensus-driven approach to facilitate the interactions of multiple entities through
+the use of a shared ledger. Beyond the financial sector, where much of the initial develop-
+ment is taking place, blockchain has the potential to revolutionize the healthcare system.
+By providing doctors, patients, researchers, and other healthcare professionals with a mech-
+anism for the controlled exchange of sensitive, permissioned data, blockchain technology can
+improve data sharing and transparency between clinical and research data systems. Any
+healthcare organization participating in a blockchain consortium would be able to share
+medical information, regardless of their native electronic health record system. Blockchain
+provides significant opportunities for healthcare organizations to deliver more efficacious
+treatments and diagnoses through increased provider data sharing and potentially safer and
+more effective remote patient monitoring through advanced technologies such as AI.
+4.1. Blockchain Architecture of Proposed RPM System
+We propose a lightweight blockchain architecture to manage the data storage and re-
+trieval operations in the remote patient monitoring system. In our system, we implement a
+lightweight blockchain architecture that aims at reducing the delay in accessing the cloud
+by the end users while maintaining the security and immutability of data at all nodes. The
+blockchain will store all healthcare-related data, such as the IoT sensor readings, lab test
+results, physicians’ decisions, commands, etc.
+In addition, the blockchain will comprise
+transactions that contain management and security-related data, such as nodes’ and users’
+registrations, access requests, smart contract results, etc.
+In the proposed architecture, cloud servers play the role of full blockchain nodes that store
+the full copy of the blockchain and participate in transaction validation, block generation,
+12
+
+and consensus. On the other hand, IoT gateways play the role of light blockchain nodes
+that store part of the blockchain. In our system, each gateway will be connected to a certain
+number of IoT networks. For example, an IoT gateway at a patient’s home will connect to
+a single IoT network that contains the IoT devices that are monitoring the patient. On the
+other hand, an IoT gateway at a hospital could connect several IoT networks, such as IoT
+devices, in several patients’ rooms. Here, the IoT devices in a certain room or Lab form a
+separate IoT subnetwork since the data produced by these devices will be linked together
+(for example, data related to a specific patient, doctor, lab, etc.).
+The IoT gateways and sensors that exist in the same IoT ecosystem (for example, home,
+hospital, health institution, etc.) form a cluster that store and manage a local blockchain.
+Each local blockchain is created as part of the full blockchain that is related to the cor-
+responding ecosystem. For example, in a certain patient’s home, a set of IoT devices are
+connected to an IoT gateway. The devices and gateway form a cluster that store and man-
+age a local blockchain that contains the blockchain blocks related to that home only. In a
+hospital, several gateways and sets of IoT devices will form a cluster that store and manage
+the blockchain of the hospital.
+In each cluster, the sensor nodes store only the blocks headers of the full blockchain,
+while the gateways store the block headers of the full blockchain in addition to the full blocks
+of the local blockchain (as illustrated in Figure 2). In addition, to avoid overwhelming the
+gateways with excessive storage as the blockchain grows, each transaction will have an expiry
+time after which it becomes obsolete (for example, when the information in the transaction
+becomes old and is no more relevant). Each gateway saves a data structure that contains,
+for each transaction, the ID of the block in which the transaction is stored (BlockID) and
+the transaction expiry date (Tex). The gateway continuously updates the data structure
+when a transaction expires. In addition, the gateway searches the data structure to detect
+any block in which all transactions have expired and deletes it. Using this approach allows
+the gateway to remove old blocks and create room in its storage for new blocks in the local
+blockchain.
+In the proposed system, IoT nodes continuously generate data and send them to the
+IoT gateway. In addition, healthcare providers (doctors, nurses, scientists, etc.) send their
+data (such as prescriptions, sensors’ configurations, lab test results, commands to activate
+actuators, data analytic results, etc.) to the nearest IoT gateway in their institution’s IoT
+cluster. The IoT gateway stores the data it receives in a temporary cache. Each small period
+(for example, every 100 ms), the IoT gateway aggregates and groups the received data into a
+blockchain block and sends it to the cloud server. Note that each block can contain multiple
+transactions. For example, the readings of a certain sensor can be aggregated into a single
+transaction. Similarly, if the doctor is sending configuration commands to the IoT sensor,
+the configuration settings of each sensor can be grouped into a transaction. Each transaction
+will be signed by the owner that created the transaction.
+4.2. Consensus Protocol
+We consider a network of cloud servers that are used by various healthcare providers
+to manage the system. As mentioned, the cloud servers act as full blockchain nodes that
+13
+
+Figure 2: The architecture of the proposed blockchain model: the cloud servers store the full blockchain,
+the IoT gateways save the local blockchain, while the IoT nodes store the block headers.
+store all the blockchain blocks. In addition, the cloud servers participate in the blockchain
+consensus protocol. Each cloud server has a unique blockchain ID. The cloud servers create
+the blockchain blocks successively based on their IDs. In other words, the server with the
+smallest ID creates the first block, followed by the server that has the second smallest ID,
+and so on. When the server that has the biggest ID creates a block, the turn goes back to
+the first server. Note that the block generation time at the IoT gateway should be adjusted
+to allow all the cloud servers to generate their blocks in order to avoid block accumulation
+at the cloud servers.
+When its turn to create the new block arrives, a cloud server CS 1 broadcasts the block
+that it received from the gateway to all the cloud servers. Each cloud server CS i verifies that
+all transactions in the block are legitimate by validating the signature of each transaction.
+Next, CS i replies with a CONFIRM message to CS 1. The confirm message contains CS i’s
+signature of the new block. However, If CS i discovers that one or more transactions in
+the block are not valid, it replies with an ERROR message. In its turn, CS 1 waits until it
+receives at least (N /2+1) CONFIRM messages before it adds the block to the blockchain
+and broadcasts its ID in a “Block Add” message to all cloud servers. Here, N is the number
+of the cloud servers.
+This mechanism allows a cloud server to add the new block after
+the majority of cloud servers confirm its validity. The ”Block Add” message contains the
+signatures that CS 1 received in the CONFIRM messages. When a cloud server CS j receives
+a ”Block Add” message, it checks the attached signatures to ensure that more than (N ÷ 2)
+cloud servers have validated and confirmed the new block, before adding it to its blockchain.
+After receiving the “Block Add” message, each cloud server adds the new block to its
+blockchain and broadcasts it to its clusters. Note that each cloud server can serve multiple
+14
+
+Block Header
+Headers Blockchain
+TX 1
+Block Body!
+TX 6
+Block
+Block
+IoT Sensors
+Full
+Blockchain
+Cloud
+IoT Gateway
+Server
+Local Blockchain
++
+Block 0
+Block 1
+Block 2
+Block k-1
+Block k
+Header
+Header
+Header
+Header
+Header
+Body
+Body
+Local BlocksFigure 3: A sample scenario of the proposed consensus algorithm.
+institutions and organizations, with each institution/organization having its own IoT cluster.
+Each gateway in a cluster examines the new block to determine if it contains transactions
+that were generated by one of the IoT networks in the cluster. If yes, the gateway stores
+the block in its local blockchain and sends it to the IoT devices that are connected to it.
+Each IoT device validates the block (by hashing it and comparing the result to the hash
+in the block header) and then stores the block header in the headers’ blockchain. Next,
+the IoT device caches the block body for a small period of time before deleting it. On the
+other hand, if the new block does not contain transactions that were generated by an IoT
+network in the cluster, the gateway validates the block, sends it to the IoT devices that are
+connected to it, extracts the block header, and adds it to the headers’ blockchain, and then
+deletes the block. Each IoT device that receives the block performs the same operations
+as the gateway. This allows the gateway and IoT devices to maintain the headers of all
+blocks in the blockchain and use these headers to validate any block from outside their local
+blockchain that they obtain from the cloud servers in the future. The proposed consensus
+protocol is illustrated in Figure 3. In the figure, gateways G1 and G2 are connected to cloud
+server CS1, while gateway G3 is connected to cloud server CS2. At a certain time, G2 creates
+a new block B1 and sends it to CS1. When its turn to generate a new block arrives, CS1
+broadcasts B1 to the cloud servers. Each cloud server confirms B1 by sending a CONFIRM
+packet to CS1. Next, CS1 sends a “Block Add” packet to the cloud servers, and each cloud
+server sends the new block to its gateways. G1 and G2 receive B1 from CS1 and add it to
+15
+
+Block B1
+Block B1
+Block Bi
+Block B1
+Send Bi
+Creation
+Broadcast :
+Commit
+ppy
+to Gateways
+Gateway Gi
+Gateway G2
+B1
+Gateway G3
+Cloud Server CS
+wait for turn
+Cloud Server CS2
+Cloud Server CS3
+i
+Cloud Server CS.
+New Block Packet
+Add B ock Packet
+* : Add Block to Local Blockchain
+Block Broadcast Packet
+^ : Add Block Header to Local Blockchain
+Block ACK Packettheir copies of the local blockchain (since B1 was generated by a gateway in CS1’s cluster),
+while G3 receives B1 from CS2 and adds its header only to the local blockchain.
+4.3. Smart Contracts and Data Management
+When an IoT device or a user requires data from the blockchain, it sends a request to the
+IoT gateway. The latter searches for the data in its local chain. If it finds it, the gateway
+authenticates the sender and verifies that it has access to the requested data. If yes, the
+gateway replies directly to the sender with the block that contains the data and the token
+that enables the sender to access the data (more about this soon). If the gateway finds
+that the required data doesn’t exist in the local blockchain, it forwards the request to the
+cloud server. The latter performs the same operation, i.e., it authenticates the requesting
+node and verifies that it has access to the requested data. If yes, the cloud server sends the
+block that contains the data and the access token to the gateway, which forwards them to
+the sender. When the latter receives the block, it validates it using the headers blockchain
+before it retrieves the required transactions from the block and decrypts it using the access
+token.
+Note that in our system, all transactions that can be accessed together are assigned an
+access token by the creator and saved into a smart contract. When the creator wants to
+grant access to the transaction to a certain node/user, the creator executes a smart contract
+function that adds the ID of the node/user to the access list of these transactions that is
+saved in the smart contract. When the node/user wants to access the transactions, it should
+authenticate itself and obtain the access token as described before. If the transactions belong
+to the local chain, the smart contract is executed by the gateway within the local chain.
+Else, the smart contract is executed by the cloud server within the full chain. The various
+subsystems and interactions in the proposed RPM platform are presented in Figure 4.
+5. Specifications of Gateways at the Fog Layer
+The fog layer is made up of IoT gateways which function primarily as a hub between
+the cloud and IoT levels [31].
+With an in-depth study of the role of the gateway in a
+smart home/hospital, where the location and mobility of things and users are confined
+to hospital premises or buildings, it can be recognized that the stationary nature of the
+gateways empowers them with the property of being non-resource constrained in terms of
+power consumption, processing power, and communication. These advantages can be used
+by allowing gateways with ample intelligence, computing power, and structured networks.
+An inter-device communication is the key task of a gateway and supports numerous
+wireless protocols.
+We broaden the function of such gateways into fog enablers by (1)
+building a distributed gateway network and (2) implementing features such as the repository
+(i.e., local data processing and storage using blockchain) to temporarily preserve data for
+analysis by sensors and users. These are important to provide local pre-processing of sensor
+information and, therefore, to be an intelligent gateway for medical services. In a smart
+gateway, the main functions are:
+16
+
+Figure 4: Interaction Model of the proposed blockchain-based remote patient monitoring system.
+5.1. Local data processing and storage
+Local data processing is a key aspect of fog computing and is performed locally so
+that intelligence is accessible at the doors.
+Based on the device architecture, fog/edge
+layers must continuously handle a large amount of information and respond to different
+conditions in a short time. In the remote patient management system, this becomes more
+important by allowing the system to respond to medical emergencies as quickly as possible.
+Gateways should store inbound information in local storage to ensure that the remote patient
+monitoring system can quickly recuperate patient medical data. In the proposed system, we
+make use of the local blockchain to achieve this objective in a secure manner. The patient
+data can be stored as blockchain transactions in an encrypted or compressed form depending
+on their context and security requirements. The gateway stores all data related to the local
+cluster in the local blockchain. In addition, when the gateway receives a blockchain block
+that contains data related to other clusters, it caches the block for a small period of time to
+allow users in the cluster who require data from the block to access it in a fast manner while
+the data is hot. Moreover, since the network bandwidth is limited between the gateway and
+the cloud, the locally cached blocks can be used to maintain a continuous data flow in the
+event of a weak or unstable connection.
+5.2. Data filtering
+Data from various medical sensors must be obtained before further processing, e.g., data
+analysis, on the fog layer. The major sources of knowledge for the assessment of the health
+status of a patient in the remote patient monitoring system [32, 33] are bio-signals, for exam-
+ple, Electroencephalography (EEG), Electrocardiogram (ECG or EKG), and Electromyog-
+raphy (EMG). They typically have complex types that have small amplitudes and varying
+17
+
+Remote Patient Monitoring Healthcare System
+Smart Contract
+Call
+Smart
+Identity
+Storage
+Communication
+Consensus
+Smart Contract
+Contract
+Management
+Management
+Protocol
+Reply
+Interface
+Patient
+Data Access
+Data
+Local
+All Block
+All Blocks
+Data
+Blocks
+Validation
+Headers
+Enrollment
+loT Gateway
+Cloud Server
+Certificate
+Smart Contract
+Healthcare Blockchain Platform
+Nurse
+Result
+Healthcare
+Registration
+Certificate
+Data
+Sensor1
+Sensor2
+Sensor3
+Sensorn
+Sensor4
+Healthcare Sensors
+Doctorfrequencies. It is important to remember that noise is often introduced to the signals during
+a patient’s body sensing in a way that distorts the accuracy of the signals. These noises
+are caused by different sources, including electromagnetic interference from other electrical
+devices, shifts in current in the electricity grid, and inappropriate attachment of sensors
+to the body of users. At the fog level, due to the proximity of the sensors, the gateway
+addresses this issue. The fog layer is digitized via different contact protocols by sensors
+(e.g., 6LoWPAN, Zigbee, etc.). While sensors are able to perform lightweight filtering to
+eliminate certain noises during the data collection process, the fog layer offers more complex
+and robust data filtering.
+5.3. Data analysis
+With local data analysis in the fog layer, the sensitivity of the device can be corrected.
+It helps the device to anticipate and diagnose situations of emergency. The developed deep
+learning module for detecting irregular cardiac conditions is implemented in the fog layer in
+our proposed RPM medical system. The deep learning module can categorize signals and
+detect abnormal conditions on the basis of the sensed ECG signal. As a result, the device
+responds more accurately, rapidly, and in real time to emergency situations. In addition,
+local input and locally sensed data analysis change the quality and reliability of the device
+in the event of the unavailability of the Internet link. Internet disconnection may occur
+regularly for the long-term monitoring of patients with chronic diseases. Fog computing, in
+this case, provides local maintenance of the system’s features. Thus, the sensed data and
+processing results can be kept locally on the fog layer and later synchronized to the cloud via
+the blockchain. Data analysis in the fog often helps the device minimize severe parameter
+processing latencies.
+5.4. Improved latency
+Agile responses and quick decision-making for acute diseases and emergencies, where
+transmission time and data processing are to be reduced, are important for a continuous
+remote control system. When raw medical data is transferred from medical sensor nodes to
+the cloud, cloud computing can trigger response latencies indefinitely if the network condition
+is not predictable. This becomes serious when streaming-based data processing, such as that
+EEG or ECG signals that are obtained from patients, is needed. Hence, deploying high-
+priority data analytics in distributed gateways in the fog later and making time-sensitive
+and critical decisions inside the local network make the remote patient monitoring system
+more predictable and robust. The processed data can then be transmitted for storage and
+further processing to the cloud.
+5.5. Sensor nodes energy efficiency
+There are various drawbacks to the processing of data at sensor nodes, as medical sensors
+are resource-restricted devices. Complicated tasks can, in certain cases, be performed suc-
+cessfully at sensor nodes but at significant energy costs. The transfer of heavy-weight tasks
+from sensors to intelligent gateways in the fog layer can be an effective solution for solving
+the above-mentioned problem, in particular when sensors do not have sufficient resources.
+18
+
+Much energy can be saved with the aid of fog computing by outsourcing tasks from medical
+sensors to intelligent gateways.
+6. Performance Evaluation
+In this section, we present the performance evaluation of the proposed RPM system. We
+have mainly evaluated the performance of the proposed blockchain module and demonstrated
+the efficiency of Fog computing in dealing with critical healthcare applications.
+6.1. Blockchain Implementation and Performance Evaluation
+The proposed blockchain model was implemented via the Hyperledger platform. Hy-
+perledger is an open-source development platform for blockchain applications. It has been
+widely used as an implementation platform by the research community and is considered
+a benchmark tool to evaluate the performance of the proposed approach against state-of-
+the-art approaches. For smart contracts, the Hyperledger tool provides easy-to-configure
+and user APIs, thus making validation easy for our research work. Furthermore, the REST-
+ful API is utilized to provide the functionality of interoperability and expose the back-end
+blockchain services to the client application through which the patients or other medical
+personnel interact with the system. The smart contacts are designed and aggregated in the
+form of .bna files known as business network archive. Hyperledger Composer [34] is used
+to implement and design the proposed medical blockchain, which aims to enhance system
+operations in terms of throughput and latency. Hyperledger Composer is an open-source
+tool used to design blockchain applications. The .bna in the designed platform consists
+of a model, query definition, transaction, and access control rules. The model file is the
+combination of participants, assets, and transactions. The participants are the user of the
+system who can interact with the system to commit transactions. Similarly, the assets are
+the medical services that are used by the system users (participants), which are stored in
+the blockchain. Likewise, transactions are operations that are used to communicate with
+assets. Moreover, transactions are also used to amend the values of assets and participants.
+Similarly, the access control rules are also defined to yield authentication and authorization
+to the users of the system. We also used the world state database to store the blockchain
+data. We specified the queries that are required to determine the interaction between the
+blockchain and the world state database. The queries are also used to fetch the user-based
+customized data from the database.
+Table 2 encapsulates the business network archive file with transactions, assets, and
+participants. The users are patients, doctors, and nurses. Similarly, the assets comprise
+patients’ medical records, sensors, vital sign readings, and other healthcare records. Lastly,
+transactions include getVitalSignReadings, AddHealthcareSensor, and DetectStatus.
+The business network archive is then used to construct a Representational state trans-
+fer (REST) Application Program Interface (API) in order to provide communication be-
+tween the client application and the back-end database. The RESTful API provides cross-
+accessibility, where the user of the system can access it from any platform with authentic
+credentials. Table 3, presents the RESTful API for the proposed medical blockchain, which
+19
+
+Table 2: Smart Contract Modeling for Proposed RPM System
+Type
+Components
+Description
+Asset
+Healthcare˙Sensor
+Healthcare sensors, such as ECG, or
+EMG etc.
+Vital Sign˙Sensing˙Data
+The vital signs of patients acquired
+from healthcare sensor.
+HealthRecord
+The patient medical information,
+such as current health condition, de-
+ployed sensors, etc.
+Participant
+Doctor
+System user.
+Patient
+System user.
+Nurse
+System user.
+Transaction
+getVitalSignReadings
+Get vital sign reading from health-
+care sensors.
+Add˙Healthcare˙Sensor
+Addition of new healthcare sensor in
+a medical blockchain platform.
+Modify˙Sensor
+Modify sensor composition.
+Detect˙Status
+Detect the patient vital sign status.
+is based on HTTP protocol. The generated RESTful API is used to expose the medical plat-
+form services to the client application. The services are related to patients, nurses, doctors,
+EMR, and other medical information. Figure 5 demonstrates how the major components of
+the proposed RPM system have interacted during the simulation study.
+Table 3: RESTful API for proposed Medical Blockchain
+Action
+Verb
+Media Type
+URI
+Patient Dashboard
+ALL
+Application/json
+/api/Patient
+Doctor Dashboard
+ALL
+Application/json
+/api/Doctor
+Nurse Dashboard
+ALL
+Application/json
+/api/Nurse
+Healthcare Sensor Dashboard
+ALL
+Application/json
+/api/Sensor
+Vital˙Sign
+Application/json
+/api/VitalSignReading
+EMR Dashboard
+ALL
+Application/json
+/api/PatientRecord
+Share patient record with healthcare personnel
+POST
+Application/json
+/api/ShareRecord
+Blockchain Network Text
+GET
+Application/json
+/api/system/ping
+Issue identity to system user
+POST
+Application/json
+/api/SystemIdentities/issue
+Get Identities
+GET
+Application/json
+/api/System/identities
+Retrieve historian records
+GET
+Application/json
+/api/System/historian
+Within our blockchain implementation, each piece of medical record has one user (owner)
+who can share the data they own with other users (doctors) at varying levels of access. Data
+sharing between users is modeled by a system where users can share data with other users
+in different groups, as well as receive data requests from other users at any access level.
+If a user responds to a request by granting data access, an access token is provided to the
+receiver in a way that allows that receiver to access the data at the specified access level only.
+Our system ensures that sensitive information is never exposed on the blockchain, including
+20
+
+Figure 5: Sequence of interactions conducted during simulation
+both private and document keys, which is necessary in order to maintain the privacy and
+security of user-controlled data.
+We evaluate the performance of the proposed blockchain model using Hyperledger Caliper
+[35]. For experimental analysis, we carried out several experiments in terms of the execution
+time when adding a new healthcare device and executing a healthcare data query. We also
+measure the average time of the proposed consensus algorithm. The execution time is the
+round-trip time which includes the total time of sending the request by the client and getting
+the response from the network. In order to evaluate the execution time, we utilized the Post-
+man tool, which is used to explore and test the RESTful APIs by simulating a customized
+user load within the network. In this study, we created three groups of devices: 150, 300,
+and 500, in order to investigate the execution time of registering a device in the proposed
+blockchain model. Furthermore, the execution time is analyzed using different statistical
+21
+
+Doctor/Nurse
+Gateway/Fog Node
+REST Server
+Patient
+(Sensor)
+(GUI)
+(Local Chain)
+(Global Chain)
+1
+Device Registration
+Device Registation
+smart
+contract
+Device ldentity and Certificate
+call
+Vital Sign (ECG Signal)
+create
+block
+New Block
+block
+mining
+New Block Broadcast
+save block
+or block
+New Block Broadcast
+header
+save block
+or block
+header
+Data Request
+check
+data
+location
+alt
+certify
+sender's
+[data in local chain]
+Secure Health Data
+certificate
+[data in global chain]
+Data Request
+certify
+ sender's
+certificate
+Secure Health Datameasures, such as the minimum, maximum, and average times. As shown in Figure 6, in
+the case of 150 users, the average, minimum, and maximum execution time to register the
+healthcare device is recorded as 2335 ms, 2257 ms, and 2795 ms, respectively. Likewise,
+the minimum, maximum, and average execution times for 300 healthcare device-group is
+are 1785 ms, 3204 ms, and 2454 ms, respectively. Finally, for 500 devices the minimum
+execution time is recorded as 2810 ms, whereas the maximum and average execution time
+is 3524 ms and 3015 ms respectively (Figure 6).
+Figure 6: Healthcare device registration execution time
+The execution time of the proposed system is also evaluated in the case of retrieving
+healthcare data from the blockchain network.
+Every healthcare device in the proposed
+platform has the HTTP client functionality which is used to send requests for vital sign
+sensing data through the IoT gateway. The request is initially processed by the IoT gateway.
+If the requested data is found in the local chain, the IoT gateway validates the device
+certificate via the local smart contract and then replies to the device with the encrypted
+data. Else, the IoT gateway forwards the request to the REST server, which performs a
+similar process. The execution time of reading the vital sign data is illustrated in Figure 7.
+The same set of device groups has been considered for the experimental evaluation, i.e., 150,
+300, and 500 devices. It is observed from the graph that the increase in the device scale in
+the proposed healthcare system will also create an impact on the execution time. However,
+the overall execution time of the network remains stable until there is high congestion in the
+network. The average execution time of vital sign sensing data in the case of 150, 300, and
+22
+
+4000
+■150 Devices
+300Devices
+500 Devices
+3500
+3000
+Execution Time (ms)
+2500
+2000
+1500
+1000
+500
+0
+Minimum
+Average
+Maximum500 devices are 2552 ms, 2525 ms, and 2775 ms, respectively, which are comparable to the
+execution times of registering a device that is shown in Figure 6.
+Figure 7: Vital signs reading execution time
+In order to evaluate the effectiveness of the proposed consensus method, we tested several
+scenarios in which we deployed five REST servers and five IoT gateways. The IoT devices
+were distributed evenly among the gateways, and each gateway was connected to a REST
+server. The servers saved all the blocks that were confirmed by the consensus protocol,
+while the IoT gateways saved the blocks of the devices that connected to them only. In
+these scenarios, we measure the consensus time of each created block, then we calculate the
+minimum, maximum, and average values for all the created blocks. The results are shown
+in Figure 8. We notice that the consensus time generally increases as the number of devices
+increases, which is logical since, with more devices, the total number of transactions increase,
+which adds more time to validate the new blocks. However, the increase in the consensus
+time is only 12.5 ms (on average) as the number of devices increases from 150 to 500, which
+proves the efficiency of the proposed consensus approach. In addition, the average consensus
+time of the system is 140 ms. In case of Ethereum and Bitcoin, it requires 10 to 19 seconds
+and 10 minutes to an hour respectively to mine a new block. Hence, the proposed consensus
+algorithm outperforms those of other blockchain platforms in terms of consensus time.
+23
+
+4000
+150Devices
+300Devices
+500Devices
+3500
+3000
+Execution Time (ms)
+2500
+2000
+1500
+1000
+500
+0
+Minimum
+Average
+MaximumFigure 8: Block consensus time
+6.2. Efficiency of the Fog computing infrastructure
+Figure 9 depicts the distributed data flow model for our proposed IoT-driven critical
+healthcare applications. According to this model, data signals generated by the IoT devices
+are pushed into the client module, an initial application interface for interacting with the IoT
+devices and actuators and receiving the user’s information, such as name, location, address,
+sex, and age of the patient.
+After pre-processing and filtering the data that is coming
+from the IoT devices, the client module forwards the data to the Data Processing module
+for further processing. Here, AI-enabled modules can execute data analytics processes for
+testing purposes. Based on the outcome of the data processing, a command is issued by
+the Data Processing module for the client module so that it can trigger physical emergency
+actions through the actuators. Next, the Data Processing module dispatches the processed
+data to the aggregator module, which simultaneously interacts with the blockchain module
+at the IoT gateway and cloud server to add the data to the blockchain and ensure data
+integrity and location-independent data access. The blockchain module interacts with the
+storage module in case the data is to be stored off-chain. Finally, The Data Processing
+module at the cloud server interacts with the blockchain module to consistently produce
+the results that are requested by the application users. Since the client module directly
+interacts with the IoT devices, it is preferable to be deployed at the IoT gateways (e.g.,
+ECG machines). For the deployment of other modules, there exist different approaches in
+the literature. For instance, cloud computation has been exploited in [36] [37] to execute the
+data analytics, aggregator, blockchain, storage, and training module. On the other hand,
+the proposed RPM system adopts Fog computing for executing these modules and utilizes
+the cloud to host the blockchain, storage, and processing modules.
+24
+
+200
+150 Devices
+300Devices500Devices
+175
+150
+(ms)
+Time (
+125
+Consensus
+100
+75
+50
+25
+0
+Minimum
+Average
+MaximumFigure 9: Data flow model for the proposed RPM system
+In this phase of performance evaluation, we demonstrate how the augmentation of Fog
+computing in remote patient monitoring improves the service latency and the energy usage
+in comparison to harnessing cloud-based resources. The experiments are conducted in an
+iFogSim [38] simulated Fog-Cloud computing environment. The computing resources within
+the simulation environment are organized in a hierarchical order, as shown in Figure 10. At
+the lower level of the simulation environment, twenty-four ECG machines (EMs) equipped
+with ECG sensors and emergency alert systems are placed. Based on the simulation design,
+an EM can connect with any of the four Fog local servers (FLSs) at the upper level. All
+FLSs are also set connected with a Fog regional server (FRS) that helps the lower-level
+computing devices to maintain seamless communication with the Cloud datacenter. Table
+4 presents the details of the simulation parameters used in the experiments. The numerical
+values have been extracted from real-world references as specified in [39] [40]. Additionally,
+Table 5 illustrates the configuration of different application modules for the simulations,
+Figure 10: Architecture of the simulated Fog-Cloud computing environment
+25
+
+Cloud
+Server
+FRS#1
+FLS#1
+FLS#6
+EM#1
+EM#2
+EM#3
+EM#4
+EM#21
+EM#22
+EM#23
+EM#24
+883
+89loT layer
+Fog layer
+Cloud layer
+Data
+600
+Global
+Signal
+Processed
+Aggregator
+Records
+Blockchain
+Data
+Module
+ECG
+Raw Data
+Module
+Sensor
+Data
+Client
+Processing
+Data
+Retrieve
+Response
+Module
+Records
+Query
+Module
+Command
+ Save
+Local
+Analytic
+Alert
+Storage
+Blockchain
+Training
+Module
+Block Metadata
+Module
+Emergency
+Module
+IndicatorTable 4: Parameters of simulated environment
+Device configuration
+Name
+Processing
+speed
+Downlink
+bandwidth
+Uplink
+bandwidth
+Memory
+capacity
+Busy
+power
+Idle
+power
+(in MIPS)
+(in MB)
+(in MB)
+(in GB)
+(in MWh)
+(in MWh)
+EM
+1000
+10
+5
+8
+1.1
+0.2
+FLS
+7000
+8
+3
+12
+1.3
+0.4
+FRS
+15000
+6
+2
+16
+1.6
+0.8
+Cloud
+40000
+3
+4
+32
+3.2
+1.4
+Sensing frequency of ECG sensors
+5 signals per second
+Simulation time
+500 seconds
+Table 5: Module configuration
+Name
+Program size
+Packet size
+RAM usage
+(in MB)
+(in KB)
+(in GB)
+Client module
+2000
+500
+1
+Data analytic module
+4000
+1500
+6
+Aggregator module
+1500
+1800
+2
+Blockchain module (periodic)
+1000
+2000
+4
+Storage module
+1000
+2000
+2
+Analytic training module
+8000
+2000
+12
+Figure 11: Performance in reducing sense-process-actuation delay
+which have been approximated based on the profiled run-time, resource utilization, and
+data communication delay of the proposed solutions in heterogeneous computing devices
+and networking context.
+The results of the simulation experiments conducted in the aforementioned computing
+26
+
+350
+ms)
+300 -
+Sense-Process-Actuation delay (in
+250
+200
+150
+100.
+S
+Proposed RPMS
+Cloud-based RPMSFigure 12: Performance in reducing energy consumption
+Figure 13: Performance in blockchain transaction retrieval
+setup demonstrate that our proposed Fog computing-based RPMS outperforms the Cloud
+computing-based RPMS both in terms of reducing sense-process-actuation delay (calculated
+using iFogSim AppLoop model on ECG sensors → client module → data analytic module →
+client module → emergency alert system data flow) and energy usage. Figure 11 indicates
+that the augmentation of Fog computing can improve the responsiveness of RPMS by 40%
+in initiating alert messages during emergency situations compared to its cloud counterpart.
+Such performance improvement happens mainly for executing the data analytics module
+closer to the sources, that consequently decreases the data transfer delay to remote cloud
+servers. Moreover, the computing devices in the Fog paradigm consume a reduced amount
+of energy than a cloud server because of their capacity constraints. Statistically, this feature
+27
+
+0.40
+0.35 -
+0.30-
+0.25
+0.20
+0.15
+0.10.
+0.05 -
+0.00
+Proposed RPMS
+Cloud-based RPMS25
+ProposedRPMS
+ Transaction Retrieval
+WCloud-basedRPMs
+20
+ime (ms)
+15
+3lockchain
+B
+0
+Transaction size = 500 KB
+Transactionsize=2000KBalso has an influence in lowering the idle energy consumption of Fog computing devices.
+Therefore, when the time-based energy consumption model (as programmed in the iFogSim
+simulator) is applied, the Fog computing-based RPMS promises to deliver its services by
+consuming around 36% less energy than its Cloud-based implementation (as shown in Figure
+12).
+On the other hand, due to executing the blockchain module at the fog devices, the delay
+required to retrieve a random blockchain transaction decreases as compared to the cloud-
+based RPMS, as shown in Figure 13. The figure illustrates that when the transaction size
+is equal to 500 KB, the proposed system requires an average of 7.16 ms to retrieve the
+transaction from the blockchain, while cloud-based RPMS needs 16.54 ms. On the other
+hand, for a 2000 KB transaction, the proposed RPMS produces a delay equal to 8.9 ms,
+while the cloud-based RPMS needs 19.34 ms.
+Hence, the proposed RPMS reduces the
+transaction retrieval delay by an average of 55.1%. This is mainly due to the cases in which
+the transaction is fetched from the local chain, which require much less end-to-end delay
+than retrieving the transaction from the global chain, due to the deployment of fog nodes
+at locations that are much nearer to the sensor nodes than the cloud servers.
+7. Security Analysis
+Having a robust architecture encryption scheme as part of a blockchain-based data-
+sharing system is particularly critical from a security perspective because most blockchain
+implementations replicate the entire transaction ledger onto each node, therefore, multiply-
+ing the potential attack surface by the number of nodes in the network. In the following,
+we discuss the security analysis which we performed on the proposed patient monitoring
+system.
+• Key attack: Elliptic curve encryption method is employed from a key pair, and an
+attacker can’t calculate the private key to address the elliptic curve logarithm problem;
+hence the security of the proposed model is ensured. Moreover, for each session, a
+temporary private key is generated for interaction among the nodes. In such a way, if
+a private key gets compromised in terms of leakage, then this will not have an impact
+on the session, as the attacker would not be able to calculate a session key for a session
+that is currently going on among the nodes; and (b) the leaked private key is of no use
+until the session is completed.
+• Replay attack: The proposed model uses an individual temporary private key that
+is different for each session agreement among the interacting nodes. It is improbable
+that a replay attack becomes successful since private keys hold a bounded lifetime.
+• Impersonation attack: This attack is executed only if the attacker has successfully
+obtained the private key. The proposed model employs an individual private key and
+elliptic curve encryption. Therefore, this attack cannot be executed.
+28
+
+• Sybil attack: there are different methods to remove the impact of Sybil attack on
+the proposed model, such as increasing the price to form a new identity. This method
+restricts attackers from obtaining fake identities, using a two-factor authentication
+mechanism and accumulating the MAC and IP addresses of the participants, which
+permits the detection of those participants who have varying identities.
+• False data injection attack: Prior to validating the records, the consensus algorithm
+is executed by the blockchain nodes. On arrival of the positive consensus, a node can
+confirm the legitimacy of the received record.
+• Tampering attack: For encryption and signing the transaction, a public key crypto-
+system is employed. This indicates that the tampering node cannot amend the transac-
+tion as it does not hold the private key of the signing node. Furthermore, the proposed
+model can handle the key attacks; therefore, the adversaries cannot exploit the private
+keys.
+• Modification attack: As explained above, this attack is impossible because the
+adversaries cannot exploit the private keys.
+• Hiding blocks attack: A record in the proposed vital sign monitoring platform
+holds a unique sequence number. It is a must for a blockchain node to provide its
+saved records if requested. If a node in the network does not offer its records, it is
+detached from the network and disallowed to interact with other nodes.
+• Man-in-the-middle attack: A mutual authentication is performed between the
+nodes in the proposed model, which employs private keys for each session agreement,
+therefore, man-in-the-middle attacks are prevented.
+• Compromisation attack: If an attacker compromises a cloud server and attempts
+to sabotage the consensus operation by sending a ”Block Add” message that contains
+an invalid block, the legitimate cloud servers will detect the attack from the invalid
+signatures in the ”Block Add” message, since the attacker will not be able to generate
+the valid signatures of the other cloud servers. If the attacker drops the block that
+it receives from the IoT gateway, the latter reports the attack to the IoT ecosystem
+administrator when it detects that its block was not added to the blockchain in due
+time. Finally, if the attacker sends a wrong reply message when it receives a new block
+from another cloud server, the attack will not have an effect as long as the number of
+legitimate cloud servers is greater than N /2.
+8. Conclusion and Future Work
+In this work, we have presented a three-layer remote patient monitoring system that
+leverages blockchain technology for better security and Fog technology for providing low-
+latency services to IoT devices and healthcare users. The most important functions that
+encompass the system components are described and evaluated. In addition, a new consensus
+29
+
+protocol that is tailored to the RPM environment is discussed and analyzed. Moreover, the
+blockchain module was implemented and tested using Hyperledger Fabric Framework, and it
+achieved low execution and consensus delays. Moreover, the augmentation of Fog computing
+can improve the responsiveness of the remote patient monitoring system by 40%.
+Several future works are being studied to enhance the proposed system. For example,
+we are planning to perform the simulations using real healthcare datasets (such as that in
+[41]). In addition, we intend to add a prediction module at the cloud layer that can predict
+a heart disease problem before its occurrence. The module would analyze the patient’s data
+from the global blockchain over an extended period to enhance prediction accuracy. Another
+enhancement would be the integration of the proposed blockchain system with a body area
+network (BAN) framework that is used to collect patient medical data in an efficient manner.
+Such integration should be carefully designed in order to secure the BAN operations without
+adding significant overhead in terms of computation and energy consumption on the BAN
+nodes. A similar system was proposed in [42]. Hence, we aim to study the literature in order
+to adjust the proposed blockchain system to make it suitable for a BAN environment.
+Another important future work is to enhance the proposed fog layer by augmenting it
+with modern technological tools that will improve its performance. For example, federated
+learning can be used by fog nodes to filter and analyze the readings of IoT devices in order
+to provide more accurate results to healthcare providers. Another important aspect is to
+design the scheduling of IoT data on the fog layer using the blockchain. For this aspect, we
+intend to adopt a previous strategy that we proposed in [43] to guarantee that a fog node
+treats data from IoT nodes fairly and provides equal opportunities for IoT nodes to save
+their data in the blockchain.
+Finally, we will study the scalability of the proposed system and its ability to support a
+large number of IoT ecosystems. For this purpose, we will design a hierarchical clustering
+framework that distributes cloud servers, fog nodes, and IoT devices into clusters based on
+their geographic locations and the deployed healthcare application. Using clustering will
+allow us to reduce the delay overhead when the application contains a huge number of
+blockchain nodes. In such a system, it is possible to execute a blockchain query in parallel
+by distributing it over the cluster heads, which would result in a reduced end-to-end delay
+between the patient and the healthcare provider.
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+
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+page_content=' Tunisia  bComputer Science and Mathematics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Lebanese American University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Beirut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Lebanon  cDepartment of Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Jeju National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Korea  dSchool of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Computing and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Curtin University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Perth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Australia  eRobotics and Internet of Things Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Prince Sultan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Riyadh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  fCalifornia State University, Dominguez Hills (CSUDH)  gISIMA, Mahdia, University of Monastir, Tunisia  Abstract  IoT has enabled the rapid growth of smart remote healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These IoT-based  remote healthcare applications deliver fast and preventive medical services to patients at risk  or with chronic diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, ensuring data security and patient privacy while exchang-  ing sensitive medical data among medical IoT devices is still a significant concern in remote  healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Altered or corrupted medical data may cause wrong treatment and  create grave health issues for patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, current remote medical applications’ ef-  ficiency and response time need to be addressed and improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Considering the need for  secure and efficient patient care, this paper proposes a lightweight Blockchain-based and Fog-  enabled remote patient monitoring system that provides a high level of security and efficient  response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Simulation results and security analysis show that the proposed lightweight  blockchain architecture fits the resource-constrained IoT devices well and is secure against  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the augmentation of Fog computing improved the responsiveness of the  remote patient monitoring system by 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Keywords:  IoT, Healthcare monitoring, Lightweight Blockchain, Fog computing,  consensus protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Introduction  Healthcare IoT networks are evolving from centralized to distributed systems to con-  nect with each other to provide patients with high-quality healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' According to pre-  ∗I am corresponding author  Email addresses: omar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='cheikhrouhou@isetsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='rnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='tn (Omar Cheikhrouhou),  khaleel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='mershad@lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='lb (Khaleel Mershad), faisal@jejunu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='kr (Faisal Jamil),  mdredowan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='mahmud@curtin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='au (Redowan Mahmud), akoubaa@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='sa (Anis Koubaa),  srahimimoosavi@csudh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='edu (Sanaz Rahimi Moosavi)  Preprint submitted to Elsevier  January 10, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='03551v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='CR]  9 Jan 2023  dictions, the current hospital-centered healthcare monitoring systems will develop first to  hospital–home-balanced in 2025 and then ultimately to home-centered in 2030 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' New  system architectures, technologies, and computing paradigms are needed to realize such  evolution, specifically in the Healthcare Internet of Things (HIoT) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Emerging tech-  nologies like IoT, blockchain, and artificial intelligence have made deploying smart remote  patient monitoring systems a fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Indeed, IoT devices permit them to sense and moni-  tor patients’ physiological parameters, hence exempting them from a long waiting queue at  a doctor’s visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' All necessary physiological parameters needed by doctors can be sensed  by the biomedical IoT devices (also known as the Internet of Medical Things devices) and  sent remotely to the doctor, allowing the latter to decide the appropriate treatment for the  patient [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The evolution of sophisticated security attacks and the rising need for individualized  healthcare has made it essential for medical institutions to embrace blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The arrival of the blockchain provides solutions to several problems that the healthcare sys-  tem has been facing for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The growing numbers of healthcare data breaches, pa-  tient privacy violations, counterfeit drugs, and many other issues are major reasons for steer-  ing the blockchain market’s growth in the healthcare industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In general, the blockchain  brings a large number of opportunities to smart healthcare, which can be summarized as  follows:  Secure access to personal health records: the decentralized blockchain system offers the  power of controlling data access to the owner of the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Smart contracts register  and authorize users to access the patient’s data according to the patient consent policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Patient Consent Management: the fundamental features of the blockchain, such as  transparency and immutability, enables healthcare applications to build trust among  patients and verify compliance with consent management policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Traceability of remote treatment: the blockchain permits healthcare applications to  create immutable and coherent electronic records (EHRs) that can be viewed by all  stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The transparency and consistency of blockchain EHRs aid in tracing the  medical history of patients to offer the appropriate treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Traceability of in-home medical kits and devices: the blockchain provides immutable  and transparent record transactions to the ownership and performance of medical kits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Reputation scores of medical devices and kits are saved in the blockchain using smart  contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Reputation-aware specialist referral services: during the treatment of a remote pa-  tient, medical referrals and expert suggestions are acquired through smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Blockchain enables healthcare providers to store these referral documents on an Inter-  Planetary File System (IPFS) server, such that an IPFS hash of the document is stored  securely in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The hash prevents the alteration of the stored document  and maintains its integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  2  Automated payments: blockchain provides digitally signed automatic payments to  guarantee non-repudiated secure transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  A complete discussion on the blockchain benefits to smart healthcare applications can  be found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Ensuring the security of the remote patient monitoring (RPM) system is a must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Since a  vulnerability in such a system could enable attackers to steal/modify sensitive information  and endanger the patient’s life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain has emerged as a promising technology that  can store and secure assets through a transparent and distributed ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In healthcare,  where patient data is a critical asset that needs to be securely managed, the blockchain could  become the right technology to address this challenge and provide a secure, transparent, and  tamper-proof management of patient healthcare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, the blockchain is a heavy  system requiring much processing and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Lightweight IoT devices would face  problems if they were to act as full blockchain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hence, a solution should be adopted to  enable IoT devices to participate in the blockchain network without affecting their limited  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The lightweight blockchain [5, 6] has been proposed to achieve this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Here, the blockchain architecture and processes are modified to assign light roles to the IoT  devices while allowing them to benefit from the blockchain services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In traditional RPM systems, patient healthcare data is stored in an Electronic Healthcare  Record (EHR) and saved in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Cloud computing provides ubiquitous access to  patients’ data through a user-centric access control model, where the user chooses which  data and to whom he/she should give access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, a cloud computing system presents  the disadvantages of high latency and, therefore, cannot fit critical healthcare application  requirements where immediate intervention is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' More precisely, real-time detection  and notification of abnormal situations must be implemented in the context of a heart disease  use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Otherwise, the patient’s life will be at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  To overcome the high latency limits of cloud computing and to fit the real-time require-  ments of most healthcare applications, we propose leveraging fog computing technology in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In our proposed architecture, fog computing will not replace cloud computing  but will cooperate via the lightweight blockchain to provide real-time and efficient service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  More precisely, we introduce the fog computing layer that will host a lightweight blockchain  application with low latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the other hand, complex AI algorithms can  be executed at the cloud computing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Currently, smart cities are moving towards adopting blockchain technology in many smart  city applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In healthcare, and especially in remote patient monitoring, the blockchain  can change the methods in which the application is executed and managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Integrating the  blockchain allows healthcare managers to guarantee the transparency of public healthcare  data and removes the need to apply trust-based mechanisms and systems to achieve this  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, the blockchain guarantees the privacy of patients’ personal data through  smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the blockchain allows for fast and direct connectivity between  healthcare officials, providers, staff, and patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Issuing blockchain transactions allows  these entities to communicate securely via the blockchain without intermediaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Finally,  the blockchain allows healthcare and smart city officials to know the origin and destination  3  of each medical resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' They can also find out how healthcare services are being used  without compromising people’s privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  To sum up, we propose a smart and secure remote patient monitoring system based  on three technology pillars: IoT, fog computing, and blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' More precisely, the key  contributions of this paper are as follows:  We propose the architecture of a smart and secure remote patient monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The proposed architecture uses IoT for patient vital signs collection and blockchain to  guarantee the privacy and security of the patient-collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The efficiency of the proposed architecture is achieved through the introduction of the  fog computing layer to provide real-time response and aggregate the patients’ collected  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  To reduce the heavy demands of traditional blockchain, we modify the blockchain  structure to include a local blockchain within the IoT ecosystems and a global chain  at the cloud layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each IoT ecosystem saves the block headers of all blockchain blocks,  the bodies of the blocks of interest to the local chain, and the smart contract functions  needed within the local chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the other hand, the global chain comprises whole  blocks and smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  We propose a lightweight consensus model that enables the fog nodes to participate  in the consensus protocol without consuming a lot of processing and energy resources  and allows IoT nodes to store only the information they need to verify the legitimacy  and integrity of the blockchain data that they obtain from fog nodes and cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The remainder of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Section 2 outlines the existing literature  on the remote patient monitoring system using blockchain and Fog Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Section 3  gives an overview of the proposed remote patient monitoring architecture with its different  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Section 4 describes the details of the proposed lightweight blockchain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Section 5 describes the fog computing layer functions and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The performance  evaluation of the system is discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Section 7 analyses the security of the  proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Finally, we conclude and give future directions in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Related Work  As our work is based on three technologies, namely the IoT, fog computing, and blockchain,  in this section, we present relevant work that uses one or more of these technologies to deploy  a healthcare solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The discussed works are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Table 1: Summary of related work  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Ref Contribution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Use case  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Used Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Pros(+)/Cons(-)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='IoT BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='FC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[7]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Cloud based remote health  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='monitoring system with sig-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='nal watermarking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ECG-based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='health monitor-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='+Providing sig-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='nal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='authentica-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='tion using water-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='marking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[8]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='hierarchical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='fog-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='computing-assisted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ar-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='chitecture for IoT health  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='monitoring system  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Arrhythmia de-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='tection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Map  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='IBM’s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='MAPE-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='computing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='healthcare  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ap-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='plication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='They developed a smart e-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Health gateway localized at  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Heart disease  ✓  ✓  +Full-system  implementation  [9]  Improved the energy con-  sumption of sensor nodes  during  data  transmission  and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Migraine disease  ✓  ✓  +Energy  con-  sumption reduc-  tion  [10]  An  Edge-Based  Architec-  ture for IoT-Healthcare ap-  plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Detect  high-  stress conditions  for workers and  athletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ✓  ✓  security  is-  sues  are  not  addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[11]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Used retraining of SDA in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='testing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ar-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='rhythmia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='add or merge features in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='anomaly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='detector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blockchain for access con-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='trol  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='arrhythmia clas-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='sification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='+High accuracy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[12] Used blockchain to secure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='remote patient monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='General  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='✓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Time issue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Key  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='manage-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ment issue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Continued on next page  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Table 1: Summary of related work  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Ref Contribution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Use case  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Used Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Pros(+)/Cons(-)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='IoT BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='FC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[13]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Remote health monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='system using Tor to min-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='imize the latency of the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='blockchain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Cardiac  Pa-  tients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Sleep  Apnoea  Pa-  tients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Epileptic  Patients  ✓  ✓  The accuracy of  the system is not  tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  [14]  HealthFog: A heart disease  analysis system based on  ensemble deep learning and  using integrated IoT and  Fog computing  Heart disease  ✓  ✓  Security  is-  sue  are  not  addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  [15]  They developed an intelli-  gent e-Health architecture  integrating  AI,  IoT,  and  cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ECG-based  arrhythmia  detection  ✓  ✓  +Hardware  im-  plementation of  AI algorithms  [16]  An IoT and fog comput-  ing architecture with par-  allelization and core allo-  cation capabilities to accel-  erate healthcare processor-  intensive services  ECG-based  arrhythmia  detection  ✓  ✓  +Response  Time  was  im-  proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  [17]  Lightweight identity man-  agement and access control  scheme for IoT devices us-  ing IOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  General  ✓  ✓  Does not sup-  port smart con-  tracts  [18]  Blockchain based architec-  ture to provide patient cen-  tric data access  Healthcare  ✓  ✓  +  Use  cluster-  ing techniques to  improve system  scalability  BC: Blockchain, FC: Fog Computing or any cloud computing related technologies  Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' [7] proposed a cloud-based architecture for ECG signal monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' To  authenticate the captured ECG signal, the authors add a watermark that will be checked on  the cloud side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the authors proposed additional services, including ECG signal  enhancement, classification, and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Azimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', [8] proposed a hierarchical computing  architecture leveraging fog and cloud computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors proposed a  methodology to partition the existing machine learning methods for fog-enabled healthcare  IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The authors in [1] developed a smart e-Health gateway localized at the edge to  provide several functions, including local storage, real-time local data processing, embedded  data mining, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' By releasing the small IoT devices from these functions, a considerable  6  amount of energy can be saved by outsourcing some loads from sensor nodes to these smart  gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In [9], the authors also proposed the integration of the IoT and cloud computing tech-  nologies to predict migraine disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors’ main contributions are the design of  low-power techniques in the radio and data processing for the sensor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the  authors proposed workload-balancing policies for cloud computing servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors in [10] proposed BodyEdge: an edge-based architecture for IoT-healthcare  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The system was implemented in two examples of edge gateway: Raspberry Pi3  and Zotac CI540 NANO Pc, and its performance was compared to cloud systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover,  as a validation example, the authors have implemented the system to detect high-stress  conditions for users in two different scenarios, namely Workers in a factory and Athletes  training in a fitness center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In [11], authors used blockchain to secure access control to patient  EHR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The authors proposed to store patient data in an off-chain database to overcome the  storage constraint of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors in [12] used a consortium blockchain based on the IBM Hyperledger platform  to secure remote patient monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In their proposed system, sensors interact with a  gateway (such as a mobile phone) that implements smart contracts for data analysis and  sends essential notifications to patients and healthcare providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain was used  to securely log transactions (such as data reads and doctor’s commands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, patient  data was stored on a local database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  They have proved that blockchain could be used  to resolve security concerns about the transfer and logging of data transactions in an IoT  healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The limitation rests in perfecting the time of the transmission of the  aggregated data sent by the gateway to the blockchain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors in [13] proposed a decentralized peer-to-peer remote health monitoring sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed architecture uses Tor hidden services for off-chain data delivery between  patients and doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The authors in [14] proposed HealthFog, a Fog-based healthcare sys-  tem that integrates Edge computing and IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Their work was motivated by latency-sensitive  healthcare applications, especially deep learning-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed system was  validated for a health disease use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In [15], an AI-driven e-health solution was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The solution integrates IoT with  cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The key difference of this solution is the distribution of the AI intelligence  across the three architecture layers, namely: the Device layer, Fog layer, and Cloud layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Moreover, the authors proposed hardware implementation of the SVM, ANN, and CNN  algorithms using digital circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The authors in [16] proposed a framework for accelerating  the response to remote patients requiring the execution of smart eHealth services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Their  proposed framework supports distributed offloading to fog servers and multicore processors’  capacity to accelerate its execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The authors in [17] proposed a blockchain-based lightweight authentication and autho-  rization scheme for IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed scheme uses distributed ledger technology  IOTA to design a lightweight and scalable mechanism for identity management and access  control of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, this solution did not support smart contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The authors  in [18] proposed a blockchain-based architecture that enables data owners to define their  desired access policies over their privacy-sensitive healthcare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The architecture used  7  two separate chains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' one for storing data transactions and one for storing access policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Several lightweight blockchain architectures have been proposed in the literature [19, 20,  21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, the ECLB protocol in [19] saves the full blockchain on edge  nodes, while the IoT nodes store what the authors call the fragmented ledger structure,  which contains the block headers and some of the transactions in each block that are needed  by the lightweight node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' A multi-layer blockchain model is proposed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain  network is divided into three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' At the first layer, ordinary IoT nodes are divided into  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' At the second layer, IoT cluster heads (CHs) store the local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', cluster) copy of  the BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The cellular base stations (BSs) store the full global BC at the third layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Nodes  in Layers 2 and 3 collaborate to create new blocks and execute the consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On  the other hand, some IoT nodes at layer one can be peers that maintain a copy of the local  BC and act as transaction endorsers or committers, while CHs and BSs act as Hyperledger  orderers who order transactions and create blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  From the study of the existing work, we note that several works address only the per-  formance and energy aspect of their proposed RPMs by adding the fog layer that manages  the computing and data processing tasks [7, 8, 1, 9, 10, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, these schemes  did not address the security aspects, and therefore, they are vulnerable to attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Other  schemes such as [11, 12, 13, 17, 18], addressed the security aspect by adopting the blockchain  technology, however, they used classical blockchain platforms and architectures that cannot  fit the resource-constrained IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In this paper, we leverage the fog computing layer  not only to improve the performance of the RPM system but also to lighten the load of  blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' More precisely, the fog layer permits the proposal of a lightweight  blockchain architecture that provides security services adaptable to resources constrained  IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the proposed consensus mechanism frees the architecture from the  burden of classical consensus algorithms such as PoW or PoS [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The Proposed RPM System Overview  This section gives an overview of the proposed RPM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It highlights its three-layer  architecture and the different communication interaction between the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' RPM System Architecture  The proposed remote patient monitoring system is a three-layer architecture as shown  in Figure 1, and which are:  The IoT devices layer: This layer, composed of biomedical sensor nodes, wearable  sensor nodes, and IoT medical devices, is responsible for collecting the vital signs of the  monitored patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These sensor readouts are collected continuously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' however, their  transmission to the gateway node located at the fog layer can be done periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The  transmission period depends on the nature of the vital sign and generally is determined  by the patient supervising doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The Fog Computing layer: This layer is responsible for the lightweight processing  of vital signs received from the IoT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, suppose the monitored vital  8  Cloud Layer  DB  DB  DB  Gateway  Gateway  Gateway  Gateway  Edge/ Fog Layer  IoT Layer  Global Blockchain  Big data Analytics  Patient  Doctor  IoMT devices  GetVitalSign  Write/Order  Local Blockchain  GiveAccess/  Audit Data  GetVitalSign/  GetNotification  cloud  server  cloud  server  cloud  server  Blockchain  update  Figure 1: The three-layer architecture of the proposed remote patient monitoring system  sign exceeds a specific threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In that case, an alert message will be triggered and  sent to the patient and the supervising doctor to make the right decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover,  the fog layer first decides which data needs to be recorded in the blockchain network  and then interacts with this latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The fog layer will also aggregate continuous sensed  data before sending it to the cloud server for permanent storage and data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Additionally, the fog layer contains IoT gateways that include the local blockchain  network, which is a subset of the global blockchain network (please refer to Section 5 for  full description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In the proposed system, the fog computing module consists of many  geographical intelligent gateways, that is, forming the fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each gateway supports  different protocols for communication and serves as a point of contact between the  sensor network and the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It collects data from different sub-networks, translates  protocols, and offers other higher-level services, including filters, data aggregation,  analysis, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The fog computing layer extends cloud computing to the edge of  the network and its facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' From cloud to end users/devices, the fog recognizes real-  time interaction, dense geographical distribution, heterogeneity, accessibility support,  pre-processing interoperability along with cloud interaction [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This enables latency  to be decreased, particularly for real-time applications such as in-house IoT monitoring  of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The fog reduces contact with the cloud, particularly in the event of a loss  of cloud connectivity, where the data is stored locally on these gateways, and patients’  data is sent to the cloud when the connection is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The Cloud Computing layer: This layer is responsible for permanent data storage  and data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Complex AI and deep learning algorithms can be implemented  at this layer for data classification, disease detection and prediction, and treatment  9  +OAEplan decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Moreover, in the proposed architecture, cloud servers play the role  of full blockchain nodes that store the full copy of the blockchain and participate in  transaction validation, block generation, and consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain records pa-  tient data and actions of patients and caregivers and permits the patients to decide to  whom they give access to their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, blockchain technology contains pieces  of code called smart contracts that can be automatically triggered when an event is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These smart contracts are a powerful tool for a remote patient monitoring  system as they can trigger an alarm and notify the doctor in an abnormal situation  (for example, when the vital sign value exceeds a specific threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, the  blockchain is used to ensure patient data privacy and the system’s security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' First,  thanks to blockchain technology, the patient will be given an anonymous identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  This permits hiding the real patient’s identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' therefore, doctors can treat his/her  data privately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, in our system, we propose to use a private blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This  type of blockchain has the advantage of restricting access to users’ data to only au-  thorized persons (such as patients, doctors, and caregivers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Furthermore, blockchain  architecture permits a patient-centric data management architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' More precisely,  the patient will decide to whom he/she shares data access (please refer to Section 4  for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Communication Models  The fog layer enables us to control access to IoT devices for medical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each  fog node manages and operates a group of medical IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This layer also interacts  with a network of fog nodes allowed by blockchain, which function together on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  All the related smart medical devices are connected with the closest blockchain-enabled  fog node, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', in an in-house monitoring scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These blockchain-enabled fog nodes are  communicated by IoT nodes and system users for authentication, authorization, and safe  communication synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' An intelligent contract with a collection of rules can also  be established on top of the fog nodes allowed by blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Furthermore, the consensus  algorithm is performed to validate the transactions and blocks for those transactions after  they are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Transaction blocks can be exchanged between cloud servers and the  blockchain-enabled fog nodes or between the fog nodes to support robust authentication,  permission, and distributed secure communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed solution mainly includes  four forms of communication:  (1) Medical caregiver-to-fog communication: Where the end user (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', a healthcare  provider) is willing to use a particular IoT system, he first sends a request for au-  thentication with a query authentication function specifying the sensor details to the  blockchain-enabled fog node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The fog node with the blockchain feature will search for  that medical attendant in the available list of approved sensor equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' A reject  message will be given when the user is not allowed to access the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Oth-  erwise, if the user is approved, the blockchain-enabled fog node issues an access token  containing Unique Identification (UID) information, length, time of access, blockchain  address for the data, user blockchain address, and blockchain address of the fog node  10  that stores the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Notice that every fog node, sensor, and the user has a  unique blockchain address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  (2) Medical sensor-to-fog communication: The sensor-to-fog correspondence has two  principle objectives in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  IoT medical services system mainly aims to  validate and authorize the clinical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The following goal is to insert a blockchain-  enabled fog connected to sensor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It helps new sensors to enlist with the mist and  ensures that all sensors are recognizable by the blockchain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In our context, each  IoT medical care system has at least one blockchain-enabled fog node close to the entire  blockchain network and is used for the enlistment, confirmation, and authorization of  IoT medical care gadgets with the same framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Initially, the gadgets will enroll  with their associated blockchain fog node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' As an exchange and blocks are made for  them, data concerning these gadgets are placed in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These blocks would  then be transported between the wide range of different blockchain fog nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Should a  system with a collecting place need confirmation and consent, the associated blockchain-  enabled fog node should be given its certifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain approves the provided  accreditation, and if there are significant requirements, the IoT gadgets for medical care  are effectively checked and authorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If the certification is not valid, the gadget is  refused and will not obtain permission to access the blockchain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  (3) Fog-to-fog communication: The main goal is to synchronize the information associ-  ated with IoT medical service confirmation and approval across all blockchain-enabled  fog nodes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Several biological or physiological parameters are obtained by medical  sensors transmitted by patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Medical IoT programs should be reliable and diligent  in supporting patients moving to a hospital or home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Typically, the mobility support  of the medical sensors from the upper layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', fog layer) should be given so that zero  reconfiguration in the sensor layer is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The strategic location and distribution  of smart gates in the fog layer can be used to provide smooth mobility for medical  sensors and relieve processing loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Fog-to-fog contact helps patients wander around  the hospital wards, ensuring their health monitoring is not disrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The patient-free  movement provides a high level of medical services using a portable patient monitoring  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Support of mobility for healthcare IoT systems is one of the most critical prob-  lems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The improvements to patients’ quality of life in such programs are essential  [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is important to encourage patients to walk into the hospital/medical facilities  knowing that monitoring their well-being is not disrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is necessary to establish  self-configuration or transfer mechanisms to ensure safe and successful data transfer be-  tween different Medical Sensor Networks (MSNs) [30] to achieve ongoing monitoring of  patients considering mobility support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, when a patient is moving across the  clinics, a data transfer mechanism is described as the process of modifying or updating  the registration of mobile sensors on its MSN base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Data handover solutions should  allow ubiquity when they need to function independently without human interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  (4) Medical sensor-to-medical sensor communication: When two clinical devices are  effectively tested and approved (both have a position with a similar system or another  one), they may create a safe link to each other and convey information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In a case,  for example, where a patient is released from a clinic but still needs to be constantly  11  monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The doctors bind the patient’s body before he/she leaves the medical cen-  ter to health tracking devices, including blood pressure monitors, pulse sensors, blood  glucose monitoring sensors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These devices sense the patient’s blood pressure, heart  rate, and glucose level and transmit them through a safe channel to the health workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  These devices can also interact with the patient’s intelligent home devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For exam-  ple, if the patient’s condition becomes severe or a fall is detected, an immediate alarm  may automatically be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The hospital-related devices must interact to check the  availability of hospital beds in a smart city to ensure a correct count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed  framework provides medical devices with access control in the IoT healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Under this mechanism, devices can only communicate with recorded and successfully  authenticated and certified devices with blockchain-enabled fog nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' A device not reg-  istered in the blockchain cannot authenticate itself or communicate with other devices  within the same healthcare ecosystem or external ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The contact between malicious  devices and legitimate devices would also be alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In what follows, we detail the proposed lightweight blockchain model and the Fog layer  functions and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The Blockchain Module Description  Blockchain technology provides a decentralized, transparent, authenticated platform that  applies a consensus-driven approach to facilitate the interactions of multiple entities through  the use of a shared ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Beyond the financial sector, where much of the initial develop-  ment is taking place, blockchain has the potential to revolutionize the healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  By providing doctors, patients, researchers, and other healthcare professionals with a mech-  anism for the controlled exchange of sensitive, permissioned data, blockchain technology can  improve data sharing and transparency between clinical and research data systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Any  healthcare organization participating in a blockchain consortium would be able to share  medical information, regardless of their native electronic health record system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Blockchain  provides significant opportunities for healthcare organizations to deliver more efficacious  treatments and diagnoses through increased provider data sharing and potentially safer and  more effective remote patient monitoring through advanced technologies such as AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Blockchain Architecture of Proposed RPM System  We propose a lightweight blockchain architecture to manage the data storage and re-  trieval operations in the remote patient monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In our system, we implement a  lightweight blockchain architecture that aims at reducing the delay in accessing the cloud  by the end users while maintaining the security and immutability of data at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The  blockchain will store all healthcare-related data, such as the IoT sensor readings, lab test  results, physicians’ decisions, commands, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In addition, the blockchain will comprise  transactions that contain management and security-related data, such as nodes’ and users’  registrations, access requests, smart contract results, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In the proposed architecture, cloud servers play the role of full blockchain nodes that store  the full copy of the blockchain and participate in transaction validation, block generation,  12  and consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the other hand, IoT gateways play the role of light blockchain nodes  that store part of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In our system, each gateway will be connected to a certain  number of IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, an IoT gateway at a patient’s home will connect to  a single IoT network that contains the IoT devices that are monitoring the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the  other hand, an IoT gateway at a hospital could connect several IoT networks, such as IoT  devices, in several patients’ rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Here, the IoT devices in a certain room or Lab form a  separate IoT subnetwork since the data produced by these devices will be linked together  (for example, data related to a specific patient, doctor, lab, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The IoT gateways and sensors that exist in the same IoT ecosystem (for example, home,  hospital, health institution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=') form a cluster that store and manage a local blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Each local blockchain is created as part of the full blockchain that is related to the cor-  responding ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, in a certain patient’s home, a set of IoT devices are  connected to an IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The devices and gateway form a cluster that store and man-  age a local blockchain that contains the blockchain blocks related to that home only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In a  hospital, several gateways and sets of IoT devices will form a cluster that store and manage  the blockchain of the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In each cluster, the sensor nodes store only the blocks headers of the full blockchain,  while the gateways store the block headers of the full blockchain in addition to the full blocks  of the local blockchain (as illustrated in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, to avoid overwhelming the  gateways with excessive storage as the blockchain grows, each transaction will have an expiry  time after which it becomes obsolete (for example, when the information in the transaction  becomes old and is no more relevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each gateway saves a data structure that contains,  for each transaction, the ID of the block in which the transaction is stored (BlockID) and  the transaction expiry date (Tex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The gateway continuously updates the data structure  when a transaction expires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, the gateway searches the data structure to detect  any block in which all transactions have expired and deletes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Using this approach allows  the gateway to remove old blocks and create room in its storage for new blocks in the local  blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  In the proposed system, IoT nodes continuously generate data and send them to the  IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, healthcare providers (doctors, nurses, scientists, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=') send their  data (such as prescriptions, sensors’ configurations, lab test results, commands to activate  actuators, data analytic results, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=') to the nearest IoT gateway in their institution’s IoT  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The IoT gateway stores the data it receives in a temporary cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each small period  (for example, every 100 ms), the IoT gateway aggregates and groups the received data into a  blockchain block and sends it to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Note that each block can contain multiple  transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, the readings of a certain sensor can be aggregated into a single  transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Similarly, if the doctor is sending configuration commands to the IoT sensor,  the configuration settings of each sensor can be grouped into a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each transaction  will be signed by the owner that created the transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Consensus Protocol  We consider a network of cloud servers that are used by various healthcare providers  to manage the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' As mentioned, the cloud servers act as full blockchain nodes that  13  Figure 2: The architecture of the proposed blockchain model: the cloud servers store the full blockchain,  the IoT gateways save the local blockchain, while the IoT nodes store the block headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  store all the blockchain blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, the cloud servers participate in the blockchain  consensus protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each cloud server has a unique blockchain ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The cloud servers create  the blockchain blocks successively based on their IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In other words, the server with the  smallest ID creates the first block, followed by the server that has the second smallest ID,  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When the server that has the biggest ID creates a block, the turn goes back to  the first server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Note that the block generation time at the IoT gateway should be adjusted  to allow all the cloud servers to generate their blocks in order to avoid block accumulation  at the cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  When its turn to create the new block arrives, a cloud server CS 1 broadcasts the block  that it received from the gateway to all the cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each cloud server CS i verifies that  all transactions in the block are legitimate by validating the signature of each transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Next, CS i replies with a CONFIRM message to CS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The confirm message contains CS i’s  signature of the new block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, If CS i discovers that one or more transactions in  the block are not valid, it replies with an ERROR message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In its turn, CS 1 waits until it  receives at least (N /2+1) CONFIRM messages before it adds the block to the blockchain  and broadcasts its ID in a “Block Add” message to all cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Here, N is the number  of the cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  This mechanism allows a cloud server to add the new block after  the majority of cloud servers confirm its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The ”Block Add” message contains the  signatures that CS 1 received in the CONFIRM messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When a cloud server CS j receives  a ”Block Add” message, it checks the attached signatures to ensure that more than (N ÷ 2)  cloud servers have validated and confirmed the new block, before adding it to its blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  After receiving the “Block Add” message, each cloud server adds the new block to its  blockchain and broadcasts it to its clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Note that each cloud server can serve multiple  14  Block Header  Headers Blockchain  TX 1  Block Body!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  TX 6  Block  Block  IoT Sensors  Full  Blockchain  Cloud  IoT Gateway  Server  Local Blockchain  +  Block 0  Block 1  Block 2  Block k-1  Block k  Header  Header  Header  Header  Header  Body  Body  Local BlocksFigure 3: A sample scenario of the proposed consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  institutions and organizations, with each institution/organization having its own IoT cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Each gateway in a cluster examines the new block to determine if it contains transactions  that were generated by one of the IoT networks in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If yes, the gateway stores  the block in its local blockchain and sends it to the IoT devices that are connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Each IoT device validates the block (by hashing it and comparing the result to the hash  in the block header) and then stores the block header in the headers’ blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Next,  the IoT device caches the block body for a small period of time before deleting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the  other hand, if the new block does not contain transactions that were generated by an IoT  network in the cluster, the gateway validates the block, sends it to the IoT devices that are  connected to it, extracts the block header, and adds it to the headers’ blockchain, and then  deletes the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each IoT device that receives the block performs the same operations  as the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This allows the gateway and IoT devices to maintain the headers of all  blocks in the blockchain and use these headers to validate any block from outside their local  blockchain that they obtain from the cloud servers in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed consensus  protocol is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In the figure, gateways G1 and G2 are connected to cloud  server CS1, while gateway G3 is connected to cloud server CS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' At a certain time, G2 creates  a new block B1 and sends it to CS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When its turn to generate a new block arrives, CS1  broadcasts B1 to the cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Each cloud server confirms B1 by sending a CONFIRM  packet to CS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Next, CS1 sends a “Block Add” packet to the cloud servers, and each cloud  server sends the new block to its gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' G1 and G2 receive B1 from CS1 and add it to  15  Block B1  Block B1  Block Bi  Block B1  Send Bi  Creation  Broadcast :  Commit  ppy  to Gateways  Gateway Gi  Gateway G2  B1  Gateway G3  Cloud Server CS  wait for turn  Cloud Server CS2  Cloud Server CS3  i  Cloud Server CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  New Block Packet  Add B ock Packet  : Add Block to Local Blockchain  Block Broadcast Packet  ^ : Add Block Header to Local Blockchain  Block ACK Packettheir copies of the local blockchain (since B1 was generated by a gateway in CS1’s cluster),  while G3 receives B1 from CS2 and adds its header only to the local blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Smart Contracts and Data Management  When an IoT device or a user requires data from the blockchain, it sends a request to the  IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The latter searches for the data in its local chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If it finds it, the gateway  authenticates the sender and verifies that it has access to the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If yes, the  gateway replies directly to the sender with the block that contains the data and the token  that enables the sender to access the data (more about this soon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If the gateway finds  that the required data doesn’t exist in the local blockchain, it forwards the request to the  cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The latter performs the same operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', it authenticates the requesting  node and verifies that it has access to the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If yes, the cloud server sends the  block that contains the data and the access token to the gateway, which forwards them to  the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When the latter receives the block, it validates it using the headers blockchain  before it retrieves the required transactions from the block and decrypts it using the access  token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Note that in our system, all transactions that can be accessed together are assigned an  access token by the creator and saved into a smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When the creator wants to  grant access to the transaction to a certain node/user, the creator executes a smart contract  function that adds the ID of the node/user to the access list of these transactions that is  saved in the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When the node/user wants to access the transactions, it should  authenticate itself and obtain the access token as described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If the transactions belong  to the local chain, the smart contract is executed by the gateway within the local chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Else, the smart contract is executed by the cloud server within the full chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The various  subsystems and interactions in the proposed RPM platform are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Specifications of Gateways at the Fog Layer  The fog layer is made up of IoT gateways which function primarily as a hub between  the cloud and IoT levels [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  With an in-depth study of the role of the gateway in a  smart home/hospital, where the location and mobility of things and users are confined  to hospital premises or buildings, it can be recognized that the stationary nature of the  gateways empowers them with the property of being non-resource constrained in terms of  power consumption, processing power, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These advantages can be used  by allowing gateways with ample intelligence, computing power, and structured networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  An inter-device communication is the key task of a gateway and supports numerous  wireless protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  We broaden the function of such gateways into fog enablers by (1)  building a distributed gateway network and (2) implementing features such as the repository  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', local data processing and storage using blockchain) to temporarily preserve data for  analysis by sensors and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These are important to provide local pre-processing of sensor  information and, therefore, to be an intelligent gateway for medical services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In a smart  gateway, the main functions are:  16  Figure 4: Interaction Model of the proposed blockchain-based remote patient monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Local data processing and storage  Local data processing is a key aspect of fog computing and is performed locally so  that intelligence is accessible at the doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Based on the device architecture, fog/edge  layers must continuously handle a large amount of information and respond to different  conditions in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In the remote patient management system, this becomes more  important by allowing the system to respond to medical emergencies as quickly as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Gateways should store inbound information in local storage to ensure that the remote patient  monitoring system can quickly recuperate patient medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In the proposed system, we  make use of the local blockchain to achieve this objective in a secure manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The patient  data can be stored as blockchain transactions in an encrypted or compressed form depending  on their context and security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The gateway stores all data related to the local  cluster in the local blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, when the gateway receives a blockchain block  that contains data related to other clusters, it caches the block for a small period of time to  allow users in the cluster who require data from the block to access it in a fast manner while  the data is hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, since the network bandwidth is limited between the gateway and  the cloud, the locally cached blocks can be used to maintain a continuous data flow in the  event of a weak or unstable connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Data filtering  Data from various medical sensors must be obtained before further processing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', data  analysis, on the fog layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The major sources of knowledge for the assessment of the health  status of a patient in the remote patient monitoring system [32, 33] are bio-signals, for exam-  ple, Electroencephalography (EEG), Electrocardiogram (ECG or EKG), and Electromyog-  raphy (EMG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' They typically have complex types that have small amplitudes and varying  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Remote Patient Monitoring Healthcare System  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Smart Contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Call  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Smart  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Identity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Storage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Communication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Consensus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Smart Contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Protocol  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Reply  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Interface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Patient  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data Access  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Local  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='All Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='All Blocks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blocks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Validation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Headers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Enrollment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='loT Gateway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Cloud Server  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Certificate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Smart Contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Healthcare Blockchain Platform  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Nurse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Result  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Healthcare  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Registration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Certificate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensor1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensor2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensor3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensorn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensor4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Healthcare Sensors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Doctorfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is important to remember that noise is often introduced to the signals during  a patient’s body sensing in a way that distorts the accuracy of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' These noises  are caused by different sources, including electromagnetic interference from other electrical  devices, shifts in current in the electricity grid, and inappropriate attachment of sensors  to the body of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' At the fog level, due to the proximity of the sensors, the gateway  addresses this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The fog layer is digitized via different contact protocols by sensors  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', 6LoWPAN, Zigbee, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' While sensors are able to perform lightweight filtering to  eliminate certain noises during the data collection process, the fog layer offers more complex  and robust data filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Data analysis  With local data analysis in the fog layer, the sensitivity of the device can be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  It helps the device to anticipate and diagnose situations of emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The developed deep  learning module for detecting irregular cardiac conditions is implemented in the fog layer in  our proposed RPM medical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The deep learning module can categorize signals and  detect abnormal conditions on the basis of the sensed ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' As a result, the device  responds more accurately, rapidly, and in real time to emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition,  local input and locally sensed data analysis change the quality and reliability of the device  in the event of the unavailability of the Internet link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Internet disconnection may occur  regularly for the long-term monitoring of patients with chronic diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Fog computing, in  this case, provides local maintenance of the system’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Thus, the sensed data and  processing results can be kept locally on the fog layer and later synchronized to the cloud via  the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Data analysis in the fog often helps the device minimize severe parameter  processing latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Improved latency  Agile responses and quick decision-making for acute diseases and emergencies, where  transmission time and data processing are to be reduced, are important for a continuous  remote control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' When raw medical data is transferred from medical sensor nodes to  the cloud, cloud computing can trigger response latencies indefinitely if the network condition  is not predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This becomes serious when streaming-based data processing, such as that  EEG or ECG signals that are obtained from patients, is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hence, deploying high-  priority data analytics in distributed gateways in the fog later and making time-sensitive  and critical decisions inside the local network make the remote patient monitoring system  more predictable and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The processed data can then be transmitted for storage and  further processing to the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Sensor nodes energy efficiency  There are various drawbacks to the processing of data at sensor nodes, as medical sensors  are resource-restricted devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Complicated tasks can, in certain cases, be performed suc-  cessfully at sensor nodes but at significant energy costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The transfer of heavy-weight tasks  from sensors to intelligent gateways in the fog layer can be an effective solution for solving  the above-mentioned problem, in particular when sensors do not have sufficient resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  18  Much energy can be saved with the aid of fog computing by outsourcing tasks from medical  sensors to intelligent gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Performance Evaluation  In this section, we present the performance evaluation of the proposed RPM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' We  have mainly evaluated the performance of the proposed blockchain module and demonstrated  the efficiency of Fog computing in dealing with critical healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Blockchain Implementation and Performance Evaluation  The proposed blockchain model was implemented via the Hyperledger platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hy-  perledger is an open-source development platform for blockchain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It has been  widely used as an implementation platform by the research community and is considered  a benchmark tool to evaluate the performance of the proposed approach against state-of-  the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For smart contracts, the Hyperledger tool provides easy-to-configure  and user APIs, thus making validation easy for our research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Furthermore, the REST-  ful API is utilized to provide the functionality of interoperability and expose the back-end  blockchain services to the client application through which the patients or other medical  personnel interact with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The smart contacts are designed and aggregated in the  form of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='bna files known as business network archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hyperledger Composer [34] is used  to implement and design the proposed medical blockchain, which aims to enhance system  operations in terms of throughput and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hyperledger Composer is an open-source  tool used to design blockchain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='bna in the designed platform consists  of a model, query definition, transaction, and access control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The model file is the  combination of participants, assets, and transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The participants are the user of the  system who can interact with the system to commit transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Similarly, the assets are  the medical services that are used by the system users (participants), which are stored in  the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Likewise, transactions are operations that are used to communicate with  assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, transactions are also used to amend the values of assets and participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Similarly, the access control rules are also defined to yield authentication and authorization  to the users of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' We also used the world state database to store the blockchain  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' We specified the queries that are required to determine the interaction between the  blockchain and the world state database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The queries are also used to fetch the user-based  customized data from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Table 2 encapsulates the business network archive file with transactions, assets, and  participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The users are patients, doctors, and nurses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Similarly, the assets comprise  patients’ medical records, sensors, vital sign readings, and other healthcare records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Lastly,  transactions include getVitalSignReadings, AddHealthcareSensor, and DetectStatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The business network archive is then used to construct a Representational state trans-  fer (REST) Application Program Interface (API) in order to provide communication be-  tween the client application and the back-end database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The RESTful API provides cross-  accessibility, where the user of the system can access it from any platform with authentic  credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Table 3, presents the RESTful API for the proposed medical blockchain, which  19  Table 2: Smart Contract Modeling for Proposed RPM System  Type  Components  Description  Asset  Healthcare˙Sensor  Healthcare sensors, such as ECG, or  EMG etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Vital Sign˙Sensing˙Data  The vital signs of patients acquired  from healthcare sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  HealthRecord  The patient medical information,  such as current health condition, de-  ployed sensors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Participant  Doctor  System user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Patient  System user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Nurse  System user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Transaction  getVitalSignReadings  Get vital sign reading from health-  care sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Add˙Healthcare˙Sensor  Addition of new healthcare sensor in  a medical blockchain platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Modify˙Sensor  Modify sensor composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Detect˙Status  Detect the patient vital sign status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  is based on HTTP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The generated RESTful API is used to expose the medical plat-  form services to the client application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The services are related to patients, nurses, doctors,  EMR, and other medical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Figure 5 demonstrates how the major components of  the proposed RPM system have interacted during the simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Table 3: RESTful API for proposed Medical Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Action  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Verb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Media Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='URI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Patient Dashboard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/Patient  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Doctor Dashboard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/Doctor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Nurse Dashboard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/Nurse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Healthcare Sensor Dashboard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Vital˙Sign  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/VitalSignReading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EMR Dashboard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/PatientRecord  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Share patient record with healthcare personnel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='POST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/ShareRecord  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blockchain Network Text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='GET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/system/ping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Issue identity to system user  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='POST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/SystemIdentities/issue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Get Identities  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='GET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/System/identities  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Retrieve historian records  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='GET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Application/json  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='/api/System/historian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Within our blockchain implementation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' each piece of medical record has one user (owner)  who can share the data they own with other users (doctors) at varying levels of access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Data  sharing between users is modeled by a system where users can share data with other users  in different groups, as well as receive data requests from other users at any access level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  If a user responds to a request by granting data access, an access token is provided to the  receiver in a way that allows that receiver to access the data at the specified access level only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Our system ensures that sensitive information is never exposed on the blockchain, including  20  Figure 5: Sequence of interactions conducted during simulation  both private and document keys, which is necessary in order to maintain the privacy and  security of user-controlled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  We evaluate the performance of the proposed blockchain model using Hyperledger Caliper  [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For experimental analysis, we carried out several experiments in terms of the execution  time when adding a new healthcare device and executing a healthcare data query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' We also  measure the average time of the proposed consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The execution time is the  round-trip time which includes the total time of sending the request by the client and getting  the response from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In order to evaluate the execution time, we utilized the Post-  man tool, which is used to explore and test the RESTful APIs by simulating a customized  user load within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In this study, we created three groups of devices: 150, 300,  and 500, in order to investigate the execution time of registering a device in the proposed  blockchain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' the execution time is analyzed using different statistical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Doctor/Nurse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Gateway/Fog Node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='REST Server  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Patient  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(Sensor)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(GUI)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(Local Chain)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(Global Chain)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Device Registration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Device Registation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='smart  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Device ldentity and Certificate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='call  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Vital Sign (ECG Signal)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='create  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='New Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='mining  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='New Block Broadcast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='save block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='or block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='New Block Broadcast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='header  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='save block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='or block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='header  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data Request  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='check  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='location  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='alt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='certify  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content="sender's  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[data in local chain]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Secure Health Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='certificate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='[data in global chain]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data Request  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='certify  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content="sender's  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='certificate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Secure Health Datameasures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' such as the minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' and average times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' As shown in Figure 6, in  the case of 150 users, the average, minimum, and maximum execution time to register the  healthcare device is recorded as 2335 ms, 2257 ms, and 2795 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Likewise,  the minimum, maximum, and average execution times for 300 healthcare device-group is  are 1785 ms, 3204 ms, and 2454 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Finally, for 500 devices the minimum  execution time is recorded as 2810 ms, whereas the maximum and average execution time  is 3524 ms and 3015 ms respectively (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Figure 6: Healthcare device registration execution time  The execution time of the proposed system is also evaluated in the case of retrieving  healthcare data from the blockchain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Every healthcare device in the proposed  platform has the HTTP client functionality which is used to send requests for vital sign  sensing data through the IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The request is initially processed by the IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  If the requested data is found in the local chain, the IoT gateway validates the device  certificate via the local smart contract and then replies to the device with the encrypted  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Else, the IoT gateway forwards the request to the REST server, which performs a  similar process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The execution time of reading the vital sign data is illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The same set of device groups has been considered for the experimental evaluation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=', 150,  300, and 500 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is observed from the graph that the increase in the device scale in  the proposed healthcare system will also create an impact on the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However,  the overall execution time of the network remains stable until there is high congestion in the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The average execution time of vital sign sensing data in the case of 150, 300, and  22  4000  ■150 Devices  300Devices  500 Devices  3500  3000  Execution Time (ms)  2500  2000  1500  1000  500  0  Minimum  Average  Maximum500 devices are 2552 ms, 2525 ms, and 2775 ms, respectively, which are comparable to the  execution times of registering a device that is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Figure 7: Vital signs reading execution time  In order to evaluate the effectiveness of the proposed consensus method, we tested several  scenarios in which we deployed five REST servers and five IoT gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The IoT devices  were distributed evenly among the gateways, and each gateway was connected to a REST  server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The servers saved all the blocks that were confirmed by the consensus protocol,  while the IoT gateways saved the blocks of the devices that connected to them only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In  these scenarios, we measure the consensus time of each created block, then we calculate the  minimum, maximum, and average values for all the created blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The results are shown  in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' We notice that the consensus time generally increases as the number of devices  increases, which is logical since, with more devices, the total number of transactions increase,  which adds more time to validate the new blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' However, the increase in the consensus  time is only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='5 ms (on average) as the number of devices increases from 150 to 500, which  proves the efficiency of the proposed consensus approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, the average consensus  time of the system is 140 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In case of Ethereum and Bitcoin, it requires 10 to 19 seconds  and 10 minutes to an hour respectively to mine a new block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hence, the proposed consensus  algorithm outperforms those of other blockchain platforms in terms of consensus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  23  4000  150Devices  300Devices  500Devices  3500  3000  Execution Time (ms)  2500  2000  1500  1000  500  0  Minimum  Average  MaximumFigure 8: Block consensus time  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Efficiency of the Fog computing infrastructure  Figure 9 depicts the distributed data flow model for our proposed IoT-driven critical  healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' According to this model, data signals generated by the IoT devices  are pushed into the client module, an initial application interface for interacting with the IoT  devices and actuators and receiving the user’s information, such as name, location, address,  sex, and age of the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  After pre-processing and filtering the data that is coming  from the IoT devices, the client module forwards the data to the Data Processing module  for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Here, AI-enabled modules can execute data analytics processes for  testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Based on the outcome of the data processing, a command is issued by  the Data Processing module for the client module so that it can trigger physical emergency  actions through the actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Next, the Data Processing module dispatches the processed  data to the aggregator module, which simultaneously interacts with the blockchain module  at the IoT gateway and cloud server to add the data to the blockchain and ensure data  integrity and location-independent data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The blockchain module interacts with the  storage module in case the data is to be stored off-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Finally, The Data Processing  module at the cloud server interacts with the blockchain module to consistently produce  the results that are requested by the application users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Since the client module directly  interacts with the IoT devices, it is preferable to be deployed at the IoT gateways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=',  ECG machines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For the deployment of other modules, there exist different approaches in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For instance, cloud computation has been exploited in [36] [37] to execute the  data analytics, aggregator, blockchain, storage, and training module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the other hand,  the proposed RPM system adopts Fog computing for executing these modules and utilizes  the cloud to host the blockchain, storage, and processing modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  24  200  150 Devices  300Devices500Devices  175  150  (ms)  Time (  125  Consensus  100  75  50  25  0  Minimum  Average  MaximumFigure 9: Data flow model for the proposed RPM system  In this phase of performance evaluation, we demonstrate how the augmentation of Fog  computing in remote patient monitoring improves the service latency and the energy usage  in comparison to harnessing cloud-based resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The experiments are conducted in an  iFogSim [38] simulated Fog-Cloud computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The computing resources within  the simulation environment are organized in a hierarchical order, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' At  the lower level of the simulation environment, twenty-four ECG machines (EMs) equipped  with ECG sensors and emergency alert systems are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Based on the simulation design,  an EM can connect with any of the four Fog local servers (FLSs) at the upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' All  FLSs are also set connected with a Fog regional server (FRS) that helps the lower-level  computing devices to maintain seamless communication with the Cloud datacenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Table  4 presents the details of the simulation parameters used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The numerical  values have been extracted from real-world references as specified in [39] [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Table 5 illustrates the configuration of different application modules for the simulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Figure 10: Architecture of the simulated Fog-Cloud computing environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Server  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='FRS#1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='FLS#1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='FLS#6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM#24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='883  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='89loT layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Fog layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Cloud layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Global  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Processed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Aggregator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Records  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='ECG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Raw Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Client  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Retrieve  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Response  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Records  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Query  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Command  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Save  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Local  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Analytic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Alert  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Storage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Training  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Block Metadata  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Emergency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='IndicatorTable 4: Parameters of simulated environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Device configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='speed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Downlink  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='bandwidth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Uplink  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='bandwidth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='capacity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Busy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Idle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in MIPS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
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+page_content='(in GB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in MWh)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in MWh)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='EM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2  FLS  7000  8  3  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4  FRS  15000  6  2  16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='8  Cloud  40000  3  4  32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Sensing frequency of ECG sensors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='5 signals per second  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Simulation time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='500 seconds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Table 5: Module configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Program size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Packet size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='RAM usage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in MB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in KB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='(in GB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Client module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Data analytic module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Aggregator module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Blockchain module (periodic)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Storage module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Analytic training module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='8000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='Figure 11: Performance in reducing sense-process-actuation delay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='which have been approximated based on the profiled run-time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' resource utilization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' and  data communication delay of the proposed solutions in heterogeneous computing devices  and networking context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  The results of the simulation experiments conducted in the aforementioned computing  26  350  ms)  300 -  Sense-Process-Actuation delay (in  250  200  150  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  S  Proposed RPMS  Cloud-based RPMSFigure 12: Performance in reducing energy consumption  Figure 13: Performance in blockchain transaction retrieval  setup demonstrate that our proposed Fog computing-based RPMS outperforms the Cloud  computing-based RPMS both in terms of reducing sense-process-actuation delay (calculated  using iFogSim AppLoop model on ECG sensors → client module → data analytic module →  client module → emergency alert system data flow) and energy usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Figure 11 indicates  that the augmentation of Fog computing can improve the responsiveness of RPMS by 40%  in initiating alert messages during emergency situations compared to its cloud counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Such performance improvement happens mainly for executing the data analytics module  closer to the sources, that consequently decreases the data transfer delay to remote cloud  servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the computing devices in the Fog paradigm consume a reduced amount  of energy than a cloud server because of their capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Statistically, this feature  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='35 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='30-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='05 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='00  Proposed RPMS  Cloud-based RPMS25  ProposedRPMS  Transaction Retrieval  WCloud-basedRPMs  20  ime (ms)  15  3lockchain  B  0  Transaction size = 500 KB  Transactionsize=2000KBalso has an influence in lowering the idle energy consumption of Fog computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Therefore, when the time-based energy consumption model (as programmed in the iFogSim  simulator) is applied, the Fog computing-based RPMS promises to deliver its services by  consuming around 36% less energy than its Cloud-based implementation (as shown in Figure  12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  On the other hand, due to executing the blockchain module at the fog devices, the delay  required to retrieve a random blockchain transaction decreases as compared to the cloud-  based RPMS, as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The figure illustrates that when the transaction size  is equal to 500 KB, the proposed system requires an average of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='16 ms to retrieve the  transaction from the blockchain, while cloud-based RPMS needs 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='54 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On the other  hand, for a 2000 KB transaction, the proposed RPMS produces a delay equal to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='9 ms,  while the cloud-based RPMS needs 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='34 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Hence, the proposed RPMS reduces the  transaction retrieval delay by an average of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This is mainly due to the cases in which  the transaction is fetched from the local chain, which require much less end-to-end delay  than retrieving the transaction from the global chain, due to the deployment of fog nodes  at locations that are much nearer to the sensor nodes than the cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Security Analysis  Having a robust architecture encryption scheme as part of a blockchain-based data-  sharing system is particularly critical from a security perspective because most blockchain  implementations replicate the entire transaction ledger onto each node, therefore, multiply-  ing the potential attack surface by the number of nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In the following,  we discuss the security analysis which we performed on the proposed patient monitoring  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Key attack: Elliptic curve encryption method is employed from a key pair, and an  attacker can’t calculate the private key to address the elliptic curve logarithm problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  hence the security of the proposed model is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, for each session, a  temporary private key is generated for interaction among the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In such a way, if  a private key gets compromised in terms of leakage, then this will not have an impact  on the session, as the attacker would not be able to calculate a session key for a session  that is currently going on among the nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' and (b) the leaked private key is of no use  until the session is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Replay attack: The proposed model uses an individual temporary private key that  is different for each session agreement among the interacting nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is improbable  that a replay attack becomes successful since private keys hold a bounded lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Impersonation attack: This attack is executed only if the attacker has successfully  obtained the private key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The proposed model employs an individual private key and  elliptic curve encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Therefore, this attack cannot be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  28  Sybil attack: there are different methods to remove the impact of Sybil attack on  the proposed model, such as increasing the price to form a new identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This method  restricts attackers from obtaining fake identities, using a two-factor authentication  mechanism and accumulating the MAC and IP addresses of the participants, which  permits the detection of those participants who have varying identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  False data injection attack: Prior to validating the records, the consensus algorithm  is executed by the blockchain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' On arrival of the positive consensus, a node can  confirm the legitimacy of the received record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Tampering attack: For encryption and signing the transaction, a public key crypto-  system is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' This indicates that the tampering node cannot amend the transac-  tion as it does not hold the private key of the signing node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Furthermore, the proposed  model can handle the key attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' therefore, the adversaries cannot exploit the private  keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Modification attack: As explained above, this attack is impossible because the  adversaries cannot exploit the private keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Hiding blocks attack: A record in the proposed vital sign monitoring platform  holds a unique sequence number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' It is a must for a blockchain node to provide its  saved records if requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If a node in the network does not offer its records, it is  detached from the network and disallowed to interact with other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Man-in-the-middle attack: A mutual authentication is performed between the  nodes in the proposed model, which employs private keys for each session agreement,  therefore, man-in-the-middle attacks are prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Compromisation attack: If an attacker compromises a cloud server and attempts  to sabotage the consensus operation by sending a ”Block Add” message that contains  an invalid block, the legitimate cloud servers will detect the attack from the invalid  signatures in the ”Block Add” message, since the attacker will not be able to generate  the valid signatures of the other cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' If the attacker drops the block that  it receives from the IoT gateway, the latter reports the attack to the IoT ecosystem  administrator when it detects that its block was not added to the blockchain in due  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Finally, if the attacker sends a wrong reply message when it receives a new block  from another cloud server, the attack will not have an effect as long as the number of  legitimate cloud servers is greater than N /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Conclusion and Future Work  In this work, we have presented a three-layer remote patient monitoring system that  leverages blockchain technology for better security and Fog technology for providing low-  latency services to IoT devices and healthcare users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The most important functions that  encompass the system components are described and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, a new consensus  29  protocol that is tailored to the RPM environment is discussed and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the  blockchain module was implemented and tested using Hyperledger Fabric Framework, and it  achieved low execution and consensus delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Moreover, the augmentation of Fog computing  can improve the responsiveness of the remote patient monitoring system by 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Several future works are being studied to enhance the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example,  we are planning to perform the simulations using real healthcare datasets (such as that in  [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In addition, we intend to add a prediction module at the cloud layer that can predict  a heart disease problem before its occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' The module would analyze the patient’s data  from the global blockchain over an extended period to enhance prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Another  enhancement would be the integration of the proposed blockchain system with a body area  network (BAN) framework that is used to collect patient medical data in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Such integration should be carefully designed in order to secure the BAN operations without  adding significant overhead in terms of computation and energy consumption on the BAN  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' A similar system was proposed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Hence, we aim to study the literature in order  to adjust the proposed blockchain system to make it suitable for a BAN environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Another important future work is to enhance the proposed fog layer by augmenting it  with modern technological tools that will improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For example, federated  learning can be used by fog nodes to filter and analyze the readings of IoT devices in order  to provide more accurate results to healthcare providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Another important aspect is to  design the scheduling of IoT data on the fog layer using the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For this aspect, we  intend to adopt a previous strategy that we proposed in [43] to guarantee that a fog node  treats data from IoT nodes fairly and provides equal opportunities for IoT nodes to save  their data in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content='  Finally, we will study the scalability of the proposed system and its ability to support a  large number of IoT ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' For this purpose, we will design a hierarchical clustering  framework that distributes cloud servers, fog nodes, and IoT devices into clusters based on  their geographic locations and the deployed healthcare application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' Using clustering will  allow us to reduce the delay overhead when the application contains a huge number of  blockchain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
+page_content=' In such a system, it is possible to execute a blockchain query in parallel  by distributing it over the cluster heads, which would result in a reduced end-to-end delay  between the patient and the healthcare provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQf9AZG/content/2301.03551v1.pdf'}
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diff --git a/AtE3T4oBgHgl3EQfTgrD/content/tmp_files/2301.04443v1.pdf.txt b/AtE3T4oBgHgl3EQfTgrD/content/tmp_files/2301.04443v1.pdf.txt
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+Entangled States are Harder to Transfer than Product States
+Tony J. G. Apollaro
+,1, ∗ Salvatore Lorenzo
+,2 Francesco
+Plastina
+,3, 4 Mirko Consiglio
+,1 and Karol ˙Zyczkowski
+5, 6
+1Department of Physics, University of Malta, Msida MSD 2080, Malta
+2Universit`a degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr`e, via Archirafi 36, I-90123 Palermo, Italy
+3Dipartimento di Fisica, Universit`a della Calabria, 87036 Arcavacata di Rende (CS), Italy
+4INFN, gruppo collegato di Cosenza, 87036 Arcavacata di Rende (CS), Italy
+5Institute of Theoretical Physics, Jagiellonian University, ul. �Lojasiewicza 11, 30–348 Krak´ow, Poland
+6Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warszawa, Poland
+The distribution of entangled states is a key task of utmost importance for many quantum infor-
+mation processing protocols. A commonly adopted setup for distributing quantum states envisages
+the creation of the state in one location, which is then sent to (possibly different) distant receivers
+through some quantum channels. While it is undoubted and, perhaps, intuitively expected that the
+distribution of entangled quantum states is less efficient than that of product states, a thorough
+quantification of this inefficiency (namely, of the difference between the quantum-state transfer fi-
+delity for entangled and factorized states) has not been performed. To this end, in this work, we
+consider n-independent amplitude-damping channels, acting in parallel, i.e., each, locally, on one
+part of an n-qubit state. We derive exact analytical results for the fidelity decrease, with respect to
+the case of product states, in the presence of entanglement in the initial state, for up to four qubits.
+Interestingly, we find that genuine multipartite entanglement has a more detrimental effect on the
+fidelity than two-qubit entanglement. Our results hint at the fact that, for larger n-qubit states,
+the difference in the average fidelity between product and entangled states increases with increasing
+single-qubit fidelity, thus making the latter a less trustworthy figure of merit.
+I.
+INTRODUCTION
+Distributing entangled states among several distant recipients is a task of paramount importance in a variety of
+quantum-information processing protocols, ranging from n-party quantum key distribution [1] to distributed quantum
+computing [2]. In many of these protocols, an n-partite entangled state is created at location S (the sender’s location),
+and its parts are distributed among m ≤ n receivers, generally at different locations (which we dub R, the receivers’
+location).
+The special case of distributing a bipartite entangled state was already considered in the seminal paper by Bose on
+quantum-state transfer (QST), where the transfer protocol is employed to send (the state of) one party of a two-qubit
+Bell state to the opposite edge of a spin chain [3]. After this first instance, a considerable amount of research, both
+theoretical and experimental, has been performed in order to improve the transfer performance and optimize the Bell
+state distribution protocol [4–10]. Moreover, with the increasing exploration (and exploitation) of the fascinating
+realm of quantum correlations by quantum technological applications, the distribution of n-partite entangled states,
+with n > 2, has become a very active research topic [11–15].
+At variance with the entanglement of two-qubits, which is the only system whose entanglement properties have been
+fully characterized both for pure and mixed states, for n > 2 there are only a handful of closed, analytical results for
+the quantification of entanglement [16], making the task of evaluating the efficiency of an entanglement distribution
+protocol very difficult to assess.
+Here, we address the distribution of an n-partite entangled state utilizing the fidelity between the sender’s and the
+receivers’ state as a figure of merit for the quality of the protocol. Although the fidelity is not a bona fide tool to
+characterize quantum resources [17, 18], it is nevertheless widely employed in constructing entanglement witnesses
+following the idea that states close to an entangled state must be entangled as well [19]. Hence, building on recent
+results [20, 21] reporting the fidelity of an n-qubits QST (n-QST) protocol for arbitrary quantum channels, we
+investigate the effect of the presence of entanglement on the n-QST fidelity, which is evaluated when each qubit is
+subject to an independent U(1)-symmetric quantum channel, e.g., an amplitude-damping channel. In particular, we
+find that the presence of entanglement in the sender state is detrimental to the efficiency of the n-QST protocol. It
+is not surprising that independent quantum channels acting on n qubits tend to destroy their quantum correlations,
+thus lowering the transmission fidelity. We are able to provide a quantification of the fidelity reduction as a function
+∗ tony.apollaro@um.edu.mt
+arXiv:2301.04443v1  [quant-ph]  11 Jan 2023
+
+2
+of different entanglement monotones . In particular, we show that genuine multipartite entanglement, as quantified,
+e.g., by the three-tangle, has a more pronounced effect on lowering the n-QST fidelity than bipartite entanglement
+between two qubits, as quantified by the concurrence [22].
+The paper is organised as follows: in Section II, we introduce our model and provide a brief recap of the n-QST
+fidelity; in Section III, we apply the developed formalism to the case of n = 2, 3, 4 qubits; finally, in Section IV, we
+draw our conclusions.
+II.
+N -QST FIDELITY FOR INDEPENDENT AMPLITUDE-DAMPING CHANNELS
+Let us consider an n-qubit quantum-state transfer protocol as depicted in Figure 1. A sender, located at position
+S, prepares an n-qubit arbitrary state and wants to transfer each party to different receivers to which the sender is
+connected by different quantum channels. Without a loss of generality, let us assume the sender state to be a pure
+state, ρS = |Ψ⟩⟨Ψ|n. The state at the receivers’ location reads
+ρR(t) = [Φ1 ⊗ Φ2 ⊗ · · · ⊗ Φn] (t) (ρS) .
+(1)
+The fidelity between the sender and the receivers’ state is given by the Uhlmann–Jozsa fidelity [23]
+F (|Ψ⟩ , ρ(t)) = ⟨Ψ| ρ |Ψ⟩ .
+(2)
+Expressing an arbitrary input state in the computational basis
+|Ψ⟩ =
+2n
+�
+i=1
+ai |i⟩ ,
+(3)
+the elements of the receivers’ density matrix read (sum over repeated indexes is assumed)
+ρR
+ij = Anm
+ij ρS
+nm
+(4)
+yielding the fidelity
+F (|Ψ⟩ , ρ) =
+d−1
+�
+ijnm=0
+a∗
+i ajana∗
+mAnm
+ij ,
+(5)
+where all of the amplitudes a refer to the initial state of the sender.
+For the case represented in Figure 1, the total map is given by the tensor products of n independent maps as in
+Equation (1). Hence, Equation (4) can be cast in the following form: [24]
+ρR
+i1i2...in;j1j2...jn = Ap1p2...pn;q1q2...qn
+i1i2...in;j1j2...jn ρS
+p1p2...pn;q1q2...qn,
+(6)
+where i, j, p, q = 0, 1 and the corresponding subscript refers to the i’s qubit, with
+Ap1p2...pn;q1q2...qn
+i1i2...in;j1j2...jn
+= Ap1;q1
+i1;j1 Ap2;q2
+i2;j2 . . . Apn;qn
+in;jn .
+(7)
+Each A in Equation (7) comes from a single qubit map connecting the sender qubit si and the receiver qubit ri,
+which, for an U(1)-symmetric channel, can be expressed as
+�
+�
+�
+ρ00
+ρ01
+ρ10
+ρ11
+�
+�
+�
+ri
+=
+�
+�
+�
+�
+1
+0
+0
+1 −
+��f ri
+si
+��2
+0 f ri
+si
+0
+0
+0
+0
+�
+f ri
+si
+�∗
+0
+0
+0
+0
+��f ri
+si
+��2
+�
+�
+�
+�
+�
+�
+�
+ρ00
+ρ01
+ρ10
+ρ11
+�
+�
+�
+si
+,
+(8)
+where f ri
+si is the transition amplitude for the excitation initially on si to reach ri. A widely used U(1)-symmetric
+quantum channel is given by the so called XXZ spin- 1
+2 Hamiltonian,
+H =
+�
+i,j
+Jij
+�
+σx
+i σx
+j + σy
+i σy
+j
+�
++ ∆ijσz
+i σz
+j + hiσz
+i
+(9)
+
+3
+FIG. 1. A quantum router. A dispatch center, encircled in red, creates an n-qubit entangled state (red spheres) with the aim
+to send each party to a different receiver (green spheres) along independent quantum channels (blue spheres).
+where σα
+i (α = x, y, x) are Pauli matrices and i, j are the position indexes on an arbitrary d-dimensional lattice.
+Assuming that each quantum channel is fully polarized, for the sender state , the map Φi reduces to an amplitude-
+damping channel [25]. In particular, for f ri
+si = 1, the map Φi entails a SWAP operation. Therefore, our formalism
+also describes entanglement swapping protocols via imperfect operations [26]. Finally, to express Equation (6) in the
+form of Equation (4), it is sufficient to express the bit strings in decimal notation.
+By making use of Equations (4), (7), and (8), it is straightforward to evaluate the average fidelity [13] of an arbitrary
+quantum state |Ψ⟩
+⟨F⟩ = 1
+Ω
+�
+Ω
+dΩ F (|Ψ⟩ , ρ(t)) ,
+(10)
+with Ω denoting the space of pure states and, with an abuse of notation, its volume and the measure of it .
+An average with respect to an n qubit system will be denoted as ⟨Fn⟩. In the case of n independent channels,
+making use of the transition amplitude f introduced in Equation (8), one arrives at the expression,
+⟨Fn⟩ =
+1
+2n + 1 +
+1
+2n (2n + 1) |1 + f|2n .
+(11)
+Notice that the average fidelity ⟨Fn⟩ ≤ �n
+i=1 ⟨F1⟩, with equality holding only for f = 0, 1. While the left-hand side of
+the latter inequality gives the average over all possible pure input states, its right-hand side, on the other hand, gives
+the average restricted to fully factorized states only, i.e., to product states of the form |Ψ⟩n = �n
+i=1 |ψ⟩i, thus not
+including the entangled states. Hence, we conclude that, when n ≥ 2, in the set of all pure input states, entangled
+states have a lower n-QST fidelity than the product state. In the next sections, we will provide a quantitative analysis
+for this intuitive observation.
+III.
+N -QST FIDELITY AS A FUNCTION OF ENTANGLEMENT
+This Section contains our main result, namely, that the presence of entanglement reduces the transfer fidelity.
+Below, we illustrate this idea separately for two, three, and four qubit transmissions. In particular, we will show that,
+in the presence of entanglement in the states to be sent, a reduction factor exists, which we dub Rn, such that the
+average fidelity for the QST of n-qubits can be generically written
+⟨Fn⟩ = ⟨F1⟩n − En Rn.
+
+4
+Here, ⟨F1⟩n gives the average fidelity for the transfer of factorized states (indeed, intuitively, qubits can be transferred
+one by one, in this case), so that the difference ⟨Fn⟩ − ⟨F1⟩n is entirely due to the fact that entangled states are
+possibly transferred. The coefficient En is an entanglement quantifier that changes with n. It is given by twice the
+square of concurrence for n = 2, while for n = 3 it is proportional to a linear combination of the invariant polynomials
+identifying the different classes of entangled states. Finally, the factor Rn (defined below for the various cases) gives
+the weight of entanglement-induced fidelity decrease, and it also enters the fidelity averaged over specific entanglement
+classes of states (for n = 3, 4).
+A.
+Two Qubits
+Adopting the (Schmidt-)parametrization of two-qubits pure states in terms of their entanglement [27], we write
+|Ψ(s)⟩ =
+�
+1 + s
+2
+|00⟩ +
+�
+1 − s
+2
+|11⟩
+(12)
+where the parameter s ∈ [−1, 1] is related to the concurrence [22] via C =
+√
+1 − s2, and every two-qubit pure state
+can be obtained from Equation (12) via local, unitary operations |Φ(s)⟩ = U1U2 |Ψ(s)⟩, with Ui ∈ SU(2) acting on
+qubit i = 1, 2. Below, we obtain the average fidelity for a two-qubit QST protocol with independent channels as a
+function of the amount of entanglement of the sender state, which is invariant under local unitaries, and which we
+denote as ⟨·⟩U ⊗2
+This average fidelity (which, to say it shortly, is averaged at fixed values of entanglement) reads
+⟨F2⟩U ⊗2 = 1
+36
+�
+3 + |f|2 + 2 |f| cos φ
+�2
+− 1
+18
+�
+|f|2 + 2 |f| cos φ
+� �
+3 − |f|2 + 2 |f| cos φ
+�
+C2
+(13)
+where we expressed the complex transition amplitude f as f = |f| eiφ.
+Following the procedure outlined by Bose [3], in order to maximise Equation (13), one sets cos φ = 1, which,
+physically, can be obtained, e.g., by a uniform magnetic field applied over the spin chain. Hence, the average two-
+qubit fidelity F2 can be cast in the form
+⟨F2⟩U ⊗2 = 1
+36
+�
+3 + |f|2 + 2 |f|
+�2
+− 1
+18
+�
+|f|2 + 2 |f|
+� �
+3 −
+�
+|f|2 + 2 |f|
+��
+C2.
+(14)
+From Equation (14), since 0 ≤ |f| ≤ 1, one can readily appreciate that the more concurrence the sender’s pure state
+contains, the lower the fidelity with the received state. Equation (14) can also be rewritten as a function of the
+single-qubit QST average fidelity
+⟨F1⟩ = 1
+6
+�
+3 + 2 |f| + |f|2�
+,
+(15)
+to read
+⟨F2⟩U ⊗2 = ⟨F1⟩2 − 2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) C2 = ⟨F1⟩2 − 2R2C2.
+(16)
+Again, as 1
+2 ≤ ⟨F1⟩ ≤ 1, the average fidelity ⟨F2⟩ decreases with the amount of concurrence of the sender state.
+From Equation (16), we see that the average 2-QST fidelity is reduced in the presence of the squared concurrence
+by a factor of
+R2 =
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) ,
+(17)
+which is reported in Figure 2 (left panel), together with the two-qubit average fidelity ⟨F2⟩, displayed for different
+values of the squared concurrence as a function of the one-qubit fidelity ⟨F1⟩ (right panel).
+B.
+Three Qubits
+Having obtained a quantitative expression giving the reduction in the transmission fidelity due to the presence
+of entanglement for two qubits, we move to the more intricate three qubit case in order to try and obtain similar
+relations.
+
+5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.6
+0.8
+1.0
+FIG. 2. (left) Reduction factor (17) for entangled states of the 2-QST average fidelity ⟨F2⟩ (16) as a function of the 1-QST
+average fidelity ⟨F1⟩. The dotted, vertical line reports the maximum of R2 attained at ⟨F⟩1 = 0.75. (left)
+1.
+Three-qubit pure-state entanglement
+Let us now consider a system of three qubits A, B, and C. A three-qubit pure state can be written in canonical
+form as [28]
+|Ψ⟩ABC = λ0 |000⟩ + λ1eiφ |100⟩ + λ2 |101⟩ + λ3 |110⟩ + λ4 |111⟩ ,
+(18)
+where λi ≥ 0, 0 ≤ φ ≤ π, and the normalisation condition reads �
+i λ2
+i = 1.
+In terms of the coefficients of the state in Equation (18), one can introduce five invariant polynomials, allowing to
+identify different entanglement classes [29]:
+J1 =
+��λ1λ4eiφ − λ2λ3
+��2 , J2 = λ2
+0λ2
+2 , J3 = λ2
+0λ2
+3
+(19a)
+J4 = λ2
+0λ2
+4 , J5 = λ2
+0
+�
+J1 + λ2
+2λ2
+3 − λ2
+1λ2
+4
+�
+.
+(19b)
+The relation between invariant polynomials and entanglement measures is given by
+C2
+jk = 4Ji,
+(20)
+for i ̸= j ̸= k = 1, 2, 3, and where now, (1, 2, 3) = (A, B, C) holds on the LHS of Equation (20) . At variance with the
+two-qubit case, no single entanglement measure can capture genuine three-partite entanglement as three qubits can
+be entangled in two inequivalent ways [30].
+One type of entanglement is quantified by the three-tangle [31],
+τ 2
+3 = 4J4 ,
+(21)
+while an inequivalent type of genuine multipartite entanglement (GME) is quantified by the so called GME concur-
+rence, CGME [32], defined, in terms of invariant polynomials, for a three-qubit pure state as:
+CGME = 4 (min {J2 + J3, J1 + J3, J1 + J2} + J4) .
+(22)
+Hence, three qubit states can be sorted in the following entanglement classes [30]:
+• Product states. All Ji = 0, resulting into A − B − C (class 1). All entanglement measures vanish.
+• Biseparable states. All Ji = 0 except i) J1 for A − BC, ii) J2 for B − AC, and iii) J3 for C − AB (class 2a).
+Only the concurrence for one single pair of qubits is different from zero.
+• W-states. CGME > 0 and τ3 = 0
+1. J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 (class 3a).
+2. J4 = 0 and √J1J2J3 = J5
+2 (class 4a).
+
+6
+• GHZ-states. CGME > 0 and τ3 > 0, with 5 possible cases:
+1. All Ji = 0 except J4 (class 2b).
+2. J1 = J2 = J5 = 0, or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0 (class 3b).
+3. J2 = J5 = 0 or J3 = J5 = 0 (class 4b).
+4. J1J4 + J1J2 + J1J3 = √J1J2J3 = J5
+2 (class 4c).
+5. √J1J2J3 = |J5|
+2
+and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0 (class 4d) ,
+where, with the notation X − Y , we indicate that subsystems X and Y do not share any type of entanglement.
+Notably, for 3-qubit pure states, a monogamy relation exists between the amount of entanglement that can be
+shared among the parties [31]
+C2
+A|BC = C2
+AB + C2
+AC + τ 2
+3 .
+(23)
+2.
+Fidelity of 3-QST
+Here, we derive the fidelity of the QST of a tree-qubit pure state. First, we average the state in Equation (18) over
+single-qubit, local operations U, and use the notation ⟨·⟩U ⊗3, in order to indicate the average fidelity at given values
+of (and, thus, as a function of) the entanglement quantifiers. Subsequently, utilizing the averages of the invariant
+polynomials obtained in Ref. [29], we derive the average fidelity of each three-qubit class and use the notation ⟨·⟩.
+The 3-QST fidelity, expressed in terms of the single-qubit average, reads
+⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+J1 + J2 + J3 + 3
+2J4
+�
+.
+(24)
+Since 1
+2 ≤ ⟨F1⟩ ≤ 1, and 0 ≤ Ji ≤ 1
+4, the second term on the right hand side of the latter equation is always
+negative, so that one sees at once that entanglement reduces the 3-QST fidelity.
+Introducing the reduction factor R3,
+R3 = ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) ,
+(25)
+we can write
+⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 R3
+�
+J1 + J2 + J3 + 3
+2J4
+�
+.
+(26)
+Now, using the averages of the invariant polynomials obtained in Ref. [29], ⟨J4⟩ =
+1
+12 and ⟨Jk⟩ =
+1
+24 (k = 1, 2, 3),
+the average fidelity ⟨F3⟩ (with average taken over the full three qubit Hilbert space) is given by
+⟨F3⟩ = ⟨F1⟩3 − 2R3
+(27)
+Here, we see that, at fixed single-particle average fidelity, entanglement is responsible for a decrease in the 3-QST
+average fidelity by twice the reduction factor R3.
+In particular, the fidelities for the canonical states belonging to the different entanglement classes read
+• class 1 (product state)
+⟨Fc1⟩U ⊗3 = ⟨F1⟩3
+(28)
+⟨Fc1⟩ = ⟨F1⟩3
+(29)
+• class 2a (biseparable states)
+⟨Fc2a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) C2
+jk
+(30)
+⟨Fc2a⟩ = ⟨F1⟩3 − R3
+3
+(31)
+where i ̸= j ̸= k = 1, 2, 3;
+
+7
+• class 2b (GHZ-states)
+⟨Fc2b⟩U ⊗3 = ⟨F1⟩3 − 3 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) τ 2
+3
+(32)
+⟨Fc2b⟩ = ⟨F1⟩3 − R3
+(33)
+• class 3a (J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 )
+⟨Fc3a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+C2
+BC + C2
+AC + C2
+AB
+�
+⟨Fc3a⟩ = ⟨F1⟩3 − R3
+(34)
+• Class 3b (J1 = J2 = J5 = 0 or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0)
+⟨Fc3b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2C2
+BC + 3τ 2
+3
+�
+⟨Fc3b⟩ = ⟨F1⟩3 − 4
+3R3
+(35)
+• Class 4a J4 = 0 and √J1J2J3 = J5
+2
+⟨Fc4a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+C2
+BC + C2
+AC + C2
+AB
+�
+⟨Fc4a⟩ = ⟨F1⟩3 − R3
+(36)
+• Class 4b (J2 = J5 = 0 or J3 = J5 = 0)
+⟨Fc4b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC
+�
++ 3τ 2
+3
+�
+⟨Fc4b⟩ = ⟨F1⟩3 − 5
+3R3
+(37)
+• Class 4c (J1J4 + J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 )
+⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC + C2
+AB
+�
++ 3τ 2
+3
+�
+⟨Fc4c⟩ = ⟨F1⟩3 − 2R3
+(38)
+• Class 4d (√J1J2J3 = |J5|
+2
+and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0)
+⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC + C2
+AB
+�
++ 3τ 2
+3
+�
+⟨Fc4d⟩ = ⟨F1⟩3 − 2R3
+(39)
+Comparing Equation (30) with Equation (32), it turns out that, for an equivalent amount of the entanglement
+monotone C2
+jk and τ 2
+3 , at fixed ⟨F1⟩ (or, equivalently, at a fixed transition amplitude f), the fidelity of the canonical
+state in class 2a is greater than that in class 2b. This is in line with our intuition that the more entangled a state
+is, the harder it is to achieve high fidelity in our parallel QST protocol, as shown in Figure 3 (right panel), where we
+plot the average fidelity for different three qubit classes. In the left panel of Figure 3 we report the reduction factor
+R3 of Equation (25) as a function of the single-particle average fidelity ⟨F1⟩.
+From the above equations of the average fidelity of the three-qubit classes, we see that the average fidelity is
+decreased, with respect to the product state class, whenever there is two-qubit concurrence or genuine multipartite
+entanglement, both as CGME and as τ3. Moreover, per equal amount of squared two-qubit concurrence C2 and genuine
+multipartite entanglement CGME, the reducing factor is respectively 2 and 3 times R3. As a consequence, we state
+that, at fixed amount of entanglement, GME states are harder to transfer than biseparable states.
+
+8
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.6
+0.7
+0.8
+0.9
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 3. (left) Reduction factor for the average fidelity in the presence of entanglement as in Equation (25). (right) Average
+fidelity for the three-qubit classes. The dotted, vertical line is at ⟨F⟩1 = 1
+2
+�
+1 +
+1
+√
+3
+�
+≃ 0.789.
+C.
+Four Qubits
+While for two and three qubits, the entanglement of pure states has been fully characterized, for four (or more)
+qubits there are infinitely many inequivalent entanglement classes [30, 33] under SLOCC operations (stochastic local
+operations and classical communication).
+Here, we consider the fidelity of specific four-qubit states averaged over random local unitaries on each qubit.
+Whereas this does not account for all entangled states within a given class, as the group of stocastic local operations
+includes deterministic local operations, SU(2) ⊆ SL(2, C) (with equality holding for pure states), the results nev-
+ertheless hint at the fact that the average fidelity decreases with the entanglement of the sender state and that the
+reduction factor depends on the type of entanglement contained in the state.
+We will consider the three irreducibly balanced states [34]: the 4-qubits GHZ-state, the cluster, and X4 :
+|GHZ4⟩ =
+1
+√
+2 (|0000⟩ + |1111⟩)
+(40a)
+|Cl4⟩ = 1
+2 (|0000⟩ + |0111⟩ + |1011⟩ + |1100⟩)
+(40b)
+|X4⟩ =
+1
+√
+6
+�√
+2 |1111⟩ + |0001⟩ + |0010⟩ + |0100⟩ + |1000⟩
+�
+(40c)
+and two additional entangled states: the product of two-Bell states and the 4-qubit W-state:
+|B2⟩ = |Φ⟩12 ⊗ |Φ⟩34
+(41a)
+|W4⟩ = 1
+2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩) .
+(41b)
+The average fidelities of the states reported in Equations (40) and (41), expressed in terms of 1-QST average fidelity,
+read
+⟨FGHZ4⟩U ⊗4 = ⟨FB2⟩U ⊗4 = ⟨F1⟩4 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+1 − 3 ⟨F⟩ + 4 ⟨F⟩2�
+(42a)
+⟨FCl4⟩U ⊗4 = ⟨FX4⟩U ⊗4 = ⟨F1⟩4 − 4 ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+(42b)
+⟨FW4⟩U ⊗4 = ⟨F1⟩4 − 3 ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) .
+(42c)
+Notice that the reduction factor for the states |Cl4⟩ , |X4⟩ , |W4⟩ is the same, although with different weights, and
+
+9
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.6
+0.7
+0.8
+0.9
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 4. (left) Reduction factor for the average fidelity in the presence of entanglement as in Equations (43). (right) Average
+fidelity for the entangled classes as reported in Equations (42). The blue and red dotted, vertical lines are, respectively, at
+⟨F⟩1 = 0.82 and ⟨F⟩1 = 0.85.
+differs from the reduction factor for the states |GHZ4⟩ , |B2⟩, reading, respectively,
+R4a = ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+1 − 3 ⟨F⟩ + 4 ⟨F⟩2�
+(43a)
+R4b = ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) .
+(43b)
+A possible reason may be that, if one considers the four-tangle as an entanglement measure, although it is not a
+measure of genuine multipartite entanglement, the first set of state has zero four-tangle, whereas for the second one
+it is non-zero. In Figure 4 (left panel) we report the reduction factors in Equation (43), while in the right panel we
+report the average fidelity of Equation (42).
+IV.
+DISCUSSION
+We have shown that the QST of an entangled n ≥ 2 quantum state across parallel, independent U(1)-symmetric
+quantum channels, as, e.g., embodied by an XXZ spin- 1
+2 Hamiltonian, leads to a lower average fidelity than that of
+the QST of a product state at fixed one-qubit QST average fidelity, or, equivalently, at fixed transition amplitude.
+For the case of n = 2, we have expressed the average fidelity reduction in terms of the squared concurrence times a
+reduction factor. Similarly, for n = 3, we obtained that the presence of entanglement, both bipartite and multipartite,
+has a detrimental effect on the average fidelity. In particular, we obtained that the reduction factor has a greater
+weight in the presence of genuine three-partite entanglement, i.e., three-tangle and GME concurrence, than in the
+presence of two-qubit squared concurrence for specific canonical classes of the three-qubit pure state. Finally, we have
+considered specific cases of 4-qubit entangled states, which, again, result in an average fidelity reduction due to the
+presence of entanglement in the initial state.
+Our work clearly shows that for entanglement distribution in a routing configuration, where parties are sent over
+independent quantum channels, the single-qubit average fidelity is not a reliable figure of merit. This calls for more
+investigations into the properties of quantum channels able to faithfully distribute multipartite entangled states.
+ACKNOWLEDGEMENTS
+TJGA acknowledges funding through the IPAS+ (Internationalisation Partnership Awards Scheme +) QUEST
+project by the MCST (The Malta Council for Science & Technology). MC acknowledges funding from the Tertiary
+Education Scholarships Scheme and from the Project QVAQT financed by the Malta Council for Science & Technology,
+for and on behalf of the Foundation for Science and Technology, through the FUSION: R&I Research Excellence
+Programme REP-2022-003.
+K ˙Z acknowledges support by Narodowe Centrum Nauki under the Quantera project
+
+10
+number 2021/03/Y/ST2/00193 and by the Foundation for Polish Science under the Team-Net project POIR.04.04.00-
+00-17C1/18-00. SL acknowledges support by MUR under PRIN Project No. 2017 SRN-BRK QUSHIP
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+
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+page_content=' 87036 Arcavacata di Rende (CS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Italy  5Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Jagiellonian University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' �Lojasiewicza 11, 30–348 Krak´ow, Poland  6Center for Theoretical Physics, Polish Academy of Sciences, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Lotnik´ow 32/46, 02-668 Warszawa, Poland  The distribution of entangled states is a key task of utmost importance for many quantum infor-  mation processing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A commonly adopted setup for distributing quantum states envisages  the creation of the state in one location, which is then sent to (possibly different) distant receivers  through some quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' While it is undoubted and, perhaps, intuitively expected that the  distribution of entangled quantum states is less efficient than that of product states, a thorough  quantification of this inefficiency (namely, of the difference between the quantum-state transfer fi-  delity for entangled and factorized states) has not been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' To this end, in this work, we  consider n-independent amplitude-damping channels, acting in parallel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', each, locally, on one  part of an n-qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' We derive exact analytical results for the fidelity decrease, with respect to  the case of product states, in the presence of entanglement in the initial state, for up to four qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Interestingly, we find that genuine multipartite entanglement has a more detrimental effect on the  fidelity than two-qubit entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Our results hint at the fact that, for larger n-qubit states,  the difference in the average fidelity between product and entangled states increases with increasing  single-qubit fidelity, thus making the latter a less trustworthy figure of merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  INTRODUCTION  Distributing entangled states among several distant recipients is a task of paramount importance in a variety of  quantum-information processing protocols, ranging from n-party quantum key distribution [1] to distributed quantum  computing [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In many of these protocols, an n-partite entangled state is created at location S (the sender’s location),  and its parts are distributed among m ≤ n receivers, generally at different locations (which we dub R, the receivers’  location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  The special case of distributing a bipartite entangled state was already considered in the seminal paper by Bose on  quantum-state transfer (QST), where the transfer protocol is employed to send (the state of) one party of a two-qubit  Bell state to the opposite edge of a spin chain [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' After this first instance, a considerable amount of research, both  theoretical and experimental, has been performed in order to improve the transfer performance and optimize the Bell  state distribution protocol [4–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Moreover, with the increasing exploration (and exploitation) of the fascinating  realm of quantum correlations by quantum technological applications, the distribution of n-partite entangled states,  with n > 2, has become a very active research topic [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  At variance with the entanglement of two-qubits, which is the only system whose entanglement properties have been  fully characterized both for pure and mixed states, for n > 2 there are only a handful of closed, analytical results for  the quantification of entanglement [16], making the task of evaluating the efficiency of an entanglement distribution  protocol very difficult to assess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Here, we address the distribution of an n-partite entangled state utilizing the fidelity between the sender’s and the  receivers’ state as a figure of merit for the quality of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Although the fidelity is not a bona fide tool to  characterize quantum resources [17, 18], it is nevertheless widely employed in constructing entanglement witnesses  following the idea that states close to an entangled state must be entangled as well [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Hence, building on recent  results [20, 21] reporting the fidelity of an n-qubits QST (n-QST) protocol for arbitrary quantum channels, we  investigate the effect of the presence of entanglement on the n-QST fidelity, which is evaluated when each qubit is  subject to an independent U(1)-symmetric quantum channel, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', an amplitude-damping channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In particular, we  find that the presence of entanglement in the sender state is detrimental to the efficiency of the n-QST protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' It  is not surprising that independent quantum channels acting on n qubits tend to destroy their quantum correlations,  thus lowering the transmission fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' We are able to provide a quantification of the fidelity reduction as a function  ∗ tony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='apollaro@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='mt  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='04443v1  [quant-ph]  11 Jan 2023  2  of different entanglement monotones .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In particular, we show that genuine multipartite entanglement, as quantified,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', by the three-tangle, has a more pronounced effect on lowering the n-QST fidelity than bipartite entanglement  between two qubits, as quantified by the concurrence [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  The paper is organised as follows: in Section II, we introduce our model and provide a brief recap of the n-QST  fidelity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' in Section III, we apply the developed formalism to the case of n = 2, 3, 4 qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' finally, in Section IV, we  draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  N -QST FIDELITY FOR INDEPENDENT AMPLITUDE-DAMPING CHANNELS  Let us consider an n-qubit quantum-state transfer protocol as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A sender, located at position  S, prepares an n-qubit arbitrary state and wants to transfer each party to different receivers to which the sender is  connected by different quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Without a loss of generality, let us assume the sender state to be a pure  state, ρS = |Ψ⟩⟨Ψ|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' The state at the receivers’ location reads  ρR(t) = [Φ1 ⊗ Φ2 ⊗ · · · ⊗ Φn] (t) (ρS) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (1)  The fidelity between the sender and the receivers’ state is given by the Uhlmann–Jozsa fidelity [23]  F (|Ψ⟩ , ρ(t)) = ⟨Ψ| ρ |Ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (2)  Expressing an arbitrary input state in the computational basis  |Ψ⟩ =  2n  �  i=1  ai |i⟩ ,  (3)  the elements of the receivers’ density matrix read (sum over repeated indexes is assumed)  ρR  ij = Anm  ij ρS  nm  (4)  yielding the fidelity  F (|Ψ⟩ , ρ) =  d−1  �  ijnm=0  a∗  i ajana∗  mAnm  ij ,  (5)  where all of the amplitudes a refer to the initial state of the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  For the case represented in Figure 1, the total map is given by the tensor products of n independent maps as in  Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Hence, Equation (4) can be cast in the following form: [24]  ρR  i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='j1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='jn = Ap1p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='q1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='qn  i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='j1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='jn ρS  p1p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='q1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='qn,  (6)  where i, j, p, q = 0, 1 and the corresponding subscript refers to the i’s qubit, with  Ap1p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='q1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='qn  i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='j1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='jn  = Ap1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='q1  i1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='j1 Ap2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='q2  i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Apn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='qn  in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='jn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (7)  Each A in Equation (7) comes from a single qubit map connecting the sender qubit si and the receiver qubit ri,  which, for an U(1)-symmetric channel, can be expressed as  �  �  �  ρ00  ρ01  ρ10  ρ11  �  �  �  ri  =  �  �  �  �  1  0  0  1 −  ��f ri  si  ��2  0 f ri  si  0  0  0  0  �  f ri  si  �∗  0  0  0  0  ��f ri  si  ��2  �  �  �  �  �  �  �  ρ00  ρ01  ρ10  ρ11  �  �  �  si  ,  (8)  where f ri  si is the transition amplitude for the excitation initially on si to reach ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A widely used U(1)-symmetric  quantum channel is given by the so called XXZ spin- 1  2 Hamiltonian,  H =  �  i,j  Jij  �  σx  i σx  j + σy  i σy  j  �  + ∆ijσz  i σz  j + hiσz  i  (9)  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A quantum router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A dispatch center, encircled in red, creates an n-qubit entangled state (red spheres) with the aim  to send each party to a different receiver (green spheres) along independent quantum channels (blue spheres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  where σα  i (α = x, y, x) are Pauli matrices and i, j are the position indexes on an arbitrary d-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Assuming that each quantum channel is fully polarized, for the sender state , the map Φi reduces to an amplitude-  damping channel [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In particular, for f ri  si = 1, the map Φi entails a SWAP operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Therefore, our formalism  also describes entanglement swapping protocols via imperfect operations [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Finally, to express Equation (6) in the  form of Equation (4), it is sufficient to express the bit strings in decimal notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  By making use of Equations (4), (7), and (8), it is straightforward to evaluate the average fidelity [13] of an arbitrary  quantum state |Ψ⟩  ⟨F⟩ = 1  Ω  �  Ω  dΩ F (|Ψ⟩ , ρ(t)) ,  (10)  with Ω denoting the space of pure states and, with an abuse of notation, its volume and the measure of it .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  An average with respect to an n qubit system will be denoted as ⟨Fn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In the case of n independent channels,  making use of the transition amplitude f introduced in Equation (8), one arrives at the expression,  ⟨Fn⟩ =  1  2n + 1 +  1  2n (2n + 1) |1 + f|2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (11)  Notice that the average fidelity ⟨Fn⟩ ≤ �n  i=1 ⟨F1⟩, with equality holding only for f = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' While the left-hand side of  the latter inequality gives the average over all possible pure input states, its right-hand side, on the other hand, gives  the average restricted to fully factorized states only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', to product states of the form |Ψ⟩n = �n  i=1 |ψ⟩i, thus not  including the entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Hence, we conclude that, when n ≥ 2, in the set of all pure input states, entangled  states have a lower n-QST fidelity than the product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In the next sections, we will provide a quantitative analysis  for this intuitive observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  N -QST FIDELITY AS A FUNCTION OF ENTANGLEMENT  This Section contains our main result, namely, that the presence of entanglement reduces the transfer fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Below, we illustrate this idea separately for two, three, and four qubit transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In particular, we will show that,  in the presence of entanglement in the states to be sent, a reduction factor exists, which we dub Rn, such that the  average fidelity for the QST of n-qubits can be generically written  ⟨Fn⟩ = ⟨F1⟩n − En Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  4  Here, ⟨F1⟩n gives the average fidelity for the transfer of factorized states (indeed, intuitively, qubits can be transferred  one by one, in this case), so that the difference ⟨Fn⟩ − ⟨F1⟩n is entirely due to the fact that entangled states are  possibly transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' The coefficient En is an entanglement quantifier that changes with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' It is given by twice the  square of concurrence for n = 2, while for n = 3 it is proportional to a linear combination of the invariant polynomials  identifying the different classes of entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Finally, the factor Rn (defined below for the various cases) gives  the weight of entanglement-induced fidelity decrease, and it also enters the fidelity averaged over specific entanglement  classes of states (for n = 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Two Qubits  Adopting the (Schmidt-)parametrization of two-qubits pure states in terms of their entanglement [27], we write  |Ψ(s)⟩ =  �  1 + s  2  |00⟩ +  �  1 − s  2  |11⟩  (12)  where the parameter s ∈ [−1, 1] is related to the concurrence [22] via C =  √  1 − s2, and every two-qubit pure state  can be obtained from Equation (12) via local, unitary operations |Φ(s)⟩ = U1U2 |Ψ(s)⟩, with Ui ∈ SU(2) acting on  qubit i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' we obtain the average fidelity for a two-qubit QST protocol with independent channels as a  function of the amount of entanglement of the sender state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' which is invariant under local unitaries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' and which we  denote as ⟨·⟩U ⊗2  This average fidelity (which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' to say it shortly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' is averaged at fixed values of entanglement) reads  ⟨F2⟩U ⊗2 = 1  36  �  3 + |f|2 + 2 |f| cos φ  �2  − 1  18  �  |f|2 + 2 |f| cos φ  � �  3 − |f|2 + 2 |f| cos φ  �  C2  (13)  where we expressed the complex transition amplitude f as f = |f| eiφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Following the procedure outlined by Bose [3], in order to maximise Equation (13), one sets cos φ = 1, which,  physically, can be obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', by a uniform magnetic field applied over the spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Hence, the average two-  qubit fidelity F2 can be cast in the form  ⟨F2⟩U ⊗2 = 1  36  �  3 + |f|2 + 2 |f|  �2  − 1  18  �  |f|2 + 2 |f|  � �  3 −  �  |f|2 + 2 |f|  ��  C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (14)  From Equation (14), since 0 ≤ |f| ≤ 1, one can readily appreciate that the more concurrence the sender’s pure state  contains, the lower the fidelity with the received state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Equation (14) can also be rewritten as a function of the  single-qubit QST average fidelity  ⟨F1⟩ = 1  6  �  3 + 2 |f| + |f|2�  ,  (15)  to read  ⟨F2⟩U ⊗2 = ⟨F1⟩2 − 2  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) C2 = ⟨F1⟩2 − 2R2C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (16)  Again, as 1  2 ≤ ⟨F1⟩ ≤ 1, the average fidelity ⟨F2⟩ decreases with the amount of concurrence of the sender state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  From Equation (16), we see that the average 2-QST fidelity is reduced in the presence of the squared concurrence  by a factor of  R2 =  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) ,  (17)  which is reported in Figure 2 (left panel), together with the two-qubit average fidelity ⟨F2⟩, displayed for different  values of the squared concurrence as a function of the one-qubit fidelity ⟨F1⟩ (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Three Qubits  Having obtained a quantitative expression giving the reduction in the transmission fidelity due to the presence  of entanglement for two qubits, we move to the more intricate three qubit case in order to try and obtain similar  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (left) Reduction factor (17) for entangled states of the 2-QST average fidelity ⟨F2⟩ (16) as a function of the 1-QST  average fidelity ⟨F1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' The dotted, vertical line reports the maximum of R2 attained at ⟨F⟩1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (left)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Three-qubit pure-state entanglement  Let us now consider a system of three qubits A, B, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A three-qubit pure state can be written in canonical  form as [28]  |Ψ⟩ABC = λ0 |000⟩ + λ1eiφ |100⟩ + λ2 |101⟩ + λ3 |110⟩ + λ4 |111⟩ ,  (18)  where λi ≥ 0, 0 ≤ φ ≤ π, and the normalisation condition reads �  i λ2  i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  In terms of the coefficients of the state in Equation (18), one can introduce five invariant polynomials, allowing to  identify different entanglement classes [29]:  J1 =  ��λ1λ4eiφ − λ2λ3  ��2 , J2 = λ2  0λ2  2 , J3 = λ2  0λ2  3  (19a)  J4 = λ2  0λ2  4 , J5 = λ2  0  �  J1 + λ2  2λ2  3 − λ2  1λ2  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (19b)  The relation between invariant polynomials and entanglement measures is given by  C2  jk = 4Ji,  (20)  for i ̸= j ̸= k = 1, 2, 3, and where now, (1, 2, 3) = (A, B, C) holds on the LHS of Equation (20) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' At variance with the  two-qubit case, no single entanglement measure can capture genuine three-partite entanglement as three qubits can  be entangled in two inequivalent ways [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  One type of entanglement is quantified by the three-tangle [31],  τ 2  3 = 4J4 ,  (21)  while an inequivalent type of genuine multipartite entanglement (GME) is quantified by the so called GME concur-  rence, CGME [32], defined, in terms of invariant polynomials, for a three-qubit pure state as:  CGME = 4 (min {J2 + J3, J1 + J3, J1 + J2} + J4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (22)  Hence, three qubit states can be sorted in the following entanglement classes [30]:  Product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' All Ji = 0, resulting into A − B − C (class 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' All entanglement measures vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Biseparable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' All Ji = 0 except i) J1 for A − BC, ii) J2 for B − AC, and iii) J3 for C − AB (class 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Only the concurrence for one single pair of qubits is different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  W-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' CGME > 0 and τ3 = 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5  2 (class 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' J4 = 0 and √J1J2J3 = J5  2 (class 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  6  GHZ-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' CGME > 0 and τ3 > 0, with 5 possible cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' All Ji = 0 except J4 (class 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' J1 = J2 = J5 = 0, or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0 (class 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' J2 = J5 = 0 or J3 = J5 = 0 (class 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' J1J4 + J1J2 + J1J3 = √J1J2J3 = J5  2 (class 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' √J1J2J3 = |J5|  2  and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0 (class 4d) ,  where, with the notation X − Y , we indicate that subsystems X and Y do not share any type of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Notably, for 3-qubit pure states, a monogamy relation exists between the amount of entanglement that can be  shared among the parties [31]  C2  A|BC = C2  AB + C2  AC + τ 2  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (23)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Fidelity of 3-QST  Here, we derive the fidelity of the QST of a tree-qubit pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' First, we average the state in Equation (18) over  single-qubit, local operations U, and use the notation ⟨·⟩U ⊗3, in order to indicate the average fidelity at given values  of (and, thus, as a function of) the entanglement quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Subsequently, utilizing the averages of the invariant  polynomials obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' [29], we derive the average fidelity of each three-qubit class and use the notation ⟨·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  The 3-QST fidelity, expressed in terms of the single-qubit average, reads  ⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 ⟨F1⟩  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩)  �  J1 + J2 + J3 + 3  2J4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (24)  Since 1  2 ≤ ⟨F1⟩ ≤ 1, and 0 ≤ Ji ≤ 1  4, the second term on the right hand side of the latter equation is always  negative, so that one sees at once that entanglement reduces the 3-QST fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Introducing the reduction factor R3,  R3 = ⟨F1⟩  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) ,  (25)  we can write  ⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 R3  �  J1 + J2 + J3 + 3  2J4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (26)  Now, using the averages of the invariant polynomials obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' [29], ⟨J4⟩ =  1  12 and ⟨Jk⟩ =  1  24 (k = 1, 2, 3),  the average fidelity ⟨F3⟩ (with average taken over the full three qubit Hilbert space) is given by  ⟨F3⟩ = ⟨F1⟩3 − 2R3  (27)  Here, we see that, at fixed single-particle average fidelity, entanglement is responsible for a decrease in the 3-QST  average fidelity by twice the reduction factor R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  In particular, the fidelities for the canonical states belonging to the different entanglement classes read  class 1 (product state)  ⟨Fc1⟩U ⊗3 = ⟨F1⟩3  (28)  ⟨Fc1⟩ = ⟨F1⟩3  (29)  class 2a (biseparable states)  ⟨Fc2a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) C2  jk  (30)  ⟨Fc2a⟩ = ⟨F1⟩3 − R3  3  (31)  where i ̸= j ̸= k = 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='class 2b (GHZ-states)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc2b⟩U ⊗3 = ⟨F1⟩3 − 3 ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩) τ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(32)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc2b⟩ = ⟨F1⟩3 − R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(33)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='class 3a (J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2 )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc3a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc3a⟩ = ⟨F1⟩3 − R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(34)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Class 3b (J1 = J2 = J5 = 0 or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc3b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + 3τ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc3b⟩ = ⟨F1⟩3 − 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(35)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Class 4a J4 = 0 and √J1J2J3 = J5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4a⟩ = ⟨F1⟩3 − R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(36)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Class 4b (J2 = J5 = 0 or J3 = J5 = 0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='+ 3τ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4b⟩ = ⟨F1⟩3 − 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(37)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Class 4c (J1J4 + J1J2 + J1J3 + J2J3 = √J1J2J3 = J5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2 )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='+ 3τ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4c⟩ = ⟨F1⟩3 − 2R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(38)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Class 4d (√J1J2J3 = |J5|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨F1⟩ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(1 − ⟨F1⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='BC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AC + C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='AB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='+ 3τ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='⟨Fc4d⟩ = ⟨F1⟩3 − 2R3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='(39)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='Comparing Equation (30) with Equation (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' it turns out that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' for an equivalent amount of the entanglement  monotone C2  jk and τ 2  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' at fixed ⟨F1⟩ (or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' at a fixed transition amplitude f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' the fidelity of the canonical  state in class 2a is greater than that in class 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' This is in line with our intuition that the more entangled a state  is, the harder it is to achieve high fidelity in our parallel QST protocol, as shown in Figure 3 (right panel), where we  plot the average fidelity for different three qubit classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In the left panel of Figure 3 we report the reduction factor  R3 of Equation (25) as a function of the single-particle average fidelity ⟨F1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  From the above equations of the average fidelity of the three-qubit classes, we see that the average fidelity is  decreased, with respect to the product state class, whenever there is two-qubit concurrence or genuine multipartite  entanglement, both as CGME and as τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Moreover, per equal amount of squared two-qubit concurrence C2 and genuine  multipartite entanglement CGME, the reducing factor is respectively 2 and 3 times R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' As a consequence, we state  that, at fixed amount of entanglement, GME states are harder to transfer than biseparable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (left) Reduction factor for the average fidelity in the presence of entanglement as in Equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (right) Average  fidelity for the three-qubit classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' The dotted, vertical line is at ⟨F⟩1 = 1  2  �  1 +  1  √  3  �  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Four Qubits  While for two and three qubits, the entanglement of pure states has been fully characterized, for four (or more)  qubits there are infinitely many inequivalent entanglement classes [30, 33] under SLOCC operations (stochastic local  operations and classical communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Here, we consider the fidelity of specific four-qubit states averaged over random local unitaries on each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Whereas this does not account for all entangled states within a given class, as the group of stocastic local operations  includes deterministic local operations, SU(2) ⊆ SL(2, C) (with equality holding for pure states), the results nev-  ertheless hint at the fact that the average fidelity decreases with the entanglement of the sender state and that the  reduction factor depends on the type of entanglement contained in the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  We will consider the three irreducibly balanced states [34]: the 4-qubits GHZ-state, the cluster, and X4 :  |GHZ4⟩ =  1  √  2 (|0000⟩ + |1111⟩)  (40a)  |Cl4⟩ = 1  2 (|0000⟩ + |0111⟩ + |1011⟩ + |1100⟩)  (40b)  |X4⟩ =  1  √  6  �√  2 |1111⟩ + |0001⟩ + |0010⟩ + |0100⟩ + |1000⟩  �  (40c)  and two additional entangled states: the product of two-Bell states and the 4-qubit W-state:  |B2⟩ = |Φ⟩12 ⊗ |Φ⟩34  (41a)  |W4⟩ = 1  2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (41b)  The average fidelities of the states reported in Equations (40) and (41), expressed in terms of 1-QST average fidelity,  read  ⟨FGHZ4⟩U ⊗4 = ⟨FB2⟩U ⊗4 = ⟨F1⟩4 − 2 ⟨F1⟩  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩)  �  1 − 3 ⟨F⟩ + 4 ⟨F⟩2�  (42a)  ⟨FCl4⟩U ⊗4 = ⟨FX4⟩U ⊗4 = ⟨F1⟩4 − 4 ⟨F1⟩2  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩)  (42b)  ⟨FW4⟩U ⊗4 = ⟨F1⟩4 − 3 ⟨F1⟩2  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (42c)  Notice that the reduction factor for the states |Cl4⟩ , |X4⟩ , |W4⟩ is the same, although with different weights, and  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (left) Reduction factor for the average fidelity in the presence of entanglement as in Equations (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' (right) Average  fidelity for the entangled classes as reported in Equations (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' The blue and red dotted, vertical lines are, respectively, at  ⟨F⟩1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='82 and ⟨F⟩1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  differs from the reduction factor for the states |GHZ4⟩ , |B2⟩, reading, respectively,  R4a = ⟨F1⟩  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩)  �  1 − 3 ⟨F⟩ + 4 ⟨F⟩2�  (43a)  R4b = ⟨F1⟩2  �  ⟨F1⟩ − 1  2  �  (1 − ⟨F1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  (43b)  A possible reason may be that, if one considers the four-tangle as an entanglement measure, although it is not a  measure of genuine multipartite entanglement, the first set of state has zero four-tangle, whereas for the second one  it is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In Figure 4 (left panel) we report the reduction factors in Equation (43), while in the right panel we  report the average fidelity of Equation (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  DISCUSSION  We have shown that the QST of an entangled n ≥ 2 quantum state across parallel, independent U(1)-symmetric  quantum channels, as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', embodied by an XXZ spin- 1  2 Hamiltonian, leads to a lower average fidelity than that of  the QST of a product state at fixed one-qubit QST average fidelity, or, equivalently, at fixed transition amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  For the case of n = 2, we have expressed the average fidelity reduction in terms of the squared concurrence times a  reduction factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Similarly, for n = 3, we obtained that the presence of entanglement, both bipartite and multipartite,  has a detrimental effect on the average fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In particular, we obtained that the reduction factor has a greater  weight in the presence of genuine three-partite entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=', three-tangle and GME concurrence, than in the  presence of two-qubit squared concurrence for specific canonical classes of the three-qubit pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Finally, we have  considered specific cases of 4-qubit entangled states, which, again, result in an average fidelity reduction due to the  presence of entanglement in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Our work clearly shows that for entanglement distribution in a routing configuration, where parties are sent over  independent quantum channels, the single-qubit average fidelity is not a reliable figure of merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' This calls for more  investigations into the properties of quantum channels able to faithfully distribute multipartite entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  TJGA acknowledges funding through the IPAS+ (Internationalisation Partnership Awards Scheme +) QUEST  project by the MCST (The Malta Council for Science & Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' MC acknowledges funding from the Tertiary  Education Scholarships Scheme and from the Project QVAQT financed by the Malta Council for Science & Technology,  for and on behalf of the Foundation for Science and Technology, through the FUSION: R&I Research Excellence  Programme REP-2022-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  K ˙Z acknowledges support by Narodowe Centrum Nauki under the Quantera project  10  number 2021/03/Y/ST2/00193 and by the Foundation for Polish Science under the Team-Net project POIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='00-  00-17C1/18-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' SL acknowledges support by MUR under PRIN Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' 2017 SRN-BRK QUSHIP  [1] Hillery, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Buˇzek, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Berthiaume, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Quantum secret sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' A 1999, 59, 1829–1834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='org/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='1829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  [2] Cuomo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='  [9] Streltsov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Bruß, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Entanglement Distribution and Quantum Discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' In Lectures on General  Quantum Correlations and their Applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='  [10] Wengerowsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Joshi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Scheidl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=' Dobrovolskiy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='  Los, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' 2020, 6, 1–5, [1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='1038/s41534-019-0238-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='3390/e20100745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' Verschelde, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content='  New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
+page_content='1088/1367-2630/12/7/075025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfTgrD/content/2301.04443v1.pdf'}
diff --git a/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/2301.01716v1.pdf.txt b/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/2301.01716v1.pdf.txt
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+arXiv:2301.01716v1  [math.OC]  4 Jan 2023
+First-order penalty methods for bilevel optimization
+Zhaosong Lu ∗
+Sanyou Mei ∗
+January 4, 2023
+Abstract
+In this paper we study a class of unconstrained and constrained bilevel optimization problems in
+which the lower-level part is a convex optimization problem, while the upper-level part is possibly
+a nonconvex optimization problem. In particular, we propose penalty methods for solving them,
+whose subproblems turn out to be a structured minimax problem and are suitably solved by a first-
+order method developed in this paper. Under some suitable assumptions, an operation complexity
+of O(ε−4 log ε−1) and O(ε−7 log ε−1), measured by their fundamental operations, is established for
+the proposed penalty methods for finding an ε-KKT solution of the unconstrained and constrained
+bilevel optimization problems, respectively.
+To the best of our knowledge, the methodology and
+results in this paper are new.
+Keywords: bilevel optimization, minimax optimization, penalty methods, first-order methods, opera-
+tion complexity
+Mathematics Subject Classification: 90C26, 90C30, 90C47, 90C99, 65K05
+1
+Introduction
+Bilevel optimization is a two-level hierarchical optimization in which partial or full decision variables in
+the upper level are also involved in the lower level. Generically, it can be written in the following form:
+min
+x,y
+f(x, y)
+s.t.
+g(x, y) ≤ 0,
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}.1
+(1)
+Bilevel optimization has found a variety of important applications, including adversarial training [36,
+37, 46], continual learning [32], hyperparameter tuning [3, 17], image reconstruction [9], meta-learning
+[4, 23, 42], neural architecture search [15, 30], reinforcement learning [20, 27], and Stackelberg games [48].
+More applications about it can be found in [2, 8, 10, 11, 12, 44] and the references therein. Theoretical
+properties including optimality conditions of (1) have been extensively studied in the literature (e.g., see
+[12, 13, 34, 47, 50]).
+Numerous methods have been developed for solving some special cases of (1). For example, constraint-
+based methods [19, 43], deterministic gradient-based methods [16, 17, 21, 35, 41, 42], and stochastic
+gradient-based methods [6, 18, 20, 24, 26] were proposed for solving (1) with g ≡ 0, ˜g ≡ 0, f, ˜f being
+smooth, and ˜f being strongly convex with respect to y. Besides, when all the functions involved in
+(1) are smooth and ˜f, ˜g are convex with respect to y, gradient-type methods were proposed by solving
+the mathematical program with equilibrium constraints (MPEC) resulting from replacing the lower-level
+optimization problem of (1) by its first-order optimality conditions (e.g., see [1, 33, 40]).
+Recently,
+difference-of-convex (DC) algorithms were developed in [51] for solving (1) with g ≡ 0, f being a DC
+function, and ˜f, ˜g being convex functions. In addition, a double penalty method [22] was proposed for
+(1), which solves a sequence of bilevel optimization problems of the form
+min
+x,y
+f(x, y) + ρkΨ(x, y)
+s.t.
+y ∈ Argmin
+z
+˜f(x, z) + ρk ˜Ψ(x, z),
+(2)
+∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email:
+zhaosong@umn.edu,
+mei00035@umn.edu). This work was partially supported by NSF Award IIS-2211491.
+1For ease of reading, throughout this paper the tilde symbol is particularly used for the functions related to the lower-level
+optimization problem. Besides, “Argmin” denotes the set of optimal solutions of the associated problem.
+1
+
+where {ρk} is a sequence of penalty parameters, and Ψ and ˜Ψ are a penalty function associated with the
+sets {(x, y)|g(x, y) ≤ 0} and {(x, z)|˜g(x, z) ≤ 0}, respectively. Though problem (2) appears to be simpler
+than (1), there is no method available for finding an approximate solution of (2) in general. Conse-
+quently, the double penalty method [22] is typically not implementable. More discussion on algorithmic
+development for bilevel optimization can be found in [2, 8, 12, 31, 45, 47]) and the references therein.
+It has long been known that the notorious challenge of bilevel optimization (1) mainly comes from the
+lower level part, which requires that the variable y be a solution of another optimization problem. Due
+to this, for the sake of simplicity, we only consider a subclass of bilevel optimization with the constraint
+g(x, y) ≤ 0 being excluded, namely,
+min
+x,y
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}.
+(3)
+Nevertheless, the results in this paper can be possibly extended to problem (1).
+The main goal of this paper is to develop a first-order penalty method for solving problem (3). Our
+key observations toward this development are: (i) problem (3) can be approximately solved as a penalty
+problem (see (49)); (ii) such a penalty problem is equivalent to a structured minimax problem (see
+(50)), which can be suitably solved by a first-order method proposed in Section 2. As a result, these
+observations lead to development of a novel first-order penalty method for solving (3) (see Sections 3
+and 4), which enjoys the following appealing features.
+• It uses only the first-order information of the problem. Specifically, its fundamental operations
+consist only of evaluations of the gradient of ˜g and the smooth component of f and ˜f and also
+the proximal operator of the nonsmooth component of f and ˜f. Thus, it is suitable for solving
+large-scale problems (see Sections 3 and 4).
+• It has theoretical guarantees on operation complexity, which is measured by the aforementioned
+fundamental operations, for finding an ε-KKT solution of (3).
+In particular, when ˜g ≡ 0, it
+enjoys an operation complexity of O(ε−4 log ε−1). Otherwise, it enjoys an operation complexity of
+O(ε−7 log ε−1) (see Theorems 4 and 6).
+• It is applicable to a broader class of problems than existing methods.
+For example, it can be
+applied to (3) with f, ˜f being nonsmooth and ˜f, ˜g being nonconvex with respect to x, which is
+however not suitable for existing methods.
+To the best of our knowledge, the methodology and results in this paper are new.
+The rest of this paper is organized as follows. In Subsection 1.1 we introduce some notation and
+terminology. In Section 2 we propose a first-order method for solving a nonconvex-concave minimax
+problem and study its complexity.
+In Sections 3 and 4, we propose first-order penalty methods for
+unconstrained and constrained bilevel optimization and study their complexity, respectively. In Section
+5 we present the proofs of the main results. Finally, we make some concluding remarks in Section 6.
+1.1
+Notation and terminology
+The following notation will be used throughout this paper.
+Let Rn denote the Euclidean space of
+dimension n and Rn
++ denote the nonnegative orthant in Rn. The standard inner product and Euclidean
+norm are denoted by ⟨·, ·⟩ and ∥ · ∥, respectively. For any v ∈ Rn, let v+ denote the nonnegative part of
+v, that is, (v+)i = max{vi, 0} for all i. For any two vectors u and v, (u; v) denotes the vector resulting
+from stacking v under u. Given a point x and a closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥ and
+IS denote the indicator function associated with S.
+A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ Lφ∥x−x′∥
+for all x, x′ ∈ S. In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x) − ∇φ(x′)∥ ≤ L∇φ∥x − x′∥ for
+all x, x′ ∈ S. For a closed convex function p : Rn → R ∪ {∞},2 the proximal operator associated with p
+is denoted by proxp, that is,
+proxp(x) = arg min
+x′∈Rn
+�1
+2∥x′ − x∥2 + p(x′)
+�
+∀x ∈ Rn.
+(4)
+2For convenience, ∞ stands for +∞.
+2
+
+Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of proxγp(x)
+as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn.
+For a lower semicontinuous function φ : Rn → R∪{∞}, its domain is the set dom φ := {x|φ(x) < ∞}.
+The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by
+φ′(x; d) = lim sup
+x′ φ
+→x, t↓0
+inf
+d′→d
+φ(x′ + td′) − φ(x′)
+t
+,
+where t ↓ 0 means both t > 0 and t → 0, and x′
+φ→ x means both x′ → x and φ(x′) → φ(x). The
+subdifferential of φ at x ∈ dom φ is the set
+∂φ(x) = {s ∈ Rn��sT d ≤ φ′(x; d) ∀d ∈ Rn}.
+We use ∂xiφ(x) to denote the subdifferential with respect to xi. In addition, for an upper semicontinuous
+function φ, its subdifferential is defined as ∂φ = −∂(−φ). If φ is locally Lipschitz continuous, the above
+definition of subdifferential coincides with the Clarke subdifferential. Besides, if φ is convex, it coincides
+with the ordinary subdifferential for convex functions. Also, if φ is continuously differentiable at x , we
+simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the gradient of φ at x. In addition, it is not hard to
+verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if φ1 is continuously differentiable at x and φ2 is lower or
+upper semicontinuous at x. See [7, 49] for more details.
+Finally, we introduce two types of approximate solutions for a general minimax problem
+Ψ∗ = min
+x max
+y
+Ψ(x, y),
+(5)
+where Ψ(·, y) : Rn → R ∪ {∞} is a lower semicontinuous function, Ψ(x, ·) : Rm → R ∪ {−∞} is an upper
+semicontinuous function, and Ψ∗ is finite.
+Definition 1. A point (xǫ, yǫ) is called an ǫ-optimal solution of the minimax problem (5) if
+max
+y
+Ψ(xǫ, y) − Ψ(xǫ, yǫ) ≤ ǫ,
+Ψ(xǫ, yǫ) − Ψ∗ ≤ ǫ.
+Definition 2. A point (x, y) is called a stationary point of the minimax problem (5) if
+0 ∈ ∂xΨ(x, y),
+0 ∈ ∂yΨ(x, y).
+In addition, for any ǫ > 0, a point (xǫ, yǫ) is called an ǫ-stationary point of the minimax problem (5) if
+dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ,
+dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ.
+2
+A first-order method for nonconvex-concave minimax prob-
+lem
+In this section, we propose a first-order method for finding an approximate stationary point of a
+nonconvex-concave minimax problem, which will be used as a subproblem solver for the penalty methods
+proposed in Sections 3 and 4. In particular, we consider the minimax problem
+H∗ = min
+x max
+y
+{H(x, y) := h(x, y) + p(x) − q(y)} .
+(6)
+Assume that problem (6) has at least one optimal solution. In addition, h, p and q satisfy the following
+assumptions.
+Assumption 1.
+(i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions and
+continuous on their domain, and moreover, their domain is compact.
+(ii) The proximal operator associated with p and q can be exactly evaluated.
+(iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p.
+3
+
+Recently, an accelerated inexact proximal point smoothing (AIPP-S) scheme was proposed in [28]
+for finding an approximate stationary point of a class of minimax composite nonconvex optimization
+problems, which includes (6) as a special case. When applied to (6), AIPP-S requires the availability of
+the oracle including exact evaluation of ∇xh(x, y) and
+arg min
+x
+�
+p(x) + 1
+2λ∥x − x′∥2
+�
+,
+arg max
+y
+�
+h(x′, y) − q(y) − 1
+2λ∥y − y′∥2
+�
+(7)
+for any λ > 0, x′ ∈ Rn and y′ ∈ Rm. Note that h is typically sophisticated and the exact solution of the
+second problem in (7) usually cannot be found. As a result, AIPP-S is generally not implementable for
+(6), though an oracle complexity of O(ǫ−5/2) was established in [28] for it to find an ǫ-stationary point
+of (6).
+In what follows, we first propose a modified optimal first-order method for solving a strongly-convex-
+strongly-concave minimax problem in Subsection 2.1. Using this method as a subproblem solver for an
+inexact proximal point scheme, we then propose a first-order method for (6) in Subsection 2.2, which
+enjoys an operation complexity of O(ǫ−5/2 log ǫ−1), measured by the amount of evaluations of ∇h and
+proximal operator of p and q, for finding an ǫ-stationary point of (6).
+2.1
+A modified optimal first-order method for strongly-convex-strongly-concave
+minimax problem
+In this subsection, we consider the strongly-convex-strongly-concave minimax problem
+¯H∗ = min
+x max
+y
+� ¯H(x, y) := ¯h(x, y) + p(x) − q(y)
+�
+,
+(8)
+where p and q satisfy Assumption 1 and ¯h satisfies the following assumption.
+Assumption 2. ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q
+for some σx, σy > 0.
+The goal of this subsection is to propose a modified optimal first-order method for finding an approx-
+imate stationary point of problem (8) and study its complexity. Before proceeding, we introduce some
+more notation below, most of which is adopted from [29].
+Let (x∗, y∗) denote the optimal solution of (8), z∗ = −σxx∗, and
+Dp = max{∥u − v∥
+��u, v ∈ dom p},
+Dq = max{∥u − v∥
+��u, v ∈ dom q},
+(9)
+¯Hlow = min
+� ¯H(x, y)|
+�
+x, y) ∈ dom p × dom q},
+(10)
+ˆh(x, y) = ¯h(x, y) − σx∥x∥2/2 + σy∥y∥2/2,
+(11)
+G(z, y) = sup
+x {⟨x, z⟩ − p(x) − ˆh(x, y) + q(y)},
+(12)
+P(z, y) = σ−1
+x ∥z∥2/2 + σy∥y∥2/2 + G(z, y),
+(13)
+ϑk = η−1
+z ∥zk − z∗∥2 + η−1
+y ∥yk − y∗∥2 + 2¯α−1(P(zk
+f, yk
+f) − P(z∗, y∗)),
+(14)
+ak
+x(x, y) = ∇xˆh(x, y) + σx(x − σ−1
+x zk
+g)/2,
+ak
+y(x, y) = −∇yˆh(x, y) + σyy + σx(y − yk
+g)/8,
+where ¯α = min
+�
+1,
+�
+8σy/σx
+�
+, ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, and yk, yk
+f, yk
+g, zk, zk
+f and zk
+g
+are generated at iteration k of Algorithm 1 below. By Assumptions 1 and 2, one can observe that Dp,
+Dq and ¯Hlow are finite.
+We are now ready to present a modified optimal first-order method for solving (8) in Algorithm 1. It is
+a slight modification of the novel optimal first-order method [29, Algorithm 4] by incorporating a forward-
+backward splitting scheme and also a verifiable termination criterion (see steps 23-25 in Algorithm 1) in
+order to find a τ-stationary point of (8) (see Definition 2) for any prescribed tolerance τ > 0.
+4
+
+Algorithm 1 A modified optimal first-order method for (8)
+Input: τ > 0, ¯z0 = z0
+f ∈ −σxdom p,3 ¯y0 = y0
+f ∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = min
+�
+1,
+�
+8σy/σx
+�
+,
+ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ =
+�
+2
+√
+5(1 + 8L∇¯h/σx)
+�−1, γx = γy =
+8σ−1
+x , and ˆζ = min{σx, σy}/L2
+∇¯h.
+1: for k = 0, 1, 2, . . . do
+2:
+(zk
+g , yk
+g) = ¯α(zk, yk) + (1 − ¯α)(zk
+f, yk
+f).
+3:
+(xk,−1, yk,−1) = (−σ−1
+x zk
+g, yk
+g).
+4:
+xk,0 = proxζγxp(xk,−1 − ζγxak
+x(xk,−1, yk,−1)).
+5:
+yk,0 = proxζγyq(yk,−1 − ζγyak
+y(xk,−1, yk,−1)).
+6:
+bk,0
+x
+=
+1
+ζγx (xk,−1 − ζγxak
+x(xk,−1, yk,−1) − xk,0).
+7:
+bk,0
+y
+=
+1
+ζγy (yk,−1 − ζγyak
+y(xk,−1, yk,−1) − yk,0).
+8:
+t = 0.
+9:
+while
+γx∥ak
+x(xk,t, yk,t) + bk,t
+x ∥2 + γy∥ak
+y(xk,t, yk,t) + bk,t
+y ∥2 > γ−1
+x ∥xk,t − xk,−1∥2 + γ−1
+y ∥yk,t − yk,−1∥2
+do
+10:
+xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak
+x(xk,t, yk,t) + bk,t
+x ).
+11:
+yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak
+y(xk,t, yk,t) + bk,t
+y ).
+12:
+xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak
+x(xk,t+1/2, yk,t+1/2)).
+13:
+yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak
+y(xk,t+1/2, yk,t+1/2)).
+14:
+bk,t+1
+x
+=
+1
+ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak
+x(xk,t+1/2, yk,t+1/2) − xk,t+1).
+15:
+bk,t+1
+y
+=
+1
+ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak
+y(xk,t+1/2, yk,t+1/2) − yk,t+1).
+16:
+t ← t + 1.
+17:
+end while
+18:
+(xk+1
+f
+, yk+1
+f
+) = (xk,t, yk,t).
+19:
+(zk+1
+f
+, wk+1
+f
+) = (∇xˆh(xk+1
+f
+, yk+1
+f
+) + bk,t
+x , −∇yˆh(xk+1
+f
+, yk+1
+f
+) + bk,t
+y ).
+20:
+zk+1 = zk + ηzσ−1
+x (zk+1
+f
+− zk) − ηz(xk+1
+f
++ σ−1
+x zk+1
+f
+).
+21:
+yk+1 = yk + ηyσy(yk+1
+f
+− yk) − ηy(wk+1
+f
++ σyyk+1
+f
+).
+22:
+xk+1 = −σ−1
+x zk+1.
+23:
+ˆxk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)).
+24:
+ˆyk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)).
+25:
+Terminate the algorithm and output (ˆxk+1, ˆyk+1) if
+∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥ ≤ τ.
+(15)
+26: end for
+The following theorem presents iteration and operation complexity of Algorithm 1 for finding a τ-
+stationary point of problem (8), whose proof is deferred to Subsection 5.1.
+Theorem 1 (Complexity of Algorithm 1). Suppose that Assumptions 1 and 2 hold. Let ¯H∗, Dp,
+Dq, ¯Hlow, and ϑ0 be defined in (8), (9), (10) and (14), σx, σy and L∇¯h be given in Assumption 2, ¯α,
+ηy, ηz, τ, ˆζ be given in Algorithm 1, and
+¯δ = (2 + ¯α−1)σxD2
+p + max{2σy, ¯ασx/4}D2
+q,
+(16)
+¯K =
+�
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+,
+(17)
+¯N =
+�
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max {1/(2σx), min {1/(2σy), 4/(¯ασx)}}
+�¯δ + 2¯α−1 � ¯H∗ − ¯Hlow
+��
+(L2
+∇¯h/ min{σx, σy} + L∇¯h)−2τ 2
+�
++
+×
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+�
+.
+(18)
+Then Algorithm 1 outputs a τ-stationary point of (8) in at most ¯K iterations.
+Moreover, the total
+3For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}.
+5
+
+number of evaluations of ∇¯h and proximal operator of p and q performed in Algorithm 1 is no more than
+¯N, respectively.
+Remark 1. It can be observed from Theorem 1 that Algorithm 1 enjoys an operation complexity of
+O(log τ−1), measured by the amount of evaluations of ∇¯h and proximal operator of p and q, for finding
+a τ-stationary point of the strongly-convex-strongly-concave minimax problem (8).
+2.2
+A first-order method for problem (6)
+In this subsection, we propose a first-order method for finding an ǫ-stationary point of problem (6) (see
+Definition 2) for any prescribed tolerance ǫ > 0. In particular, we first add a perturbation to the max
+part of (6) for obtaining an approximation of (6), which is given as follows:
+min
+x max
+y
+�
+h(x, y) + p(x) − q(y) −
+ǫ
+4Dq
+∥y − ˆy0∥2
+�
+(19)
+for some ˆy0 ∈ dom q. We then apply an inexact proximal point method [25] to (19), which consists of
+approximately solving a sequence of subproblems
+min
+x max
+y
+{Hk(x, y) := hk(x, y) + p(x) − q(y)} ,
+(20)
+where
+hk(x, y) = h(x, y) − ǫ∥y − ˆy0∥2/(4Dq) + L∇h∥x − xk∥2.
+(21)
+By Assumption 1, one can observe that (i) hk is L∇h-strongly convex in x and ǫ/(2Dq)-strongly concave
+in y on dom p × dom q; (ii) hk is (3L∇h + ǫ/(2Dq))-smooth on dom p × dom q. Consequently, problem
+(20) is a special case of (8) and we can apply Algorithm 1 to solve (20). The resulting first-order method
+for (6) is presented in Algorithm 2.
+Algorithm 2 A first-order method for problem (6)
+Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1).
+1: for k = 0, 1, 2, . . . do
+2:
+Call Algorithm 1 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h + ǫ/(2Dq),
+¯z0 = z0
+f ← −σxxk, ¯y0 = y0
+f ← yk, and denote its output by (xk+1, yk+1), where hk is given in (21).
+3:
+Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if
+∥xk+1 − xk∥ ≤ ǫ/(4L∇h).
+(22)
+4: end for
+Remark 2. It can be observed from step 2 of Algorithm 2 that (xk+1, yk+1) results from applying Algo-
+rithm 1 to the subproblem (20). As will be shown in Lemma 2, (xk+1, yk+1) is an ǫk-stationary point of
+(20).
+We next study complexity of Algorithm 2 for finding an ǫ-stationary point of problem (6). Before
+proceeding, we define
+Hlow := min {H(x, y)|(x, y) ∈ dom p × dom q} .
+(23)
+By Assumption 1, one can observe that Hlow is finite.
+The following theorem presents iteration and operation complexity of Algorithm 2 for finding an
+ǫ-stationary point of problem (6), whose proof is deferred to Subsection 5.2.
+Theorem 2 (Complexity of Algorithm 2). Suppose that Assumption 1 holds. Let H∗, H Dp, Dq,
+and Hlow be defined in (6), (9) and (23), L∇h be given in Assumption 1, ǫ, ǫ0 and ˆx0 be given in
+6
+
+Algorithm 2, and
+α = min
+�
+1,
+�
+4ǫ/(DqL∇h)
+�
+,
+(24)
+δ = (2 + α−1)L∇hD2
+p + max {ǫ/Dq, αL∇h/4} D2
+q,
+(25)
+K =
+�
+16(max
+y
+H(ˆx0, y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)ǫ−2 − 1
+�
++
+,
+(26)
+N =
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇hǫ−1
+�
+×
+�
+(K + 1)
+�
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+0
+�
++
++ K + 1 + 2K log(K + 1)
+�
+.
+(27)
+Then Algorithm 2 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (6) in at most K + 1 outer
+iterations that satisfies
+max
+y
+H(xǫ, y) ≤ max
+y
+H(ˆx0, y) + ǫDq/4 + 2ǫ2
+0
+�
+L−1
+∇h + 4D2
+qL∇hǫ−2�
+.
+(28)
+Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed in Algo-
+rithm 2 is no more than N, respectively.
+Remark 3. Since ǫ0 ∈ (0, ǫ/2], one can observe from Theorem 2 that α = O(ǫ1/2), δ = O(ǫ−1/2),
+K = O(ǫ−2), and N = O(ǫ−5/2 log(ǫ−1
+0 ǫ−1). Consequently, Algorithm 2 enjoys an operation complexity
+of O(ǫ−5/2 log(ǫ−1
+0 ǫ−1)), measured by the amount of evaluations of ∇h and proximal operator of p and
+q, for finding an ǫ-stationary point of the nonconvex-concave minimax problem (6).
+3
+Unconstrained bilevel optimization
+In this section, we consider an unconstrained bilevel optimization problem4
+f ∗ = min
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+˜f(x, z).
+(29)
+Assume that problem (29) has at least one optimal solution. In addition, f and ˜f satisfy the following
+assumptions.
+Assumption 3.
+(i) f(x, y) = f1(x, y)+f2(x) and ˜f(x, y) = ˜f1(x, y)+ ˜f2(y) are continuous on X ×Y,
+where f2 : Rn → R ∪ {∞} and ˜f2 : Rm → R ∪ {∞} are proper closed convex functions, ˜f1(x, ·) is
+convex for any given x ∈ X, and f1, ˜f1 are respectively L∇f1- and L∇ ˜f1-smooth on X × Y with
+X := dom f2 and Y := dom ˜f2.
+(ii) The proximal operator associated with f2 and ˜f2 can be exactly evaluated.
+(iii) The sets X and Y (namely, dom f2 and dom ˜f2) are compact.
+For notational convenience, we define
+Dx := max{∥u − v∥
+��u, v ∈ X},
+Dy := max{∥u − v∥
+��u, v ∈ Y},
+(30)
+˜fhi := max{ ˜f(x, y)|(x, y) ∈ X × Y},
+˜flow := min{ ˜f(x, y)|(x, y) ∈ X × Y},
+(31)
+flow := min{f(x, y)|(x, y) ∈ X × Y}.
+(32)
+4For convenience, problem (29) is referred to as an unconstrained bilevel optimization problem since its lower level part
+does not have an explicit constraint. Strictly speaking, it can be a constrained bilevel optimization problem. For example,
+when part of f and/or ˜f is the indicator function of a closed convex set, (29) is essentially a constrained bilevel optimization
+problem.
+7
+
+By Assumption 3, one can observe that Dx, Dy, ˜fhi, ˜flow and flow are finite.
+The goal of this subsection is to propose penalty methods for solving problem for solving (29). To
+this end, we observe that problem (29) can be viewed as
+min
+x,y {f(x, y)| ˜f(x, y) ≤ min
+z
+˜f(x, z)}.
+(33)
+Notice that ˜f(x, y) − minz ˜f(x, z) ≥ 0 for all x, y. Consequently, a natural penalty problem associated
+with (33) is
+min
+x,y f(x, y) + ρ( ˜f(x, y) − min
+z
+˜f(x, z)),
+(34)
+where ρ > 0 is a penalty parameter. We further observe that (34) is equivalent to the minimax problem
+min
+x,y max
+z
+Pρ(x, y, z),
+where
+Pρ(x, y, z) := f(x, y) + ρ( ˜f(x, y) − ˜f(x, z)).
+(35)
+In view of Assumption 3(i), Pρ can be rewritten as
+Pρ(x, y, z) =
+�
+f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z)
+�
++
+�
+f2(x) + ρ ˜f2(y) − ρ ˜f2(z)
+�
+.
+(36)
+By this and Assumption 3, one can observe that Pρ enjoys the following nice properties.
+• Pρ is the sum of smooth function f1(x, y)+ ρ ˜f1(x, y)− ρ ˜f1(x, z) with Lipschitz continuous gradient
+and possibly nonsmooth function f2(x)+ρ ˜f2(y)−ρ ˜f2(z) with exactly computable proximal operator.
+• Pρ is nonconvex in (x, y) but concave in z.
+Thanks to the nice structure of Pρ, an approximate stationary point of the minimax problem (35) can
+be found by Algorithm 2 proposed in Subsection 2.2.
+Based on the above observations, we are now ready to propose penalty methods for the unconstrained
+bilevel optimization problem (29) by solving either a sequence of or a single minimax problem in the
+form of (35). In particular, we first propose an ideal penalty method for (29) by solving a sequence of
+minimax problems (see Algorithm 3). Then we propose a practical penalty method for (29) by finding
+an approximate stationary point of a single minimax problem (see Algorithm 4).
+Algorithm 3 An ideal penalty method for problem (29)
+Input: positive sequences {ρk} and {ǫk} with limk→∞(ρk, ǫk) = (∞, 0).
+1: for k = 0, 1, 2, . . . do
+2:
+Find an ǫk-optimal solution (xk, yk, zk) of problem (35) with ρ = ρk.
+3: end for
+The following theorem states a convergence result of Algorithm 3, whose proof is deferred to Section
+5.3.
+Theorem 3 (Convergence of Algorithm 3). Suppose that Assumption 3 holds and that {(xk, yk, zk)}
+is generated by Algorithm 3. Then any accumulation point of {(xk, yk)} is an optimal solution of problem
+(29).
+Notice that (35) is a nonconvex-concave minimax problem. It is typically hard to find an ǫ-optimal
+solution of (35) for an arbitrary ǫ > 0. Consequently, Algorithm 3 is not implementable in general. We
+next propose a practical penalty method for problem (29) by finding an approximate stationary point of
+a single minimax problem (35) with a suitable choice of ρ.
+Algorithm 4 A practical penalty method for problem (29)
+Input: ε ∈ (0, 1/4], ρ = ε−1, (x0, y0) ∈ X × Y with ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε.
+1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε3/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ε−1L∇ ˜
+f1 to
+find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (35) with ρ = ε−1.
+2: Output: (xǫ, yǫ).
+8
+
+Remark 4. (i) The initial point (x0, y0) of Algorithm 4 can be found by an additional procedure. Indeed,
+one can first choose any x0 ∈ X and then apply accelerated proximal gradient method [38] to the problem
+miny ˜f(x0, y) for finding y0 ∈ Y such that ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε; (ii) As seen from Theorem 2,
+an ǫ-stationary point of (35) can be successfully found in step 1 of Algorithm 4 by applying Algorithm 2
+to (35); (iii) For the sake of simplicity, a single subproblem of the form (35) with static penalty and
+tolerance parameters is solved in Algorithm 4. Nevertheless, Algorithm 4 can be modified into a perhaps
+practically more efficient algorithm by solving a sequence of subproblems of the form (35) with dynamic
+penalty and tolerance parameters instead.
+In order to characterize the approximate solution found by Algorithm 4, we next introduce a termi-
+nology called an ε-KKT solution of problem (29).
+Recall that problem (29) can be viewed as problem (33). In the spirit of classical constrained opti-
+mization, one would naturally be interested in a KKT solution (x, y) of (33) or equivalently (29), namely,
+(x, y) satisfies ˜f(x, y) ≤ minz ˜f(x, z) and moreover (x, y) is a stationary point of the problem
+min
+x′,y′ f(x′, y′) + ρ
+� ˜f(x′, y′) − min
+z′
+˜f(x′, z′)
+�
+(37)
+for some ρ ≥ 0. Yet, due to the sophisticated problem structure, characterizing a stationary point of (37)
+is generally difficult. On another hand, notice that problem (37) is equivalent to the minimax problem
+min
+x′,y′ max
+z′
+f(x′, y′) + ρ( ˜f(x′, y′) − ˜f(x′, z′)),
+whose stationary point (x, y, z) according to Definition 2 satisfies
+0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z); 0),
+0 ∈ ρ∂z ˜f(x, z).
+(38)
+Based on this observation, we are instead interested in a (weak) KKT solution of problem (29) and its
+inexact counterpart that are defined below.
+Definition 3. The pair (x, y) is said to be a KKT solution of problem (29) if there exists (z, ρ) ∈ Rm×R+
+such that (38) and ˜f(x, y) ≤ minz′ ˜f(x, z′) hold. In addition, for any ε > 0, (x, y) is said to be an ε-KKT
+solution of problem (29) if there exists (z, ρ) ∈ Rm × R+ such that
+dist
+�
+0, ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂z ˜f(x, z)
+�
+≤ ε,
+˜f(x, y) − min
+z′
+˜f(x, z′) ≤ ε.
+We are now ready to present a theorem regarding operation complexity of Algorithm 4, measured by
+the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT
+solution of (29), whose proof is deferred to Subsection 5.3.
+Theorem 4 (Complexity of Algorithm 4). Suppose that Assumption 3 holds. Let f ∗, f, ˜f, Dx, Dy,
+˜fhi, ˜flow and flow be defined in (29), (30), (31) and (32), L∇f1 and L∇ ˜
+f1 be given in Assumption 3, ε,
+ρ, x0, y0 and zǫ be given in Algorithm 4, and
+�L = L∇f1 + 2ε−1L∇ ˜
+f1, ˆα = min
+�
+1,
+�
+4ε/(Dy�L)
+�
+,
+(39)
+ˆδ = (2 + ˆα−1)(D2
+x + D2
+y)�L + max
+�
+ε/Dy, ˆα�L/4
+�
+D2
+y,
+�C =
+4 max
+�
+1
+2�L, min
+�
+Dy
+ε ,
+4
+ˆα�L
+�� �
+ˆδ + 2ˆα−1(f ∗ − flow + ε−1( ˜fhi − ˜flow) + εDy/4 + �L(D2
+x + D2
+y))
+�
+�
+(3�L + ε/(2Dy))2/ min{�L, ε/(2Dy)} + 3�L + ε/(2Dy)
+�−2
+ε3
+,
+�K =
+�
+16(1 + f(x0, y0) − flow + εDy/4)�Lε−2 + 32(1 + 4D2
+y�L2ε−2)ε − 1
+�
++ ,
+�
+N =
+��
+96
+√
+2(1 + (24�L + 4ε/Dy)�L−1)
+�
++ 2
+�
+max
+�
+2,
+�
+Dy�Lε−1
+�
+× (( �
+K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)).
+9
+
+Then Algorithm 4 outputs an approximate solution (xǫ, yǫ) of (29) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε,
+(40)
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ε
+�
+1 + f(x0, y0) − flow + 2ε3(�L−1 + 4D2
+y�Lε−2) + Dyε/4
+�
+,
+(41)
+after at most �
+N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, respectively.
+Remark 5. One can observe from Theorem 4 that �L = O(ε−1), ˆα = O(ε), ˆδ = O(ε−2), �C = O(ε−11),
+�K = O(ε−3), and �
+N = O(ε−4 log ε−1). Consequently, Algorithm 4 enjoys an operation complexity of
+O(ε−4 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2,
+for finding an O(ε)-KKT solution (xǫ, yǫ) of (29) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε,
+˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z) = O(ε),
+where zǫ is given in Algorithm 4 and ρ = ε−1.
+4
+Constrained bilevel optimization
+In this section, we consider a constrained bilevel optimization problem5
+f ∗ = min
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0},
+(42)
+where f and ˜f satisfy Assumption 3. Recall from Assumption 3 that X = dom f2 and Y = dom ˜f2. We
+now make some additional assumptions for problem (42).
+Assumption 4.
+(i) f and ˜f are Lf- and L ˜
+f-Lipschitz continuous on X × Y, respectively.
+(ii) ˜g : Rn × Rm → Rl is L∇˜g-smooth and L˜g-Lipschitz continuous on X × Y.
+(iii) ˜gi(x, ·) is convex and there exists ˆzx ∈ Y for each x ∈ X such that ˜gi(x, ˆzx) < 0 for all i = 1, 2, . . ., l
+and G := min{−˜gi(x, ˆzx)|x ∈ X, i = 1, . . . , l} > 0.6
+For notational convenience, we define
+˜f ∗(x) := min
+z { ˜f(x, z)|˜g(x, z) ≤ 0},
+(43)
+˜f ∗
+hi := sup{ ˜f ∗(x)|x ∈ X},
+(44)
+˜ghi := max{∥˜g(x, y)∥
+��(x, y) ∈ X × Y},
+(45)
+It then follows from Assumption 4(ii) that
+∥∇˜g(x, y)∥ ≤ L˜g
+∀(x, y) ∈ X × Y.
+(46)
+In addition, by Assumptions 3 and 4 and the compactness of X and Y, one can observe that ˜ghi and G
+are finite. Besides, as will be shown in Lemma 6(ii), ˜f ∗
+hi is finite.
+The goal of this subsection is to propose penalty methods for solving problem (42). To this end, let
+us introduce a penalty function for the lower level optimization problem y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}
+of (42), which is given by
+�Pµ(x, z) = ˜f(x, z) + µ ∥[˜g(x, z)]+∥2
+(47)
+5For convenience, problem (42) is referred to as a constrained bilevel optimization problem since its lower level part has
+at least one explicit constraint.
+6The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for the lower
+level part of (42), that is, there exists ˆzx ∈ Y such that ˜g(x, ˆzx) < 0 for each x ∈ X. Indeed, if G > 0, Assumption 4(iii)
+clearly holds. Otherwise, one can solve the perturbed counterpart of (42) with ˜g(x, z) being replaced by ˜g(x, z) − ǫ for
+some suitable ǫ > 0 instead, which satisfies Assumption 4(iii).
+10
+
+for a penalty parameter µ > 0. Observe that problem (42) can be approximately solved as the uncon-
+strained bilevel optimization problem
+f ∗
+µ = min
+x,y
+�
+f(x, y)|y ∈ Argmin
+z
+�Pµ(x, z)
+�
+.
+(48)
+Further, by the study in Section 3, problem (48) can be approximately solved as the penalty problem
+min
+x,y f(x, y) + ρ
+�
+�Pµ(x, y) − min
+z
+�Pµ(x, z)
+�
+(49)
+for some suitable ρ > 0. One can also observe that problem (49) is equivalent to the minimax problem
+min
+x,y max
+z
+Pρ,µ(x, y, z),
+where
+Pρ,µ(x, y, z) := f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z)).
+(50)
+In view of (47), (50) and Assumption 3(i), Pρ,µ can be rewritten as
+Pρ,µ(x, y, z) =
+�
+f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2 �
++
+�
+f2(x) + ρ ˜f2(y) − ρ ˜f2(z)
+�
+.
+(51)
+By this and Assumptions 3 and 4, one can observe that Pρ,µ enjoys the following nice properties.
+• Pρ,µ is the sum of smooth function f1(x, y)+ρ ˜f1(x, y)+ρµ ∥[˜g(x, y)]+∥2−ρ ˜f1(x, z)−ρµ ∥[˜g(x, z)]+∥2
+with Lipschitz continuous gradient and possibly nonsmooth function f2(x) + ρ ˜f2(y) − ρ ˜f2(z) with
+exactly computable proximal operator;
+• Pρ,µ is nonconvex in (x, y) but concave in z.
+Due to the nice structure of Pρ,µ, an approximate stationary point of the minimax problem (50) can be
+found by Algorithm 2 proposed in Subsection 2.2.
+Based on the above observations, we are now ready to propose penalty methods for the constrained
+bilevel optimization problem (42) by solving a sequence of or a single minimax problem of the form (50).
+In particular, we first propose an ideal penalty method for (42) by solving a sequence of minimax problems
+(see Algorithm 5). Then we propose a practical penalty method for (42) by finding an approximate
+stationary point of a single minimax problem (see Algorithm 6).
+Algorithm 5 An ideal penalty method for problem (42)
+Input: positive sequences {ρk}, {µk} and {ǫk} with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).
+1: for k = 0, 1, 2, . . . do
+2:
+Find an ǫk-optimal solution (xk, yk, zk) of problem (50) with (ρ, µ) = (ρk, µk).
+3: end for
+To study convergence of Algorithm 5, we make the following error bound assumption on the solution
+set of the lower level optimization problem of (42). This type of error bounds has been considered in the
+context of set-value mappings in the literature (e.g., see [14]).
+Assumption 5. There exists a non-decreasing function ω : R+ → R+ with limθ↓0 ω(θ) = 0 and ¯θ > 0
+such that dist(z, Sθ(x)) ≤ ω(θ) for all x ∈ X, z ∈ S0(x) and θ ∈ [0, ¯θ], where
+Sθ(x) := Argmin
+z
+{ ˜f(x, z) : ∥[˜g(x, z)]+∥ ≤ θ}.
+We are now ready to state a convergence result of Algorithm 5, whose proof is deferred to Section
+5.4.
+Theorem 5 (Convergence of Algorithm 5). Suppose that Assumptions 3-5 hold and that {(xk, yk, zk)}
+is generated by Algorithm 5. Then any accumulation point of {(xk, yk)} is an optimal solution of problem
+(42).
+Notice that (50) is a nonconvex-concave minimax problem. It is generally hard to find an ǫ-optimal
+solution of (50) for an arbitrary ǫ > 0. As a result, Algorithm 5 is generally not implementable. We next
+propose a practical penalty method for problem (42) by finding an approximate stationary point of (50)
+with a suitable choice of ρ and µ.
+11
+
+Algorithm 6 A practical penalty method for problem (42)
+Input: ε ∈ (0, 1/4], ρ = ε−1, µ = ε−2, (x0, y0) ∈ X × Y with �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.
+1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε5/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ρL∇ ˜f1 +
+4ρµ(˜ghiL∇˜g +L2
+˜g) to find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (50) with ρ = ε−1 and µ = ε−2.
+2: Output: (xǫ, yǫ).
+Remark 6. (i) The initial point (x0, y0) of Algorithm 6 can be found by the similar procedure as described
+in Remark 4 with ˜f being replaced by �Pµ; (ii) As seen from Theorem 2, an ǫ-stationary point of (50)
+can be successfully found in step 1 of Algorithm 6 by applying Algorithm 2 to (50); (iii) For the sake of
+simplicity, a single subproblem of the form (50) with static penalty and tolerance parameters is solved in
+Algorithm 6. Nevertheless, Algorithm 6 can be modified into a perhaps practically more efficient algorithm
+by solving a sequence of subproblems of the form (50) with dynamic penalty and tolerance parameters
+instead.
+In order to characterize the approximate solution found by Algorithm 6, we next introduce a termi-
+nology called an ε-KKT solution of problem (42).
+By the definition of ˜f ∗ in (43), problem (42) can be viewed as
+min
+x,y {f(x, y)| ˜f(x, y) ≤ ˜f ∗(x), ˜g(x, y) ≤ 0}.
+(52)
+Its associated Lagrangian function is given by
+L(x, y, ρ, λ) = f(x, y) + ρ( ˜f(x, y) − ˜f ∗(x)) + ⟨λ, ˜g(x, y)⟩.
+(53)
+In the spirit of classical constrained optimization, one would naturally be interested in a KKT solution
+(x, y) of (52) or equivalently (42), namely, (x, y) satisfies
+˜f(x, y) ≤ ˜f ∗(x),
+˜g(x, y) ≤ 0,
+ρ( ˜f(x, y) − ˜f ∗(x)) = 0,
+⟨λ, ˜g(x, y)⟩ = 0,
+(54)
+and moreover (x, y) is a stationary point of the problem
+min
+x′,y′ L(x′, y′, ρ, λ)
+(55)
+for some ρ ≥ 0 and λ ∈ Rl
++. Yet, due to the sophisticated problem structure, characterizing a stationary
+point of (55) is generally difficult. On another hand, notice from Lemma 6 and (53) that problem (55)
+is equivalent to the minimax problem
+min
+x′,y′,˜λ′ max
+z′
+�
+f(x′, y′) + ρ
+� ˜f(x′, y′) − ˜f(x′, z′) − ⟨˜λ′, ˜g(x′, z′)⟩
+�
++ ⟨λ, ˜g(x′, y′)⟩ + IRl
++(˜λ′)
+�
+,
+whose stationary point (x, y, ˜λ, z) according to Definition 2 satisfies
+0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ; 0) + ∇˜g(x, y)λ,
+(56)
+0 ∈ ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ),
+(57)
+˜λ ∈ Rl
++,
+˜g(x, z) ≤ 0,
+⟨˜λ, ˜g(x, z)⟩ = 0.
+(58)
+Based on this observation and also the fact that (54) is equivalent to
+˜f(x, y) = ˜f ∗(x),
+˜g(x, y) ≤ 0,
+⟨λ, ˜g(x, y)⟩ = 0,
+(59)
+we are instead interested in a (weak) KKT solution of problem (42) and its inexact counterpart that are
+defined below.
+Definition 4. The pair (x, y) is said to be a KKT solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈
+Rm × R+ × Rl
++ × Rl
++ such that (56)-(59) hold. In addition, for any ε > 0, (x, y) is said to be an ε-KKT
+solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈ Rm × R+ × Rl
++ × Rl
++ such that
+dist
+�
+0, ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ; 0) + ∇˜g(x, y)λ
+�
+≤ ε,
+dist
+�
+0, ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ)
+�
+≤ ε,
+∥[˜g(x, z)]+∥ ≤ ε,
+|⟨˜λ, ˜g(x, z)⟩| ≤ ε,
+| ˜f(x, y) − ˜f ∗(x)| ≤ ε,
+∥[˜g(x, y)]+∥ ≤ ε,
+|⟨λ, ˜g(x, y)⟩| ≤ ε,
+where ˜f ∗ is defined in (43).
+12
+
+We are now ready to present an operation complexity of Algorithm 6, measured by the amount of
+evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution of
+(42), whose proof is deferred to Subsection 5.4.
+Theorem 6 (Complexity of Algorithm 6). Suppose that Assumptions 3 and 4 hold. Let f ∗, f, ˜f,
+˜g, Dx, Dy, ˜fhi, ˜flow, flow, ˜f ∗, ˜f ∗
+hi, and ˜ghi be defined in (29), (30), (31), (32), (43), (44) and (45),
+L∇f1, L∇ ˜
+f1, L ˜
+f, L∇˜g, L˜g and G be given in Assumptions 3 and 4, ε, ρ, µ, x0, y0 and zǫ be given in
+Algorithm 6, and
+˜λ = 2ε−1[˜g(xǫ, zǫ)]+,
+ˆλ = 2ε−3[˜g(xǫ, yǫ)]+,
+(60)
+�L = L∇f1 + 2ε−1L∇ ˜
+f1 + 4ε−3(˜ghiL∇˜g + L2
+˜g),
+(61)
+˜α = min
+�
+1,
+�
+4ε/(Dy�L)
+�
+, ˜δ = (2 + ˜α−1)(D2
+x + D2
+y)�L + max
+�
+ε/Dy, ˜α�L/4
+�
+D2
+y,
+�C =
+4 max{1/(2�L), min{Dyε−1, 4/(˜α�L)}}
+[(3�L + ε/(2Dy))2/ min{�L, ε/(2Dy)} + 3�L + ε/(2Dy)]−2ε5
+×
+�
+˜δ + 2˜α−1[f ∗ − flow + 2ε−1( ˜fhi − ˜flow) + ε−3˜g2
+hi + εDy/4 + �L(D2
+x + D2
+y)]
+�
+,
+�K =
+�
+32(1 + f(x0, y0) − flow + εDy/4)�Lε−2 + 32ε3 �
+1 + 4D2
+y�L2ε−2�
+− 1
+�
++ ,
+�
+N =
+��
+96
+√
+2
+�
+1 + (24�L + 4ε/Dy)�L−1��
++ 2
+�
+max
+�
+2,
+�
+Dy�Lε−1
+�
+× [( �K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)].
+Then Algorithm 6 outputs an approximate solution (xǫ, yǫ) of (42) satisfying
+dist
+�
+∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ε,
+(62)
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ε,
+(63)
+∥[˜g(xǫ, zǫ)]+∥ ≤ ε2G−1Dy(ε2 + L ˜
+f)/2,
+(64)
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ ε2G−2D2
+y(ρ−1ǫ + L ˜
+f)2/2,
+(65)
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max
+�
+ε
+�
+1 + f(x0, y0) − flow + 2ε5(�L−1 + 4D2
+y�Lε−2) + Dyε/4
+�
+,
+ε2G−2D2
+yL ˜
+f(ε2 + εLf + L ˜
+f)/2
+�
+,
+(66)
+∥[˜g(xǫ, yǫ)]+∥ ≤ ε2G−1Dy(ε2 + εLf + L ˜
+f)/2,
+(67)
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ εG−2D2
+y(ε2 + εLf + L ˜
+f)2/2,
+(68)
+after at most �
+N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, respectively.
+Remark 7. One can observe from Theorem 6 that �L = O(ε−3), ˜α = O(ε2), ˜δ = O(ε−5), �C = O(ε−23),
+�K = O(ε−5), and �
+N = O(ε−7 log ε−1). Consequently, Algorithm 6 enjoys an operation complexity of
+O(ε−7 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2
+and ˜f2, for finding an O(ε)-KKT solution (xǫ, yǫ) of (42) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ε,
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ε,
+∥[˜g(xǫ, zǫ)]+∥ = O(ε2),
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| = O(ε2),
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| = O(ε),
+∥[˜g(xǫ, yǫ)]+∥ = O(ε2),
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| = O(ε),
+where ˜f ∗ is defined in (43), ˆλ, ˜λ ∈ Rl
++ are defined in (60), zǫ is given in Algorithm 6 and ρ = ε−1.
+5
+Proof of the main results
+In this section we provide a proof of our main results presented in Sections 2, 3 and 4, which are
+particularly Theorems 1-6.
+13
+
+5.1
+Proof of the main results in Subsection 2.1
+In this subsection we prove Theorem 1. Before proceeding, we establish a lemma below.
+Lemma 1. Suppose that Assumptions 1 and 2 hold. Let ¯H∗, ¯Hlow, ϑ0 and ¯δ be defined in (8), (10),
+(14) and (16), and ¯α be given in Algorithm 1. Then we have
+ϑ0 ≤ ¯δ + 2¯α−1 � ¯H∗ − ¯Hlow
+�
+.
+(69)
+Proof. By (8), (10), (11) and (12), one has
+G(¯z0, ¯y0)
+(12)
+=
+sup
+x
+�
+⟨x, ¯z0⟩ − p(x) − ˆh(x, ¯y0) + q(¯y0)
+�
+(11)
+=
+max
+x∈dom p
+�
+⟨x, ¯z0⟩ − p(x) − ¯h(x, ¯y0) + σx
+2 ∥x∥2 − σy
+2 ∥¯y0∥2 + q(¯y0)
+�
+(8)(10)
+≤
+max
+x∈dom p
+�
+⟨x, ¯z0⟩ + σx
+2 ∥x∥2�
+− σy
+2 ∥¯y0∥2 − ¯Hlow
+=
+max
+x∈dom p
+σx
+2 ∥x + σ−1
+x ¯z0∥2 − σ−1
+x
+2 ∥¯z0∥2 − σy
+2 ∥¯y0∥2 − ¯Hlow
+≤ σxD2
+p
+2
+− σ−1
+x
+2 ∥¯z0∥2 − σy
+2 ∥¯y0∥2 − ¯Hlow,
+(70)
+where the last inequality follows from (9) and the fact that z0 ∈ −σxdom p.
+Recall that (x∗, y∗) is the optimal solution of (8) and z∗ = −σxx∗. It follows from (8), (11) and (12)
+that
+G(z∗, y∗)
+(12)
+=
+sup
+x
+�
+⟨x, z∗⟩ − p(x) − ˆh(x, y∗) + q(y∗)
+�
+≥ ⟨x∗, z∗⟩ − p(x∗) − ˆh(x∗, y∗) + q(y∗)
+(11)
+= ⟨x∗, z∗⟩ + σx
+2 ∥x∗∥2 − σy
+2 ∥y∗∥2 − p(x∗) − ¯h(x∗, y∗) + q(y∗)
+=
+− σ−1
+x
+2 ∥z∗∥2 − σy
+2 ∥y∗∥2 − ¯H∗,
+where the last equality follows from (8), the definition of (x∗, y∗), and z∗ = −σxx∗. This together with
+(13) and (70) implies that
+P(¯z0, ¯y0) − P(z∗, y∗) = σ−1
+x
+2 ∥¯z0∥2 + σy
+2 ∥¯y0∥2 + G(¯z0, ¯y0) − σ−1
+x
+2 ∥z∗∥2 − σy
+2 ∥y∗∥2 − G(z∗, y∗)
+≤ σxD2
+p/2 − ¯Hlow + ¯H∗.
+Notice from Algorithm 1 that z0 = z0
+f = ¯z0 ∈ −σxdom p and y0 = y0
+f = ¯y0 ∈ dom q.
+By these,
+z∗ = −σxx∗, (9), (14), and the above inequality, one has
+ϑ0
+(14)
+= η−1
+z ∥¯z0 − z∗∥2 + η−1
+y ∥¯y0 − y∗∥2 + 2¯α−1(P(¯z0, ¯y0) − P(z∗, y∗))
+≤ η−1
+z σ2
+xD2
+p + η−1
+y D2
+q + 2¯α−1 �
+σxD2
+p/2 − ¯Hlow + ¯H∗�
+= η−1
+z σ2
+xD2
+p + ¯α−1σxD2
+p + η−1
+y D2
+q + 2¯α−1 � ¯H∗ − ¯Hlow
+�
+.
+Hence, the conclusion follows from this, (16), ηz = σx/2 and ηy = min {1/(2σy), 4/(¯ασx)}.
+We are now ready to prove Theorem 1.
+Proof of Theorem 1. Suppose for contradiction that Algorithm 1 runs for more than ¯K outer itera-
+tions, where ¯K is given in (17). By this and Algorithm 1, one can assert that (15) does not hold for
+k = ¯K − 1. On the other hand, by (17) and [29, Theorem 3], one has
+∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥ ≤ (ˆζ−1 + L∇¯h)−1τ/2,
+(71)
+where (x∗, y∗) is the optimal solution of problem (8) and ˆζ is an input of Algorithm 1. Notice from
+Algorithm 1 that (ˆx ¯
+K, ˆy ¯
+K) results from the forward-backward splitting (FBS) step applied to the strongly
+monotone inclusion problem 0 ∈ (∇x¯h(x, y), −∇y¯h(x, y)) + (∂p(x), ∂q(y)) at the point (x ¯
+K, y ¯
+K). It then
+14
+
+follows from this, ˆζ = min{σx, σy}/L2
+∇¯h (see Algorithm 1), and the contraction property of FBS [5,
+Corollary 2.5] that ∥(ˆx ¯
+K, ˆy ¯
+K) − (x∗, y∗)∥ ≤ ∥(x ¯
+K, y ¯
+K) − (x∗, y∗)∥. Using this and (71), we have
+∥ˆζ−1(x
+¯
+K − ˆx
+¯
+K, ˆy
+¯
+K − y
+¯
+K) − (∇¯h(x
+¯
+K, y
+¯
+K) − ∇¯h(ˆx
+¯
+K, ˆy
+¯
+K))∥
+≤ ˆζ−1∥(x
+¯
+K, y
+¯
+K) − (ˆx
+¯
+K, ˆy
+¯
+K)∥ + ∥∇¯h(x
+¯
+K, y
+¯
+K) − ∇¯h(ˆx
+¯
+K, ˆy
+¯
+K)∥
+≤ (ˆζ−1 + L∇¯h)∥(x
+¯
+K, y
+¯
+K) − (ˆx
+¯
+K, ˆy
+¯
+K)∥
+≤ (ˆζ−1 + L∇¯h)(∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥ + ∥(ˆx
+¯
+K, ˆy
+¯
+K) − (x∗, y∗)∥)
+≤ 2(ˆζ−1 + L∇¯h)∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥
+(71)
+≤ τ,
+where the second inequality uses the fact that ¯h is L∇¯h-smooth on dom p × dom q. It follows that (15)
+holds for k = ¯K − 1, which contradicts the above assertion. Hence, Algorithm 1 must terminate in at
+most ¯K outer iterations.
+We next show that the output of Algorithm 1 is a τ-stationary point of (8). To this end, suppose
+that Algorithm 1 terminates at some iteration k at which (15) is satisfied. Then by (4) and the definition
+of ˆxk+1 and ˆyk+1 (see steps 23 and 24 of Algorithm 1), one has
+0 ∈ ˆζ∂p(ˆxk+1) + ˆxk+1 − xk+1 + ˆζ∇x¯h(xk+1, yk+1),
+0 ∈ ˆζ∂q(ˆyk+1) + ˆyk+1 − yk+1 − ˆζ∇y¯h(xk+1, yk+1),
+which yield
+ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂p(ˆxk+1), ˆζ−1(yk+1 − ˆyk+1) + ∇y¯h(xk+1, yk+1) ∈ ∂q(ˆyk+1).
+These together with the definition of ¯H in (8) imply that
+∇x¯h(ˆxk+1, ˆyk+1) + ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂x ¯H(ˆxk+1, ˆyk+1),
+∇y¯h(ˆxk+1, ˆyk+1) − ˆζ−1(yk+1 − ˆyk+1) − ∇y¯h(xk+1, yk+1) ∈ ∂y ¯H(ˆxk+1, ˆyk+1).
+Using these and (15), we obtain
+dist(0, ∂x ¯H(ˆxk+1, ˆyk+1))2 + dist(0, ∂y ¯H(ˆxk+1, ˆyk+1))2
+≤ ∥ˆζ−1(xk+1 − ˆxk+1) + ∇x¯h(ˆxk+1, ˆyk+1) − ∇x¯h(xk+1, yk+1)∥2
++ ∥ˆζ−1(ˆyk+1 − yk+1) + ∇y¯h(ˆxk+1, ˆyk+1) − ∇y¯h(xk+1, yk+1)∥2
+= ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥2 (15)
+≤ τ2,
+which implies that dist(0, ∂x ¯H(ˆxk+1, ˆyk+1)) ≤ τ and dist(0, ∂y ¯H(ˆxk+1, ˆyk+1)) ≤ τ. It then follows from
+these and Definition 2 that the output (ˆxk+1, ˆyk+1) of Algorithm 1 is a τ-stationary point of (8).
+Finally, we show that the total number of evaluations of ∇¯h and proximal operator of p and q
+performed in Algorithm 1 is no more than ¯N, respectively.
+Indeed, notice from Algorithm 1 that
+¯α = min
+�
+1,
+�
+8σy/σx
+�
+, which implies that 2/¯α = max{2,
+�
+σx/(2σy)} and ¯α ≤
+�
+8σy/σx. By these,
+one has
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+≤ max
+�
+2,
+� σx
+2σy
+,
+�
+8σy
+σx
+σx
+4σy
+�
+= max
+�
+2,
+� σx
+2σy
+�
+.
+(72)
+In addition, by [29, Lemma 4], the number of inner iterations performed in each outer iteration of
+Algorithm 1 is at most
+T =
+�
+48
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
+− 1.
+Then one can observe that the number of evaluations of ∇¯h and proximal operator of p and q performed
+15
+
+in Algorithm 1 is at most
+(2T + 3) ¯K ≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+� �
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+(72)
+≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+� �
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+�
+×
+�
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max{1/(2σx), min {1/(2σy), 4/(¯ασx)}} ϑ0
+(L2
+∇¯h/ min{σx, σy} + L∇¯h)−2τ 2
+�
++
+(69)(18)
+≤
+¯N,
+where the second last inequality follows from the definition of ηy, ηz and ˆζ in Algorithm 1. Hence, the
+conclusion holds as desired.
+5.2
+Proof of the main results in Subsection 2.2
+In this subsection we prove Theorem 2.
+Before proceeding, let {(xk, yk)}k∈K denote all the iterates
+generated by Algorithm 2, where K is a subset of consecutive nonnegative integers starting from 0. Also,
+we define K − 1 = {k − 1 : k ∈ K}. We first establish two lemmas and then use them to prove Theorem
+2 subsequently.
+Lemma 2. Suppose that Assumption 1 holds. Let {(xk, yk)}k∈K be generated by Algorithm 2, H∗, Dp,
+Dq, Hlow, α, δ be defined in (6), (9), (23), (24) and (25), L∇h be given in Assumption 1, ǫ, ǫk be given
+in Algorithm 2, and
+Nk =
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+×
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+× log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+�
++
+.
+(73)
+Then for all 0 ≤ k ∈ K−1, (xk+1, yk+1) is an ǫk-stationary point of (20). Moreover, the total number of
+evaluations of ∇h and proximal operator of p and q performed at iteration k of Algorithm 2 for generating
+(xk+1, yk+1) is no more than Nk, respectively.
+Proof. Let (x∗, y∗) be an optimal solution of (6). Recall that H, Hk and hk are given in (6), (20) and
+(21), respectively. Then we have
+Hk,∗ := min
+x max
+y
+Hk(x, y) = min
+x max
+y
+�
+H(x, y) −
+ǫ
+4Dq
+∥y − ˆy0∥2 + L∇h∥x − xk∥2
+�
+≤ max
+y {H(x∗, y) + L∇h∥x∗ − xk∥2}
+(6)(9)
+≤
+H∗ + L∇hD2
+p.
+(74)
+Moreover, by (9) and (23), one has
+Hk,low :=
+min
+(x,y)∈dom p×dom q Hk(x, y) =
+min
+(x,y)∈dom p×dom q
+�
+H(x, y) −
+ǫ
+4Dq
+∥y − ˆy0∥2 + L∇h∥x − xk∥2
+�
+(23)
+≥ Hlow − max
+y∈dom q
+ǫ
+4Dq
+∥y − ˆy0∥2 (9)
+≥ Hlow − ǫDq/4.
+(75)
+In addition, by Assumption 1 and the definition of hk in (21), it is not hard to verify that hk(x, y) is
+L∇h-strongly-convex in x, ǫ/(2Dq)-strongly-concave in y, and (3L∇h + ǫ/(2Dq))-smooth on its domain.
+Also, recall from Remark 2 that (xk+1, yk+1) results from applying Algorithm 1 to problem (20). The
+conclusion of this lemma then follows by using (74) and (75) and applying Theorem 1 to (20) with τ = ǫk,
+σx = L∇h, σy = ǫ/(2Dq), L∇¯h = 3L∇h + ǫ/(2Dq), ¯α = α, ¯δ = δ, ¯Hlow = Hk,low, and ¯H∗ = Hk,∗.
+16
+
+Lemma 3. Suppose that Assumption 1 holds. Let {xk}k∈K be generated by Algorithm 2, H, H∗ and
+Dq be defined in (6) and (9), L∇h be given in Assumption 1, and ǫ, ǫ0 and ˆx0 be given in Algorithm 2.
+Then for all 0 ≤ K ∈ K − 1, we have
+min
+0≤k≤K ∥xk+1 − xk∥ ≤ maxy H(ˆx0, y) − H∗ + ǫDq/4
+L∇h(K + 1)
++ 2ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)
+L2
+∇h(K + 1)
+,
+(76)
+max
+y
+H(xK+1, y) ≤ max
+y
+H(ˆx0, y) + ǫDq/4 + 2ǫ2
+0
+�
+L−1
+∇h + 4D2
+qL∇hǫ−2�
+.
+(77)
+Proof. For convenience of the proof, let
+H∗
+ǫ (x) = max
+y
+�
+H(x, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+,
+(78)
+H∗
+k(x) = max
+y
+Hk(x, y),
+yk+1
+∗
+= arg max
+y
+Hk(xk+1, y).
+(79)
+One can observe from these, (20) and (21) that
+H∗
+k(x) = H∗
+ǫ (x) + L∇h∥x − xk∥2.
+(80)
+By this and Assumption 1, one can also see that H∗
+k is L∇h-strongly convex on dom p. In addition,
+recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of problem (20) for all 0 ≤ k ∈ K − 1.
+It then follows from Definition 2 that there exist some u ∈ ∂xHk(xk+1, yk+1) and v ∈ ∂yHk(xk+1, yk+1)
+with ∥u∥ ≤ ǫk and ∥v∥ ≤ ǫk.
+Also, by (79), one has 0 ∈ ∂yHk(xk+1, yk+1
+∗
+), which together with
+v ∈ ∂yHk(xk+1, yk+1) and ǫ/(2Dq)-strong concavity of Hk(xk+1, ·), implies that ⟨−v, yk+1 − yk+1
+∗
+⟩ ≥
+ǫ∥yk+1 − yk+1
+∗
+∥2/(2Dq). This and ∥v∥ ≤ ǫk yield
+∥yk+1 − yk+1
+∗
+∥ ≤ 2ǫkDq/ǫ.
+(81)
+In addition, by u ∈ ∂xHk(xk+1, yk+1), (20) and (21), one has
+u ∈ ∇xh(xk+1, yk+1) + ∂p(xk+1) + 2L∇h(xk+1 − xk).
+(82)
+Also, observe from (20), (21) and (79) that
+∂H∗
+k(xk+1) = ∇xh(xk+1, yk+1
+∗
+) + ∂p(xk+1) + 2L∇h(xk+1 − xk),
+which together with (82) yields
+u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1) ∈ ∂H∗
+k(xk+1).
+By this and L∇h-strong convexity of H∗
+k, one has
+H∗
+k(xk) ≥ H∗
+k(xk+1) + ⟨u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + L∇h∥xk − xk+1∥2/2. (83)
+Using this, (80), (81), (83), ∥u∥ ≤ ǫk, and the Lipschitz continuity of ∇h, we obtain
+H∗
+ǫ (xk) − H∗
+ǫ (xk+1)
+(80)
+= H∗
+k(xk) − H∗
+k(xk+1) + L∇h∥xk − xk+1∥2
+(83)
+≥ ⟨u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + 3L∇h∥xk − xk+1∥2/2
+≥
+�
+− ∥u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥∥xk − xk+1∥ + L∇h∥xk − xk+1∥2/2
+�
++ L∇h∥xk − xk+1∥2
+≥ −(2L∇h)−1∥u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥2 + L∇h∥xk − xk+1∥2
+≥ −L−1
+∇h∥u∥2 − L−1
+∇h∥∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥2 + L∇h∥xk − xk+1∥2
+≥ −L−1
+∇hǫ2
+k − L∇h∥yk+1 − yk+1
+∗
+∥2 + L∇h∥xk − xk+1∥2
+(81)
+≥ −(L−1
+∇h + 4D2
+qL∇hǫ−2)ǫ2
+k + L∇h∥xk − xk+1∥2,
+where the second and fourth inequalities follow from Cauchy-Schwartz inequality, and the third inequal-
+ity is due to Young’s inequality, and the fifth inequality follows from L∇h-Lipschitz continuity of ∇h.
+Summing up the above inequality for k = 0, 1, . . ., K yields
+L∇h
+K
+�
+k=0
+∥xk − xk+1∥2 ≤ H∗
+ǫ (x0) − H∗
+ǫ (xK+1) + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k.
+(84)
+17
+
+In addition, it follows from (6), (9) and (78) that
+H∗
+ǫ (xK+1) = max
+y
+�
+H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+≥ min
+x max
+y
+H(x, y) − ǫDq/4 = H∗ − ǫDq/4,
+H∗
+ǫ (x0) = max
+y
+�
+H(x0, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+≤ max
+y
+H(x0, y).
+(85)
+These together with (84) yield
+L∇h(K + 1) min
+0≤k≤K ∥xk+1 − xk∥2 ≤ L∇h
+K
+�
+k=0
+∥xk − xk+1∥2
+≤ max
+y
+H(x0, y) − H∗ + ǫDq/4 + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k,
+which together with x0 = ˆx0, ǫk = ǫ0(k + 1)−1 and �K
+k=0(k + 1)−2 < 2 implies that (76) holds.
+Finally, we show that (77) holds. Indeed, it follows from (9), (78), (84), (85), ǫk = ǫ0(k + 1)−1, and
+�K
+k=0(k + 1)−2 < 2 that
+max
+y
+H(xK+1, y)
+(9)
+≤
+max
+y
+�
+H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
++ ǫDq/4
+(78)
+= H∗
+ǫ (xK+1) + ǫDq/4
+(84)
+≤ H∗
+ǫ (x0) + ǫDq/4 + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k
+(85)
+≤
+max
+y
+H(x0, y) + ǫDq/4 + 2ǫ2
+0(L−1
+∇h + 4D2
+qL∇hǫ−2).
+It then follows from this and x0 = ˆx0 that (77) holds.
+We are now ready to prove Theorem 2.
+Proof of Theorem 2. Suppose for contradiction that Algorithm 2 runs for more than K + 1 outer
+iterations, where K is given in (26). By this and Algorithm 2, one can then assert that (22) does not
+hold for all 0 ≤ k ≤ K. On the other hand, by (26) and (76), one has
+min
+0≤k≤K ∥xk+1 − xk∥2
+(76)
+≤
+maxy H(ˆx0, y) − H∗ + ǫDq/4
+L∇h(K + 1)
++ 2ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)
+L2
+∇h(K + 1)
+(26)
+≤
+ǫ2
+16L2
+∇h
+,
+which implies that there exists some 0 ≤ k ≤ K such that ∥xk+1 − xk∥ ≤ ǫ/(4L∇h), and thus (22) holds
+for such k, which contradicts the above assertion. Hence, Algorithm 2 must terminate in at most K + 1
+outer iterations.
+Suppose that Algorithm 2 terminates at some iteration 0 ≤ k ≤ K, namely, (22) holds for such k. We
+next show that its output (xǫ, yǫ) = (xk+1, yk+1) is an ǫ-stationary point of (6) and moreover it satisfies
+(28). Indeed, recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of (20), namely, it satisfies
+dist(0, ∂xHk(xk+1, yk+1)) ≤ ǫk and dist(0, ∂yHk(xk+1, yk+1)) ≤ ǫk. By these, (6), (20) and (21), there
+exists (u, v) such that
+u ∈ ∂xH(xk+1, yk+1) + 2L∇h(xk+1 − xk),
+∥u∥ ≤ ǫk,
+v ∈ ∂yH(xk+1, yk+1) − ǫ(yk+1 − ˆy0)/(2Dq),
+∥v∥ ≤ ǫk.
+It then follows that u−2L∇h(xk+1−xk) ∈ ∂xH(xk+1, yk+1) and v+ǫ(yk+1−ˆy0)/(2Dq) ∈ ∂yH(xk+1, yk+1).
+These together with (9), (22), and ǫk ≤ ǫ0 ≤ ǫ/2 (see Algorithm 2) imply that
+dist
+�
+0, ∂xH(xk+1, yk+1)
+�
+≤ ∥u − 2L∇h(xk+1 − xk)∥ ≤ ∥u∥ + 2L∇h∥xk+1 − xk∥
+(22)
+≤ ǫk + ǫ/2 ≤ ǫ,
+dist
+�
+0, ∂yH(xk+1, yk+1)
+�
+≤ ∥v + ǫ(yk+1 − ˆy0)/(2Dq)∥ ≤ ∥v∥ + ǫ∥yk+1 − ˆy0∥/(2Dq)
+(9)
+≤ ǫk + ǫ/2 ≤ ǫ.
+Hence, the output (xk+1, yk+1) of Algorithm 2 is an ǫ-stationary point of (6). In addition, (28) holds
+due to Lemma 3.
+18
+
+Recall from Lemma 2 that the number of evaluations of ∇h and proximal operator of p and q
+performed at iteration k of Algorithm 2 is at most Nk, respectively, where Nk is defined in (73). Also,
+one can observe from the above proof and the definition of K that |K| ≤ K + 2. It then follows that the
+total number of evaluations of ∇h and proximal operator of p and q in Algorithm 2 is respectively no
+more than �|K|−2
+k=0
+Nk. Consequently, to complete the rest of the proof of Theorem 2, it suffices to show
+that �|K|−2
+k=0
+Nk ≤ N, where N is given in (27). Indeed, by (27), (73) and |K| ≤ K + 2, one has
+|K|−2
+�
+k=0
+Nk
+(73)
+≤
+K
+�
+k=0
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+×
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+× log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+�
++
+≤
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+×
+K
+�
+k=0
+
+
+
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+
+
++
++ 1
+
+
+≤
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+×
+�
+(K + 1)
+�
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+0
+�
++
++ K + 1 + 2
+K
+�
+k=0
+log(k + 1)
+�
+(27)
+≤ N,
+where the last inequality is due to (27) and �K
+k=0 log(k + 1) ≤ K log(K + 1). This completes the proof
+of Theorem 2.
+5.3
+Proof of the main results in Section 3
+In this subsection we prove Theorems 3 and 4. We first establish a lemma below, which will be used to
+prove Theorem 3 subsequently.
+Lemma 4. Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (35) for
+some ǫ > 0. Let f, ˜f, f ∗, flow and ρ be given in (29), (32) and (35), respectively. Then we have
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1(f ∗ − flow + 2ǫ),
+f(xǫ, yǫ) ≤ f ∗ + 2ǫ.
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-optimal solution of (35), it follows from Definition 1 that
+max
+z
+Pρ(xǫ, yǫ, z) ≤ Pρ(xǫ, yǫ, zǫ) + ǫ,
+Pρ(xǫ, yǫ, zǫ) ≤ min
+x,y max
+z
+Pρ(x, y, z) + ǫ.
+Summing up these inequalities yields
+max
+z
+Pρ(xǫ, yǫ, z) ≤ min
+x,y max
+z
+Pρ(x, y, z) + 2ǫ.
+(86)
+Let (x∗, y∗) be an optimal solution of (29).
+It then follows that f(x∗, y∗) = f ∗ and ˜f(x∗, y∗) =
+minz ˜f(x∗, z). By these and the definition of Pρ in (35), one has
+max
+z
+Pρ(x∗, y∗, z) = f(x∗, y∗) + ρ( ˜f(x∗, y∗) − min
+z
+˜f(x∗, z)) = f(x∗, y∗) = f ∗,
+which implies that
+min
+x,y max
+z
+Pρ(x, y, z) ≤ max
+z
+Pρ(x∗, y∗, z) = f ∗.
+(87)
+19
+
+It then follows from (35), (86) and (87) that
+f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z))
+(35)
+= max
+z
+Pρ(xǫ, yǫ, z)
+(86)(87)
+≤
+f ∗ + 2ǫ,
+which together with ˜f(xǫ, yǫ) − minz ˜f(xǫ, z) ≥ 0 implies that
+f(xǫ, yǫ) ≤ f ∗ + 2ǫ,
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1 (f ∗ − f(xǫ, yǫ) + 2ǫ) .
+The conclusion of this lemma directly follows from these and (32).
+We are now ready to prove Theorem 3.
+Proof of Theorem 3. Let {(xk, yk, zk)} be generated by Algorithm 3 with limk→∞(ρk, ǫk) = (∞, 0).
+By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =
+(x∗, y∗). we now show that (x∗, y∗) is an optimal solution of problem (29). Indeed, since (xk, yk, zk)
+is an ǫk-optimal solution of (35) with ρ = ρk, it follows from Lemma 4 with (ρ, ǫ) = (ρk, ǫk) and
+(xǫ, yǫ) = (xk, yk) that
+˜f(xk, yk) ≤ min
+z
+˜f(xk, z) + ρ−1
+k (f ∗ − flow + 2ǫk),
+f(xk, yk) ≤ f ∗ + 2ǫk.
+By the continuity of f and ˜f, limk→∞(xk, yk) = (x∗, y∗), limk→∞(ρk, ǫk) = (∞, 0), and taking limits as
+k → ∞ on both sides of the above relations, we obtain that ˜f(x∗, y∗) ≤ minz ˜f(x∗, z) and f(x∗, y∗) ≤ f ∗,
+which clearly imply that y∗ ∈ Argminz ˜f(x∗, z) and f(x∗, y∗) = f ∗. Hence, (x∗, y∗) is an optimal solution
+of (29) as desired.
+We next prove Theorem 4. Before proceeding, we establish a lemma below, which will be used to
+prove Theorem 4 subsequently.
+Lemma 5. Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35). Let Dy,
+flow, ˜f, ρ, and Pρ be given in (30), (32) and (35), respectively. Then we have
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ǫ,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ǫ,
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1(max
+z
+Pρ(xǫ, yǫ, z) − flow).
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35), it follows from Definition 2 that
+dist
+�
+0, ∂x,yPρ(xǫ, yǫ, zǫ)
+�
+≤ ǫ,
+dist
+�
+0, ∂zPρ(xǫ, yǫ, zǫ)
+�
+≤ ǫ.
+Using these and the definition of Pρ in (35), we have
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ǫ,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε.
+In addition, by (35), we have
+f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z)) = max
+z
+Pρ(xǫ, yǫ, z),
+which along with (32) implies that
+˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z) ≤ ρ−1(max
+z
+Pρ(xǫ, yǫ, z) − flow).
+This completes the proof of this lemma.
+We are now ready to prove Theorem 4.
+Proof of Theorem 4. Observe from (36) that problem (35) can be viewed as
+min
+x,y max
+z
+{Pρ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,
+where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z), p(x, y) = f2(x) + ρ ˜f2(y), and q(z) = ρ ˜f2(z). Hence,
+problem (35) is in the form of (6) with H = Pρ. By Assumption 3 and ρ = ε−1, one can see that h is
+20
+
+�L-smooth on its domain, where �L is given in (39). Also, notice from Algorithm 4 that ǫ0 = ε3/2 ≤ ε/2
+due to ε ∈ (0, 1/4]. Consequently, Algorithm 2 can be suitably applied to problem (35) with ρ = ε−1 for
+finding an ǫ-stationary point (xǫ, yǫ, zǫ) of it.
+In addition, notice from Algorithm 4 that ˜f(x0, y0) ≤ miny ˜f(x0, y)+ε. Using this, (35) and ρ = ε−1,
+we obtain
+max
+z
+Pρ(x0, y0, z) = f(x0, y0) + ρ( ˜f(x0, y0) − min
+z
+˜f(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.
+(88)
+By this and (28) with H = Pρ, ǫ = ε, ǫ0 = ε3/2, ˆx0 = (x0, y0), Dq = Dy, and L∇h = �L, one has
+Pρ(xǫ, yǫ, zǫ) ≤ max
+z
+Pρ(x0, y0, z) + εDy/4 + 2ε3(�L−1 + 4D2
+y�Lε−2)
+(88)
+≤ 1 + f(x0, y0) + εDy/4 + 2ε3(�L−1 + 4D2
+y�Lε−2).
+It then follows from this and Lemma 5 with ǫ = ε and ρ = ε−1 that (xǫ, yǫ, zǫ) satisfies (40) and (41).
+We next show that at most �
+N evaluations of ∇f1, ∇ ˜f1, and proximal operator of f2 and ˜f2 are
+respectively performed in Algorithm 4. Indeed, by (31), (32) and (35), one has
+min
+x,y max
+z
+Pρ(x, y, z)
+(35)
+= min
+x,y {f(x, y) + ρ( ˜f(x, y) − min
+z
+˜f(x, z))} ≥
+min
+(x,y)∈X ×Y f(x, y)
+(32)
+= flow,
+(89)
+min
+(x,y,z)∈X ×Y×Y Pρ(x, y, z)
+(35)
+=
+min
+(x,y,z)∈X ×Y×Y{f(x, y) + ρ( ˜f(x, y) − ˜f(x, z))}
+(31)(32)
+≥
+flow + ρ( ˜flow − ˜fhi).
+(90)
+For convenience of the rest proof, let
+H = Pρ,
+H∗ = min
+x,y max
+z
+Pρ(x, y, z),
+Hlow = min{Pρ(x, y, z)|(x, y, z) ∈ X × Y × Y}.
+(91)
+In view of these, (87), (88), (89), (90), and ρ = ε−1, we obtain that
+max
+z
+H(x0, y0, z)
+(88)
+≤ f(x0, y0) + 1,
+flow
+(89)
+≤ H∗ (87)
+≤ f ∗,
+Hlow
+(90)
+≥ flow + ρ( ˜flow − ˜fhi) = flow + ε−1( ˜flow − ˜fhi).
+Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =
+�
+D2x + D2y, Dq = Dy, ǫ0 = ε3/2, L∇h = �L,
+α = ˆα, δ = ˆδ, and H, H∗, Hlow given in (91), we can conclude that Algorithm 4 performs at most
+�
+N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2 respectively for finding an approximate
+solution (xǫ, yǫ) of problem (29) satisfying (40) and (41).
+5.4
+Proof of the main results in Section 4
+In this subsection we prove Theorems 5 and 6. Before proceeding, we define
+r = G−1Dy(ρ−1ǫ + L ˜
+f),
+B+
+r = {λ ∈ Rl
++ : ∥λ∥ ≤ r},
+(92)
+where Dy is defined in (30), G is given in Assumption 4(iii), and ǫ and ρ are given in Algorithm 6. In
+addition, one can observe from (43) and (47) that
+min
+z
+�Pµ(x, z) ≤ ˜f ∗(x)
+∀x ∈ X,
+(93)
+which will be frequently used later.
+We next establish several technical lemmas that will be used to prove Theorem 5 subsequently.
+Lemma 6. Suppose that Assumptions 3 and 4 hold. Let Dy, L ˜
+f, G, ˜f ∗, ˜f ∗
+hi and B+
+r be given in (30),
+(43), (44), (92) and Assumption 4, respectively. Then the following statements hold.
+(i) ∥λ∗∥ ≤ G−1L ˜
+fDy and λ∗ ∈ B+
+r for all λ∗ ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the set of
+optimal Lagrangian multipliers of problem (43) for any x ∈ X.
+21
+
+(ii) The function ˜f ∗ is Lipschitz continuous on X and ˜f ∗
+hi is finite.
+(iii) It holds that
+˜f ∗(x) = max
+λ
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+∀x ∈ X,
+(94)
+where IRl
++(·) is the indicator function associated with Rl
++.
+Proof. (i) Let x ∈ X and λ∗ ∈ Λ∗(x) be arbitrarily chosen, and let z∗ ∈ Y be such that (z∗, λ∗) is a pair
+of primal-dual optimal solutions of (43). It then follows that
+z∗ ∈ Argmin
+z
+˜f(x, z) + ⟨λ∗, ˜g(x, z)⟩,
+⟨λ∗, ˜g(x, z∗)⟩ = 0,
+˜g(x, z∗) ≤ 0,
+λ∗ ≥ 0.
+The first relation above yields
+˜f(x, z∗) + ⟨λ∗, ˜g(x, z∗)⟩ ≤ ˜f(x, ˆzx) + ⟨λ∗, ˜g(x, ˆzx)⟩,
+where ˆzx is given in Assumption 4(iii). By this and ⟨λ∗, ˜g(x, z∗)⟩ = 0, one has
+⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗),
+which together with λ∗ ≥ 0, (30) and Assumption 4 implies that
+G
+l
+�
+i=1
+λ∗
+i ≤ ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗) ≤ L ˜
+f∥ˆzx − z∗∥ ≤ L ˜
+fDy,
+(95)
+where the first inequality is due to Assumption 4(iii), and the third inequality follows from (30) and L ˜
+f-
+Lipschitz continuity of ˜f (see Assumption 4(i)). By (92), (95) and λ∗ ≥ 0, we have ∥λ∗∥ ≤ �l
+i=1 λ∗
+i ≤
+G−1L ˜
+fDy and λ∗ ∈ B+
+r .
+(ii) Recall from Assumptions 3(i) and 4(iii) that ˜f(x, ·) and ˜gi(x, ·), i = 1, . . . , l, are convex for any
+given x ∈ X. Using this, (43) and the first statement of this lemma, we observe that
+˜f ∗(x) = min
+z
+max
+λ∈B+
+r
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩
+∀x ∈ X.
+(96)
+Notice from Assumption 4 that ˜f and ˜g are Lipschitz continuous on their domain. Then it is not hard to
+observe that max{ ˜f(x, z)+⟨λ, ˜g(x, z)⟩|λ ∈ B+
+r } is a Lipschitz continuous function of (x, z) on its domain.
+By this and (96), one can easily verify that ˜f ∗ is Lipschitz continuous on X. In addition, the finiteness
+of ˜f ∗
+hi follows from (44), the continuity of ˜f ∗, and the compactness of X.
+(iii) One can observe from (43) that for all x ∈ X,
+˜f ∗(x) = min
+z
+max
+λ
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ) ≥ max
+λ
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+where the inequality follows from the weak duality. In addition, it follows from Assumption 3 that the
+domain of ˜f(x, ·) is compact for all x ∈ X. By this, (96) and the strong duality, one has
+˜f ∗(x) = max
+λ∈B+
+r
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+∀x ∈ X,
+which together with the above inequality implies that (94) holds.
+Lemma 7. Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-optimal solution of
+problem (50) for some ǫ > 0. Let flow, f, �Pµ, f ∗
+µ, ρ and µ be given in (32), (42), (47), (48) and (50),
+respectively. Then we have
+�Pµ(xǫ, yǫ) ≤ min
+z
+�Pµ(xǫ, z) + ρ−1(f ∗
+µ − flow + 2ǫ),
+f(xǫ, yǫ) ≤ f ∗
+µ + 2ǫ.
+(97)
+Proof. The proof follows from the same argument as the one for Lemma 4 with f ∗ and ˜f being replaced
+by f ∗
+µ and �Pµ, respectively.
+Lemma 8. Suppose that Assumptions 3-5 hold. Let ˜flow, f ∗, ˜f ∗
+hi, f ∗
+µ be defined in (31), (42), (44) and
+(48), and Lf, ω and ¯θ be given in Assumptions 4 and 5. Suppose that µ ≥ ( ˜f ∗
+hi − ˜flow)/¯θ2. Then we have
+f ∗
+µ ≤ f ∗ + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
+.
+(98)
+22
+
+Proof. Let x ∈ X, y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} and z∗ ∈ Argminz �Pµ(x, z) be arbitrarily chosen.
+One can easily see from (47) and (93) that ˜f(x, z∗) + µ ∥[˜g(x, z∗)]+∥2 ≤ ˜f ∗(x), which together with (31)
+and (44) implies that
+∥[˜g(x, z∗)]+∥2 ≤ µ−1( ˜f ∗
+hi − ˜flow).
+(99)
+Since µ ≥ ( ˜f ∗
+hi− ˜flow)/¯θ2, it follows from (99) that ∥[˜g(x, z∗)]+∥ ≤ ¯θ. By this relation, y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤
+0} and Assumption 5, there exists some ˆz∗ such that
+∥y − ˆz∗∥ ≤ ω(∥[˜g(x, z∗)]+∥),
+ˆz∗ ∈ Argmin
+z
+�
+˜f(x, z)
+�� ∥[˜g(x, z)]+∥ ≤ ∥[˜g(x, z∗)]+∥
+�
+.
+(100)
+In view of (47), z∗ ∈ Argminz �Pµ(x, z) and the second relation in (100), one can observe that ˆz∗ ∈
+Argminz �Pµ(x, z), which along with (48) yields f(x, ˆz∗) ≥ f ∗
+µ. Also, using (100) and Lf-Lipschitz conti-
+nuity of f (see Assumption 4), we have
+f(x, y) − f(x, ˆz∗) ≥ −Lf∥y − ˆz∗∥
+(100)
+≥
+−Lfω(∥[˜g(x, z∗)]+∥).
+Taking minimum over x ∈ X and y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} on both sides of this relation, and
+using (42), (99), f(x, ˆz∗) ≥ f ∗
+µ and the monotonicity of ω, we can conclude that (98) holds.
+Lemma 9. Suppose that Assumptions 3-5 hold. Let ˜flow, flow, f, ˜f, f ∗, ˜f ∗, ˜f ∗
+hi, ρ and µ be given in
+(31), (32), (42), (43), (44) and (50), and Lf, ω and ¯θ be given in Assumptions 4 and 5, respectively.
+Suppose that µ ≥ ( ˜f ∗
+hi − ˜flow)/¯θ2 and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (50) for some ǫ > 0.
+Then we have
+f(xǫ, yǫ) ≤ f ∗ + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ,
+˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1�
+f ∗ − flow + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ
+�
+,
+∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1�
+˜f ∗(xǫ) − ˜flow + ρ−1�
+f ∗ − flow + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ
+��
+.
+Proof. By (47), (93), and the first relation in (97), one has
+˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)
+=
+�Pµ(xǫ, yǫ)
+(93)(97)
+≤
+˜f ∗(xǫ) + ρ−1(f ∗
+µ − flow + 2ǫ).
+It then follows from this and (31) that
+˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1(f ∗
+µ − flow + 2ǫ),
+∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1( ˜f ∗(xǫ) − ˜flow + ρ−1(f ∗
+µ − flow + 2ǫ)).
+In addition, recall from (97) that f(xǫ, yǫ) ≤ f ∗
+µ + 2ǫ. The conclusion of this lemma then follows from
+these three relations and (98).
+We are now ready to prove Theorem 5.
+Proof of Theorem 5. Let {(xk, yk, zk)} be generated by Algorithm 5 with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).
+By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =
+(x∗, y∗). We now show that (x∗, y∗) is an optimal solution of problem (42). Indeed, since (xk, yk, zk) is
+an ǫk-optimal solution of (50) with (ρ, µ) = (ρk, µk) and limk→∞ µk = ∞, it follows from Lemma 9 with
+(ρ, µ, ǫ) = (ρk, µk, ǫk) and (xǫ, yǫ) = (xk, yk) that for all sufficiently large k, one has
+f(xk, yk) ≤ f ∗ + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk,
+˜f(xk, yk) ≤ ˜f ∗(xk) + ρ−1
+k
+�
+f ∗ − flow + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk
+�
+,
+��[˜g(xk, yk)]+
+��2 ≤ µ−1
+k
+�
+˜f ∗(xk) − ˜flow + ρ−1
+k
+�
+f ∗ − flow + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk
+��
+.
+By the continuity of f, ˜f and ˜f ∗ (see Assumption 3(i) and Lemma 6(ii)), limk→∞(xk, yk) = (x∗, y∗),
+limk→∞(ρk, µk, ǫk) = (∞, ∞, 0), limθ↓0 ω(θ) = 0, and taking limits as k → ∞ on both sides of the above
+relations, we obtain that f(x∗, y∗) ≤ f ∗, ˜f(x∗, y∗) ≤ ˜f ∗(x∗) and [˜g(x∗, y∗)]+ = 0, which along with
+(42) and (43) imply that f(x∗, y∗) = f ∗ and y∗ ∈ Argminz{ ˜f(x∗, z)|˜g(x∗, z) ≤ 0}. Hence, (x∗, y∗) is an
+optimal solution of (42) as desired.
+23
+
+We next prove Theorem 6. Before proceeding, we establish several technical lemmas below, which
+will be used to prove Theorem 6 subsequently.
+Lemma 10. Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-stationary point of
+problem (50) for some ǫ > 0. Let Dy, ˜g, ρ, µ, Lf, L ˜
+f and G be given in (30), (42), (50) and Assumption
+4, respectively. Then we have
+∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f),
+(101)
+∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜
+f).
+(102)
+Proof. We first prove (101). Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition
+2 that dist(0, ∂zPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ. Also, by (47) and (50), one has
+Pρ,µ(x, y, z) = f(x, y) + ρ( ˜f(x, y) + µ ∥[˜g(x, y)]+∥2) − ρ( ˜f(x, z) + µ ∥[˜g(x, z)]+∥2).
+(103)
+Using these relations, we have
+dist
+�
+0, ∂z ˜f(xǫ, zǫ) + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)
+�
+≤ ρ−1ǫ.
+Hence, there exists s ∈ ∂z ˜f(xǫ, zǫ) such that
+���s + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)
+��� ≤ ρ−1ǫ.
+(104)
+Let ˆzxǫ and G be given in Assumption 4(iii). It then follows that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for
+all i. Notice that [˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i and ∥zǫ − ˆzxǫ∥ ≤ Dy due to (30). Using these, (104),
+and the convexity of ˜f(xǫ, ·) and ˜gi(xǫ, ·) for all i, we have
+˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) + 2µG
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+ ≤ ˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) − 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+˜gi(xǫ, ˆzxǫ)
+≤ ˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+(˜gi(xǫ, zǫ) − ˜gi(xǫ, ˆzxǫ))
+≤ ⟨s, zǫ − ˆzxǫ⟩ + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+⟨∇z˜gi(xǫ, zǫ), zǫ − ˆzxǫ⟩
+= ⟨s + 2µ
+l
+�
+i=1
+[˜g(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ), zǫ − ˆzxǫ⟩ ≤ ρ−1Dyǫ,
+(105)
+where the first inequality is due to −˜gi(xǫ, ˆzxǫ) ≥ G for all i, the second inequality follows from
+[˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i, the third inequality is due to s ∈ ∂z ˜f(xǫ, zǫ) and the convexity of
+˜f(xǫ, ·) and ˜gi(xǫ, ·) for all i, and the last inequality follows from (30) and (104). In view of (30), (105),
+and L ˜
+f-Lipschitz continuity of ˜f(x, y) (see Assumption 4), one has
+∥[˜g(xǫ, zǫ)]+∥ ≤
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+
+(105)
+≤
+(2µG)−1(ρ−1Dyǫ + ˜f(xǫ, ˆzxǫ) − ˜f(xǫ, zǫ))
+≤ (2µG)−1(ρ−1Dyǫ + L ˜
+f∥ˆzxǫ − zǫ∥)
+(30)
+≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f).
+Hence, (101) holds.
+We next prove (102). Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition 2
+that dist(0, ∂yPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ. This together with (103) implies that
+dist
+�
+0, ∂yf(xǫ, yǫ) + ρ∂y ˜f(xǫ, yǫ) + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+
+�
+≤ ǫ.
+Hence, there exists s ∈ ∂yf(xǫ, yǫ) and ˜s ∈ ∂y ˜f(xǫ, yǫ) such that
+∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ ≤ ǫ.
+(106)
+24
+
+Let ¯
+A(xǫ, yǫ) = {i|˜gi(xǫ, yǫ) > 0, 1 ≤ i ≤ l}, ˆzxǫ and G be given in Assumption 4(iii). It then follows
+that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for all i. Using these and the convexity of ˜gi(xǫ, ·) for all i, we
+have
+⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩ =
+�
+i∈ ¯
+A(xǫ,yǫ)
+⟨∇y˜gi(xǫ, yǫ), yǫ − ˆzxǫ⟩[gi(xǫ, yǫ)]+
+≥
+�
+i∈ ¯
+A(xǫ,yǫ)
+(˜gi(xǫ, yǫ) − ˜gi(xǫ, ˆzxǫ))[˜gi(xǫ, yǫ)]+
+≥
+�
+i∈ ¯
+A(xǫ,yǫ)
+G[˜gi(xǫ, yǫ)]+ = G
+l
+�
+i=1
+[˜gi(xǫ, yǫ)]+ ≥ G ∥[˜g(xǫ, yǫ)]+∥ ,
+(107)
+where the first inequality follows from the convexity of ˜g(xǫ, ·) and the second inequality is due to
+−˜gi(xǫ, ˆzxǫ) ≥ G. It then follows from this, (106) and (107) that
+Dyǫ ≥ ∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ · ∥yǫ − ˆzxǫ∥
+≥ ⟨s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩
+= ⟨s + ρ˜s, yǫ − ˆzxǫ⟩ + 2ρµ⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩
+(107)
+≥ − (∥s∥ + ρ∥˜s∥) ∥yǫ − ˆzxǫ∥ + 2ρµG ∥[˜g(xǫ, yǫ)]+∥
+≥ −(Lf + ρL ˜
+f)Dy + 2ρµG ∥[˜g(xǫ, yǫ)]+∥ ,
+(108)
+where the last inequality follows from ∥yǫ − ˆzxǫ∥ ≤ Dy and the fact that ∥s∥ ≤ Lf and ∥˜s∥ ≤ L ˜
+f, which
+are due to (30), s ∈ ∂yf(xǫ, yǫ), ˜s ∈ ∂y ˜f(xǫ, yǫ) and Assumption 4(i). By (108), one can immediately see
+that (102) holds.
+Lemma 11. Suppose that Assumptions 3 and 4 hold. Let f, ˜f, ˜g, Dy, flow, ˜f ∗ and Pρ,µ be given in (29),
+(30), (32), (43) and (50), Lf, L ˜
+f and G be given in Assumptions 3 and 4, (xǫ, yǫ, zǫ) be an ǫ-stationary
+point of (50) for some ǫ > 0, and
+˜λ = 2µ[˜g(xǫ, zǫ)]+,
+ˆλ = 2ρµ[˜g(xǫ, yǫ)]+.
+(109)
+Then we have
+dist
+�
+∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ǫ,
+(110)
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ǫ,
+(111)
+∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f),
+(112)
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ (2µ)−1G−2D2
+y(ρ−1ǫ + L ˜
+f)2,
+(113)
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max
+�
+ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − flow), (2µ)−1G−2D2
+yL ˜
+f(ρ−1ǫ + ρ−1Lf + L ˜
+f)
+�
+,
+(114)
+∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜
+f),
+(115)
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ (2µ)−1ρG−2D2
+y(ρ−1ǫ + ρ−1Lf + L ˜
+f)2.
+(116)
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it easily follows from (103), (109) and Definition
+2 that (110) and (111) hold. Also, it follows from (101) and (102) that (112) and (115) hold. In addition,
+in view of (109), (112) and (115), one has
+|⟨˜λ, ˜g(xǫ, zǫ)⟩|
+(109)
+=
+2µ ∥[˜g(xǫ, zǫ)]+∥2 (112)
+≤
+(2µ)−1G−2D2
+y(ρ−1ǫ + L ˜
+f)2,
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩|
+(109)
+=
+2ρµ ∥[˜g(xǫ, yǫ)]∥+∥2 (115)
+≤
+(2µ)−1ρG−2D2
+y(ρ−1ǫ + ρ−1Lf + L ˜
+f)2,
+and hence (113) and (116) hold. Also, observe from the definition of Pρ,µ in (50) that
+�Pµ(xǫ, yǫ) − min
+z
+�Pµ(xǫ, z) = ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ)).
+25
+
+Using this, (32), (47) and (93), we obtain that
+˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)
+=
+�Pµ(xǫ, yǫ) =
+min
+z
+�Pµ(xǫ, z) + ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ))
+(32)(93)
+≤
+˜f ∗(xǫ) + ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − flow).
+(117)
+On the other hand, let λ∗ ∈ Rl
++ be an optimal Lagrangian multiplier of problem (43) with x = xǫ. It
+then follows from Lemma 6(i) that ∥λ∗∥ ≤ G−1L ˜
+fDy. Using these and (115), we have
+˜f ∗(xǫ) = min
+y
+�
+˜f(xǫ, y) + ⟨λ∗, ˜g(xǫ, y)⟩
+�
+≤ ˜f(xǫ, yǫ) + ⟨λ∗, ˜g(xǫ, yǫ)⟩
+≤ ˜f(xǫ, yǫ) + ∥λ∗∥∥[˜g(xǫ, yǫ)]+∥ ≤ ˜f(xǫ, yǫ) + (2µ)−1G−2D2
+yL ˜
+f(ρ−1ǫ + ρ−1Lf + L ˜
+f).
+By this and (117), one can see that (114) holds.
+We are now ready to prove Theorem 6.
+Proof of Theorem 6. Observe from (51) that problem (50) can be viewed as
+min
+x,y max
+z
+{Pρ,µ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,
+where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2, p(x, y) = f2(x) +
+ρ ˜f2(y) and q(z) = ρ ˜f2(z). Hence, problem (50) is in the form of (6) with H = Pρ,µ. By Assumption 3,
+(45), (46), ρ = ε−1 and µ = ε−2, one can see that h is �L-smooth on its domain, where �L is given in (61).
+Also, notice from Algorithm 6 that ǫ0 = ε5/2 ≤ ε/2 = ǫ/2 due to ε ∈ (0, 1/4]. Consequently, Algorithm 2
+can be suitably applied to problem (50) with ρ = ε−1 and µ = ε−2 for finding an ǫ-stationary point
+(xǫ, yǫ, zǫ) of it.
+In addition, notice from Algorithm 6 that �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε. Using this, (50) and
+ρ = ε−1, we obtain
+max
+z
+Pρ,µ(x0, y0, z)
+(50)
+= f(x0, y0) + ρ( �Pµ(x0, y0) − min
+z
+�Pµ(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1. (118)
+By this and (28) with H = Pρ,µ, ǫ = ε, ǫ0 = ε5/2, ˆx0 = (x0, y0), Dq = Dy and L∇h = �L, one has
+Pρ,µ(xǫ, yǫ, zǫ) ≤
+max
+z
+Pρ,µ(x0, y0, z) + εDy/4 + 2ε5(�L−1 + 4D2
+y�Lε−2)
+(118)
+≤
+1 + f(x0, y0) + εDy/4 + 2ε5(�L−1 + 4D2
+y�Lε−2).
+It then follows from this and Lemma 11 with ǫ = ε, ρ = ε−1 and µ = ε−2 that (xǫ, yǫ, zǫ) satisfies the
+relations (62)-(68).
+We next show that at most �
+N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 are
+respectively performed in Algorithm 6. Indeed, by (31), (32), (45), (47) and (50), one has
+min
+x,y max
+z
+Pρ,µ(x, y, z)
+(50)
+= min
+x,y {f(x, y) + ρ( �Pµ(x, y) − min
+z
+�Pµ(x, z))} ≥
+min
+(x,y)∈X ×Y f(x, y)
+(32)
+= flow, (119)
+min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}
+(50)
+= min{f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z))|(x, y, z) ∈ X × Y × Y}
+(47)
+= min{f(x, y) + ρ( ˜f(x, y) + µ∥[˜g(x, y)]+∥2 − ˜f(x, z) − µ∥[˜g(x, z)]+∥2)|(x, y, z) ∈ X × Y × Y}
+≥ flow + ρ( ˜flow − ˜fhi) − ρµ˜g2
+hi,
+(120)
+where the last inequality follows from (31), (32) and (45). In addition, let (x∗, y∗) be an optimal solution
+of (42). It then follows that f(x∗, y∗) = f ∗ and [˜g(x∗, y∗)]+ = 0. By these, (31), (47) and (50), one has
+min
+x,y max
+z
+Pρ,µ(x, y, z) ≤ max
+z
+Pρ,µ(x∗, y∗, z)
+(50)
+= f(x∗, y∗) + ρ
+�
+�Pµ(x∗, y∗) − min
+z
+�Pµ(x∗, z)
+�
+(47)
+= f(x∗, y∗) + ρ( ˜f(x∗, y∗) + µ∥[˜g(x∗, y∗)]+∥2 − min
+z { ˜f(x∗, z) + µ∥[˜g(x∗, z)]+∥2})
+(31)
+≤ f ∗ + ρ( ˜fhi − ˜flow).
+(121)
+26
+
+For convenience of the rest proof, let
+H = Pρ,µ,
+H∗ = min
+x,y max
+z
+Pρ,µ(x, y, z),
+Hlow = min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}.
+(122)
+In view of these, (118), (119), (120), (121), ρ = ε−1 and µ = ε−2, we obtain that
+max
+z
+H(x0, y0, z)
+(118)
+≤ f(x0, y0) + 1,
+flow
+(119)
+≤
+H∗ (121)
+≤
+f ∗ + ρ( ˜fhi − ˜flow) = f ∗ + ε−1( ˜fhi − ˜flow),
+Hlow
+(120)
+≥
+flow + ρ( ˜flow − ˜fhi) − ρµ˜g2
+hi = flow + ε−1( ˜flow − ˜fhi) − ε−3˜g2
+hi.
+Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =
+�
+D2x + D2y, Dq = Dy, ǫ0 = ε5/2, L∇h = �L,
+α = ˜α, δ = ˜δ, and H, H∗, Hlow given in (122), we can conclude that Algorithm 6 performs at most �
+N
+evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 for finding an approximate solution
+(xǫ, yǫ) of problem (42) satisfying (62)-(68).
+6
+Concluding remarks
+For the sake of simplicity, first-order penalty methods are proposed only for problem (3) in this paper.
+It would be interesting to extend them to problem (1) by using a standard technique (e.g., see [39]) for
+handling the constraint g(x, y) ≤ 0. In addition, a single subproblem with static penalty and tolerance
+parameters is solved in our methods (Algorithms 4 and 6), which may be conservative in practice. To
+make the methods possibly practically more efficient, it would be natural to modify them by solving
+a sequence of subproblems with dynamic penalty and tolerance parameters instead. These along with
+numerical experiments will be left for the future research.
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diff --git a/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt b/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c63b71fc2dde3d73ed2c5e62c715f9bf6c20fd16
--- /dev/null
+++ b/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt
@@ -0,0 +1,1322 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf,len=1321
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='01716v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='OC]  4 Jan 2023  First-order penalty methods for bilevel optimization  Zhaosong Lu ∗  Sanyou Mei ∗  January 4, 2023  Abstract  In this paper we study a class of unconstrained and constrained bilevel optimization problems in  which the lower-level part is a convex optimization problem, while the upper-level part is possibly  a nonconvex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In particular, we propose penalty methods for solving them,  whose subproblems turn out to be a structured minimax problem and are suitably solved by a first-  order method developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Under some suitable assumptions, an operation complexity  of O(ε−4 log ε−1) and O(ε−7 log ε−1), measured by their fundamental operations, is established for  the proposed penalty methods for finding an ε-KKT solution of the unconstrained and constrained  bilevel optimization problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  To the best of our knowledge, the methodology and  results in this paper are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Keywords: bilevel optimization, minimax optimization, penalty methods, first-order methods, opera-  tion complexity  Mathematics Subject Classification: 90C26, 90C30, 90C47, 90C99, 65K05  1  Introduction  Bilevel optimization is a two-level hierarchical optimization in which partial or full decision variables in  the upper level are also involved in the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Generically, it can be written in the following form:  min  x,y  f(x, y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  g(x, y) ≤ 0,  y ∈ Argmin  z  { ˜f(x, z)|˜g(x, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1  (1)  Bilevel optimization has found a variety of important applications, including adversarial training [36,  37, 46], continual learning [32], hyperparameter tuning [3, 17], image reconstruction [9], meta-learning  [4, 23, 42], neural architecture search [15, 30], reinforcement learning [20, 27], and Stackelberg games [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  More applications about it can be found in [2, 8, 10, 11, 12, 44] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Theoretical  properties including optimality conditions of (1) have been extensively studied in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', see  [12, 13, 34, 47, 50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Numerous methods have been developed for solving some special cases of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For example, constraint-  based methods [19, 43], deterministic gradient-based methods [16, 17, 21, 35, 41, 42], and stochastic  gradient-based methods [6, 18, 20, 24, 26] were proposed for solving (1) with g ≡ 0, ˜g ≡ 0, f, ˜f being  smooth, and ˜f being strongly convex with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Besides, when all the functions involved in  (1) are smooth and ˜f, ˜g are convex with respect to y, gradient-type methods were proposed by solving  the mathematical program with equilibrium constraints (MPEC) resulting from replacing the lower-level  optimization problem of (1) by its first-order optimality conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', see [1, 33, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Recently,  difference-of-convex (DC) algorithms were developed in [51] for solving (1) with g ≡ 0, f being a DC  function, and ˜f, ˜g being convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, a double penalty method [22] was proposed for  (1), which solves a sequence of bilevel optimization problems of the form  min  x,y  f(x, y) + ρkΨ(x, y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  y ∈ Argmin  z  ˜f(x, z) + ρk ˜Ψ(x, z),  (2)  ∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email:  zhaosong@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='edu,  mei00035@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This work was partially supported by NSF Award IIS-2211491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1For ease of reading, throughout this paper the tilde symbol is particularly used for the functions related to the lower-level  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Besides, “Argmin” denotes the set of optimal solutions of the associated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1  where {ρk} is a sequence of penalty parameters, and Ψ and ˜Ψ are a penalty function associated with the  sets {(x, y)|g(x, y) ≤ 0} and {(x, z)|˜g(x, z) ≤ 0}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Though problem (2) appears to be simpler  than (1), there is no method available for finding an approximate solution of (2) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Conse-  quently, the double penalty method [22] is typically not implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' More discussion on algorithmic  development for bilevel optimization can be found in [2, 8, 12, 31, 45, 47]) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It has long been known that the notorious challenge of bilevel optimization (1) mainly comes from the  lower level part, which requires that the variable y be a solution of another optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Due  to this, for the sake of simplicity, we only consider a subclass of bilevel optimization with the constraint  g(x, y) ≤ 0 being excluded, namely,  min  x,y  f(x, y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  y ∈ Argmin  z  { ˜f(x, z)|˜g(x, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (3)  Nevertheless, the results in this paper can be possibly extended to problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The main goal of this paper is to develop a first-order penalty method for solving problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Our  key observations toward this development are: (i) problem (3) can be approximately solved as a penalty  problem (see (49));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (ii) such a penalty problem is equivalent to a structured minimax problem (see  (50)), which can be suitably solved by a first-order method proposed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' As a result, these  observations lead to development of a novel first-order penalty method for solving (3) (see Sections 3  and 4), which enjoys the following appealing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It uses only the first-order information of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Specifically, its fundamental operations  consist only of evaluations of the gradient of ˜g and the smooth component of f and ˜f and also  the proximal operator of the nonsmooth component of f and ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Thus, it is suitable for solving  large-scale problems (see Sections 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It has theoretical guarantees on operation complexity, which is measured by the aforementioned  fundamental operations, for finding an ε-KKT solution of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In particular, when ˜g ≡ 0, it  enjoys an operation complexity of O(ε−4 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Otherwise, it enjoys an operation complexity of  O(ε−7 log ε−1) (see Theorems 4 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It is applicable to a broader class of problems than existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  For example, it can be  applied to (3) with f, ˜f being nonsmooth and ˜f, ˜g being nonconvex with respect to x, which is  however not suitable for existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  To the best of our knowledge, the methodology and results in this paper are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1 we introduce some notation and  terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In Section 2 we propose a first-order method for solving a nonconvex-concave minimax  problem and study its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In Sections 3 and 4, we propose first-order penalty methods for  unconstrained and constrained bilevel optimization and study their complexity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In Section  5 we present the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Finally, we make some concluding remarks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1  Notation and terminology  The following notation will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Let Rn denote the Euclidean space of  dimension n and Rn  + denote the nonnegative orthant in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The standard inner product and Euclidean  norm are denoted by ⟨·, ·⟩ and ∥ · ∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For any v ∈ Rn, let v+ denote the nonnegative part of  v, that is, (v+)i = max{vi, 0} for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For any two vectors u and v, (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' v) denotes the vector resulting  from stacking v under u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Given a point x and a closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥ and  IS denote the indicator function associated with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ Lφ∥x−x′∥  for all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x) − ∇φ(x′)∥ ≤ L∇φ∥x − x′∥ for  all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For a closed convex function p : Rn → R ∪ {∞},2 the proximal operator associated with p  is denoted by proxp, that is,  proxp(x) = arg min  x′∈Rn  �1  2∥x′ − x∥2 + p(x′)  �  ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (4)  2For convenience, ∞ stands for +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2  Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of proxγp(x)  as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  For a lower semicontinuous function φ : Rn → R∪{∞}, its domain is the set dom φ := {x|φ(x) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by  φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' d) = lim sup  x′ φ  →x, t↓0  inf  d′→d  φ(x′ + td′) − φ(x′)  t  ,  where t ↓ 0 means both t > 0 and t → 0, and x′  φ→ x means both x′ → x and φ(x′) → φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The  subdifferential of φ at x ∈ dom φ is the set  ∂φ(x) = {s ∈ Rn��sT d ≤ φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' d) ∀d ∈ Rn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We use ∂xiφ(x) to denote the subdifferential with respect to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, for an upper semicontinuous  function φ, its subdifferential is defined as ∂φ = −∂(−φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' If φ is locally Lipschitz continuous, the above  definition of subdifferential coincides with the Clarke subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Besides, if φ is convex, it coincides  with the ordinary subdifferential for convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, if φ is continuously differentiable at x , we  simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the gradient of φ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, it is not hard to  verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if φ1 is continuously differentiable at x and φ2 is lower or  upper semicontinuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' See [7, 49] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Finally, we introduce two types of approximate solutions for a general minimax problem  Ψ∗ = min  x max  y  Ψ(x, y),  (5)  where Ψ(·, y) : Rn → R ∪ {∞} is a lower semicontinuous function, Ψ(x, ·) : Rm → R ∪ {−∞} is an upper  semicontinuous function, and Ψ∗ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' A point (xǫ, yǫ) is called an ǫ-optimal solution of the minimax problem (5) if  max  y  Ψ(xǫ, y) − Ψ(xǫ, yǫ) ≤ ǫ,  Ψ(xǫ, yǫ) − Ψ∗ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' A point (x, y) is called a stationary point of the minimax problem (5) if  0 ∈ ∂xΨ(x, y),  0 ∈ ∂yΨ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In addition, for any ǫ > 0, a point (xǫ, yǫ) is called an ǫ-stationary point of the minimax problem (5) if  dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ,  dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2  A first-order method for nonconvex-concave minimax prob-  lem  In this section, we propose a first-order method for finding an approximate stationary point of a  nonconvex-concave minimax problem, which will be used as a subproblem solver for the penalty methods  proposed in Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In particular, we consider the minimax problem  H∗ = min  x max  y  {H(x, y) := h(x, y) + p(x) − q(y)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (6)  Assume that problem (6) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, h, p and q satisfy the following  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions and  continuous on their domain, and moreover, their domain is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (ii) The proximal operator associated with p and q can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3  Recently, an accelerated inexact proximal point smoothing (AIPP-S) scheme was proposed in [28]  for finding an approximate stationary point of a class of minimax composite nonconvex optimization  problems, which includes (6) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' When applied to (6), AIPP-S requires the availability of  the oracle including exact evaluation of ∇xh(x, y) and  arg min  x  �  p(x) + 1  2λ∥x − x′∥2  �  ,  arg max  y  �  h(x′, y) − q(y) − 1  2λ∥y − y′∥2  �  (7)  for any λ > 0, x′ ∈ Rn and y′ ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Note that h is typically sophisticated and the exact solution of the  second problem in (7) usually cannot be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' As a result, AIPP-S is generally not implementable for  (6), though an oracle complexity of O(ǫ−5/2) was established in [28] for it to find an ǫ-stationary point  of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In what follows, we first propose a modified optimal first-order method for solving a strongly-convex-  strongly-concave minimax problem in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using this method as a subproblem solver for an  inexact proximal point scheme, we then propose a first-order method for (6) in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2, which  enjoys an operation complexity of O(ǫ−5/2 log ǫ−1), measured by the amount of evaluations of ∇h and  proximal operator of p and q, for finding an ǫ-stationary point of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1  A modified optimal first-order method for strongly-convex-strongly-concave  minimax problem  In this subsection, we consider the strongly-convex-strongly-concave minimax problem  ¯H∗ = min  x max  y  � ¯H(x, y) := ¯h(x, y) + p(x) − q(y)  �  ,  (8)  where p and q satisfy Assumption 1 and ¯h satisfies the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q  for some σx, σy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The goal of this subsection is to propose a modified optimal first-order method for finding an approx-  imate stationary point of problem (8) and study its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before proceeding, we introduce some  more notation below, most of which is adopted from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Let (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y∗) denote the optimal solution of (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z∗ = −σxx∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and  Dp = max{∥u − v∥  ��u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' v ∈ dom p},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Dq = max{∥u − v∥  ��u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' v ∈ dom q},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (9)  ¯Hlow = min  � ¯H(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y)|  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) ∈ dom p × dom q},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (10)  ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) = ¯h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) − σx∥x∥2/2 + σy∥y∥2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (11)  G(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) = sup  x {⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z⟩ − p(x) − ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + q(y)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (12)  P(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) = σ−1  x ∥z∥2/2 + σy∥y∥2/2 + G(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (13)  ϑk = η−1  z ∥zk − z∗∥2 + η−1  y ∥yk − y∗∥2 + 2¯α−1(P(zk  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk  f) − P(z∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y∗)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (14)  ak  x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) = ∇xˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + σx(x − σ−1  x zk  g)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ak  y(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) = −∇yˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + σyy + σx(y − yk  g)/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  where ¯α = min  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  8σy/σx  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηz = σx/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηy = min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 4/(¯ασx)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk  g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zk  f and zk  g  are generated at iteration k of Algorithm 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By Assumptions 1 and 2, one can observe that Dp,  Dq and ¯Hlow are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to present a modified optimal first-order method for solving (8) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It is  a slight modification of the novel optimal first-order method [29, Algorithm 4] by incorporating a forward-  backward splitting scheme and also a verifiable termination criterion (see steps 23-25 in Algorithm 1) in  order to find a τ-stationary point of (8) (see Definition 2) for any prescribed tolerance τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  Algorithm 1 A modified optimal first-order method for (8)  Input: τ > 0, ¯z0 = z0  f ∈ −σxdom p,3 ¯y0 = y0  f ∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = min  �  1,  �  8σy/σx  �  ,  ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ =  �  2  √  5(1 + 8L∇¯h/σx)  �−1, γx = γy =  8σ−1  x , and ˆζ = min{σx, σy}/L2  ∇¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' do  2:  (zk  g , yk  g) = ¯α(zk, yk) + (1 − ¯α)(zk  f, yk  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3:  (xk,−1, yk,−1) = (−σ−1  x zk  g, yk  g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4:  xk,0 = proxζγxp(xk,−1 − ζγxak  x(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5:  yk,0 = proxζγyq(yk,−1 − ζγyak  y(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  6:  bk,0  x  =  1  ζγx (xk,−1 − ζγxak  x(xk,−1, yk,−1) − xk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  7:  bk,0  y  =  1  ζγy (yk,−1 − ζγyak  y(xk,−1, yk,−1) − yk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  8:  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  9:  while  γx∥ak  x(xk,t, yk,t) + bk,t  x ∥2 + γy∥ak  y(xk,t, yk,t) + bk,t  y ∥2 > γ−1  x ∥xk,t − xk,−1∥2 + γ−1  y ∥yk,t − yk,−1∥2  do  10:  xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak  x(xk,t, yk,t) + bk,t  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  11:  yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak  y(xk,t, yk,t) + bk,t  y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  12:  xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak  x(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  13:  yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak  y(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  14:  bk,t+1  x  =  1  ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak  x(xk,t+1/2, yk,t+1/2) − xk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  15:  bk,t+1  y  =  1  ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak  y(xk,t+1/2, yk,t+1/2) − yk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  16:  t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  17:  end while  18:  (xk+1  f  , yk+1  f  ) = (xk,t, yk,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  19:  (zk+1  f  , wk+1  f  ) = (∇xˆh(xk+1  f  , yk+1  f  ) + bk,t  x , −∇yˆh(xk+1  f  , yk+1  f  ) + bk,t  y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  20:  zk+1 = zk + ηzσ−1  x (zk+1  f  − zk) − ηz(xk+1  f  + σ−1  x zk+1  f  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  21:  yk+1 = yk + ηyσy(yk+1  f  − yk) − ηy(wk+1  f  + σyyk+1  f  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  22:  xk+1 = −σ−1  x zk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  23:  ˆxk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  24:  ˆyk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  25:  Terminate the algorithm and output (ˆxk+1, ˆyk+1) if  ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥ ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (15)  26: end for  The following theorem presents iteration and operation complexity of Algorithm 1 for finding a τ-  stationary point of problem (8), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 1 (Complexity of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let ¯H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯Hlow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and ϑ0 be defined in (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (10) and (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' σy and L∇¯h be given in Assumption 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ηy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆζ be given in Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and  ¯δ = (2 + ¯α−1)σxD2  p + max{2σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯ασx/4}D2  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (16)  ¯K =  �  max  � 2  ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯ασx  4σy  �  log 4 max{ηzσ−2  x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηy}ϑ0  (ˆζ−1 + L∇¯h)−2τ 2  �  +  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (17)  ¯N =  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  � σx  2σy  �  log 4 max {1/(2σx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 4/(¯ασx)}}  �¯δ + 2¯α−1 � ¯H∗ − ¯Hlow  ��  (L2  ∇¯h/ min{σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' σy} + L∇¯h)−2τ 2  �  +  ×  ��  96  √  2  �  1 + 8L∇¯hσ−1  x  ��  + 2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (18)  Then Algorithm 1 outputs a τ-stationary point of (8) in at most ¯K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Moreover, the total  3For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5  number of evaluations of ∇¯h and proximal operator of p and q performed in Algorithm 1 is no more than  ¯N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It can be observed from Theorem 1 that Algorithm 1 enjoys an operation complexity of  O(log τ−1), measured by the amount of evaluations of ∇¯h and proximal operator of p and q, for finding  a τ-stationary point of the strongly-convex-strongly-concave minimax problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2  A first-order method for problem (6)  In this subsection, we propose a first-order method for finding an ǫ-stationary point of problem (6) (see  Definition 2) for any prescribed tolerance ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In particular, we first add a perturbation to the max  part of (6) for obtaining an approximation of (6), which is given as follows:  min  x max  y  �  h(x, y) + p(x) − q(y) −  ǫ  4Dq  ∥y − ˆy0∥2  �  (19)  for some ˆy0 ∈ dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We then apply an inexact proximal point method [25] to (19), which consists of  approximately solving a sequence of subproblems  min  x max  y  {Hk(x, y) := hk(x, y) + p(x) − q(y)} ,  (20)  where  hk(x, y) = h(x, y) − ǫ∥y − ˆy0∥2/(4Dq) + L∇h∥x − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (21)  By Assumption 1, one can observe that (i) hk is L∇h-strongly convex in x and ǫ/(2Dq)-strongly concave  in y on dom p × dom q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (ii) hk is (3L∇h + ǫ/(2Dq))-smooth on dom p × dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, problem  (20) is a special case of (8) and we can apply Algorithm 1 to solve (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The resulting first-order method  for (6) is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Algorithm 2 A first-order method for problem (6)  Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' do  2:  Call Algorithm 1 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h + ǫ/(2Dq),  ¯z0 = z0  f ← −σxxk, ¯y0 = y0  f ← yk, and denote its output by (xk+1, yk+1), where hk is given in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3:  Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if  ∥xk+1 − xk∥ ≤ ǫ/(4L∇h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (22)  4: end for  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It can be observed from step 2 of Algorithm 2 that (xk+1, yk+1) results from applying Algo-  rithm 1 to the subproblem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' As will be shown in Lemma 2, (xk+1, yk+1) is an ǫk-stationary point of  (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next study complexity of Algorithm 2 for finding an ǫ-stationary point of problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before  proceeding, we define  Hlow := min {H(x, y)|(x, y) ∈ dom p × dom q} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (23)  By Assumption 1, one can observe that Hlow is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The following theorem presents iteration and operation complexity of Algorithm 2 for finding an  ǫ-stationary point of problem (6), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 2 (Complexity of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' H Dp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  and Hlow be defined in (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (9) and (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L∇h be given in Assumption 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ0 and ˆx0 be given in  6  Algorithm 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and  α = min  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  4ǫ/(DqL∇h)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (24)  δ = (2 + α−1)L∇hD2  p + max {ǫ/Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' αL∇h/4} D2  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (25)  K =  �  16(max  y  H(ˆx0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2  0(1 + 4D2  qL2  ∇hǫ−2)ǫ−2 − 1  �  +  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (26)  N =  ��  96  √  2  �  1 + (24L∇h + 4ǫ/Dq) L−1  ∇h  ��  + 2  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  DqL∇hǫ−1  �  ×  �  (K + 1)  �  log  4 max  �  1  2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min  �  Dq  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  αL∇h  �� �  δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2  p)  �  [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2  0  �  +  + K + 1 + 2K log(K + 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (27)  Then Algorithm 2 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (6) in at most K + 1 outer  iterations that satisfies  max  y  H(xǫ, y) ≤ max  y  H(ˆx0, y) + ǫDq/4 + 2ǫ2  0  �  L−1  ∇h + 4D2  qL∇hǫ−2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (28)  Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed in Algo-  rithm 2 is no more than N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since ǫ0 ∈ (0, ǫ/2], one can observe from Theorem 2 that α = O(ǫ1/2), δ = O(ǫ−1/2),  K = O(ǫ−2), and N = O(ǫ−5/2 log(ǫ−1  0 ǫ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 2 enjoys an operation complexity  of O(ǫ−5/2 log(ǫ−1  0 ǫ−1)), measured by the amount of evaluations of ∇h and proximal operator of p and  q, for finding an ǫ-stationary point of the nonconvex-concave minimax problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3  Unconstrained bilevel optimization  In this section, we consider an unconstrained bilevel optimization problem4  f ∗ = min  f(x, y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  y ∈ Argmin  z  ˜f(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (29)  Assume that problem (29) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, f and ˜f satisfy the following  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (i) f(x, y) = f1(x, y)+f2(x) and ˜f(x, y) = ˜f1(x, y)+ ˜f2(y) are continuous on X ×Y,  where f2 : Rn → R ∪ {∞} and ˜f2 : Rm → R ∪ {∞} are proper closed convex functions, ˜f1(x, ·) is  convex for any given x ∈ X, and f1, ˜f1 are respectively L∇f1- and L∇ ˜f1-smooth on X × Y with  X := dom f2 and Y := dom ˜f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (ii) The proximal operator associated with f2 and ˜f2 can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (iii) The sets X and Y (namely, dom f2 and dom ˜f2) are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  For notational convenience, we define  Dx := max{∥u − v∥  ��u, v ∈ X},  Dy := max{∥u − v∥  ��u, v ∈ Y},  (30)  ˜fhi := max{ ˜f(x, y)|(x, y) ∈ X × Y},  ˜flow := min{ ˜f(x, y)|(x, y) ∈ X × Y},  (31)  flow := min{f(x, y)|(x, y) ∈ X × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (32)  4For convenience, problem (29) is referred to as an unconstrained bilevel optimization problem since its lower level part  does not have an explicit constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Strictly speaking, it can be a constrained bilevel optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For example,  when part of f and/or ˜f is the indicator function of a closed convex set, (29) is essentially a constrained bilevel optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  7  By Assumption 3, one can observe that Dx, Dy, ˜fhi, ˜flow and flow are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The goal of this subsection is to propose penalty methods for solving problem for solving (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' To  this end, we observe that problem (29) can be viewed as  min  x,y {f(x, y)| ˜f(x, y) ≤ min  z  ˜f(x, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (33)  Notice that ˜f(x, y) − minz ˜f(x, z) ≥ 0 for all x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, a natural penalty problem associated  with (33) is  min  x,y f(x, y) + ρ( ˜f(x, y) − min  z  ˜f(x, z)),  (34)  where ρ > 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We further observe that (34) is equivalent to the minimax problem  min  x,y max  z  Pρ(x, y, z),  where  Pρ(x, y, z) := f(x, y) + ρ( ˜f(x, y) − ˜f(x, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (35)  In view of Assumption 3(i), Pρ can be rewritten as  Pρ(x, y, z) =  �  f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z)  �  +  �  f2(x) + ρ ˜f2(y) − ρ ˜f2(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (36)  By this and Assumption 3, one can observe that Pρ enjoys the following nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Pρ is the sum of smooth function f1(x, y)+ ρ ˜f1(x, y)− ρ ˜f1(x, z) with Lipschitz continuous gradient  and possibly nonsmooth function f2(x)+ρ ˜f2(y)−ρ ˜f2(z) with exactly computable proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Pρ is nonconvex in (x, y) but concave in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Thanks to the nice structure of Pρ, an approximate stationary point of the minimax problem (35) can  be found by Algorithm 2 proposed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Based on the above observations, we are now ready to propose penalty methods for the unconstrained  bilevel optimization problem (29) by solving either a sequence of or a single minimax problem in the  form of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In particular, we first propose an ideal penalty method for (29) by solving a sequence of  minimax problems (see Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we propose a practical penalty method for (29) by finding  an approximate stationary point of a single minimax problem (see Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Algorithm 3 An ideal penalty method for problem (29)  Input: positive sequences {ρk} and {ǫk} with limk→∞(ρk, ǫk) = (∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' do  2:  Find an ǫk-optimal solution (xk, yk, zk) of problem (35) with ρ = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3: end for  The following theorem states a convergence result of Algorithm 3, whose proof is deferred to Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 3 (Convergence of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 3 holds and that {(xk, yk, zk)}  is generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then any accumulation point of {(xk, yk)} is an optimal solution of problem  (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Notice that (35) is a nonconvex-concave minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It is typically hard to find an ǫ-optimal  solution of (35) for an arbitrary ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 3 is not implementable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We  next propose a practical penalty method for problem (29) by finding an approximate stationary point of  a single minimax problem (35) with a suitable choice of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Algorithm 4 A practical penalty method for problem (29)  Input: ε ∈ (0, 1/4], ρ = ε−1, (x0, y0) ∈ X × Y with ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε3/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ε−1L∇ ˜  f1 to  find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (35) with ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2: Output: (xǫ, yǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  8  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (i) The initial point (x0, y0) of Algorithm 4 can be found by an additional procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed,  one can first choose any x0 ∈ X and then apply accelerated proximal gradient method [38] to the problem  miny ˜f(x0, y) for finding y0 ∈ Y such that ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (ii) As seen from Theorem 2,  an ǫ-stationary point of (35) can be successfully found in step 1 of Algorithm 4 by applying Algorithm 2  to (35);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (iii) For the sake of simplicity, a single subproblem of the form (35) with static penalty and  tolerance parameters is solved in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Nevertheless, Algorithm 4 can be modified into a perhaps  practically more efficient algorithm by solving a sequence of subproblems of the form (35) with dynamic  penalty and tolerance parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In order to characterize the approximate solution found by Algorithm 4, we next introduce a termi-  nology called an ε-KKT solution of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Recall that problem (29) can be viewed as problem (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In the spirit of classical constrained opti-  mization, one would naturally be interested in a KKT solution (x, y) of (33) or equivalently (29), namely,  (x, y) satisfies ˜f(x, y) ≤ minz ˜f(x, z) and moreover (x, y) is a stationary point of the problem  min  x′,y′ f(x′, y′) + ρ  � ˜f(x′, y′) − min  z′  ˜f(x′, z′)  �  (37)  for some ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Yet, due to the sophisticated problem structure, characterizing a stationary point of (37)  is generally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' On another hand, notice that problem (37) is equivalent to the minimax problem  min  x′,y′ max  z′  f(x′, y′) + ρ( ˜f(x′, y′) − ˜f(x′, z′)),  whose stationary point (x, y, z) according to Definition 2 satisfies  0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0),  0 ∈ ρ∂z ˜f(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (38)  Based on this observation, we are instead interested in a (weak) KKT solution of problem (29) and its  inexact counterpart that are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The pair (x, y) is said to be a KKT solution of problem (29) if there exists (z, ρ) ∈ Rm×R+  such that (38) and ˜f(x, y) ≤ minz′ ˜f(x, z′) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, for any ε > 0, (x, y) is said to be an ε-KKT  solution of problem (29) if there exists (z, ρ) ∈ Rm × R+ such that  dist  �  0, ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0)  �  ≤ ε,  dist  �  0, ρ∂z ˜f(x, z)  �  ≤ ε,  ˜f(x, y) − min  z′  ˜f(x, z′) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to present a theorem regarding operation complexity of Algorithm 4, measured by  the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT  solution of (29), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 4 (Complexity of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ˜fhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜flow and flow be defined in (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (31) and (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L∇f1 and L∇ ˜  f1 be given in Assumption 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y0 and zǫ be given in Algorithm 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and  �L = L∇f1 + 2ε−1L∇ ˜  f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆα = min  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  4ε/(Dy�L)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (39)  ˆδ = (2 + ˆα−1)(D2  x + D2  y)�L + max  �  ε/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆα�L/4  �  D2  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �C =  4 max  �  1  2�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min  �  Dy  ε ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  ˆα�L  �� �  ˆδ + 2ˆα−1(f ∗ − flow + ε−1( ˜fhi − ˜flow) + εDy/4 + �L(D2  x + D2  y))  �  �  (3�L + ε/(2Dy))2/ min{�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ε/(2Dy)} + 3�L + ε/(2Dy)  �−2  ε3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �K =  �  16(1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y0) − flow + εDy/4)�Lε−2 + 32(1 + 4D2  y�L2ε−2)ε − 1  �  + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  N =  ��  96  √  2(1 + (24�L + 4ε/Dy)�L−1)  �  + 2  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  Dy�Lε−1  �  × (( �  K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  9  Then Algorithm 4 outputs an approximate solution (xǫ, yǫ) of (29) satisfying  dist  �  0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0)  �  ≤ ε,  dist  �  0, ρ∂ ˜f(xǫ, zǫ)  �  ≤ ε,  (40)  ˜f(xǫ, yǫ) ≤ min  z  ˜f(xǫ, z) + ε  �  1 + f(x0, y0) − flow + 2ε3(�L−1 + 4D2  y�Lε−2) + Dyε/4  �  ,  (41)  after at most �  N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' One can observe from Theorem 4 that �L = O(ε−1), ˆα = O(ε), ˆδ = O(ε−2), �C = O(ε−11),  �K = O(ε−3), and �  N = O(ε−4 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 4 enjoys an operation complexity of  O(ε−4 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2,  for finding an O(ε)-KKT solution (xǫ, yǫ) of (29) satisfying  dist  �  0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0)  �  ≤ ε,  dist  �  0, ρ∂ ˜f(xǫ, zǫ)  �  ≤ ε,  ˜f(xǫ, yǫ) − min  z  ˜f(xǫ, z) = O(ε),  where zǫ is given in Algorithm 4 and ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  Constrained bilevel optimization  In this section, we consider a constrained bilevel optimization problem5  f ∗ = min  f(x, y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  y ∈ Argmin  z  { ˜f(x, z)|˜g(x, z) ≤ 0},  (42)  where f and ˜f satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Recall from Assumption 3 that X = dom f2 and Y = dom ˜f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We  now make some additional assumptions for problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (i) f and ˜f are Lf- and L ˜  f-Lipschitz continuous on X × Y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (ii) ˜g : Rn × Rm → Rl is L∇˜g-smooth and L˜g-Lipschitz continuous on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (iii) ˜gi(x, ·) is convex and there exists ˆzx ∈ Y for each x ∈ X such that ˜gi(x, ˆzx) < 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', l  and G := min{−˜gi(x, ˆzx)|x ∈ X, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' , l} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='6  For notational convenience, we define  ˜f ∗(x) := min  z { ˜f(x, z)|˜g(x, z) ≤ 0},  (43)  ˜f ∗  hi := sup{ ˜f ∗(x)|x ∈ X},  (44)  ˜ghi := max{∥˜g(x, y)∥  ��(x, y) ∈ X × Y},  (45)  It then follows from Assumption 4(ii) that  ∥∇˜g(x, y)∥ ≤ L˜g  ∀(x, y) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (46)  In addition, by Assumptions 3 and 4 and the compactness of X and Y, one can observe that ˜ghi and G  are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Besides, as will be shown in Lemma 6(ii), ˜f ∗  hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The goal of this subsection is to propose penalty methods for solving problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' To this end, let  us introduce a penalty function for the lower level optimization problem y ∈ Argmin  z  { ˜f(x, z)|˜g(x, z) ≤ 0}  of (42), which is given by  �Pµ(x, z) = ˜f(x, z) + µ ∥[˜g(x, z)]+∥2  (47)  5For convenience, problem (42) is referred to as a constrained bilevel optimization problem since its lower level part has  at least one explicit constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  6The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for the lower  level part of (42), that is, there exists ˆzx ∈ Y such that ˜g(x, ˆzx) < 0 for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed, if G > 0, Assumption 4(iii)  clearly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Otherwise, one can solve the perturbed counterpart of (42) with ˜g(x, z) being replaced by ˜g(x, z) − ǫ for  some suitable ǫ > 0 instead, which satisfies Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  10  for a penalty parameter µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Observe that problem (42) can be approximately solved as the uncon-  strained bilevel optimization problem  f ∗  µ = min  x,y  �  f(x, y)|y ∈ Argmin  z  �Pµ(x, z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (48)  Further, by the study in Section 3, problem (48) can be approximately solved as the penalty problem  min  x,y f(x, y) + ρ  �  �Pµ(x, y) − min  z  �Pµ(x, z)  �  (49)  for some suitable ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' One can also observe that problem (49) is equivalent to the minimax problem  min  x,y max  z  Pρ,µ(x, y, z),  where  Pρ,µ(x, y, z) := f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (50)  In view of (47), (50) and Assumption 3(i), Pρ,µ can be rewritten as  Pρ,µ(x, y, z) =  �  f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2 �  +  �  f2(x) + ρ ˜f2(y) − ρ ˜f2(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (51)  By this and Assumptions 3 and 4, one can observe that Pρ,µ enjoys the following nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Pρ,µ is the sum of smooth function f1(x, y)+ρ ˜f1(x, y)+ρµ ∥[˜g(x, y)]+∥2−ρ ˜f1(x, z)−ρµ ∥[˜g(x, z)]+∥2  with Lipschitz continuous gradient and possibly nonsmooth function f2(x) + ρ ˜f2(y) − ρ ˜f2(z) with  exactly computable proximal operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Pρ,µ is nonconvex in (x, y) but concave in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Due to the nice structure of Pρ,µ, an approximate stationary point of the minimax problem (50) can be  found by Algorithm 2 proposed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Based on the above observations, we are now ready to propose penalty methods for the constrained  bilevel optimization problem (42) by solving a sequence of or a single minimax problem of the form (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In particular, we first propose an ideal penalty method for (42) by solving a sequence of minimax problems  (see Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we propose a practical penalty method for (42) by finding an approximate  stationary point of a single minimax problem (see Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Algorithm 5 An ideal penalty method for problem (42)  Input: positive sequences {ρk}, {µk} and {ǫk} with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' do  2:  Find an ǫk-optimal solution (xk, yk, zk) of problem (50) with (ρ, µ) = (ρk, µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  3: end for  To study convergence of Algorithm 5, we make the following error bound assumption on the solution  set of the lower level optimization problem of (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This type of error bounds has been considered in the  context of set-value mappings in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' There exists a non-decreasing function ω : R+ → R+ with limθ↓0 ω(θ) = 0 and ¯θ > 0  such that dist(z, Sθ(x)) ≤ ω(θ) for all x ∈ X, z ∈ S0(x) and θ ∈ [0, ¯θ], where  Sθ(x) := Argmin  z  { ˜f(x, z) : ∥[˜g(x, z)]+∥ ≤ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to state a convergence result of Algorithm 5, whose proof is deferred to Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 5 (Convergence of Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3-5 hold and that {(xk, yk, zk)}  is generated by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then any accumulation point of {(xk, yk)} is an optimal solution of problem  (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Notice that (50) is a nonconvex-concave minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It is generally hard to find an ǫ-optimal  solution of (50) for an arbitrary ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' As a result, Algorithm 5 is generally not implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We next  propose a practical penalty method for problem (42) by finding an approximate stationary point of (50)  with a suitable choice of ρ and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  11  Algorithm 6 A practical penalty method for problem (42)  Input: ε ∈ (0, 1/4], ρ = ε−1, µ = ε−2, (x0, y0) ∈ X × Y with �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε5/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ρL∇ ˜f1 +  4ρµ(˜ghiL∇˜g +L2  ˜g) to find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (50) with ρ = ε−1 and µ = ε−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  2: Output: (xǫ, yǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (i) The initial point (x0, y0) of Algorithm 6 can be found by the similar procedure as described  in Remark 4 with ˜f being replaced by �Pµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (ii) As seen from Theorem 2, an ǫ-stationary point of (50)  can be successfully found in step 1 of Algorithm 6 by applying Algorithm 2 to (50);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (iii) For the sake of  simplicity, a single subproblem of the form (50) with static penalty and tolerance parameters is solved in  Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Nevertheless, Algorithm 6 can be modified into a perhaps practically more efficient algorithm  by solving a sequence of subproblems of the form (50) with dynamic penalty and tolerance parameters  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In order to characterize the approximate solution found by Algorithm 6, we next introduce a termi-  nology called an ε-KKT solution of problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By the definition of ˜f ∗ in (43), problem (42) can be viewed as  min  x,y {f(x, y)| ˜f(x, y) ≤ ˜f ∗(x), ˜g(x, y) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (52)  Its associated Lagrangian function is given by  L(x, y, ρ, λ) = f(x, y) + ρ( ˜f(x, y) − ˜f ∗(x)) + ⟨λ, ˜g(x, y)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (53)  In the spirit of classical constrained optimization, one would naturally be interested in a KKT solution  (x, y) of (52) or equivalently (42), namely, (x, y) satisfies  ˜f(x, y) ≤ ˜f ∗(x),  ˜g(x, y) ≤ 0,  ρ( ˜f(x, y) − ˜f ∗(x)) = 0,  ⟨λ, ˜g(x, y)⟩ = 0,  (54)  and moreover (x, y) is a stationary point of the problem  min  x′,y′ L(x′, y′, ρ, λ)  (55)  for some ρ ≥ 0 and λ ∈ Rl  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Yet, due to the sophisticated problem structure, characterizing a stationary  point of (55) is generally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' On another hand, notice from Lemma 6 and (53) that problem (55)  is equivalent to the minimax problem  min  x′,y′,˜λ′ max  z′  �  f(x′, y′) + ρ  � ˜f(x′, y′) − ˜f(x′, z′) − ⟨˜λ′, ˜g(x′, z′)⟩  �  + ⟨λ, ˜g(x′, y′)⟩ + IRl  +(˜λ′)  �  ,  whose stationary point (x, y, ˜λ, z) according to Definition 2 satisfies  0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0) + ∇˜g(x, y)λ,  (56)  0 ∈ ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ),  (57)  ˜λ ∈ Rl  +,  ˜g(x, z) ≤ 0,  ⟨˜λ, ˜g(x, z)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (58)  Based on this observation and also the fact that (54) is equivalent to  ˜f(x, y) = ˜f ∗(x),  ˜g(x, y) ≤ 0,  ⟨λ, ˜g(x, y)⟩ = 0,  (59)  we are instead interested in a (weak) KKT solution of problem (42) and its inexact counterpart that are  defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The pair (x, y) is said to be a KKT solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈  Rm × R+ × Rl  + × Rl  + such that (56)-(59) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, for any ε > 0, (x, y) is said to be an ε-KKT  solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈ Rm × R+ × Rl  + × Rl  + such that  dist  �  0, ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0) + ∇˜g(x, y)λ  �  ≤ ε,  dist  �  0, ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ)  �  ≤ ε,  ∥[˜g(x, z)]+∥ ≤ ε,  |⟨˜λ, ˜g(x, z)⟩| ≤ ε,  | ˜f(x, y) − ˜f ∗(x)| ≤ ε,  ∥[˜g(x, y)]+∥ ≤ ε,  |⟨λ, ˜g(x, y)⟩| ≤ ε,  where ˜f ∗ is defined in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  12  We are now ready to present an operation complexity of Algorithm 6, measured by the amount of  evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution of  (42), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Theorem 6 (Complexity of Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜fhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜f ∗  hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and ˜ghi be defined in (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (44) and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  L∇f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L∇ ˜  f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L ˜  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L∇˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' L˜g and G be given in Assumptions 3 and 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y0 and zǫ be given in  Algorithm 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and  ˜λ = 2ε−1[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ˆλ = 2ε−3[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (60)  �L = L∇f1 + 2ε−1L∇ ˜  f1 + 4ε−3(˜ghiL∇˜g + L2  ˜g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (61)  ˜α = min  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  4ε/(Dy�L)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜δ = (2 + ˜α−1)(D2  x + D2  y)�L + max  �  ε/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜α�L/4  �  D2  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �C =  4 max{1/(2�L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min{Dyε−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 4/(˜α�L)}}  [(3�L + ε/(2Dy))2/ min{�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ε/(2Dy)} + 3�L + ε/(2Dy)]−2ε5  ×  �  ˜δ + 2˜α−1[f ∗ − flow + 2ε−1( ˜fhi − ˜flow) + ε−3˜g2  hi + εDy/4 + �L(D2  x + D2  y)]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �K =  �  32(1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y0) − flow + εDy/4)�Lε−2 + 32ε3 �  1 + 4D2  y�L2ε−2�  − 1  �  + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  N =  ��  96  √  2  �  1 + (24�L + 4ε/Dy)�L−1��  + 2  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  Dy�Lε−1  �  × [( �K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Then Algorithm 6 outputs an approximate solution (xǫ, yǫ) of (42) satisfying  dist  �  ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0) + ∇˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)ˆλ  �  ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (62)  dist  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ρ(∂z ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) + ∇z˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)˜λ)  �  ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (63)  ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+∥ ≤ ε2G−1Dy(ε2 + L ˜  f)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (64)  |⟨˜λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)⟩| ≤ ε2G−2D2  y(ρ−1ǫ + L ˜  f)2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (65)  | ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ) − ˜f ∗(xǫ)| ≤ max  �  ε  �  1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y0) − flow + 2ε5(�L−1 + 4D2  y�Lε−2) + Dyε/4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ε2G−2D2  yL ˜  f(ε2 + εLf + L ˜  f)/2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (66)  ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+∥ ≤ ε2G−1Dy(ε2 + εLf + L ˜  f)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (67)  |⟨ˆλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)⟩| ≤ εG−2D2  y(ε2 + εLf + L ˜  f)2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (68)  after at most �  N evaluations of ∇f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ∇ ˜f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ∇˜g and proximal operator of f2 and ˜f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' One can observe from Theorem 6 that �L = O(ε−3), ˜α = O(ε2), ˜δ = O(ε−5), �C = O(ε−23),  �K = O(ε−5), and �  N = O(ε−7 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 6 enjoys an operation complexity of  O(ε−7 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2  and ˜f2, for finding an O(ε)-KKT solution (xǫ, yǫ) of (42) satisfying  dist  �  0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0) + ∇˜g(xǫ, yǫ)ˆλ  �  ≤ ε,  dist  �  0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)  �  ≤ ε,  ∥[˜g(xǫ, zǫ)]+∥ = O(ε2),  |⟨˜λ, ˜g(xǫ, zǫ)⟩| = O(ε2),  | ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| = O(ε),  ∥[˜g(xǫ, yǫ)]+∥ = O(ε2),  |⟨ˆλ, ˜g(xǫ, yǫ)⟩| = O(ε),  where ˜f ∗ is defined in (43), ˆλ, ˜λ ∈ Rl  + are defined in (60), zǫ is given in Algorithm 6 and ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5  Proof of the main results  In this section we provide a proof of our main results presented in Sections 2, 3 and 4, which are  particularly Theorems 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1  Proof of the main results in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='1  In this subsection we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before proceeding, we establish a lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let ¯H∗, ¯Hlow, ϑ0 and ¯δ be defined in (8), (10),  (14) and (16), and ¯α be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  ϑ0 ≤ ¯δ + 2¯α−1 � ¯H∗ − ¯Hlow  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (69)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (11) and (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' one has  G(¯z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯y0)  (12)  =  sup  x  �  ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯z0⟩ − p(x) − ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯y0) + q(¯y0)  �  (11)  =  max  x∈dom p  �  ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯z0⟩ − p(x) − ¯h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯y0) + σx  2 ∥x∥2 − σy  2 ∥¯y0∥2 + q(¯y0)  �  (8)(10)  ≤  max  x∈dom p  �  ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯z0⟩ + σx  2 ∥x∥2�  − σy  2 ∥¯y0∥2 − ¯Hlow  =  max  x∈dom p  σx  2 ∥x + σ−1  x ¯z0∥2 − σ−1  x  2 ∥¯z0∥2 − σy  2 ∥¯y0∥2 − ¯Hlow  ≤ σxD2  p  2  − σ−1  x  2 ∥¯z0∥2 − σy  2 ∥¯y0∥2 − ¯Hlow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (70)  where the last inequality follows from (9) and the fact that z0 ∈ −σxdom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Recall that (x∗, y∗) is the optimal solution of (8) and z∗ = −σxx∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It follows from (8), (11) and (12)  that  G(z∗, y∗)  (12)  =  sup  x  �  ⟨x, z∗⟩ − p(x) − ˆh(x, y∗) + q(y∗)  �  ≥ ⟨x∗, z∗⟩ − p(x∗) − ˆh(x∗, y∗) + q(y∗)  (11)  = ⟨x∗, z∗⟩ + σx  2 ∥x∗∥2 − σy  2 ∥y∗∥2 − p(x∗) − ¯h(x∗, y∗) + q(y∗)  =  − σ−1  x  2 ∥z∗∥2 − σy  2 ∥y∗∥2 − ¯H∗,  where the last equality follows from (8), the definition of (x∗, y∗), and z∗ = −σxx∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This together with  (13) and (70) implies that  P(¯z0, ¯y0) − P(z∗, y∗) = σ−1  x  2 ∥¯z0∥2 + σy  2 ∥¯y0∥2 + G(¯z0, ¯y0) − σ−1  x  2 ∥z∗∥2 − σy  2 ∥y∗∥2 − G(z∗, y∗)  ≤ σxD2  p/2 − ¯Hlow + ¯H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Notice from Algorithm 1 that z0 = z0  f = ¯z0 ∈ −σxdom p and y0 = y0  f = ¯y0 ∈ dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By these,  z∗ = −σxx∗, (9), (14), and the above inequality, one has  ϑ0  (14)  = η−1  z ∥¯z0 − z∗∥2 + η−1  y ∥¯y0 − y∗∥2 + 2¯α−1(P(¯z0, ¯y0) − P(z∗, y∗))  ≤ η−1  z σ2  xD2  p + η−1  y D2  q + 2¯α−1 �  σxD2  p/2 − ¯Hlow + ¯H∗�  = η−1  z σ2  xD2  p + ¯α−1σxD2  p + η−1  y D2  q + 2¯α−1 � ¯H∗ − ¯Hlow  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Hence, the conclusion follows from this, (16), ηz = σx/2 and ηy = min {1/(2σy), 4/(¯ασx)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose for contradiction that Algorithm 1 runs for more than ¯K outer itera-  tions, where ¯K is given in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By this and Algorithm 1, one can assert that (15) does not hold for  k = ¯K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' On the other hand, by (17) and [29, Theorem 3], one has  ∥(x  ¯  K, y  ¯  K) − (x∗, y∗)∥ ≤ (ˆζ−1 + L∇¯h)−1τ/2,  (71)  where (x∗, y∗) is the optimal solution of problem (8) and ˆζ is an input of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Notice from  Algorithm 1 that (ˆx ¯  K, ˆy ¯  K) results from the forward-backward splitting (FBS) step applied to the strongly  monotone inclusion problem 0 ∈ (∇x¯h(x, y), −∇y¯h(x, y)) + (∂p(x), ∂q(y)) at the point (x ¯  K, y ¯  K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then  14  follows from this, ˆζ = min{σx, σy}/L2  ∇¯h (see Algorithm 1), and the contraction property of FBS [5,  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='5] that ∥(ˆx ¯  K, ˆy ¯  K) − (x∗, y∗)∥ ≤ ∥(x ¯  K, y ¯  K) − (x∗, y∗)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using this and (71),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' we have  ∥ˆζ−1(x  ¯  K − ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K − y  ¯  K) − (∇¯h(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − ∇¯h(ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K))∥  ≤ ˆζ−1∥(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − (ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K)∥ + ∥∇¯h(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − ∇¯h(ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K)∥  ≤ (ˆζ−1 + L∇¯h)∥(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − (ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K)∥  ≤ (ˆζ−1 + L∇¯h)(∥(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y∗)∥ + ∥(ˆx  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆy  ¯  K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y∗)∥)  ≤ 2(ˆζ−1 + L∇¯h)∥(x  ¯  K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y  ¯  K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y∗)∥  (71)  ≤ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  where the second inequality uses the fact that ¯h is L∇¯h-smooth on dom p × dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It follows that (15)  holds for k = ¯K − 1, which contradicts the above assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, Algorithm 1 must terminate in at  most ¯K outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next show that the output of Algorithm 1 is a τ-stationary point of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' To this end, suppose  that Algorithm 1 terminates at some iteration k at which (15) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then by (4) and the definition  of ˆxk+1 and ˆyk+1 (see steps 23 and 24 of Algorithm 1), one has  0 ∈ ˆζ∂p(ˆxk+1) + ˆxk+1 − xk+1 + ˆζ∇x¯h(xk+1, yk+1),  0 ∈ ˆζ∂q(ˆyk+1) + ˆyk+1 − yk+1 − ˆζ∇y¯h(xk+1, yk+1),  which yield  ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂p(ˆxk+1), ˆζ−1(yk+1 − ˆyk+1) + ∇y¯h(xk+1, yk+1) ∈ ∂q(ˆyk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  These together with the definition of ¯H in (8) imply that  ∇x¯h(ˆxk+1, ˆyk+1) + ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂x ¯H(ˆxk+1, ˆyk+1),  ∇y¯h(ˆxk+1, ˆyk+1) − ˆζ−1(yk+1 − ˆyk+1) − ∇y¯h(xk+1, yk+1) ∈ ∂y ¯H(ˆxk+1, ˆyk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Using these and (15), we obtain  dist(0, ∂x ¯H(ˆxk+1, ˆyk+1))2 + dist(0, ∂y ¯H(ˆxk+1, ˆyk+1))2  ≤ ∥ˆζ−1(xk+1 − ˆxk+1) + ∇x¯h(ˆxk+1, ˆyk+1) − ∇x¯h(xk+1, yk+1)∥2  + ∥ˆζ−1(ˆyk+1 − yk+1) + ∇y¯h(ˆxk+1, ˆyk+1) − ∇y¯h(xk+1, yk+1)∥2  = ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥2 (15)  ≤ τ2,  which implies that dist(0, ∂x ¯H(ˆxk+1, ˆyk+1)) ≤ τ and dist(0, ∂y ¯H(ˆxk+1, ˆyk+1)) ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows from  these and Definition 2 that the output (ˆxk+1, ˆyk+1) of Algorithm 1 is a τ-stationary point of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Finally, we show that the total number of evaluations of ∇¯h and proximal operator of p and q  performed in Algorithm 1 is no more than ¯N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Indeed, notice from Algorithm 1 that  ¯α = min  �  1,  �  8σy/σx  �  , which implies that 2/¯α = max{2,  �  σx/(2σy)} and ¯α ≤  �  8σy/σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By these,  one has  max  � 2  ¯α, ¯ασx  4σy  �  ≤ max  �  2,  � σx  2σy  ,  �  8σy  σx  σx  4σy  �  = max  �  2,  � σx  2σy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (72)  In addition, by [29, Lemma 4], the number of inner iterations performed in each outer iteration of  Algorithm 1 is at most  T =  �  48  √  2  �  1 + 8L∇¯hσ−1  x  ��  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Then one can observe that the number of evaluations of ∇¯h and proximal operator of p and q performed  15  in Algorithm 1 is at most  (2T + 3) ¯K ≤  ��  96  √  2  �  1 + 8L∇¯hσ−1  x  ��  + 2  � �  max  � 2  ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ¯ασx  4σy  �  log 4 max{ηzσ−2  x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηy}ϑ0  (ˆζ−1 + L∇¯h)−2τ 2  �  +  (72)  ≤  ��  96  √  2  �  1 + 8L∇¯hσ−1  x  ��  + 2  � �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  � σx  2σy  �  log 4 max{ηzσ−2  x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηy}ϑ0  (ˆζ−1 + L∇¯h)−2τ 2  �  +  ≤  ��  96  √  2  �  1 + 8L∇¯hσ−1  x  ��  + 2  �  ×  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  � σx  2σy  �  log 4 max{1/(2σx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 4/(¯ασx)}} ϑ0  (L2  ∇¯h/ min{σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' σy} + L∇¯h)−2τ 2  �  +  (69)(18)  ≤  ¯N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  where the second last inequality follows from the definition of ηy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ηz and ˆζ in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, the  conclusion holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2  Proof of the main results in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='2  In this subsection we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Before proceeding, let {(xk, yk)}k∈K denote all the iterates  generated by Algorithm 2, where K is a subset of consecutive nonnegative integers starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also,  we define K − 1 = {k − 1 : k ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We first establish two lemmas and then use them to prove Theorem  2 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let {(xk, yk)}k∈K be generated by Algorithm 2, H∗, Dp,  Dq, Hlow, α, δ be defined in (6), (9), (23), (24) and (25), L∇h be given in Assumption 1, ǫ, ǫk be given  in Algorithm 2, and  Nk =  ��  96  √  2  �  1 + (24L∇h + 4ǫ/Dq) L−1  ∇h  ��  + 2  �  ×  �  max  �  2,  �  DqL∇h  ǫ  �  × log  4 max  �  1  2L∇h , min  �  Dq  ǫ ,  4  αL∇h  �� �  δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2  p)  �  [(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2  k  �  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (73)  Then for all 0 ≤ k ∈ K−1, (xk+1, yk+1) is an ǫk-stationary point of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Moreover, the total number of  evaluations of ∇h and proximal operator of p and q performed at iteration k of Algorithm 2 for generating  (xk+1, yk+1) is no more than Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let (x∗, y∗) be an optimal solution of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Recall that H, Hk and hk are given in (6), (20) and  (21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  Hk,∗ := min  x max  y  Hk(x, y) = min  x max  y  �  H(x, y) −  ǫ  4Dq  ∥y − ˆy0∥2 + L∇h∥x − xk∥2  �  ≤ max  y {H(x∗, y) + L∇h∥x∗ − xk∥2}  (6)(9)  ≤  H∗ + L∇hD2  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (74)  Moreover, by (9) and (23), one has  Hk,low :=  min  (x,y)∈dom p×dom q Hk(x, y) =  min  (x,y)∈dom p×dom q  �  H(x, y) −  ǫ  4Dq  ∥y − ˆy0∥2 + L∇h∥x − xk∥2  �  (23)  ≥ Hlow − max  y∈dom q  ǫ  4Dq  ∥y − ˆy0∥2 (9)  ≥ Hlow − ǫDq/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (75)  In addition, by Assumption 1 and the definition of hk in (21), it is not hard to verify that hk(x, y) is  L∇h-strongly-convex in x, ǫ/(2Dq)-strongly-concave in y, and (3L∇h + ǫ/(2Dq))-smooth on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Also, recall from Remark 2 that (xk+1, yk+1) results from applying Algorithm 1 to problem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The  conclusion of this lemma then follows by using (74) and (75) and applying Theorem 1 to (20) with τ = ǫk,  σx = L∇h, σy = ǫ/(2Dq), L∇¯h = 3L∇h + ǫ/(2Dq), ¯α = α, ¯δ = δ, ¯Hlow = Hk,low, and ¯H∗ = Hk,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  16  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let {xk}k∈K be generated by Algorithm 2, H, H∗ and  Dq be defined in (6) and (9), L∇h be given in Assumption 1, and ǫ, ǫ0 and ˆx0 be given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Then for all 0 ≤ K ∈ K − 1, we have  min  0≤k≤K ∥xk+1 − xk∥ ≤ maxy H(ˆx0, y) − H∗ + ǫDq/4  L∇h(K + 1)  + 2ǫ2  0(1 + 4D2  qL2  ∇hǫ−2)  L2  ∇h(K + 1)  ,  (76)  max  y  H(xK+1, y) ≤ max  y  H(ˆx0, y) + ǫDq/4 + 2ǫ2  0  �  L−1  ∇h + 4D2  qL∇hǫ−2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (77)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' For convenience of the proof, let  H∗  ǫ (x) = max  y  �  H(x, y) − ǫ∥y − ˆy0∥2/(4Dq)  �  ,  (78)  H∗  k(x) = max  y  Hk(x, y),  yk+1  ∗  = arg max  y  Hk(xk+1, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (79)  One can observe from these, (20) and (21) that  H∗  k(x) = H∗  ǫ (x) + L∇h∥x − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (80)  By this and Assumption 1, one can also see that H∗  k is L∇h-strongly convex on dom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition,  recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of problem (20) for all 0 ≤ k ∈ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows from Definition 2 that there exist some u ∈ ∂xHk(xk+1, yk+1) and v ∈ ∂yHk(xk+1, yk+1)  with ∥u∥ ≤ ǫk and ∥v∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Also, by (79), one has 0 ∈ ∂yHk(xk+1, yk+1  ∗  ), which together with  v ∈ ∂yHk(xk+1, yk+1) and ǫ/(2Dq)-strong concavity of Hk(xk+1, ·), implies that ⟨−v, yk+1 − yk+1  ∗  ⟩ ≥  ǫ∥yk+1 − yk+1  ∗  ∥2/(2Dq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This and ∥v∥ ≤ ǫk yield  ∥yk+1 − yk+1  ∗  ∥ ≤ 2ǫkDq/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (81)  In addition, by u ∈ ∂xHk(xk+1, yk+1), (20) and (21), one has  u ∈ ∇xh(xk+1, yk+1) + ∂p(xk+1) + 2L∇h(xk+1 − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (82)  Also, observe from (20), (21) and (79) that  ∂H∗  k(xk+1) = ∇xh(xk+1, yk+1  ∗  ) + ∂p(xk+1) + 2L∇h(xk+1 − xk),  which together with (82) yields  u + ∇xh(xk+1, yk+1  ∗  ) − ∇xh(xk+1, yk+1) ∈ ∂H∗  k(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By this and L∇h-strong convexity of H∗  k, one has  H∗  k(xk) ≥ H∗  k(xk+1) + ⟨u + ∇xh(xk+1, yk+1  ∗  ) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + L∇h∥xk − xk+1∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (83)  Using this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (80),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (81),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (83),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ∥u∥ ≤ ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and the Lipschitz continuity of ∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' we obtain  H∗  ǫ (xk) − H∗  ǫ (xk+1)  (80)  = H∗  k(xk) − H∗  k(xk+1) + L∇h∥xk − xk+1∥2  (83)  ≥ ⟨u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1  ∗  ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' xk − xk+1⟩ + 3L∇h∥xk − xk+1∥2/2  ≥  �  − ∥u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1  ∗  ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1)∥∥xk − xk+1∥ + L∇h∥xk − xk+1∥2/2  �  + L∇h∥xk − xk+1∥2  ≥ −(2L∇h)−1∥u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1  ∗  ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1)∥2 + L∇h∥xk − xk+1∥2  ≥ −L−1  ∇h∥u∥2 − L−1  ∇h∥∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1  ∗  ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk+1)∥2 + L∇h∥xk − xk+1∥2  ≥ −L−1  ∇hǫ2  k − L∇h∥yk+1 − yk+1  ∗  ∥2 + L∇h∥xk − xk+1∥2  (81)  ≥ −(L−1  ∇h + 4D2  qL∇hǫ−2)ǫ2  k + L∇h∥xk − xk+1∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  where the second and fourth inequalities follow from Cauchy-Schwartz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and the third inequal-  ity is due to Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and the fifth inequality follows from L∇h-Lipschitz continuity of ∇h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Summing up the above inequality for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', K yields  L∇h  K  �  k=0  ∥xk − xk+1∥2 ≤ H∗  ǫ (x0) − H∗  ǫ (xK+1) + (L−1  ∇h + 4D2  qL∇hǫ−2)  K  �  k=0  ǫ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (84)  17  In addition, it follows from (6), (9) and (78) that  H∗  ǫ (xK+1) = max  y  �  H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)  �  ≥ min  x max  y  H(x, y) − ǫDq/4 = H∗ − ǫDq/4,  H∗  ǫ (x0) = max  y  �  H(x0, y) − ǫ∥y − ˆy0∥2/(4Dq)  �  ≤ max  y  H(x0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (85)  These together with (84) yield  L∇h(K + 1) min  0≤k≤K ∥xk+1 − xk∥2 ≤ L∇h  K  �  k=0  ∥xk − xk+1∥2  ≤ max  y  H(x0, y) − H∗ + ǫDq/4 + (L−1  ∇h + 4D2  qL∇hǫ−2)  K  �  k=0  ǫ2  k,  which together with x0 = ˆx0, ǫk = ǫ0(k + 1)−1 and �K  k=0(k + 1)−2 < 2 implies that (76) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Finally, we show that (77) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed, it follows from (9), (78), (84), (85), ǫk = ǫ0(k + 1)−1, and  �K  k=0(k + 1)−2 < 2 that  max  y  H(xK+1, y)  (9)  ≤  max  y  �  H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)  �  + ǫDq/4  (78)  = H∗  ǫ (xK+1) + ǫDq/4  (84)  ≤ H∗  ǫ (x0) + ǫDq/4 + (L−1  ∇h + 4D2  qL∇hǫ−2)  K  �  k=0  ǫ2  k  (85)  ≤  max  y  H(x0, y) + ǫDq/4 + 2ǫ2  0(L−1  ∇h + 4D2  qL∇hǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows from this and x0 = ˆx0 that (77) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose for contradiction that Algorithm 2 runs for more than K + 1 outer  iterations, where K is given in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By this and Algorithm 2, one can then assert that (22) does not  hold for all 0 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' On the other hand, by (26) and (76), one has  min  0≤k≤K ∥xk+1 − xk∥2  (76)  ≤  maxy H(ˆx0, y) − H∗ + ǫDq/4  L∇h(K + 1)  + 2ǫ2  0(1 + 4D2  qL2  ∇hǫ−2)  L2  ∇h(K + 1)  (26)  ≤  ǫ2  16L2  ∇h  ,  which implies that there exists some 0 ≤ k ≤ K such that ∥xk+1 − xk∥ ≤ ǫ/(4L∇h), and thus (22) holds  for such k, which contradicts the above assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, Algorithm 2 must terminate in at most K + 1  outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Suppose that Algorithm 2 terminates at some iteration 0 ≤ k ≤ K, namely, (22) holds for such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We  next show that its output (xǫ, yǫ) = (xk+1, yk+1) is an ǫ-stationary point of (6) and moreover it satisfies  (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed, recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of (20), namely, it satisfies  dist(0, ∂xHk(xk+1, yk+1)) ≤ ǫk and dist(0, ∂yHk(xk+1, yk+1)) ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By these, (6), (20) and (21), there  exists (u, v) such that  u ∈ ∂xH(xk+1, yk+1) + 2L∇h(xk+1 − xk),  ∥u∥ ≤ ǫk,  v ∈ ∂yH(xk+1, yk+1) − ǫ(yk+1 − ˆy0)/(2Dq),  ∥v∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows that u−2L∇h(xk+1−xk) ∈ ∂xH(xk+1, yk+1) and v+ǫ(yk+1−ˆy0)/(2Dq) ∈ ∂yH(xk+1, yk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  These together with (9), (22), and ǫk ≤ ǫ0 ≤ ǫ/2 (see Algorithm 2) imply that  dist  �  0, ∂xH(xk+1, yk+1)  �  ≤ ∥u − 2L∇h(xk+1 − xk)∥ ≤ ∥u∥ + 2L∇h∥xk+1 − xk∥  (22)  ≤ ǫk + ǫ/2 ≤ ǫ,  dist  �  0, ∂yH(xk+1, yk+1)  �  ≤ ∥v + ǫ(yk+1 − ˆy0)/(2Dq)∥ ≤ ∥v∥ + ǫ∥yk+1 − ˆy0∥/(2Dq)  (9)  ≤ ǫk + ǫ/2 ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Hence, the output (xk+1, yk+1) of Algorithm 2 is an ǫ-stationary point of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, (28) holds  due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  18  Recall from Lemma 2 that the number of evaluations of ∇h and proximal operator of p and q  performed at iteration k of Algorithm 2 is at most Nk, respectively, where Nk is defined in (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also,  one can observe from the above proof and the definition of K that |K| ≤ K + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows that the  total number of evaluations of ∇h and proximal operator of p and q in Algorithm 2 is respectively no  more than �|K|−2  k=0  Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, to complete the rest of the proof of Theorem 2, it suffices to show  that �|K|−2  k=0  Nk ≤ N, where N is given in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' by (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (73) and |K| ≤ K + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' one has  |K|−2  �  k=0  Nk  (73)  ≤  K  �  k=0  ��  96  √  2  �  1 + (24L∇h + 4ǫ/Dq) L−1  ∇h  ��  + 2  �  ×  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  DqL∇h  ǫ  �  × log  4 max  �  1  2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min  �  Dq  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  αL∇h  �� �  δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2  p)  �  [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2  k  �  +  ≤  ��  96  √  2  �  1 + (24L∇h + 4ǫ/Dq) L−1  ∇h  ��  + 2  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  DqL∇h  ǫ  �  ×  K  �  k=0  \uf8eb  \uf8ed  \uf8eb  \uf8edlog  4 max  �  1  2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min  �  Dq  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  αL∇h  �� �  δ + 2α−1(H∗ − hlow + ǫDq/4 + L∇hD2  p)  �  [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2  k  \uf8f6  \uf8f8  +  + 1  \uf8f6  \uf8f8  ≤  ��  96  √  2  �  1 + (24L∇h + 4ǫ/Dq) L−1  ∇h  ��  + 2  �  max  �  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  �  DqL∇h  ǫ  �  ×  �  (K + 1)  �  log  4 max  �  1  2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' min  �  Dq  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  4  αL∇h  �� �  δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2  p)  �  [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2  0  �  +  + K + 1 + 2  K  �  k=0  log(k + 1)  �  (27)  ≤ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  where the last inequality is due to (27) and �K  k=0 log(k + 1) ≤ K log(K + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This completes the proof  of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='3  Proof of the main results in Section 3  In this subsection we prove Theorems 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We first establish a lemma below, which will be used to  prove Theorem 3 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (35) for  some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let f, ˜f, f ∗, flow and ρ be given in (29), (32) and (35), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  ˜f(xǫ, yǫ) ≤ min  z  ˜f(xǫ, z) + ρ−1(f ∗ − flow + 2ǫ),  f(xǫ, yǫ) ≤ f ∗ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-optimal solution of (35), it follows from Definition 1 that  max  z  Pρ(xǫ, yǫ, z) ≤ Pρ(xǫ, yǫ, zǫ) + ǫ,  Pρ(xǫ, yǫ, zǫ) ≤ min  x,y max  z  Pρ(x, y, z) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Summing up these inequalities yields  max  z  Pρ(xǫ, yǫ, z) ≤ min  x,y max  z  Pρ(x, y, z) + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (86)  Let (x∗, y∗) be an optimal solution of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows that f(x∗, y∗) = f ∗ and ˜f(x∗, y∗) =  minz ˜f(x∗, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By these and the definition of Pρ in (35), one has  max  z  Pρ(x∗, y∗, z) = f(x∗, y∗) + ρ( ˜f(x∗, y∗) − min  z  ˜f(x∗, z)) = f(x∗, y∗) = f ∗,  which implies that  min  x,y max  z  Pρ(x, y, z) ≤ max  z  Pρ(x∗, y∗, z) = f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (87)  19  It then follows from (35), (86) and (87) that  f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min  z  ˜f(xǫ, z))  (35)  = max  z  Pρ(xǫ, yǫ, z)  (86)(87)  ≤  f ∗ + 2ǫ,  which together with ˜f(xǫ, yǫ) − minz ˜f(xǫ, z) ≥ 0 implies that  f(xǫ, yǫ) ≤ f ∗ + 2ǫ,  ˜f(xǫ, yǫ) ≤ min  z  ˜f(xǫ, z) + ρ−1 (f ∗ − f(xǫ, yǫ) + 2ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The conclusion of this lemma directly follows from these and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let {(xk, yk, zk)} be generated by Algorithm 3 with limk→∞(ρk, ǫk) = (∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =  (x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' we now show that (x∗, y∗) is an optimal solution of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed, since (xk, yk, zk)  is an ǫk-optimal solution of (35) with ρ = ρk, it follows from Lemma 4 with (ρ, ǫ) = (ρk, ǫk) and  (xǫ, yǫ) = (xk, yk) that  ˜f(xk, yk) ≤ min  z  ˜f(xk, z) + ρ−1  k (f ∗ − flow + 2ǫk),  f(xk, yk) ≤ f ∗ + 2ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By the continuity of f and ˜f, limk→∞(xk, yk) = (x∗, y∗), limk→∞(ρk, ǫk) = (∞, 0), and taking limits as  k → ∞ on both sides of the above relations, we obtain that ˜f(x∗, y∗) ≤ minz ˜f(x∗, z) and f(x∗, y∗) ≤ f ∗,  which clearly imply that y∗ ∈ Argminz ˜f(x∗, z) and f(x∗, y∗) = f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, (x∗, y∗) is an optimal solution  of (29) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before proceeding, we establish a lemma below, which will be used to  prove Theorem 4 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let Dy,  flow, ˜f, ρ, and Pρ be given in (30), (32) and (35), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  dist  �  0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0)  �  ≤ ǫ,  dist  �  0, ρ∂ ˜f(xǫ, zǫ)  �  ≤ ǫ,  ˜f(xǫ, yǫ) ≤ min  z  ˜f(xǫ, z) + ρ−1(max  z  Pρ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35), it follows from Definition 2 that  dist  �  0, ∂x,yPρ(xǫ, yǫ, zǫ)  �  ≤ ǫ,  dist  �  0, ∂zPρ(xǫ, yǫ, zǫ)  �  ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Using these and the definition of Pρ in (35), we have  dist  �  0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0)  �  ≤ ǫ,  dist  �  0, ρ∂ ˜f(xǫ, zǫ)  �  ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In addition, by (35), we have  f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min  z  ˜f(xǫ, z)) = max  z  Pρ(xǫ, yǫ, z),  which along with (32) implies that  ˜f(xǫ, yǫ) − min  z  ˜f(xǫ, z) ≤ ρ−1(max  z  Pρ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  This completes the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Observe from (36) that problem (35) can be viewed as  min  x,y max  z  {Pρ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,  where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z), p(x, y) = f2(x) + ρ ˜f2(y), and q(z) = ρ ˜f2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence,  problem (35) is in the form of (6) with H = Pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By Assumption 3 and ρ = ε−1, one can see that h is  20  �L-smooth on its domain, where �L is given in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, notice from Algorithm 4 that ǫ0 = ε3/2 ≤ ε/2  due to ε ∈ (0, 1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 2 can be suitably applied to problem (35) with ρ = ε−1 for  finding an ǫ-stationary point (xǫ, yǫ, zǫ) of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In addition, notice from Algorithm 4 that ˜f(x0, y0) ≤ miny ˜f(x0, y)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using this, (35) and ρ = ε−1,  we obtain  max  z  Pρ(x0, y0, z) = f(x0, y0) + ρ( ˜f(x0, y0) − min  z  ˜f(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (88)  By this and (28) with H = Pρ, ǫ = ε, ǫ0 = ε3/2, ˆx0 = (x0, y0), Dq = Dy, and L∇h = �L, one has  Pρ(xǫ, yǫ, zǫ) ≤ max  z  Pρ(x0, y0, z) + εDy/4 + 2ε3(�L−1 + 4D2  y�Lε−2)  (88)  ≤ 1 + f(x0, y0) + εDy/4 + 2ε3(�L−1 + 4D2  y�Lε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows from this and Lemma 5 with ǫ = ε and ρ = ε−1 that (xǫ, yǫ, zǫ) satisfies (40) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next show that at most �  N evaluations of ∇f1, ∇ ˜f1, and proximal operator of f2 and ˜f2 are  respectively performed in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed, by (31), (32) and (35), one has  min  x,y max  z  Pρ(x, y, z)  (35)  = min  x,y {f(x, y) + ρ( ˜f(x, y) − min  z  ˜f(x, z))} ≥  min  (x,y)∈X ×Y f(x, y)  (32)  = flow,  (89)  min  (x,y,z)∈X ×Y×Y Pρ(x, y, z)  (35)  =  min  (x,y,z)∈X ×Y×Y{f(x, y) + ρ( ˜f(x, y) − ˜f(x, z))}  (31)(32)  ≥  flow + ρ( ˜flow − ˜fhi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (90)  For convenience of the rest proof, let  H = Pρ,  H∗ = min  x,y max  z  Pρ(x, y, z),  Hlow = min{Pρ(x, y, z)|(x, y, z) ∈ X × Y × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (91)  In view of these, (87), (88), (89), (90), and ρ = ε−1, we obtain that  max  z  H(x0, y0, z)  (88)  ≤ f(x0, y0) + 1,  flow  (89)  ≤ H∗ (87)  ≤ f ∗,  Hlow  (90)  ≥ flow + ρ( ˜flow − ˜fhi) = flow + ε−1( ˜flow − ˜fhi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =  �  D2x + D2y, Dq = Dy, ǫ0 = ε3/2, L∇h = �L,  α = ˆα, δ = ˆδ, and H, H∗, Hlow given in (91), we can conclude that Algorithm 4 performs at most  �  N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2 respectively for finding an approximate  solution (xǫ, yǫ) of problem (29) satisfying (40) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='4  Proof of the main results in Section 4  In this subsection we prove Theorems 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before proceeding, we define  r = G−1Dy(ρ−1ǫ + L ˜  f),  B+  r = {λ ∈ Rl  + : ∥λ∥ ≤ r},  (92)  where Dy is defined in (30), G is given in Assumption 4(iii), and ǫ and ρ are given in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In  addition, one can observe from (43) and (47) that  min  z  �Pµ(x, z) ≤ ˜f ∗(x)  ∀x ∈ X,  (93)  which will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next establish several technical lemmas that will be used to prove Theorem 5 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let Dy, L ˜  f, G, ˜f ∗, ˜f ∗  hi and B+  r be given in (30),  (43), (44), (92) and Assumption 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (i) ∥λ∗∥ ≤ G−1L ˜  fDy and λ∗ ∈ B+  r for all λ∗ ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the set of  optimal Lagrangian multipliers of problem (43) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  21  (ii) The function ˜f ∗ is Lipschitz continuous on X and ˜f ∗  hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (iii) It holds that  ˜f ∗(x) = max  λ  min  z  ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl  +(λ)  ∀x ∈ X,  (94)  where IRl  +(·) is the indicator function associated with Rl  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (i) Let x ∈ X and λ∗ ∈ Λ∗(x) be arbitrarily chosen, and let z∗ ∈ Y be such that (z∗, λ∗) is a pair  of primal-dual optimal solutions of (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows that  z∗ ∈ Argmin  z  ˜f(x, z) + ⟨λ∗, ˜g(x, z)⟩,  ⟨λ∗, ˜g(x, z∗)⟩ = 0,  ˜g(x, z∗) ≤ 0,  λ∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  The first relation above yields  ˜f(x, z∗) + ⟨λ∗, ˜g(x, z∗)⟩ ≤ ˜f(x, ˆzx) + ⟨λ∗, ˜g(x, ˆzx)⟩,  where ˆzx is given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By this and ⟨λ∗, ˜g(x, z∗)⟩ = 0, one has  ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗),  which together with λ∗ ≥ 0, (30) and Assumption 4 implies that  G  l  �  i=1  λ∗  i ≤ ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗) ≤ L ˜  f∥ˆzx − z∗∥ ≤ L ˜  fDy,  (95)  where the first inequality is due to Assumption 4(iii), and the third inequality follows from (30) and L ˜  f-  Lipschitz continuity of ˜f (see Assumption 4(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By (92), (95) and λ∗ ≥ 0, we have ∥λ∗∥ ≤ �l  i=1 λ∗  i ≤  G−1L ˜  fDy and λ∗ ∈ B+  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (ii) Recall from Assumptions 3(i) and 4(iii) that ˜f(x, ·) and ˜gi(x, ·), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' , l, are convex for any  given x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using this, (43) and the first statement of this lemma, we observe that  ˜f ∗(x) = min  z  max  λ∈B+  r  ˜f(x, z) + ⟨λ, ˜g(x, z)⟩  ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (96)  Notice from Assumption 4 that ˜f and ˜g are Lipschitz continuous on their domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then it is not hard to  observe that max{ ˜f(x, z)+⟨λ, ˜g(x, z)⟩|λ ∈ B+  r } is a Lipschitz continuous function of (x, z) on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By this and (96), one can easily verify that ˜f ∗ is Lipschitz continuous on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, the finiteness  of ˜f ∗  hi follows from (44), the continuity of ˜f ∗, and the compactness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (iii) One can observe from (43) that for all x ∈ X,  ˜f ∗(x) = min  z  max  λ  ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl  +(λ) ≥ max  λ  min  z  ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl  +(λ)  where the inequality follows from the weak duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, it follows from Assumption 3 that the  domain of ˜f(x, ·) is compact for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By this, (96) and the strong duality, one has  ˜f ∗(x) = max  λ∈B+  r  min  z  ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl  +(λ)  ∀x ∈ X,  which together with the above inequality implies that (94) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-optimal solution of  problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let flow, f, �Pµ, f ∗  µ, ρ and µ be given in (32), (42), (47), (48) and (50),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  �Pµ(xǫ, yǫ) ≤ min  z  �Pµ(xǫ, z) + ρ−1(f ∗  µ − flow + 2ǫ),  f(xǫ, yǫ) ≤ f ∗  µ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (97)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The proof follows from the same argument as the one for Lemma 4 with f ∗ and ˜f being replaced  by f ∗  µ and �Pµ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3-5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let ˜flow, f ∗, ˜f ∗  hi, f ∗  µ be defined in (31), (42), (44) and  (48), and Lf, ω and ¯θ be given in Assumptions 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that µ ≥ ( ˜f ∗  hi − ˜flow)/¯θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  f ∗  µ ≤ f ∗ + Lfω  ��  µ−1( ˜f ∗  hi − ˜flow)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (98)  22  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let x ∈ X, y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} and z∗ ∈ Argminz �Pµ(x, z) be arbitrarily chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  One can easily see from (47) and (93) that ˜f(x, z∗) + µ ∥[˜g(x, z∗)]+∥2 ≤ ˜f ∗(x), which together with (31)  and (44) implies that  ∥[˜g(x, z∗)]+∥2 ≤ µ−1( ˜f ∗  hi − ˜flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (99)  Since µ ≥ ( ˜f ∗  hi− ˜flow)/¯θ2, it follows from (99) that ∥[˜g(x, z∗)]+∥ ≤ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By this relation, y ∈ Argmin  z  { ˜f(x, z)|˜g(x, z) ≤  0} and Assumption 5, there exists some ˆz∗ such that  ∥y − ˆz∗∥ ≤ ω(∥[˜g(x, z∗)]+∥),  ˆz∗ ∈ Argmin  z  �  ˜f(x, z)  �� ∥[˜g(x, z)]+∥ ≤ ∥[˜g(x, z∗)]+∥  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (100)  In view of (47), z∗ ∈ Argminz �Pµ(x, z) and the second relation in (100), one can observe that ˆz∗ ∈  Argminz �Pµ(x, z), which along with (48) yields f(x, ˆz∗) ≥ f ∗  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, using (100) and Lf-Lipschitz conti-  nuity of f (see Assumption 4), we have  f(x, y) − f(x, ˆz∗) ≥ −Lf∥y − ˆz∗∥  (100)  ≥  −Lfω(∥[˜g(x, z∗)]+∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Taking minimum over x ∈ X and y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} on both sides of this relation, and  using (42), (99), f(x, ˆz∗) ≥ f ∗  µ and the monotonicity of ω, we can conclude that (98) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3-5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let ˜flow, flow, f, ˜f, f ∗, ˜f ∗, ˜f ∗  hi, ρ and µ be given in  (31), (32), (42), (43), (44) and (50), and Lf, ω and ¯θ be given in Assumptions 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Suppose that µ ≥ ( ˜f ∗  hi − ˜flow)/¯θ2 and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Then we have  f(xǫ, yǫ) ≤ f ∗ + Lfω  ��  µ−1( ˜f ∗  hi − ˜flow)  �  + 2ǫ,  ˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1�  f ∗ − flow + Lfω  ��  µ−1( ˜f ∗  hi − ˜flow)  �  + 2ǫ  �  ,  ∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1�  ˜f ∗(xǫ) − ˜flow + ρ−1�  f ∗ − flow + Lfω  ��  µ−1( ˜f ∗  hi − ˜flow)  �  + 2ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By (47), (93), and the first relation in (97), one has  ˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)  =  �Pµ(xǫ, yǫ)  (93)(97)  ≤  ˜f ∗(xǫ) + ρ−1(f ∗  µ − flow + 2ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows from this and (31) that  ˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1(f ∗  µ − flow + 2ǫ),  ∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1( ˜f ∗(xǫ) − ˜flow + ρ−1(f ∗  µ − flow + 2ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In addition, recall from (97) that f(xǫ, yǫ) ≤ f ∗  µ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' The conclusion of this lemma then follows from  these three relations and (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let {(xk, yk, zk)} be generated by Algorithm 5 with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =  (x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We now show that (x∗, y∗) is an optimal solution of problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' since (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zk) is  an ǫk-optimal solution of (50) with (ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' µ) = (ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' µk) and limk→∞ µk = ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' it follows from Lemma 9 with  (ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫ) = (ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ǫk) and (xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ) = (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk) that for all sufficiently large k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' one has  f(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk) ≤ f ∗ + Lfω  ��  µ−1  k ( ˜f ∗  hi − ˜flow)  �  + 2ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ˜f(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk) ≤ ˜f ∗(xk) + ρ−1  k  �  f ∗ − flow + Lfω  ��  µ−1  k ( ˜f ∗  hi − ˜flow)  �  + 2ǫk  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  ��[˜g(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yk)]+  ��2 ≤ µ−1  k  �  ˜f ∗(xk) − ˜flow + ρ−1  k  �  f ∗ − flow + Lfω  ��  µ−1  k ( ˜f ∗  hi − ˜flow)  �  + 2ǫk  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By the continuity of f, ˜f and ˜f ∗ (see Assumption 3(i) and Lemma 6(ii)), limk→∞(xk, yk) = (x∗, y∗),  limk→∞(ρk, µk, ǫk) = (∞, ∞, 0), limθ↓0 ω(θ) = 0, and taking limits as k → ∞ on both sides of the above  relations, we obtain that f(x∗, y∗) ≤ f ∗, ˜f(x∗, y∗) ≤ ˜f ∗(x∗) and [˜g(x∗, y∗)]+ = 0, which along with  (42) and (43) imply that f(x∗, y∗) = f ∗ and y∗ ∈ Argminz{ ˜f(x∗, z)|˜g(x∗, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, (x∗, y∗) is an  optimal solution of (42) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  23  We next prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Before proceeding, we establish several technical lemmas below, which  will be used to prove Theorem 6 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-stationary point of  problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let Dy, ˜g, ρ, µ, Lf, L ˜  f and G be given in (30), (42), (50) and Assumption  4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Then we have  ∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜  f),  (101)  ∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (102)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' We first prove (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition  2 that dist(0, ∂zPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, by (47) and (50), one has  Pρ,µ(x, y, z) = f(x, y) + ρ( ˜f(x, y) + µ ∥[˜g(x, y)]+∥2) − ρ( ˜f(x, z) + µ ∥[˜g(x, z)]+∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (103)  Using these relations, we have  dist  �  0, ∂z ˜f(xǫ, zǫ) + 2µ  l  �  i=1  [˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)  �  ≤ ρ−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Hence, there exists s ∈ ∂z ˜f(xǫ, zǫ) such that  ���s + 2µ  l  �  i=1  [˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)  ��� ≤ ρ−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (104)  Let ˆzxǫ and G be given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for  all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Notice that [˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i and ∥zǫ − ˆzxǫ∥ ≤ Dy due to (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (104),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  and the convexity of ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ·) and ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ·) for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' we have  ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ) + 2µG  l  �  i=1  [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+ ≤ ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ) − 2µ  l  �  i=1  [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ)  ≤ ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ) + 2µ  l  �  i=1  [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+(˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) − ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ))  ≤ ⟨s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ − ˆzxǫ⟩ + 2µ  l  �  i=1  [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+⟨∇z˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ − ˆzxǫ⟩  = ⟨s + 2µ  l  �  i=1  [˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+∇z˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ − ˆzxǫ⟩ ≤ ρ−1Dyǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (105)  where the first inequality is due to −˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˆzxǫ) ≥ G for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' the second inequality follows from  [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ)]+˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) ≥ 0 for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' the third inequality is due to s ∈ ∂z ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' zǫ) and the convexity of  ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ·) and ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ·) for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' and the last inequality follows from (30) and (104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In view of (30), (105),  and L ˜  f-Lipschitz continuity of ˜f(x, y) (see Assumption 4), one has  ∥[˜g(xǫ, zǫ)]+∥ ≤  l  �  i=1  [˜gi(xǫ, zǫ)]+  (105)  ≤  (2µG)−1(ρ−1Dyǫ + ˜f(xǫ, ˆzxǫ) − ˜f(xǫ, zǫ))  ≤ (2µG)−1(ρ−1Dyǫ + L ˜  f∥ˆzxǫ − zǫ∥)  (30)  ≤ (2µG)−1Dy(ρ−1ǫ + L ˜  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Hence, (101) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next prove (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition 2  that dist(0, ∂yPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' This together with (103) implies that  dist  �  0, ∂yf(xǫ, yǫ) + ρ∂y ˜f(xǫ, yǫ) + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+  �  ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Hence, there exists s ∈ ∂yf(xǫ, yǫ) and ˜s ∈ ∂y ˜f(xǫ, yǫ) such that  ∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (106)  24  Let ¯  A(xǫ, yǫ) = {i|˜gi(xǫ, yǫ) > 0, 1 ≤ i ≤ l}, ˆzxǫ and G be given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows  that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using these and the convexity of ˜gi(xǫ, ·) for all i, we  have  ⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩ =  �  i∈ ¯  A(xǫ,yǫ)  ⟨∇y˜gi(xǫ, yǫ), yǫ − ˆzxǫ⟩[gi(xǫ, yǫ)]+  ≥  �  i∈ ¯  A(xǫ,yǫ)  (˜gi(xǫ, yǫ) − ˜gi(xǫ, ˆzxǫ))[˜gi(xǫ, yǫ)]+  ≥  �  i∈ ¯  A(xǫ,yǫ)  G[˜gi(xǫ, yǫ)]+ = G  l  �  i=1  [˜gi(xǫ, yǫ)]+ ≥ G ∥[˜g(xǫ, yǫ)]+∥ ,  (107)  where the first inequality follows from the convexity of ˜g(xǫ, ·) and the second inequality is due to  −˜gi(xǫ, ˆzxǫ) ≥ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows from this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (106) and (107) that  Dyǫ ≥ ∥s + ρ˜s + 2ρµ∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+∥ · ∥yǫ − ˆzxǫ∥  ≥ ⟨s + ρ˜s + 2ρµ∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ − ˆzxǫ⟩  = ⟨s + ρ˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ − ˆzxǫ⟩ + 2ρµ⟨∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ − ˆzxǫ⟩  (107)  ≥ − (∥s∥ + ρ∥˜s∥) ∥yǫ − ˆzxǫ∥ + 2ρµG ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+∥  ≥ −(Lf + ρL ˜  f)Dy + 2ρµG ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ)]+∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (108)  where the last inequality follows from ∥yǫ − ˆzxǫ∥ ≤ Dy and the fact that ∥s∥ ≤ Lf and ∥˜s∥ ≤ L ˜  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' which  are due to (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' s ∈ ∂yf(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' ˜s ∈ ∂y ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' yǫ) and Assumption 4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By (108), one can immediately see  that (102) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Let f, ˜f, ˜g, Dy, flow, ˜f ∗ and Pρ,µ be given in (29),  (30), (32), (43) and (50), Lf, L ˜  f and G be given in Assumptions 3 and 4, (xǫ, yǫ, zǫ) be an ǫ-stationary  point of (50) for some ǫ > 0, and  ˜λ = 2µ[˜g(xǫ, zǫ)]+,  ˆλ = 2ρµ[˜g(xǫ, yǫ)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (109)  Then we have  dist  �  ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' 0) + ∇˜g(xǫ, yǫ)ˆλ  �  ≤ ǫ,  (110)  dist  �  0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)  �  ≤ ǫ,  (111)  ∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜  f),  (112)  |⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ (2µ)−1G−2D2  y(ρ−1ǫ + L ˜  f)2,  (113)  | ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max  �  ρ−1(max  z  Pρ,µ(xǫ, yǫ, z) − flow), (2µ)−1G−2D2  yL ˜  f(ρ−1ǫ + ρ−1Lf + L ˜  f)  �  ,  (114)  ∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜  f),  (115)  |⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ (2µ)−1ρG−2D2  y(ρ−1ǫ + ρ−1Lf + L ˜  f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (116)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it easily follows from (103), (109) and Definition  2 that (110) and (111) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, it follows from (101) and (102) that (112) and (115) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition,  in view of (109), (112) and (115), one has  |⟨˜λ, ˜g(xǫ, zǫ)⟩|  (109)  =  2µ ∥[˜g(xǫ, zǫ)]+∥2 (112)  ≤  (2µ)−1G−2D2  y(ρ−1ǫ + L ˜  f)2,  |⟨ˆλ, ˜g(xǫ, yǫ)⟩|  (109)  =  2ρµ ∥[˜g(xǫ, yǫ)]∥+∥2 (115)  ≤  (2µ)−1ρG−2D2  y(ρ−1ǫ + ρ−1Lf + L ˜  f)2,  and hence (113) and (116) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Also, observe from the definition of Pρ,µ in (50) that  �Pµ(xǫ, yǫ) − min  z  �Pµ(xǫ, z) = ρ−1(max  z  Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  25  Using this, (32), (47) and (93), we obtain that  ˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)  =  �Pµ(xǫ, yǫ) =  min  z  �Pµ(xǫ, z) + ρ−1(max  z  Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ))  (32)(93)  ≤  ˜f ∗(xǫ) + ρ−1(max  z  Pρ,µ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (117)  On the other hand, let λ∗ ∈ Rl  + be an optimal Lagrangian multiplier of problem (43) with x = xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It  then follows from Lemma 6(i) that ∥λ∗∥ ≤ G−1L ˜  fDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using these and (115), we have  ˜f ∗(xǫ) = min  y  �  ˜f(xǫ, y) + ⟨λ∗, ˜g(xǫ, y)⟩  �  ≤ ˜f(xǫ, yǫ) + ⟨λ∗, ˜g(xǫ, yǫ)⟩  ≤ ˜f(xǫ, yǫ) + ∥λ∗∥∥[˜g(xǫ, yǫ)]+∥ ≤ ˜f(xǫ, yǫ) + (2µ)−1G−2D2  yL ˜  f(ρ−1ǫ + ρ−1Lf + L ˜  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  By this and (117), one can see that (114) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We are now ready to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Observe from (51) that problem (50) can be viewed as  min  x,y max  z  {Pρ,µ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,  where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2, p(x, y) = f2(x) +  ρ ˜f2(y) and q(z) = ρ ˜f2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Hence, problem (50) is in the form of (6) with H = Pρ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By Assumption 3,  (45), (46), ρ = ε−1 and µ = ε−2, one can see that h is �L-smooth on its domain, where �L is given in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Also, notice from Algorithm 6 that ǫ0 = ε5/2 ≤ ε/2 = ǫ/2 due to ε ∈ (0, 1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Consequently, Algorithm 2  can be suitably applied to problem (50) with ρ = ε−1 and µ = ε−2 for finding an ǫ-stationary point  (xǫ, yǫ, zǫ) of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  In addition, notice from Algorithm 6 that �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Using this, (50) and  ρ = ε−1, we obtain  max  z  Pρ,µ(x0, y0, z)  (50)  = f(x0, y0) + ρ( �Pµ(x0, y0) − min  z  �Pµ(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (118)  By this and (28) with H = Pρ,µ, ǫ = ε, ǫ0 = ε5/2, ˆx0 = (x0, y0), Dq = Dy and L∇h = �L, one has  Pρ,µ(xǫ, yǫ, zǫ) ≤  max  z  Pρ,µ(x0, y0, z) + εDy/4 + 2ε5(�L−1 + 4D2  y�Lε−2)  (118)  ≤  1 + f(x0, y0) + εDy/4 + 2ε5(�L−1 + 4D2  y�Lε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It then follows from this and Lemma 11 with ǫ = ε, ρ = ε−1 and µ = ε−2 that (xǫ, yǫ, zǫ) satisfies the  relations (62)-(68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  We next show that at most �  N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 are  respectively performed in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' by (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (47) and (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' one has  min  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='y max  z  Pρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z)  (50)  = min  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='y {f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + ρ( �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) − min  z  �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z))} ≥  min  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='y)∈X ×Y f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y)  (32)  = flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (119)  min{Pρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z)|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z) ∈ X × Y × Y}  (50)  = min{f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + ρ( �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) − �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z))|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z) ∈ X × Y × Y}  (47)  = min{f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + ρ( ˜f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y) + µ∥[˜g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y)]+∥2 − ˜f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z) − µ∥[˜g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z)]+∥2)|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' z) ∈ X × Y × Y}  ≥ flow + ρ( ˜flow − ˜fhi) − ρµ˜g2  hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (120)  where the last inequality follows from (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' (32) and (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, let (x∗, y∗) be an optimal solution  of (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' It then follows that f(x∗, y∗) = f ∗ and [˜g(x∗, y∗)]+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' By these, (31), (47) and (50), one has  min  x,y max  z  Pρ,µ(x, y, z) ≤ max  z  Pρ,µ(x∗, y∗, z)  (50)  = f(x∗, y∗) + ρ  �  �Pµ(x∗, y∗) − min  z  �Pµ(x∗, z)  �  (47)  = f(x∗, y∗) + ρ( ˜f(x∗, y∗) + µ∥[˜g(x∗, y∗)]+∥2 − min  z { ˜f(x∗, z) + µ∥[˜g(x∗, z)]+∥2})  (31)  ≤ f ∗ + ρ( ˜fhi − ˜flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (121)  26  For convenience of the rest proof, let  H = Pρ,µ,  H∗ = min  x,y max  z  Pρ,µ(x, y, z),  Hlow = min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  (122)  In view of these, (118), (119), (120), (121), ρ = ε−1 and µ = ε−2, we obtain that  max  z  H(x0, y0, z)  (118)  ≤ f(x0, y0) + 1,  flow  (119)  ≤  H∗ (121)  ≤  f ∗ + ρ( ˜fhi − ˜flow) = f ∗ + ε−1( ˜fhi − ˜flow),  Hlow  (120)  ≥  flow + ρ( ˜flow − ˜fhi) − ρµ˜g2  hi = flow + ε−1( ˜flow − ˜fhi) − ε−3˜g2  hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =  �  D2x + D2y, Dq = Dy, ǫ0 = ε5/2, L∇h = �L,  α = ˜α, δ = ˜δ, and H, H∗, Hlow given in (122), we can conclude that Algorithm 6 performs at most �  N  evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 for finding an approximate solution  (xǫ, yǫ) of problem (42) satisfying (62)-(68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  6  Concluding remarks  For the sake of simplicity, first-order penalty methods are proposed only for problem (3) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  It would be interesting to extend them to problem (1) by using a standard technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=', see [39]) for  handling the constraint g(x, y) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' In addition, a single subproblem with static penalty and tolerance  parameters is solved in our methods (Algorithms 4 and 6), which may be conservative in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' To  make the methods possibly practically more efficient, it would be natural to modify them by solving  a sequence of subproblems with dynamic penalty and tolerance parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' These along with  numerical experiments will be left for the future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
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+page_content=' Torr, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Vedaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
+page_content=' Meta-learning with differentiable closed-form  solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}
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+	
+Machine learning prediction of the MJO extends beyond one month 
+Tamaki Suematsu1*, Kengo Nakai2*, Tsuyoshi Yoneda3, Daisuke Takasuka4,5, Takuya Jinno6,3, 
+Yoshitaka Saiki7, Hiroaki Miura6 
+ 
+1RIKEN Center for Computational Science; Kobe, Hyogo, 650-0047, Japan. 
+*Tamaki Suematsu. E-mail: tamaki.suematsu@riken.jp 
+2Faculty of Marine Technology, Tokyo University of Marine Science and Technology; Tokyo 135-8533, 
+Japan. 
+*Kengo Nakai. E-mail: knakai0@kaiyodai.ac.jp 
+3Graduate School of Economics, Hitotsubashi University; Kunitachi, Tokyo, 186-8601; Japan. 
+4Atmosphere and Ocean Research Institute, The University of Tokyo; Kashiwa, Chiba, 277-0882, Japan. 
+5Japan Agency for Marine-Earth Science and Technology; Yokohama, Kanagawa, 236-0001, Japan. 
+6Graduate School of Science, The University of Tokyo; Bunkyo-ku, Tokyo, 113-0033, Japan. 
+7Graduate School of Business Administration, Hitotsubashi University; Kunitachi, Tokyo, 186-8601, 
+Japan. 
+*These authors contributed equally to this work 
+ Abstract 
+The prediction of the Madden-Julian Oscillation (MJO), a massive tropical weather 
+event with vast global socio-economic impacts1,2, has been infamously difficult with 
+physics-based weather prediction models3–5. Here we construct a machine learning 
+model using reservoir computing technique that forecasts the real-time multivariate 
+MJO index (RMM)6, a macroscopic variable that represents the state of the MJO. 
+The training data was refined by developing a novel filter that extracts the 
+recurrency of MJO signals from the raw atmospheric data and selecting a suitable 
+
+2	
+time-delay coordinate of the RMM. The model demonstrated the skill to forecast 
+the state of MJO events for a month from the pre-developmental stages. Best-
+performing cases predicted the RMM sequence over two months, which exceeds the 
+expected inherent predictability limit of the MJO.   
+ 
+Main text 
+The Madden–Julian Oscillation (MJO) 7 is a massive cluster of convective activities in the tropics that 
+spans thousands of kilometers traveling slowly eastward from the Indian Ocean to the central Pacific in 
+approximately 20 to 60 days. It has far reaching influence on the global weather 1,8 and is recognized to 
+be one of the most important sources of predictability in extended-range weather forecast longer than 
+weeks 9,10. However, simulation of the MJO by physics-based dynamical numerical models (hereafter 
+dynamical models) has been shown to be notoriously difficult 3,11,12. It has only been since the mid 2000s 
+that the predictability of the MJO by dynamical models 13,14 exceeded that of empirical statistical models, 
+such as atmospheric-only linear inverse models, at two to three weeks 15. The current forecast skills of 
+dynamical models for MJO prediction lie between two to five weeks 16, which falls short of the expected 
+inherent predictability of the MJO estimated from multi-model ensemble studies at six to seven weeks 17. 
+Weather forecasts by dynamical models require physical parameterizations that incorporate the mean 
+effects of the sub-grid scale processes on the evolution of the grid-scale flows. However, parameterizations 
+are prone to empirical tuning and are inevitable sources of uncertainty of the dynamical models 18–20 
+because we have yet to determine the correct theoretical formulations and parameters of the statistical mean 
+states of microscopic processes. The difficulties of reducing the ambiguities in parameterizations continue 
+to be a major limiting factor for improving dynamical models and for identifying processes essential for 
+successful weather predictions. In contrast, machine learning models with relatively small number of neural 
+networks trained only by the time series of macroscopic variables has the potential to implicitly incorporate 
+
+3	
+the influence of microscopic variables on the macroscopic variables and to eliminate the parameterizations 
+that unsatisfactorily replicate the multiscale interactions between the unresolved and the resolved processes.  
+The effectiveness of the machine learning methods has been demonstrated in the fields of atmospheric 
+and climate science. Considerable progress has been achieved in areas for forecasting phenomena with 
+large socio-economic impacts such as the El Niño Southern Oscillation 21,22, Asian summer monsoons 23, 
+and hurricanes 24 at incomparably small computational costs compared to the dynamical models of the 
+atmosphere and the ocean. However, phenomena at the intraseasonal time scale have been difficult to 
+predict with the use of machine learning methods because complex interactions between processes with 
+various spatio-temporal scales that range from the convective to seasonal scales play important role in 
+determining its time evolution. Particularly with regards to the MJO forecasts, machine learning models 
+have been outperformed by dynamical models 25,26. 
+Here, we employ the reservoir computing method to advance the machine learning prediction of the MJO. 
+The reservoir computing method is a brain-inspired machine-learning technique that constructs a data-
+driven dynamical model (hereafter reservoir computing models) 27–31. By training on a time series of 
+macroscopic variables of high-dimensional dynamics, the method can efficiently predict time series and 
+frequency spectra of its chaotic behaviors 32,33. For example, it is useful for predicting the statistical 
+quantities of a chaotic fluid flow, which cannot be calculated directly from a numerical simulation of the 
+Navier–Stokes equation due to its high computational cost 34. In this study, we construct a reservoir 
+computing MJO prediction model, trained only by the time series of a macroscopic variable, with a 
+performance competitive with the state-of-the-art physics based dynamical models. Our results 
+demonstrate that the inherent predictability of some MJO cases is longer than have been expected from 
+studies by dynamical models 17.   
+
+4	
+ 
+Fig. 1. Schematic picture of reservoir computing. (A) In the training phase, the input data 𝒖(𝑡)	for time 
+𝑡 is fed to the reservoir state vector 𝒓(𝑡) through input weight matrix 𝑾!" and the output weight matrix 
+𝑾#$%	 is determined by reservoir computing. (B) In the prediction phase, the time evolution of 𝒖(𝑡) for 
+time 𝑡 + Δ	𝑡 is predicted as 𝒖*(𝑡 + Δ	𝑡) from the 𝑾∗
+#$% determined in the training phase.  
+A reservoir is a recurrent neural network whose internal parameters are adjusted to fit the data in the 
+training process 27,28. It is trained by feeding an input time series and fitting a linear function of the high 
+dimensional reservoir state vector to the desired output time series (Fig. 1). The construction of a reservoir 
+computing model simply assumes recurrent and deterministic property of the input time series and does 
+not involve any physical knowledge of the input data. The reservoir computing model of this study is 
+described by: 
+
+ATraining phase
+Reservoir
+statevector
+r(t)
+Training
+Input data
+Outputdata
+()n
+(+)n
+Win
+r(t+△t)
+W
+out
+B Prediction phase
+Reservoir
+statevector
+r(t
+Predicted
+Input data
+outputdata
+u(t)
+(+)
+Determined
+Win
+r(t+△t)
+W
+Nout5	
++𝒓(𝑡 + Δ	𝑡) = (1 − 𝛼)	𝒓(𝑡) + 𝛼 tanh4𝑨	𝒓	(𝑡) + 𝑾!"	𝒖(𝑡)6	
+𝒖*(𝑡 + Δ	𝑡) = 𝑾∗
+#$%	𝒓(𝑡 + Δ	𝑡)
+				
+(1) 
+where 𝒖(𝑡) ∈	ℝ(	  is both the input variable vector, 𝒓(𝑡) ∈	ℝ)	(𝑁 ≫ 𝑀)	 is the reservoir state vector, 
+𝑨 ∈	ℝ)×) , 𝑾!"	 ∈	ℝ)×(  , and 𝑾∗
+#$% ∈	ℝ(×)   are reservoir, input, and output weight matrices, 
+respectively, 𝛼	(0 < 𝛼 ≤ 1)		is the coefficient that adjusts the nonlinearity of the dynamics of 𝒓, and Δ	𝑡 
+is the time step. We define tanh(𝐪) = (tanh(q+) , tanh(q,) , … tanh(q)))- , for a vector 𝐪 =
+(q+, q,, … q))-	, where 𝑇 represents the transpose of a vector. 𝑾∗
+#$% is determined to satisfy 𝒖(𝑡) ≈
+𝑾#$%		𝒓(𝑡) using the training data 𝒖(𝑡), where 𝑾#$%	is the output weight matrix in the training phase. 
+Further details on the construction of the reservoir computing model are provided in the supplementary 
+materials. In the prediction phase, the predicted variable 𝒖*(𝑡 + Δ	𝑡) is obtained from 𝒖(𝑡) and 𝒓	(𝑡), using 
+eqn. (1) with fixed 𝑨, 𝑾!"	, and 𝑾∗#$%. A successful training will give 𝒖*(𝑡 + Δ	𝑡) that approximates the 
+desired unmeasured quantity 𝒖(𝑡 + Δ	𝑡).  
+The objective of our reservoir computing model is to predict the sequence of the Realtime Multi-variate 
+MJO (RMM) index 6, which is widely accepted as the standard proxy for diagnosing the state of an MJO1. 
+It captures the signals of the MJO as an envelope of convective activities coupled to planetary-scale 
+circulation from the leading pair of principal components (RMM1, RMM2) of the equatorial outgoing 
+longwave radiation and zonal winds at 850 hPa and 200 hPa. The RMM calculated from data without 
+smoothing in time 6 has been applied to machine learning prediction of the MJO 25,26; however, their 
+machine learning predictions were susceptible to degradation ascribed to noises in unsmoothed data from 
+atmospheric variabilities outside of the MJO timescale. Moreover, signals at time scales longer than the 
+MJO needs to be removed from the training data for the machine learning to identify recurring patterns. 
+Thus, to refine the RMM time series to train our reservoir computing model for MJO prediction, signals 
+outside of the MJO frequency range were removed from the raw data by an application of a filter that 
+approximately retains signals only between 20 days and 120 days frequency range 35. 
+
+6	
+The Lanczos filter 36, which is conventionally used to filter MJO signals, cannot be employed as the filter 
+to generate the training data for the machine learning. This is because the Lanczos filter, whose weights 
+are symmetric in time, requires data from both the past and the future to calculate a filtered value at a 
+certain point in time (Fig. 2 A). To resolve this problem, we design a novel filter, applicable for real-time 
+use, that does not require data from the future. The filter Ψ.!,.",0 is defined as: 
+Ψ.!,.",0	(𝑡)	= F.!,."	(𝑡)	
+sin(𝑡
+𝑐 − 𝜋)	
+(𝑡
+𝑐 − 𝜋)
+,	 
+where 
+F.!,."	(𝑡)	=		K
+sin L 𝑡
+𝑟1N
+𝑡
+	−	
+sin L 𝑡
+𝑟2N
+𝑡
+						(𝑡 ≤ 0)
+																		0																								(𝑡 > 0)
+				,										 
+and 𝑐 is a parameter that adjusts the center of the weights. We set the parameters as (𝑟1, 𝑟2, 𝑐) = (
+,3
+4 ,
++,3
+4 , 14) to remove the signals at frequencies lower than 120 days and higher than 20 days. The shape of 
+the filter function in real-space and in Fourier space is compared against that of the Lanczos filter in Fig. 2 
+A, B. In contrast to the Lanczos filter, the weights of the new filter vanish at 𝑡 = 0 and require only the 
+data from the past. The asymmetric weights of the new filter make it suitable for its application to real-
+time use such as filtering the input variable data for machine learning predictions. Due to the asymmetry, 
+the center of the weight of the new filter shifts backward only by approximately eight days. Hereafter, this 
+filter will be referred to as the real-time band-pass filter (RB filter). The RMM time series is calculated 
+from data filtered by the RB filter in this study (see methods for details). 
+
+7	
+ 
+Fig. 2. Comparison of Real-time Band-pass filter and Lanczos filter. The shape of the Lanczos (red) 
+and RB (blue) filters are shown in (A) real space and in (B) Fourier space. (C) Sample trajectories, from 
+1st December 2018 (indicated by circles) to 9th January 2019, for the original Wheeler and Hendon 2004 
+RMM index (WH04, grey), and RMM index filtered by Lanczos (red) and RB (blue) filters and RMM2 
+replaced by 12-day time-delay coordinate of RMM1. The axis for both RMM2 and 12-day time-delay 
+coordinate of RMM1 is labeled as RMM2 for brevity.  
+Furthermore, the MJO prediction is refined by employing the RMM phase space spanned by RMM1 and 
+its delay coordinate to diagnose the state of the MJO. That is, we replace RMM2 with the delay coordinate 
+of RMM1 and eliminate the model prediction of RMM2. This enhances the recurrency of the input data 
+and contributes to the robustness of the trained model. The modification is founded on the expectation that 
+RMM2 can be reconstructed from the delay coordinate of RMM1, since RMM1 and RMM2 are orthogonal 
+by definition and the trajectories of the projections of MJO events on the RMM phase space are near 
+circular. The delay time of the delay coordinate is chosen at 12 days, when the auto-correlation of RMM1 
+crosses zero for the first time. The correlation coefficient of RMM2 and 12-day delay coordinate variable 
+of RMM1 is 0.75. The trajectories of the RB-filtered and Lanczos filtered RMM sequences with RMM2 
+replaced by the 12-day delay coordinate of RMM1 is compared with the original Wheeler and Hendon 
+RMM (WH04) 6 in Fig. 2C. We confirm that the RB filter removes signals from slow variabilities and 
+noises as effectively as the Lanczos filter and that the trajectory of the RMM sequence on the phase space 
+with RMM2 replaced by the 12-day delay coordinate of RMM1 is similar to that of the WH04 RMM on 
+phase space spanned by RMM1 and RMM2. Thus, we focus on the RMM phase space spanned by RMM1 
+
+8	
+and its 12-day delay coordinate, which we will refer to as the machine learning RMM (ML-RMM) phase 
+space. The relevance of ML-RMM phase space to the conventional one spanned by RMM1 and RMM2 is 
+further discussed in the supplementary materials (Fig. S1). We will denote RMM1 and its time-delay 
+coordinate at time 𝑡 as 𝑎(𝑡) ≔ RMM1(𝑡) and 𝑏(𝑡): = 𝑎(𝑡 − 12).  
+It is known that a chaotic attractor can be reconstructed by some observable variables and its delay 
+coordinates 37,38. For the construction of a reservoir computing model, it is efficient to take the delay 
+coordinate variable with an appropriate delay time as the input when the number of observable variable is 
+smaller than the effective dimension of the attractor 33. A suitable delay time and dimension of the delay 
+coordinate of RMM1 is inferred by computing its auto-correlation function. Thus, an M-dimensional 
+delay coordinate vector of RMM1 is introduced as the input and output variable vector 𝒖 in Eq. (1): 
+𝒖(𝑡) = 4RMM1(𝑡), RMM1(𝑡 − 1Δ	𝜏), … , RMM1(𝑡 − (𝑀 − 1)Δ	𝜏)6,	 
+where ∆τ is the delay time, and (Δ	𝜏, 𝑀) = (6, 7). The pair of parameters are chosen so that the behavior 
+of 𝒖(𝑡) is deterministic and has recurrency, which are essential properties for successful modelling. The 
+reservoir model (Eq. (1)) of the RMM1 time sequence is constructed by determining 𝑾∗
+#$%. The time series 
+of the RMM1 data from 30th December 1986 to 29th December 2011 was used as the training data. The 
+optimal reservoir computing model was selected from evaluation of test cases of RMM1 forecasts 
+initialized from every day between 8th April 2014 and 6th July 2014. The selected model is used throughout 
+this study for all predictions. 
+The predicted variables at time t initialized from time p are denoted as 𝑎\(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝). We note that 
+𝑏^(𝑡, 𝑝) is predicted by the reservoir computing model simultaneously with 𝑎\(𝑡, 𝑝). The relationship 
+𝑏^(𝑡, 𝑝) =	𝑎\(𝑡 − 12, 𝑝) would hold only in an ideal case in which the model learns the delay coordinate of 
+the RMM1 perfectly. The reference time series in this case are 𝑎(𝑡) and 𝑏(𝑡). We compare the time series 
+of predicted variables against the reference time series using the bivariate correlation coefficient (COR) 
+16,39, defined by the equation:  
+
+9	
+COR(𝑞) ≔	
+∑
+L𝑎(𝑝 + 𝑞)𝑎\(𝑝 + 𝑞, 𝑝) + 	𝑏(𝑝 + 𝑞)𝑏^(𝑝 + 𝑞, 𝑝)N
+)
+56+
+c∑
+((𝑎(𝑝 + 𝑞))𝟐 + 4𝑏(𝑝 + 𝑞)6
+𝟐)
+)
+56+
++	c∑
+((𝑎\(𝑝 + 𝑞, 𝑝))𝟐 + (𝑏^(𝑝 + 𝑞, 𝑝))𝟐)
+)
+56+
+			,	 
+where 𝑞 is the forecast lead time. The COR corresponds to a covariance between the actual vector 
+(𝑎(𝑡), 𝑏(𝑡)) and the predicted vector  (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)), and is conventionally used to evaluate the MJO 
+prediction skills of dynamical and statistical models 40,41. Here, the 𝑁 = 2010 is the number of 
+predictions initialized for all days between 28th July 2014 and 28th January 2020. In Fig. 3, we show the 
+time series of COR(𝑞) for all predictions and three cases, the details of which will be described next. The 
+COR(𝑞) stays above 0.5 for 28 days for all predictions. The threshold value 0.5 is customarily adopted 
+for MJO prediction skill score 16. This signifies that the expectancy of the skill score of the model is at 
+three weeks for all days, including periods devoid of MJO activity. The forecast skill was evaluated as 
+three weeks in consideration of the approximate 8-day shift by the RB filter as discussed above.  
+It is customary to evaluate the skill of MJO predictions from the forecasts of periods when MJO events 
+are identified 14. We reevaluate the forecast skill of the reservoir model following the custom. Here, the 
+MJO events were identified as continuous sequences from phase 2 to phase 7 on the RMM phase space 
+spanned by RMM1 and its delay coordinate of 12 days 8,35 (See methods for details). For the predictions 
+initialized on three, five, and seven days before the onsets of MJO events, the COR remains above 0.5 for 
+38 days for all three cases. Considering the 8-day shift by the RB filter, this signifies that the constructed 
+model has the potential to skillfully forecast the time evolutions of the MJO events for 30 days. 
+Counterintuitively, the COR decays below 0.5 faster for the forecasts initialized three days before the 
+MJO onsets than those for five and seven days before the MJO onsets. We note however, that this is 
+consistent with the fact that the predictions reach the terminations of MJO events faster for predictions 
+that are initiated closer to the onsets.  
+The performance of the MJO prediction on individual cases are examined to illuminate the similarity 
+between the predicted and the actual trajectories of the RMM. Figure 4 compares the actual and predicted 
+
+10	
+trajectories on the ML-RMM phase, prediction errors, and the phase difference for the 10th (A, B, C), 26th 
+(D, E, F), and 50th (G, H, I) best performing cases in terms of mean error over the first 60 days of the 
+prediction. The three samples are chosen so that there are no overlaps in the forecast lead times. The errors 
+are measured by the distance between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted  (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) vectors. 
+The phase difference is evaluated from the cosine of the angle between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) 
+(cos(𝜃5(𝑡))). In all three cases, the error remains below 1.4, the threshold of the root mean square errors 
+of the predicted RMM adopted to evaluate the skills of climate simulations 42, well beyond two months (>
+75 days). The prediction also stays in phase (cos(𝜃5(𝑡)) > 0.7) for nearly two months (58, 83, and 76 
+days for the 10th, 26th, and 50th best case, respectively). We note that the rapid increases in phase differences 
+occur when the amplitude of the RMM1 decreases. This is reasonable considering that the RMM phases 
+become physically meaningless with diminishment of its amplitude. These results indicate that our 
+reservoir computing model can predict the state of some MJO events well beyond the estimated inherent 
+predictability limit of 7 weeks from dynamical model studies 17. This inference is supported by cases of 
+RMM1 predictions that skillfully forecast RMM1 phases for longer than 120 days (see Fig. S2).  
+
+11	
+ 
+  
+Fig. 3. Bivariate correlation coefficient. The mean bivariate correlation coefficient (COR) as a function 
+of forecast lead time (days) for all 2010 predictions (red), and for predictions initialized on 3 (navy), 5 
+(blue), and 7 (light blue) days before onsets of MJO events. The dash-dot and the dotted lines indicate 28th 
+and the 38th day in the forecast lead time, respectively.   
+We constructed a computationally inexpensive machine learning model, using the reservoir computing 
+technique, that is capable of month-long forecasts of the state of the MJO. This prediction skill is superior 
+to that of preceding machine learning methods and is matched only by physics-based dynamical models 
+that inevitably demand the state-of-the-art supercomputers 13,14,40,43. It is remarkable that our model was 
+trained only by the macroscopic time series of the RMM1. This signifies that intricate information of the 
+atmospheric and oceanic states that influences the MJO 44,45 were implicitly incorporated into the reservoir 
+state variables of the neural network. The extended prediction skill of our reservoir model is attributed to 
+the refinement of the training data. The signals from slow variability and high frequency noise were filtered 
+out from the input data with the RB filter to restrict the degrees of freedom of the training. This was 
+
+12	
+essential because it was necessary for the model to efficiently learn from merely 26 years of RMM1 data 
+with a limited number (< 100) of MJO events. To further enhance the efficacy of the reservoir computing, 
+we introduced the delay coordinate variable of RMM1 to employ suitably correlated variables as our 
+training data 33. It is of interest how the extension of the training data with accumulation of observational 
+data in the future will enhance the performance of the reservoir model.  
+The best performing forecasts by our reservoir model predicted the RMM time series for more than two 
+months. These results indicate that some MJO events are inherently predictable beyond the potential 
+predictability estimates made from dynamical model studies at seven weeks 17. This implies a possibility 
+for significant improvements in dynamical models to extend their lead time in MJO prediction, which is 
+crucial for reliable global weather forecasts. However, observations suggest that global warming alters the 
+characteristics of the MJO 8, meaning that the applicability of machine learning models trained on historical 
+data for MJO predictions could be undermined by climate change in the future. Furthermore, the reservoir 
+model of this study can only forecast the RMM sequence and cannot directly assess the impact of the MJO 
+on the midlatitude weather. Thus, dynamical models are expected to continue to be an imperative tool for 
+predicting and understanding the behaviors of our atmosphere and it is important to make the efforts to 
+exploit machine learning weather predictions to advance the dynamical models. 
+ 
+
+13	
+ 
+Fig. 4. Samples of best performing cases of RMM1 predictions and their errors. The (A, B, C) 10th, 
+(D, E, F) 26th, and (G, H, I) 50th best performing cases of RMM1 predictions initialized from 19th June 
+2019, 14th April 2018, and 9th December 2015 (indicated by the red dots), respectively. (A, D, G) The 
+trajectories of the actual (red) and the predicted (blue) RMM1 (𝑎(𝑡)	and 𝑎\(𝑡, 𝑝)) and its time-delay 
+coordinate (𝑏(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝)) are shown on the RMM phase space and (B, E, H) as a function of the 
+
+14	
+forecast lead time with the errors shown as the width of the gray shade. (C, F, I) The time series of 
+prediction errors measured by the cosines of the angles between  (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) 
+(cos(𝜃5(𝑡))	). The gray lines at ±0.7 in panels B, E, H indicate the threshold for the error = 1.4 and 
+cos(𝜃5(𝑡)) = 0.7 in panels C, F, I. 
+Acknowledgments 
+Funding: 	
+Japan Society for the Promotion of Science KAKENHI Grants, 21K13991 and 20H05730 (TS) 
+Japan Society for the Promotion of Science KAKENHI Grants, 22K17965 (KN) 
+Japan Science and Technology Agency PRESTO, 22724051(KN) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20H01819 (TY) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20H05728 and 22H01297 (DT) 
+MEXT Program for Promoting Researches on the Supercomputer Fugaku, hp210166 and hp220167 (DT) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20J11246 (TJ)  
+Japan Society for the Promotion of Science KAKENHI Grants, 19KK0067 and 21K18584 (YS) 
+"Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures" in Japan, 
+jh210027and jh220007, (YS) 
+HPCI System Research Project, hp210072 (YS) 
+Japan Society for the Promotion of Science KAKENHI Grants, 16H04048 and 20H05729 (HM) 
+ 
+Author contributions: 
+Conceptualization: KN, TS, DT, TY, HM, YS 
+Methodology: KN, TY, TS, HM, YS 
+Investigation: KN, TS, DT, HM, YS 
+Visualization: TS, KN 
+
+15	
+Funding acquisition: TS, HM, KN, DT, TJ, TY, YS 
+Supervision: HM, YS 
+Writing – original draft: TS, KN, HM, YS  
+Writing – review and editing: TS, HM, KN, DT, TJ, TY, YS 
+ 
+Competing interests: Authors declare that they have no competing interests.  
+ 
+Data availability  
+NOAA-OLR data are available at https://www.psl.noaa.gov/data/gridded/data.olrcdr.interp.html . 
+NCEP-DOE reanalysis data for zonal wind data are available at 
+https://psl.noaa.gov/data/gridded/data.ncep.reanalysis2.html . 
+The original Wheeler and Hendon 2004 RMM time series are available at 
+http://www.bom.gov.au/climate/mjo/ .  
+ 
+Code availability 
+All source codes of our reservoir model, filter-function of the RB filter, input and output data of the 
+reservoir computing, and the list of MJO events will be provided via zenodo before publication of this 
+work. 
+ 
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+2659–2674 (2007). 
+45. 
+Suematsu, T. & Miura, H. Changes in the Eastward Movement Speed of the Madden–Julian 
+
+20	
+Oscillation with Fluctuation in the Walker Circulation. J. Clim. 35, 211–225 (2022). 
+46. 
+Lukoševičius, M. & Jaeger, H. Reservoir computing approaches to recurrent neural network training. 
+Comput. Sci. Rev. 3, 127–149 (2009). 
+47. 
+Willoughby, R. A. Solutions of Ill-Posed Problems (A. N. Tikhonov and V. Y. Arsenin). SIAM Rev. 
+21, 266–267 (1979). 
+48. 
+Pathak, J., Lu, Z., Hunt, B. R., Girvan, M. & Ott, E. Using machine learning to replicate chaotic 
+attractors and calculate Lyapunov exponents from data. Chaos An Interdiscip. J. Nonlinear Sci. 27, 
+121102 (2017). 
+49. 
+Liebmann, B. & Smith, C. Description of a complete (interpolated) outgoing longwave radiation 
+dataset. Bull. Amer. Met. Soc. 77, 1275–1277 (1996). 
+50. 
+Kanamitsu, M. et al. NCEP–DOE AMIP-II Reanalysis (R-2). Bull. Am. Meteorol. Soc. 83, 1631–
+1644 (2002). 
+ 
+ 
+ 
+
+21	
+Methods  
+ 
+Reservoir computing technique  
+The method of determining the output weight matrix  𝑾∗
+#$%
+ of our reservoir computing machine learning 
+model is described. The time development of the reservoir state vector 𝒓(𝑙	Δ	𝑡) is determined by: 
+𝒓(𝑡 + Δ	𝑡) = (1 − 𝛼)	𝒓(𝑡) + 𝛼 tanh4𝑨	𝒓	(𝑡) + 𝑾!"	𝒖(𝑡)6	,
+(1 − M) 
+where {𝒖(𝑙	Δ	𝑡)}	(−𝐿3 ≤ 𝑙	 ≤ 𝐿) is the training time series data, 𝐿3 is the transient time, and 𝐿 is the time 
+length to determine 𝑾∗
+#$%
+ . For given random matrices  𝑨 and 𝑾!"	, we determine  𝑾#$%	 so that the 
+following quadratic form takes the minimum: 
+lm𝑾#$%		𝒓	(𝑙Δ𝑡) − 𝒖4(𝑙 + 1)Δ𝑡6m
+, + 𝛽[𝑇𝑟	(𝑾#$%	𝑾#$%	
+-
+)],
+8
+963
+	
+(2 − M) 
+where ‖𝒒‖, =	𝒒𝑻𝒒 for a vector 𝒒.  The minimizer is 
+𝑾∗
+#$% = 𝛿𝑼𝛿𝑹-(𝛿𝑹𝛿𝑹- + 𝛽	𝑰)2+		
+(3 − M) 
+where 𝑰 is the 𝑁 × 𝑁 identity matrix, 𝛿𝑹 and 𝛿𝑼 are the matrices whose 𝑙-th column is 𝒓	(𝑙Δ𝑡) and  
+𝒖	(𝑙Δ𝑡), respectively. (see Lukosevivcius and Jaeger (2009)46 P.140 and Tikhonov and Arsenin (1977)47 
+Chapter 1 for details).  
+Note that 𝑨 is chosen to have a maximum eigenvalue 𝜌	(|𝜌| < 1) in order for eqn. (2-M) to satisfy the 
+so called echo state property. It is known that addition of noises to the training time series data is potentially 
+useful in the construction of a data-driven model 22. Further details on the reservoir computing can be found 
+in preceding literatures 32–34,48. 
+The set of parameter values used to construct the reservoir computing model is shown in Table 1. We 
+determine 𝑾#$%	 using the training time series data 𝒖, which in this case is the RMM1 data from 30th 
+December 1986 (𝑡 = 0) to 29th December 2011 (𝑡 = 9131). The optimal reservoir computing model was 
+
+22	
+selected based on predictions of the RMM1 initialized every day between 8th April 2014 and 16th July 2014 
+by using 𝑾#$%	  for a given 𝑨  and 𝑾!"	 . We selected a model with the smallest prediction error 
+max
+;∈[+,>]|𝑢+(𝑡) − 𝑢\+(𝑡)| and max
+;∈[+,@3]|𝑢+(𝑡) − 𝑢\+(𝑡)|, where 𝑢+(𝑡) is the first component of 𝒖 and 𝑢\+(𝑡) is the 
+predicted variables of 𝑢+(𝑡). 
+ 
+parameter 
+value 
+𝑀 
+Dimension of input and output variables  
+7 
+𝑁 
+Dimension of reservoir state vector 
+1000 
+Δt 
+Time step of the model 
+1 (day) 
+ρ 
+Maximal eigenvalue of 𝑨  
+0.8 
+α 
+Nonlinearity degree in the model 
+0.7 
+β 
+Regularization parameter  
+0.01 
+Δτ 
+Delay-time for input and output variables  
+6 (day)  
+ 
+Table 1. The list of parameters and their values for the selected reservoir computing model 
+ 
+ 
+MJO detection method  
+ 
+The RMM is calculated from the combined empirical orthogonal functions of the outgoing 
+longwave radiation data from National Oceanic and Atmospheric Administration 49, and zonal wind data 
+from National Centers for Environmental Prediction-Department of Energy reanalysis 50.  With the 
+exception of replacing RMM2 with the 12-day time delay coordinate of RMM1, the orientation of RMM1 
+and definitions of the RMM phases follow the convention introduced by Wheeler and Hendon 6. The MJO 
+events were identified from time sequences that were projected on to the RMM index from phase 2 to phase 
+7, while satisfying the following four conditions employed by Suematsu and Miura (2018) 35: (1) Phases 
+do not skip forward nor recede backward by more than one phase. (2) The average amplitude is greater 
+than the critical value of 0.8. (3) Period of consecutive days with amplitude below 0.8 is less than 15. (4) 
+Transition from phase 2 to phase 7 takes 20 to 90 days.  
+ 
+
+23	
+Supplementary materials 
+ 
+Validity of the Machine Learning-RMM Phase Space  
+ 
+ 
+The relevance of employing the RMM phase space spanned by RMM1 and its delay coordinate, the 
+machine learning RMM (ML-RMM) phase space, to describe the MJO instead of that spanned by RMM1 
+and RMM2 is discussed. Conventionally, MJO events are identified using RMM phase space spanned by 
+the first two orthogonal functions, RMM1 and RMM2, of 20 - 120 day Lanczos bandpass filtered 36 
+outgoing longwave radiation and zonal winds at 850hPa and 200 hPa. Figure S1 compares the composites 
+of 1979 – 2020 December to February outgoing longwave radiation for each of the RMM phases spanned 
+by the conventional RMM1 and RMM2 with ML-RMM phase space.  
+The composites indicate that the definition of the ML-RMM phases (Fig. S1 B) can capture the 
+characteristic of the MJO convection to shift eastward from the Indian Ocean to the Western Pacific as 
+the conventional method (Fig. S1 A). We note however, that compared to the conventional method, the 
+convective signals over the Indian Ocean in the ML-RMM phase 2 is weaker. This may be a caveat to our 
+method that arises from replacing the RMM2 with the delay coordinate of RMM1, since the structure of 
+the eigenvector of RMM2 reflects the state of the atmosphere with deep convection over the Indian 
+Ocean (see Fig. 1 in 30). Despite the abovementioned concern, the method employed in this study is 
+capable of adequately tracking MJO events on the RMM phase space (Fig. 2C).   
+ 
+
+24	
+ 
+ 
+ 
+ 
+Fig. S1. Composites of 1979 – 2020 December to February outgoing longwave radiation for each of the 
+RMM phases on the (A) conventional RMM1 and RMM2 phase space calculated from 20-120 days 
+Lanczos bandpass filtered data and (B) on the ML-RMM calculated from the 20-120 days RB filtered 
+data. 
+ 
+ 
+
+25	
+Examples of best performing cases in terms of phase prediction 
+The best performing prediction cases in terms of ML-RMM phase predictions are examined. Figure S2 
+shows the best three cases evaluated by the first day the cosine of the phase difference between the actual 
+(𝑎(𝑡), 𝑏(𝑡)) and the predicted (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝))	 vector, cos(𝜃5(𝑡)), becomes less than 0.7. The first (Fig. 
+S2. A, D), second (Fig. S2. B, E) and third (Fig. S2.C, F) best performing cases are the predictions of ML-
+RMM initiated on 10th October 2017, 26th March 2019, and 18th April 2018, respectively. In all three 
+cases, the predictions stay in phase (cos(𝜃5(𝑡)) > 0.7) for longer than 120 days. However, there is a 
+tendency for the amplitudes to be underestimated in these cases, which leads to growth in error as measured 
+by the distance between (𝑎(𝑡), 𝑏(𝑡))  and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) from early stages of the predictions. Thus, 
+while the long predictability of the RMM phases over 120 days suggest the possibility of predicting the 
+MJO over a season (three months), overcoming the difficulty of accurately predicting the RMM phase and 
+amplitude simultaneously remains a challenge.   
+ 
+Fig. S2. The three best performing prediction of ML-RMM time series in terms of phase 
+prediction. (A, B, C) Predictions of ML-RMM initialized from (A, D) 2nd October 2017, (B, E) 18th 
+March 2019, and  (C, F) 10th April 2018, which are the three best prediction cases measured by the 
+cosine of the phase difference between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted vectors (cos(𝜃5(𝑡))).  
+
+26	
+(D, E, F) show the time evolution of the cos(𝜃5(𝑡)). The width of the grey shades in A, B, C indicates 
+the error measured by the distance between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)). 
+ 
+
diff --git a/CdAzT4oBgHgl3EQfTvxb/content/tmp_files/load_file.txt b/CdAzT4oBgHgl3EQfTvxb/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0a8966ef0921269fc8f3033cca642b36574f151b
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+++ b/CdAzT4oBgHgl3EQfTvxb/content/tmp_files/load_file.txt
@@ -0,0 +1,770 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf,len=769
+page_content='Machine learning prediction of the MJO extends beyond one month  Tamaki Suematsu1*, Kengo Nakai2*, Tsuyoshi Yoneda3, Daisuke Takasuka4,5, Takuya Jinno6,3,  Yoshitaka Saiki7, Hiroaki Miura6  1RIKEN Center for Computational Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Kobe, Hyogo, 650-0047, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Tamaki Suematsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E-mail: tamaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='suematsu@riken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='jp  2Faculty of Marine Technology, Tokyo University of Marine Science and Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Tokyo 135-8533,  Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Kengo Nakai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E-mail: knakai0@kaiyodai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='jp  3Graduate School of Economics, Hitotsubashi University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Kunitachi, Tokyo, 186-8601;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  4Atmosphere and Ocean Research Institute, The University of Tokyo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Kashiwa, Chiba, 277-0882, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  5Japan Agency for Marine-Earth Science and Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Yokohama, Kanagawa, 236-0001, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  6Graduate School of Science, The University of Tokyo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bunkyo-ku, Tokyo, 113-0033, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  7Graduate School of Business Administration, Hitotsubashi University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Kunitachi, Tokyo, 186-8601,  Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  These authors contributed equally to this work  Abstract  The prediction of the Madden-Julian Oscillation (MJO), a massive tropical weather  event with vast global socio-economic impacts1,2, has been infamously difficult with  physics-based weather prediction models3–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Here we construct a machine learning  model using reservoir computing technique that forecasts the real-time multivariate  MJO index (RMM)6, a macroscopic variable that represents the state of the MJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The training data was refined by developing a novel filter that extracts the  recurrency of MJO signals from the raw atmospheric data and selecting a suitable  2  time-delay coordinate of the RMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The model demonstrated the skill to forecast  the state of MJO events for a month from the pre-developmental stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Best-  performing cases predicted the RMM sequence over two months, which exceeds the  expected inherent predictability limit of the MJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Main text  The Madden–Julian Oscillation (MJO) 7 is a massive cluster of convective activities in the tropics that  spans thousands of kilometers traveling slowly eastward from the Indian Ocean to the central Pacific in  approximately 20 to 60 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It has far reaching influence on the global weather 1,8 and is recognized to  be one of the most important sources of predictability in extended-range weather forecast longer than  weeks 9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' However, simulation of the MJO by physics-based dynamical numerical models (hereafter  dynamical models) has been shown to be notoriously difficult 3,11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It has only been since the mid 2000s  that the predictability of the MJO by dynamical models 13,14 exceeded that of empirical statistical models,  such as atmospheric-only linear inverse models, at two to three weeks 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The current forecast skills of  dynamical models for MJO prediction lie between two to five weeks 16, which falls short of the expected  inherent predictability of the MJO estimated from multi-model ensemble studies at six to seven weeks 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Weather forecasts by dynamical models require physical parameterizations that incorporate the mean  effects of the sub-grid scale processes on the evolution of the grid-scale flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' However, parameterizations  are prone to empirical tuning and are inevitable sources of uncertainty of the dynamical models 18–20  because we have yet to determine the correct theoretical formulations and parameters of the statistical mean  states of microscopic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The difficulties of reducing the ambiguities in parameterizations continue  to be a major limiting factor for improving dynamical models and for identifying processes essential for  successful weather predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In contrast, machine learning models with relatively small number of neural  networks trained only by the time series of macroscopic variables has the potential to implicitly incorporate  3  the influence of microscopic variables on the macroscopic variables and to eliminate the parameterizations  that unsatisfactorily replicate the multiscale interactions between the unresolved and the resolved processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The effectiveness of the machine learning methods has been demonstrated in the fields of atmospheric  and climate science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Considerable progress has been achieved in areas for forecasting phenomena with  large socio-economic impacts such as the El Niño Southern Oscillation 21,22, Asian summer monsoons 23,  and hurricanes 24 at incomparably small computational costs compared to the dynamical models of the  atmosphere and the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' However, phenomena at the intraseasonal time scale have been difficult to  predict with the use of machine learning methods because complex interactions between processes with  various spatio-temporal scales that range from the convective to seasonal scales play important role in  determining its time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Particularly with regards to the MJO forecasts, machine learning models  have been outperformed by dynamical models 25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Here, we employ the reservoir computing method to advance the machine learning prediction of the MJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The reservoir computing method is a brain-inspired machine-learning technique that constructs a data-  driven dynamical model (hereafter reservoir computing models) 27–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' By training on a time series of  macroscopic variables of high-dimensional dynamics, the method can efficiently predict time series and  frequency spectra of its chaotic behaviors 32,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' For example, it is useful for predicting the statistical  quantities of a chaotic fluid flow, which cannot be calculated directly from a numerical simulation of the  Navier–Stokes equation due to its high computational cost 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In this study, we construct a reservoir  computing MJO prediction model, trained only by the time series of a macroscopic variable, with a  performance competitive with the state-of-the-art physics based dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Our results  demonstrate that the inherent predictability of some MJO cases is longer than have been expected from  studies by dynamical models 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Schematic picture of reservoir computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (A) In the training phase, the input data 𝒖(𝑡) for time  𝑡 is fed to the reservoir state vector 𝒓(𝑡) through input weight matrix 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" and the output weight matrix  𝑾#$%  is determined by reservoir computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (B) In the prediction phase, the time evolution of 𝒖(𝑡) for  time 𝑡 + Δ 𝑡 is predicted as 𝒖*(𝑡 + Δ 𝑡) from the 𝑾∗  #$% determined in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  A reservoir is a recurrent neural network whose internal parameters are adjusted to fit the data in the  training process 27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It is trained by feeding an input time series and fitting a linear function of the high  dimensional reservoir state vector to the desired output time series (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The construction of a reservoir  computing model simply assumes recurrent and deterministic property of the input time series and does  not involve any physical knowledge of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The reservoir computing model of this study is  described by:  ATraining phase  Reservoir  statevector  r(t)  Training  Input data  Outputdata  ()n  (+)n  Win  r(t+△t)  W  out  B Prediction phase  Reservoir  statevector  r(t  Predicted  Input data  outputdata  u(t)  (+)  Determined  Win  r(t+△t)  W  Nout5  +𝒓(𝑡 + Δ 𝑡) = (1 − 𝛼) 𝒓(𝑡) + 𝛼 tanh4𝑨 𝒓 (𝑡) + 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" 𝒖(𝑡)6  𝒖*(𝑡 + Δ 𝑡) = 𝑾∗  #$% 𝒓(𝑡 + Δ 𝑡)  (1)  where 𝒖(𝑡) ∈ ℝ(   is both the input variable vector, 𝒓(𝑡) ∈ ℝ) (𝑁 ≫ 𝑀)  is the reservoir state vector,  𝑨 ∈ ℝ)×) , 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" ∈ ℝ)×(  , and 𝑾∗  #$% ∈ ℝ(×)   are reservoir, input, and output weight matrices,  respectively, 𝛼 (0 < 𝛼 ≤ 1)  is the coefficient that adjusts the nonlinearity of the dynamics of 𝒓, and Δ 𝑡  is the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We define tanh(𝐪) = (tanh(q+) , tanh(q,) , … tanh(q)))- , for a vector 𝐪 =  (q+, q,, … q))- , where 𝑇 represents the transpose of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 𝑾∗  #$% is determined to satisfy 𝒖(𝑡) ≈  𝑾#$%  𝒓(𝑡) using the training data 𝒖(𝑡), where 𝑾#$% is the output weight matrix in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Further details on the construction of the reservoir computing model are provided in the supplementary  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In the prediction phase, the predicted variable 𝒖*(𝑡 + Δ 𝑡) is obtained from 𝒖(𝑡) and 𝒓 (𝑡), using  eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (1) with fixed 𝑨, 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" , and 𝑾∗#$%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A successful training will give 𝒖*(𝑡 + Δ 𝑡) that approximates the  desired unmeasured quantity 𝒖(𝑡 + Δ 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The objective of our reservoir computing model is to predict the sequence of the Realtime Multi-variate  MJO (RMM) index 6, which is widely accepted as the standard proxy for diagnosing the state of an MJO1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  It captures the signals of the MJO as an envelope of convective activities coupled to planetary-scale  circulation from the leading pair of principal components (RMM1, RMM2) of the equatorial outgoing  longwave radiation and zonal winds at 850 hPa and 200 hPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The RMM calculated from data without  smoothing in time 6 has been applied to machine learning prediction of the MJO 25,26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' however, their  machine learning predictions were susceptible to degradation ascribed to noises in unsmoothed data from  atmospheric variabilities outside of the MJO timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Moreover, signals at time scales longer than the  MJO needs to be removed from the training data for the machine learning to identify recurring patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Thus, to refine the RMM time series to train our reservoir computing model for MJO prediction, signals  outside of the MJO frequency range were removed from the raw data by an application of a filter that  approximately retains signals only between 20 days and 120 days frequency range 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  6  The Lanczos filter 36, which is conventionally used to filter MJO signals, cannot be employed as the filter  to generate the training data for the machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This is because the Lanczos filter, whose weights  are symmetric in time, requires data from both the past and the future to calculate a filtered value at a  certain point in time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 2 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' To resolve this problem, we design a novel filter, applicable for real-time  use, that does not require data from the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The filter Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' ",0 is defined as:  Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' ",0 (𝑡) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" (𝑡)  sin(𝑡  𝑐 − 𝜋)  (𝑡  𝑐 − 𝜋)  ,  where  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" (𝑡) =  K  sin L 𝑡  𝑟1N  𝑡  −  sin L 𝑡  𝑟2N  𝑡  (𝑡 ≤ 0)  0                        (𝑡 > 0)  ,  and 𝑐 is a parameter that adjusts the center of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We set the parameters as (𝑟1, 𝑟2, 𝑐) = (  ,3  4 ,  +,3  4 , 14) to remove the signals at frequencies lower than 120 days and higher than 20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The shape of  the filter function in real-space and in Fourier space is compared against that of the Lanczos filter in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 2  A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In contrast to the Lanczos filter, the weights of the new filter vanish at 𝑡 = 0 and require only the  data from the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The asymmetric weights of the new filter make it suitable for its application to real-  time use such as filtering the input variable data for machine learning predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Due to the asymmetry,  the center of the weight of the new filter shifts backward only by approximately eight days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Hereafter, this  filter will be referred to as the real-time band-pass filter (RB filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The RMM time series is calculated  from data filtered by the RB filter in this study (see methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Comparison of Real-time Band-pass filter and Lanczos filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The shape of the Lanczos (red)  and RB (blue) filters are shown in (A) real space and in (B) Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (C) Sample trajectories, from  1st December 2018 (indicated by circles) to 9th January 2019, for the original Wheeler and Hendon 2004  RMM index (WH04, grey), and RMM index filtered by Lanczos (red) and RB (blue) filters and RMM2  replaced by 12-day time-delay coordinate of RMM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The axis for both RMM2 and 12-day time-delay  coordinate of RMM1 is labeled as RMM2 for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Furthermore, the MJO prediction is refined by employing the RMM phase space spanned by RMM1 and  its delay coordinate to diagnose the state of the MJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' That is, we replace RMM2 with the delay coordinate  of RMM1 and eliminate the model prediction of RMM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This enhances the recurrency of the input data  and contributes to the robustness of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The modification is founded on the expectation that  RMM2 can be reconstructed from the delay coordinate of RMM1, since RMM1 and RMM2 are orthogonal  by definition and the trajectories of the projections of MJO events on the RMM phase space are near  circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The delay time of the delay coordinate is chosen at 12 days, when the auto-correlation of RMM1  crosses zero for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The correlation coefficient of RMM2 and 12-day delay coordinate variable  of RMM1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The trajectories of the RB-filtered and Lanczos filtered RMM sequences with RMM2  replaced by the 12-day delay coordinate of RMM1 is compared with the original Wheeler and Hendon  RMM (WH04) 6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We confirm that the RB filter removes signals from slow variabilities and  noises as effectively as the Lanczos filter and that the trajectory of the RMM sequence on the phase space  with RMM2 replaced by the 12-day delay coordinate of RMM1 is similar to that of the WH04 RMM on  phase space spanned by RMM1 and RMM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Thus, we focus on the RMM phase space spanned by RMM1  8  and its 12-day delay coordinate, which we will refer to as the machine learning RMM (ML-RMM) phase  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The relevance of ML-RMM phase space to the conventional one spanned by RMM1 and RMM2 is  further discussed in the supplementary materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We will denote RMM1 and its time-delay  coordinate at time 𝑡 as 𝑎(𝑡) ≔ RMM1(𝑡) and 𝑏(𝑡): = 𝑎(𝑡 − 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  It is known that a chaotic attractor can be reconstructed by some observable variables and its delay  coordinates 37,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' For the construction of a reservoir computing model, it is efficient to take the delay  coordinate variable with an appropriate delay time as the input when the number of observable variable is  smaller than the effective dimension of the attractor 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A suitable delay time and dimension of the delay  coordinate of RMM1 is inferred by computing its auto-correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Thus, an M-dimensional  delay coordinate vector of RMM1 is introduced as the input and output variable vector 𝒖 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (1):  𝒖(𝑡) = 4RMM1(𝑡), RMM1(𝑡 − 1Δ 𝜏), … , RMM1(𝑡 − (𝑀 − 1)Δ 𝜏)6,  where ∆τ is the delay time, and (Δ 𝜏, 𝑀) = (6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The pair of parameters are chosen so that the behavior  of 𝒖(𝑡) is deterministic and has recurrency, which are essential properties for successful modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The  reservoir model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (1)) of the RMM1 time sequence is constructed by determining 𝑾∗  #$%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The time series  of the RMM1 data from 30th December 1986 to 29th December 2011 was used as the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The  optimal reservoir computing model was selected from evaluation of test cases of RMM1 forecasts  initialized from every day between 8th April 2014 and 6th July 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The selected model is used throughout  this study for all predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The predicted variables at time t initialized from time p are denoted as 𝑎\\(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We note that  𝑏^(𝑡, 𝑝) is predicted by the reservoir computing model simultaneously with 𝑎\\(𝑡, 𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The relationship  𝑏^(𝑡, 𝑝) = 𝑎\\(𝑡 − 12, 𝑝) would hold only in an ideal case in which the model learns the delay coordinate of  the RMM1 perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The reference time series in this case are 𝑎(𝑡) and 𝑏(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We compare the time series  of predicted variables against the reference time series using the bivariate correlation coefficient (COR)  16,39, defined by the equation:  9  COR(𝑞) ≔  ∑  L𝑎(𝑝 + 𝑞)𝑎\\(𝑝 + 𝑞, 𝑝) +  𝑏(𝑝 + 𝑞)𝑏^(𝑝 + 𝑞, 𝑝)N  )  56+  c∑  ((𝑎(𝑝 + 𝑞))𝟐 + 4𝑏(𝑝 + 𝑞)6  𝟐)  )  56+  + c∑  ((𝑎\\(𝑝 + 𝑞, 𝑝))𝟐 + (𝑏^(𝑝 + 𝑞, 𝑝))𝟐)  )  56+  ,  where 𝑞 is the forecast lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The COR corresponds to a covariance between the actual vector  (𝑎(𝑡), 𝑏(𝑡)) and the predicted vector  (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)), and is conventionally used to evaluate the MJO  prediction skills of dynamical and statistical models 40,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Here, the 𝑁 = 2010 is the number of  predictions initialized for all days between 28th July 2014 and 28th January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 3, we show the  time series of COR(𝑞) for all predictions and three cases, the details of which will be described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The  COR(𝑞) stays above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='5 for 28 days for all predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The threshold value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='5 is customarily adopted  for MJO prediction skill score 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This signifies that the expectancy of the skill score of the model is at  three weeks for all days, including periods devoid of MJO activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The forecast skill was evaluated as  three weeks in consideration of the approximate 8-day shift by the RB filter as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  It is customary to evaluate the skill of MJO predictions from the forecasts of periods when MJO events  are identified 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We reevaluate the forecast skill of the reservoir model following the custom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Here, the  MJO events were identified as continuous sequences from phase 2 to phase 7 on the RMM phase space  spanned by RMM1 and its delay coordinate of 12 days 8,35 (See methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' For the predictions  initialized on three, five, and seven days before the onsets of MJO events, the COR remains above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='5 for  38 days for all three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Considering the 8-day shift by the RB filter, this signifies that the constructed  model has the potential to skillfully forecast the time evolutions of the MJO events for 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Counterintuitively, the COR decays below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='5 faster for the forecasts initialized three days before the  MJO onsets than those for five and seven days before the MJO onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We note however, that this is  consistent with the fact that the predictions reach the terminations of MJO events faster for predictions  that are initiated closer to the onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The performance of the MJO prediction on individual cases are examined to illuminate the similarity  between the predicted and the actual trajectories of the RMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Figure 4 compares the actual and predicted  10  trajectories on the ML-RMM phase, prediction errors, and the phase difference for the 10th (A, B, C), 26th  (D, E, F), and 50th (G, H, I) best performing cases in terms of mean error over the first 60 days of the  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The three samples are chosen so that there are no overlaps in the forecast lead times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The errors  are measured by the distance between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted  (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The phase difference is evaluated from the cosine of the angle between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝))  (cos(𝜃5(𝑡))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In all three cases, the error remains below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='4, the threshold of the root mean square errors  of the predicted RMM adopted to evaluate the skills of climate simulations 42, well beyond two months (>  75 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The prediction also stays in phase (cos(𝜃5(𝑡)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7) for nearly two months (58, 83, and 76  days for the 10th, 26th, and 50th best case, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We note that the rapid increases in phase differences  occur when the amplitude of the RMM1 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This is reasonable considering that the RMM phases  become physically meaningless with diminishment of its amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' These results indicate that our  reservoir computing model can predict the state of some MJO events well beyond the estimated inherent  predictability limit of 7 weeks from dynamical model studies 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This inference is supported by cases of  RMM1 predictions that skillfully forecast RMM1 phases for longer than 120 days (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bivariate correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The mean bivariate correlation coefficient (COR) as a function  of forecast lead time (days) for all 2010 predictions (red), and for predictions initialized on 3 (navy), 5  (blue), and 7 (light blue) days before onsets of MJO events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The dash-dot and the dotted lines indicate 28th  and the 38th day in the forecast lead time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  We constructed a computationally inexpensive machine learning model, using the reservoir computing  technique, that is capable of month-long forecasts of the state of the MJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This prediction skill is superior  to that of preceding machine learning methods and is matched only by physics-based dynamical models  that inevitably demand the state-of-the-art supercomputers 13,14,40,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It is remarkable that our model was  trained only by the macroscopic time series of the RMM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This signifies that intricate information of the  atmospheric and oceanic states that influences the MJO 44,45 were implicitly incorporated into the reservoir  state variables of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The extended prediction skill of our reservoir model is attributed to  the refinement of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The signals from slow variability and high frequency noise were filtered  out from the input data with the RB filter to restrict the degrees of freedom of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This was  12  essential because it was necessary for the model to efficiently learn from merely 26 years of RMM1 data  with a limited number (< 100) of MJO events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' To further enhance the efficacy of the reservoir computing,  we introduced the delay coordinate variable of RMM1 to employ suitably correlated variables as our  training data 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It is of interest how the extension of the training data with accumulation of observational  data in the future will enhance the performance of the reservoir model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The best performing forecasts by our reservoir model predicted the RMM time series for more than two  months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' These results indicate that some MJO events are inherently predictable beyond the potential  predictability estimates made from dynamical model studies at seven weeks 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This implies a possibility  for significant improvements in dynamical models to extend their lead time in MJO prediction, which is  crucial for reliable global weather forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' However, observations suggest that global warming alters the  characteristics of the MJO 8, meaning that the applicability of machine learning models trained on historical  data for MJO predictions could be undermined by climate change in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Furthermore, the reservoir  model of this study can only forecast the RMM sequence and cannot directly assess the impact of the MJO  on the midlatitude weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Thus, dynamical models are expected to continue to be an imperative tool for  predicting and understanding the behaviors of our atmosphere and it is important to make the efforts to  exploit machine learning weather predictions to advance the dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Samples of best performing cases of RMM1 predictions and their errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The (A, B, C) 10th,  (D, E, F) 26th, and (G, H, I) 50th best performing cases of RMM1 predictions initialized from 19th June  2019, 14th April 2018, and 9th December 2015 (indicated by the red dots), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (A, D, G) The  trajectories of the actual (red) and the predicted (blue) RMM1 (𝑎(𝑡) and 𝑎\\(𝑡, 𝑝)) and its time-delay  coordinate (𝑏(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝)) are shown on the RMM phase space and (B, E, H) as a function of the  14  forecast lead time with the errors shown as the width of the gray shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (C, F, I) The time series of  prediction errors measured by the cosines of the angles between  (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝))  (cos(𝜃5(𝑡)) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The gray lines at ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7 in panels B, E, H indicate the threshold for the error = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='4 and  cos(𝜃5(𝑡)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7 in panels C, F, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Acknowledgments  Funding:  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 21K13991 and 20H05730 (TS)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 22K17965 (KN)  Japan Science and Technology Agency PRESTO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 22724051(KN)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 20H01819 (TY)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 20H05728 and 22H01297 (DT)  MEXT Program for Promoting Researches on the Supercomputer Fugaku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' hp210166 and hp220167 (DT)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 20J11246 (TJ)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 19KK0067 and 21K18584 (YS)  "Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures" in Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  jh210027and jh220007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (YS)  HPCI System Research Project,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' hp210072 (YS)  Japan Society for the Promotion of Science KAKENHI Grants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 16H04048 and 20H05729 (HM)  Author contributions:  Conceptualization: KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Methodology: KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Investigation: KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Visualization: TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' KN  15  Funding acquisition: TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Supervision: HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Writing – original draft: TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Writing – review and editing: TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' HM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' KN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' TY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' YS  Competing interests: Authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Data availability  NOAA-OLR data are available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='psl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='gov/data/gridded/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='olrcdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='html .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  NCEP-DOE reanalysis data for zonal wind data are available at  https://psl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='gov/data/gridded/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='ncep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='reanalysis2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='html .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The original Wheeler and Hendon 2004 RMM time series are available at  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='bom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='au/climate/mjo/ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Code availability  All source codes of our reservoir model, filter-function of the RB filter, input and output data of the  reservoir computing, and the list of MJO events will be provided via zenodo before publication of this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Madden–Julian Oscillation: Bridging Weather and Climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 94,  1849–1870 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Roxy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Twofold expansion of the Indo-Pacific warm pool warps the MJO life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Nature 575, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Ahn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' MJO Propagation Across the Maritime Continent: Are CMIP6 Models Better Than  CMIP5 Models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 47, e2020GL087250 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Ahn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' MJO simulation in CMIP5 climate models: MJO skill metrics and process-oriented  diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 49, 4023–4045 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Vertical structure and physical processes of the Madden-Julian oscillation: Exploring  key model physics in climate simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 120, 4718–4748 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Wheeler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Hendon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' An All-Season Real-Time Multivariate MJO Index: Development  of an Index for Monitoring and Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 132, 1917–1932 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Madden, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Julian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Detection of a 40–50 Day Oscillation in the Zonal Wind in the  Tropical Pacific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 28, 702–708 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Roxy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Twofold expansion of the Indo-Pacific warm pool warps the MJO life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Nature 575, 647–651 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Waliser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Predictability and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Intraseasonal Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 389–423 (2005)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='1007/3-540-27250-X_12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Tseng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Maloney, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Barnes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The Consistency of MJO Teleconnection Patterns on  Interannual Time Scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 33, 3471–3486 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Ahn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' MJO simulation in CMIP5 climate models: MJO skill metrics and process-oriented  diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 49, 4023–4045 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Vertical structure and physical processes of the Madden-Julian oscillation: Exploring  key model physics in climate simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 120, 4718–4748 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Miura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Satoh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Nasuno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Noda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Oouchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A Madden-Julian Oscillation Event  Realistically Simulated by a Global Cloud-Resolving Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Science (80-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 318, 1763–1765 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  17  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Miyakawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Madden–Julian Oscillation prediction skill of a new-generation global model  demonstrated using a supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 5, (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Cavanaugh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The skill of atmospheric linear inverse models in hindcasting the Madden–  Julian Oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 44, 897–906 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Vitart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Waliser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Prediction of the Madden–Julian Oscillation: A Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  31, 9425–9443 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Neena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Waliser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Predictability of the Madden–Julian  Oscillation in the Intraseasonal Variability Hindcast Experiment (ISVHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 27, 4531–4543  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Randall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Wielicki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Measurements, Models, and Hypotheses in the Atmospheric  Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 78, 399–406 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Mauritsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Tuning the climate of a global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Earth Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 4, 0–01 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Mauritsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Roeckner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Tuning the MPI-ESM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='2 Global Climate Model to Improve the Match  With Instrumental Record Warming by Lowering Its Climate Sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Earth Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  12, e2019MS002037 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Gilani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Harlim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A Bayesian Machine Learning Algorithm for Predicting ENSO  Using Short Observational Time Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 48, e2021GL093704 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Ham, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Deep learning for multi-year ENSO forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nature 573, 568–  572 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Mitsui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Boers, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Seasonal prediction of Indian summer monsoon onset with echo state  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 16, 074024 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Weyn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Durran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Caruana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Cresswell-Clay, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Sub-Seasonal Forecasting With a  18  Large Ensemble of Deep-Learning Weather Prediction Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Earth Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 13,  e2021MS002502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Silini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Barreiro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Masoller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Machine learning prediction of the Madden-Julian oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  npj Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 4, 1–7 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Hagos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Leung, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Balaguru, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' An Observationally Trained Markov Model for  MJO Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 49, e2021GL095663 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Jaeger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The “echo state” approach to analysing and training recurrent neural networks-with an  erratum note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bonn, Ger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Ger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' GMD Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
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+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Jaeger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Haas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in  Wireless Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Science (80-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 304, 78–80 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Reservoir observers: Model-free inference of unmeasured variables in chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Chaos An Interdiscip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 27, 041102 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Hunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Ott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Attractor reconstruction by machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Chaos An Interdiscip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 28, 061104 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Arcomano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A Machine Learning-Based Global Atmospheric Forecast Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 47, e2020GL087776 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Nakai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Saiki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Machine-learning inference of fluid variables from data using reservoir  computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E 98, 023111 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Nakai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Saiki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Machine-learning construction of a model for a macroscopic fluid variable  using the delay-coordinate of a scalar observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' - Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S 14, 1079 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Kobayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Nakai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Saiki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Tsutsumi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dynamical system analysis of a data-driven  model constructed by reservoir computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E 104, 044215 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  19  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Suematsu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Miura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Zonal SST Difference as a Potential Environmental Factor Supporting  the Longevity of the Madden–Julian Oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 31, 7549–7564 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Duchon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Lanczos Filtering in One and Two Dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 18, 1016–1022  (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
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+page_content='  Sauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Yorke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rigorous verification of trajectories for the computer simulation of  dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nonlinearity 4, 961 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Takens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Detecting strange attractors in turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' in Dynamical Systems and Turbulence,  Warwick 1980 (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Young, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=') 366–381 (Springer Berlin Heidelberg, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Brunet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Derome, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Forecast Skill of the Madden–Julian Oscillation in Two Canadian  Atmospheric Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 136, 4130–4149 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Vitart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Evolution of ECMWF sub-seasonal forecast skill scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 140,  1889–1899 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Vitart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Waliser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Prediction of the Madden–Julian Oscillation: A Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  31, 9425–9443 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Rashid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Hendon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Wheeler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Alves, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Prediction of the Madden-Julian  oscillation with the POAMA dynamical prediction system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 36, 649–661 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Slingo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Ambitious partnership needed for reliable climate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  2022 126 12, 499–503 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Pohl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Matthews, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Observed Changes in the Lifetime and Amplitude of the Madden–Julian  Oscillation Associated with Interannual ENSO Sea Surface Temperature Anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 20,  2659–2674 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Suematsu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Miura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Changes in the Eastward Movement Speed of the Madden–Julian  20  Oscillation with Fluctuation in the Walker Circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 35, 211–225 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Lukoševičius, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Jaeger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Reservoir computing approaches to recurrent neural network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 3, 127–149 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Willoughby, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Solutions of Ill-Posed Problems (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Tikhonov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Arsenin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' SIAM Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  21, 266–267 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Pathak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Hunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=', Girvan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Ott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Using machine learning to replicate chaotic  attractors and calculate Lyapunov exponents from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Chaos An Interdiscip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 27,  121102 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Liebmann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' & Smith, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Description of a complete (interpolated) outgoing longwave radiation  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 77, 1275–1277 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Kanamitsu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' NCEP–DOE AMIP-II Reanalysis (R-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 83, 1631–  1644 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  21  Methods  Reservoir computing technique  The method of determining the output weight matrix  𝑾∗  #$%  of our reservoir computing machine learning  model is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The time development of the reservoir state vector 𝒓(𝑙 Δ 𝑡) is determined by:  𝒓(𝑡 + Δ 𝑡) = (1 − 𝛼) 𝒓(𝑡) + 𝛼 tanh4𝑨 𝒓 (𝑡) + 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" 𝒖(𝑡)6 ,  (1 − M)  where {𝒖(𝑙 Δ 𝑡)} (−𝐿3 ≤ 𝑙  ≤ 𝐿) is the training time series data, 𝐿3 is the transient time, and 𝐿 is the time  length to determine 𝑾∗  #$%  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' For given random matrices  𝑨 and 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" , we determine  𝑾#$%  so that the  following quadratic form takes the minimum:  lm𝑾#$%  𝒓 (𝑙Δ𝑡) − 𝒖4(𝑙 + 1)Δ𝑡6m  , + 𝛽[𝑇𝑟 (𝑾#$% 𝑾#$% )],  8  963  (2 − M)  where ‖𝒒‖, = 𝒒𝑻𝒒 for a vector 𝒒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The minimizer is  𝑾∗  #$% = 𝛿𝑼𝛿𝑹-(𝛿𝑹𝛿𝑹- + 𝛽 𝑰)2+  (3 − M)  where 𝑰 is the 𝑁 × 𝑁 identity matrix, 𝛿𝑹 and 𝛿𝑼 are the matrices whose 𝑙-th column is 𝒓 (𝑙Δ𝑡) and  𝒖 (𝑙Δ𝑡), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (see Lukosevivcius and Jaeger (2009)46 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='140 and Tikhonov and Arsenin (1977)47  Chapter 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Note that 𝑨 is chosen to have a maximum eigenvalue 𝜌 (|𝜌| < 1) in order for eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (2-M) to satisfy the  so called echo state property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' It is known that addition of noises to the training time series data is potentially  useful in the construction of a data-driven model 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Further details on the reservoir computing can be found  in preceding literatures 32–34,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The set of parameter values used to construct the reservoir computing model is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We  determine 𝑾#$%  using the training time series data 𝒖, which in this case is the RMM1 data from 30th  December 1986 (𝑡 = 0) to 29th December 2011 (𝑡 = 9131).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The optimal reservoir computing model was  22  selected based on predictions of the RMM1 initialized every day between 8th April 2014 and 16th July 2014  by using 𝑾#$%   for a given 𝑨  and 𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We selected a model with the smallest prediction error  max  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='∈[+,>]|𝑢+(𝑡) − 𝑢\\+(𝑡)| and max  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='∈[+,@3]|𝑢+(𝑡) − 𝑢\\+(𝑡)|, where 𝑢+(𝑡) is the first component of 𝒖 and 𝑢\\+(𝑡) is the  predicted variables of 𝑢+(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  parameter  value  𝑀  Dimension of input and output variables  7  𝑁  Dimension of reservoir state vector  1000  Δt  Time step of the model  1 (day)  ρ  Maximal eigenvalue of 𝑨  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='8  α  Nonlinearity degree in the model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7  β  Regularization parameter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='01  Δτ  Delay-time for input and output variables  6 (day)  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The list of parameters and their values for the selected reservoir computing model  MJO detection method  The RMM is calculated from the combined empirical orthogonal functions of the outgoing  longwave radiation data from National Oceanic and Atmospheric Administration 49, and zonal wind data  from National Centers for Environmental Prediction-Department of Energy reanalysis 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' With the  exception of replacing RMM2 with the 12-day time delay coordinate of RMM1, the orientation of RMM1  and definitions of the RMM phases follow the convention introduced by Wheeler and Hendon 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The MJO  events were identified from time sequences that were projected on to the RMM index from phase 2 to phase  7, while satisfying the following four conditions employed by Suematsu and Miura (2018) 35: (1) Phases  do not skip forward nor recede backward by more than one phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (2) The average amplitude is greater  than the critical value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (3) Period of consecutive days with amplitude below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='8 is less than 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (4)  Transition from phase 2 to phase 7 takes 20 to 90 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  23  Supplementary materials  Validity of the Machine Learning-RMM Phase Space  The relevance of employing the RMM phase space spanned by RMM1 and its delay coordinate, the  machine learning RMM (ML-RMM) phase space, to describe the MJO instead of that spanned by RMM1  and RMM2 is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Conventionally, MJO events are identified using RMM phase space spanned by  the first two orthogonal functions, RMM1 and RMM2, of 20 - 120 day Lanczos bandpass filtered 36  outgoing longwave radiation and zonal winds at 850hPa and 200 hPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Figure S1 compares the composites  of 1979 – 2020 December to February outgoing longwave radiation for each of the RMM phases spanned  by the conventional RMM1 and RMM2 with ML-RMM phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  The composites indicate that the definition of the ML-RMM phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S1 B) can capture the  characteristic of the MJO convection to shift eastward from the Indian Ocean to the Western Pacific as  the conventional method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S1 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' We note however, that compared to the conventional method, the  convective signals over the Indian Ocean in the ML-RMM phase 2 is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' This may be a caveat to our  method that arises from replacing the RMM2 with the delay coordinate of RMM1, since the structure of  the eigenvector of RMM2 reflects the state of the atmosphere with deep convection over the Indian  Ocean (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 1 in 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Despite the abovementioned concern, the method employed in this study is  capable of adequately tracking MJO events on the RMM phase space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  24  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Composites of 1979 – 2020 December to February outgoing longwave radiation for each of the  RMM phases on the (A) conventional RMM1 and RMM2 phase space calculated from 20-120 days  Lanczos bandpass filtered data and (B) on the ML-RMM calculated from the 20-120 days RB filtered  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  25  Examples of best performing cases in terms of phase prediction  The best performing prediction cases in terms of ML-RMM phase predictions are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Figure S2  shows the best three cases evaluated by the first day the cosine of the phase difference between the actual  (𝑎(𝑡), 𝑏(𝑡)) and the predicted (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝))  vector, cos(𝜃5(𝑡)), becomes less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The first (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' A, D), second (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' B, E) and third (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='C, F) best performing cases are the predictions of ML-  RMM initiated on 10th October 2017, 26th March 2019, and 18th April 2018, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' In all three  cases, the predictions stay in phase (cos(𝜃5(𝑡)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='7) for longer than 120 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' However, there is a  tendency for the amplitudes to be underestimated in these cases, which leads to growth in error as measured  by the distance between (𝑎(𝑡), 𝑏(𝑡))  and (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) from early stages of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' Thus,  while the long predictability of the RMM phases over 120 days suggest the possibility of predicting the  MJO over a season (three months), overcoming the difficulty of accurately predicting the RMM phase and  amplitude simultaneously remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The three best performing prediction of ML-RMM time series in terms of phase  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' (A, B, C) Predictions of ML-RMM initialized from (A, D) 2nd October 2017, (B, E) 18th  March 2019, and  (C, F) 10th April 2018, which are the three best prediction cases measured by the  cosine of the phase difference between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted vectors (cos(𝜃5(𝑡))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content='  26  (D, E, F) show the time evolution of the cos(𝜃5(𝑡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
+page_content=' The width of the grey shades in A, B, C indicates  the error measured by the distance between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfTvxb/content/2301.01254v1.pdf'}
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+STABILIZED WEIGHTED REDUCED ORDER METHODS FOR
+PARAMETRIZED ADVECTION-DOMINATED OPTIMAL CONTROL
+PROBLEMS GOVERNED BY PARTIAL DIFFERENTIAL EQUATIONS WITH
+RANDOM INPUTS
+FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3
+Abstract. In this work, we analyze Parametrized Advection-Dominated distributed Optimal
+Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simula-
+tions are initially based on a finite element method (FEM) discretization; moreover, a space-time
+approach is considered when dealing with unsteady cases. To overcome numerical instabilities that
+can occur in the optimality system for high values of the P´eclet number, we consider a Streamline
+Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach.
+We com-
+bine this method with the ROM framework in order to consider two possibilities of stabilization:
+Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parame-
+ters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach
+to deal with the issue of uncertainty quantification. Several quadrature techniques are used to
+derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo
+methods. We compare all the approaches analyzing relative errors between the FEM and ROM
+solutions and the computational efficiency based on the speedup-index.
+1. Introduction
+Here we present a numerical study concerning stabilized Parametrized Advection-Dominated Op-
+timal Control Problems (OCP(µ)s) with random inputs in a Reduced Order Methods (ROMs)
+framework. As a matter of fact, engineering and scientific applications often need very fast evalu-
+ations of the numerical solutions for many parameters that characterize the problem, for instance
+in real-time scenarios. A solution to these many-query situations can be to exploit the parameter
+dependence of the OCP(µ)s using ROMs [6, 24, 41, 40, 39]. This process is divided in two stages.
+The former is the offline phase, when many numerical solutions for different values of parameters
+are collected considering a first discretization of the OCP(µ), such a finite element method (FEM)
+one, called the high-fidelity or truth approximation. Then all parameter-independent components
+are calculated and stored, and reduced spaces are built. The latter is the online phase, when all
+parameter-dependent parts and, then, the reduced solutions are computed. More precisely, to deal
+with the randomness which is hidden in the parameters, we consider a modification of the Proper
+Orthogonal Decomposition (POD) that takes into account the probability distribution of the random
+inputs: the weighted POD (wPOD) [61, 60]. We apply this procedure in a partitioned approach,
+following good results shown in literature [30, 34, 53, 63]. As this algorithm aims to minimize the
+expectation of the square error between the truth and the ROM solutions, we can identify different
+types of weighted ROMs (wROMs) [11, 15, 13, 16, 17, 18, 49, 59, 61, 60] based on the chosen quad-
+rature rules. In this work, we will exploit Monte-Carlo and Quasi Monte-Carlo procedures, tensor
+rules based on Gauss-Jacobi and Clenshaw-Curtis quadrature techniques, and Smolyak isotropic
+sparse grids.
+1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland,
+email: fabio.zoccolan@epfl.ch
+2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy.
+email: maria.strazzullo@polito.it
+3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy.
+email: gianluigi.rozza@sissa.it
+1
+arXiv:2301.01975v1  [math.NA]  5 Jan 2023
+
+2
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+The optimization problem will always concern a linear-quadratic cost functional. We use FEM
+as the truth approximation, both for steady and unsteady problems. At a first level, FEM approx-
+imations of stochastic steady OCP(µ)s have been already presented, for example, in [26] consid-
+ering stochastic PDEs. In the parabolic case, we discretize time-dependency via a space-time ap-
+proach [25, 50, 51]. Concerning stabilization, we considered the Streamline Upwind Petrov–Galerkin
+(SUPG) [9, 28, 38] suitably combined with the ROM setting in an optimize-then-discretize approach.
+We exploit two possibilities: when stabilization only occurs in the offline phase, Offline-Only stabi-
+lization or when SUPG is applied in both phases, Offline-Online stabilization.
+Stabilized Advection-Dominated problems in a ROM framework without control are studied,
+for instance, in [37, 59], both for steady and unsteady cases. Instead, in [11] wROMs for generic
+OCP(µ)s are applied to experiments concerning environmental sciences. Instead, SUPG Advection-
+Dominated distributed OCP(µ)s are analyzed in a deterministic context in [63], both for elliptic and
+parabolic experiments. To the best of our knowledge, this is the first time that stabilized Advection-
+Dominated OCP(µ)s with random inputs are analyzed in a ROM context, both for elliptic and
+parabolic problems.
+This work is arranged as follows. In Section 2, we introduce linear-quadratic optimal control
+theory for PDEs and its FEM discretization. Section 3 firstly concerns the basic theory of SUPG
+stabilization for Advection-Dominated PDEs in an optimize-then-discretize approach [19], then the
+space-time procedure that will be used is presented. wROMs features will be illustrated in Sec-
+tion 4. Section 5 will concern numerical simulations. Two Advection-Dominated problems under
+distributed control and random inputs will be analyzed: the Graetz-Poiseuille Problem under ge-
+ometrical parametrization and the Propagating Front in a Square Problem. We will compare the
+wPOD procedures through relative errors between the FEM and the ROM solutions and computa-
+tional time considering the speedup-index. Finally, conclusions follow in Section 6.
+2. Problem Formulation for Random Input Optimal Control Problems
+2.1. Mathematical Setting. Let Ω be an open and bounded regular domain in R2, where ΓN and
+ΓD will indicate the portions of the boundary ∂Ω where Neumann and Dirichlet boundary conditions
+are imposed, respectively. With the symbol Ωobs ⊆ Ω the observation domain will be indicated, i.e.
+the subset of the domain where we seek the state variable to be similar to a desired solution profile
+yd ∈ Y , with Y Hilbert space, in a sense that will be specified later. For time-dependent problems we
+will also take into account the time interval (0, T) ⊂ R+. Let us consider a compact set P ⊆ Rp, for
+natural number p. We will call P and as the parameter space and a p-vector µ ∈ P is the parameter
+of our Parametric OCP(µ)s. As the setting is completely general, for instance µ can characterize
+our yd or geometrical and physical properties of the problem. Furthermore, we denote with B(Q, R)
+the space of linear continuous operators between Banach spaces Q and R.
+The triplet (A, F, P) will denote a complete probability space, composed by A, which is the set
+of outcomes ω ∈ A, F, that is a σ-algebra of events, and P : F → [0, 1] with P(A) = 1, which is
+the chosen probability measure. As dealing with random input OCP(µ)s, the parameter µ will be
+a real-valued random vector. In detail, µ : (A, F) → (Rp, B) is a measurable function, where B is
+the Borel σ-algebra on Rp. The distribution function of µ : A → P ⊂ Rp, being P the image of µ,
+is defined as Pµ : P → [0, 1] such that
+(1)
+∀µ ∈ P,
+Pµ(µ) = P(ω ∈ A : µ(ω) ≤ µ).
+Let dPµ(µ) denote the distribution measure of µ, i.e.,
+(2)
+∀H ⊂ P,
+P(µ ∈ H) =
+�
+H
+dPµ(µ).
+We assume that µ admits a Lebesgue density, i.e. dPµ(µ) is absolutely continuous with respect
+to the Lebesgue measure dµ. This practically means that there exists a probability density func-
+tion ρµ : P → R+ such that ρµ(µ)dµ = dPµ(µ). It is worth to notice that the measure space
+(P, B(P), ρµ(µ)dµ) is isometric to (A, F, P) under the random vector µ.
+The aim of this work is to analyze random input OCP(µ)s from the numerical point of view.
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+3
+Problem 2.1.1 (Random Input Parametric Optimal Control Problem). Consider the state equation
+E : Y × U → Q, with Y, U, and Q real Banach spaces, satisfying a set of boundary and/or initial
+conditions, and a real functional J : Y ×U → R. Then for Pµ-a.e. find the pair
+�
+y(µ), u(µ)
+�
+∈ X :=
+Y × U that minimizes cost functional J (y(µ), u(µ); µ) under the constraint E(y(µ), u(µ); µ) = 0.
+Let Xad be the set of all couples (y, u) solutions of E: we will only consider the case of full
+admissibility, i.e. when Xad = Y × U. Problem 2.1.1 looks for minimizers among all state-control
+pairs such that:
+min
+(y(µ),u(µ))∈Y ×U J (y(µ), u(µ); µ) s.t. E(y(µ), u(µ); µ) = 0.
+This can be achieved through the research of the critical points of the Lagrangian operator
+L : Y × U × Q∗ → R defined as:
+(3)
+L(y(µ), u(µ), p(µ); µ) = J (y(µ), u(µ); µ) + ⟨p(µ), E(y(µ), u(µ); µ)⟩Q∗Q,
+where p(µ) is a Lagrange multiplier belonging to the adjoint space Q∗, the dual space of Q. For
+the sake of notation we write y := y(µ), u := u(µ) and p := p(µ). In case that Pµ is the uniform
+distribution with support in P, then Problem 2.1.1 is called to be deterministic problem. In this
+work linear-quadratic problems will be involved.
+Definition 2.1.2 (Linear-Quadratic OCP(µ). Let us consider the bilinear forms m : Z × Z → R
+and n : U × U → R, which are symmetric and continuous, where Z is a Banach space called the the
+observation space. Fix α > 0, a constant called the penalization parameter and consider a quadratic
+objective functional J of the form
+(4)
+J (y, u; µ) = 1
+2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α
+2 n(u(µ), u(µ)),
+where O : Y → Z is a linear and bounded operator called the observation map and zd(µ) ∈ Z is
+the observed desired solution profile. Consider an affine map E defined as
+(5)
+E(y, u; µ) = A(µ)y + C(µ)u − f(µ),
+∀
+�
+y(µ), u(µ)
+�
+∈ Y × U,
+where A(µ) ∈ B(Y, Q), C(µ) ∈ B(U, Q) and f(µ) ∈ Q.
+Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem.
+For Linear-Quadratic OCP(µ)s well-posedness of Problem 2.1.2 yields [7, 8]. More precisely, the
+reader can refer to [10] to a comparison between the Lagrangian approach for the full-admissibility
+case and the adjoint one. Via the functional derivative of L, we obtain a optimality system to be
+solved to find the unique solution. In this case, this reads as finding (y, u, p) ∈ Y × U × Q∗ that
+satisfies [10],
+(6)
+�
+�
+�
+�
+�
+DyL(y, u, p; µ)(¯y) = 0 =⇒ m(Oy, O¯y; µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd; µ) ,
+∀¯y ∈ Y,
+DuL(y, u, p; µ)(¯u) = 0 =⇒ αn(u, ¯u; µ) + ⟨C∗(µ)p, ¯u⟩U ∗U = 0,
+∀¯u ∈ U,
+DpL(y, u, p; µ)(¯p) = 0 =⇒ ⟨¯p, A(µ)y + C(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q,
+∀¯p ∈ Q∗.
+In system (6), the first equation is called the adjoint equation, the second one is the gradient
+equation and the last one is state equation.
+Remark 2.1.3 (Notation). For the sake of notation, when Hilbert spaces will be taken into account,
+bilinear forms A, B and their adjoint counterparts will be indicate uniquely as:
+⟨A(µ)y, p⟩QQ∗ := a(y, p; µ)
+⟨C(µ)u, p⟩QQ∗ := c(u, p; µ).
+Remark 2.1.4 (Parabolic Problems). Concerning unsteady problems, one must add more hypotheses
+to the mathematical setting of Linear-Quadratic OCP(µ)ss to reach well-posedness. We will consider
+the following Hilbert spaces Y = L2(0, T; Y ), Y∗ = L2 (0, T; Y ∗), Z = L2 (0, T; Z), U = L2(0, T; U)
+with respective norms given by
+(7) ∥y∥2
+Y :=
+T
+�
+0
+∥y∥2
+Y dt,
+∥y∥2
+Y∗ :=
+T
+�
+0
+∥y∥2
+Y ∗dt,
+∥z∥2
+Z :=
+T
+�
+0
+∥z∥2
+Zdt,
+and
+∥u∥2
+U :=
+T
+�
+0
+∥u∥2
+Udt.
+
+4
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Then we define the Hilbert space Yt := {y ∈ Y
+s.t.
+∂ty ∈ Y∗}. For parabolic problems we will
+also consider the case of full-admissibility as Xad = Yt × U [5, 54, 55].
+2.2. High-Fidelity Discretization. In this work, the discretization of the optimality sistem (6)
+follows an one shot or all-at-once approach [25, 50, 51]. When we will consider Advection-Dominated
+OCP(µ)s, a stabilization technique will be also needed. Therefore, a SUPG method will be applied
+in a optimize-then-discretize approach, as we will see in Section 3.
+This means that firstly the
+optimality conditions are computed obtaining system (6) and then we discretize and stabilize it.
+Concerning numerical implementation, we use a FEM discretization for all three variables, where
+Th is a regular triangularization on Ω. Its elements K are triangles and the parameter h denotes the
+mesh size, i.e. the maximum diameter of an element of the chosen grid. In addition, we define
+Ωh := int
+� �
+K∈Th
+K
+�
+,
+as a quasi-uniform mesh for Ω. Considering Pr(K) as the space of polynomials of degree at most
+equal to r defined on K and defining
+XN ,r =
+�
+qN ∈ C(¯Ω) : qN
+|K ∈ Pr(K), ∀K ∈ Th
+�
+we set Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and
+�
+QN �∗ = Q∗ ∩ XN ,r. In this work, the numerical
+implementation will always made by a P1-FEM approximation and the same triangulation Th for
+Y N , U N , and
+�
+QN �∗. A similar discussion can be made for time-dependent problem, as we will
+see in Section 3.2. This first discretization procedure will be indicated as the truth or high-fidelity
+approximation.
+From now on, Y, U, Q will be always Hilbert spaces and we will consider the Identity restricted to
+our observation domain Ωobs as the Observation function O for both steady and unsteady problems.
+Therefore, Z = Y for steady problems and Z = Y for unsteady ones are assumed. Our desired state
+will be denoted by yd and with the same symbol will also indicate its FEM discretization.
+3. SUPG stabilization for Advection-Dominated OCP(µ)s
+In this work we only deal with Advection-Diffusion equations.
+Definition 3.0.1 (Advection-Diffusion Equations). Let us take into account the following problem:
+(8)
+T(µ)y := −γ(µ)∆y + η(µ) · ∇y = f(µ) in Ω ⊂ R2,
+where suitable boundary conditions are applied on ∂Ω. In addition, we require that:
+• the diffusion coefficient γ : Ω → R is uniformly bounded, i.e. there exists γmax, γmin > 0 such
+that
+(9)
+P
+�
+ω ∈ A : γmin < γ(x; µ) < γmax ∀x ∈ Ω
+�
+= 1.
+• the advection field η : Ω → R2 belongs to (L∞(Ω))2 for a.e. µ ∈ P. More precisely, for a.e.
+µ the following inequality holds: 0 ≥ div η(x) ≥ −ϑ, ∀x ∈ Ω, with ϑ ∈ R+
+0 ;
+• f : Ω → R is an L2(Ω)-function for a.e. µ; in addition, f has bounded second moments with
+respect to the integral along A and Ω.
+With these hypotheses, Problem (8) is called Advection-Diffusion problem and the operator T(µ)y :=
+−γ(µ)∆y + η(µ) · ∇y is said the Advection-Diffusion operator.
+For more details regarding the well-posedness and theoretical results of Stochastic Advection-
+Diffusion OCP(µ)s, we refer to [14, 13].
+Definition 3.0.2 (P´eclet number and Advection-Dominated problem). Let us consider the FEM
+discretization related to an Advection-Diffusion problem and its regular triangulation Th. For any
+element K ∈ Th, the local P´eclet number is defined as [42, 38]:
+(10)
+PeK(x) := |η(x)|hK
+2γ(x)
+∀x ∈ K,
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+5
+where hK is the diameter of K. If PeK(x) > 1 ∀x ∈ K, ∀K ∈ Th, we say to study an Advection-
+Dominated problem.
+3.1. Setting for Stabilized Advection-Dominated OCP(µ)s - Steady case. Numerical in-
+stabilities can appear, when dealing with Advection-Dominated OCP(µ)s, i.e. when |η(µ)| ≫ γ. In
+order to adjust this unpleasant behaviour without modifying the mesh size, we use the Streamline
+upwind/Petrov Galerkin (SUPG) method [9, 27, 28, 42] in a optimize-then-discretize approach [19].
+This assures the strongly consistency of the optimality system [19]. For the sake of simplicity, we
+define our Advection-Dominated problem on H1
+0(Ω) and we do not indicate the parameter depen-
+dence. We denote with T ∗ the adjoint operator related to T. This last operator can be split into its
+symmetric and skew-symmetric parts as T = TS + TSS [42], where:
+(11)
+symmetric part: TSy = −γ∆y − 1
+2(div η)y,
+∀y ∈ H1
+0(Ω),
+skew-symmetric part: TSSy = η · ∇y + 1
+2(div η)y,
+∀y ∈ H1
+0(Ω).
+This two parts can be immediately recovered using the formulae:
+(12)
+TS = T + T ∗
+2
+,
+TSS = T − T ∗
+2
+.
+After having considered FEM spaces, the stabilization occurs in the bilinear and linear terms involved
+in the state and the adjoint equations. Instead, the gradient equation is left unstabilized [19]. We
+recall that we use distributed control.
+We defined the stabilized bilinear form as and cs, and the stabilized forcing term Fs as
+(13)
+as
+�
+yN , qN ; µ
+�
+:= a
+�
+yN , qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+TyN , hK
+|η| TSSqN
+�
+K
+,
+yN , qN ∈ Y N ,
+cs
+�
+uN , qN ; µ
+�
+:= −
+�
+Ω
+uN qN −
+�
+K∈Th
+δK
+�
+uN , hK
+|η| TSSqN
+�
+K
+,
+uN ∈ U N , qN ∈ Y N ,
+Fs
+�
+qN ; µ
+�
+:= F
+�
+qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+f(µ), hK
+|η| TSSqN
+�
+K
+,
+∀qN ∈ Y N .
+where δK is a local positive dimensionless parameter related to the element K ∈ Th, consequently
+it can be different for each triangle, and (·, ·)K is the inner scalar product in L2(K).
+In (13)
+a
+�
+yN , qN ; µ
+�
+=
+�
+TyN , qN �
+L2(Ω) and F
+�
+qN ; µ
+�
+=
+�
+f, qN �
+L2(Ω), where f collects all forcing and
+lifting terms of the problem.
+For the remaining conditions of the optimality system, we will always consider m and n form as
+the L2(Ωobs) and the L2(Ω) products for steady problems. The adjoint equation is an Advection-
+Dominated equation, too, where the advective term has opposite sign with respect to the state one:
+indeed, T ∗ = TS − TSS from (12). We use the next SUPG forms for zN ∈ Y N :
+(14)
+a∗
+s
+�
+zN , pN ; µ
+�
+:= a∗ �
+zN , pN ; µ
+�
++
+�
+K∈Th
+δa
+K
+�
+(TS − TSS)pN , hK
+|η| (−TSS) zN
+�
+K
+,
+�
+yN − yd, zN ; µ
+�
+s :=
+�
+Ωobs
+(yN − yd)zN dx +
+�
+K∈Th|Ωobs
+δa
+K
+�
+yN − yd, hK
+|η| (−TSS) zN
+�
+K
+,
+where a∗ is the adjoint form of a, δa
+K is the positive stabilization parameter of the stabilized adjoint
+equation.
+In our numerical experiments, we will always consider δK = δa
+K.
+Finally, the SUPG
+optimality system for a steady OCP(µ) reads as:
+(15)
+discretized adjoint equation:
+a∗
+s
+�
+zN , pN ; µ
+�
++
+�
+yN − yd, zN ; µ
+�
+s = 0, ∀zN ∈ Y N ,
+discretized gradient equation:
+c∗�
+vN , pN ; µ
+�
++ αn
+�
+uN , vN ; µ
+�
+= 0, ∀vN ∈ U N ,
+discretized state equation:
+as
+�
+yN , qN ; µ
+�
++ cs
+�
+uN , qN ; µ
+�
+= Fs(qN ; µ), ∀qN ∈
+�
+QN �∗ .
+
+6
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+We denote with Ks and KT
+s the stiffness matrices related to the stabilized forms as and a∗
+s,
+respectively, M is the not-stabilized mass matrix related to n, instead, Ms is the stabilized mass
+matrix related to m after stabilization, Cs is the matrix linked to stable form cs, the block CT refers
+to c, and fs is the vector that contains the coefficients of the stabilized force term as components.
+Moreover, we consider with the symbol y, u and p as the vectors of coefficients of yN , uN and pN ,
+expressed in terms of the nodal basis of Y N , U N , (QN )∗, respectively. Finally, the discretized block
+system related to (15) is:
+(16)
+�
+�
+Ms
+0
+KT
+s
+0
+αM
+CT
+Ks
+Cs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+Msyd
+0
+fs
+�
+� .
+3.2. Setting for Stabilized Advection-Dominated OCP(µ)s - Unsteady case. We show the
+SUPG approach for time-dependent OCP(µ)s proposed in [63]. A classical implicit Euler discretiza-
+tion is applied to all forms including time-derivatives [3, 25, 50, 54, 55, 56]. We divide the time
+interval (0, T) in Nt sub-intervals of equal length ∆t := tj − tj−1, j ∈ {1, . . . , Nt}. Starting from
+this framework, a discretization along time is done, where each discrete instant of time is considered
+as a steady-state Advection-Dominated equation in a space-time approach [25, 50, 51, 54, 55, 56].
+In addition, the SUPG stabilization occurs for time-dependent forms, too. The general scheme is
+described as follows.
+Let us firstly define the discrete vectors y =
+�
+yT
+1 , . . . , yT
+Nt
+�T , u =
+�
+uT
+1 , . . . , uT
+Nt
+�T and p =
+�
+pT
+1 , . . . , pT
+Nt
+�T , where yi ∈ Y N , ui ∈ U N and pi ∈ (QN )∗ for 1 ≤ j ≤ Nt.
+Also here, yj, uj
+and pj indicate the column vectors containing the coefficients of the FEM discretization for state,
+control and adjoint, respectively (unlike the steady case, there are not denoted in bold style). This
+implies Ntot = 3 × Nt × N as the global dimension of the block system.
+We express all other
+terms in based of the respective nodal basis. The vector representing the initial condition for the
+state variable is y0 =
+�
+yT
+0 , 0T , . . . , 0T �T , where 0 is the zero vector in RN , yd =
+�
+yT
+d1, . . . , yT
+dNt
+�T
+is the vector including discrete time components of the discretized desired solution profile; instead,
+f s =
+�
+f T
+s1, . . . , f T
+sNt
+�T
+corresponds to the stabilized forcing term. We recall that Y, U, Q are Hilbert
+Spaces and, for the sake of simplicity, we assume Y N ≡ (QN )∗. So now we can see locally the time
+block discretization.
+• Adjoint equation: this equation is discretized backward in time using the forward Euler
+method, which is equal to a backward Euler with respect to time T − t, for t ∈ (0, T) [21].
+Firstly, we add a stabilized term to the form related to ∂tp and a∗ defined as:
+s∗ �
+zN , pN (t); µ
+�
+=
+�
+K∈Th
+δK
+�
+−∂tpN (t) + T ∗pN (t), −hK
+|η| TSSzN
+�
+K
+,
+where we define the form
+(17)
+m∗
+s
+�
+pN , zN ; µ
+�
+=
+�
+pN , zN �
+L2(Ω) −
+�
+K∈Th
+δK
+�
+pN , hK
+|η| TSSzN
+�
+K
+.
+Then, the time discretization is: for each j ∈ {Nt − 1, Nt − 2, ..., 1}, find pN
+j ∈ Y N
+s.t.
+(18)
+1
+∆tm∗
+s
+�
+pN
+j (µ) − pN
+j+1(µ), zN ; µ
+�
++ a∗
+s
+�
+zN , pN
+j (µ); µ
+�
+= −
+�
+yN
+j − ydj, zN ; µ
+�
+s
+∀zN ∈ Y N ,
+Considering M T
+s as the matrix inherent to m∗
+s, the block subsystem reads
+M T
+s pj = M T
+s pj+1 + ∆t
+�
+−M T
+s yj − KT
+s pj + M T
+s ydj
+�
+for j ∈ {Nt − 1, Nt − 2, . . . , 1} .
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+7
+Finally, we derive the following block system:
+�
+����
+M T
+s + ∆tKT
+s
+−M T
+s
+...
+...
+M T
+s + ∆tKT
+s
+−M T
+s
+M T
+s + ∆tKT
+s
+�
+����
+�
+��
+�
+AT
+s
+p +
+�
+�����
+∆tM T
+s y1
+...
+...
+∆tM T
+s yNt
+�
+�����
+=
+�
+�����
+∆tM T
+s yd1
+...
+...
+∆tM T
+s ydNt
+�
+�����
+.
+Setting the diagonal block matrix MT
+s ∈ RN ·Nt×RN ·Nt with diagonal entries [M T
+s , . . . , M T
+s ],
+the adjoint system to be solved is: ∆tMT
+s y + AT
+s p = ∆tMT
+s yd.
+• Gradient equation. We seek the solution of α∆tMuj+∆tCT pj = 0, ∀j ∈ {1, 2, . . . , Nt} ,
+which is equal to the following block system:
+(19)
+α∆t
+�
+����
+M
+M
+...
+...
+M
+�
+����
+�
+��
+�
+M
+�
+����
+u1
+u2
+...
+uNt
+�
+���� +∆t
+�
+����
+CT
+0
+· · ·
+CT
+...
+CT
+�
+����
+�
+��
+�
+CT
+�
+����
+p1
+p2
+...
+pNt
+�
+���� =
+�
+����
+0
+0
+...
+0
+�
+���� .
+More compactly, we solve α∆tMu+∆tCT p = 0.
+• State equation. A backward Euler method is used for a discretization forward in time. The
+stabilized term related to ∂ty and the bilinear form a is [29, 37, 58]:
+s
+�
+yN (t), qN ; µ
+�
+=
+�
+K∈Th
+δK
+�
+∂tyN (t) + TyN (t), hK
+|η| TSSqN
+�
+K
+,
+where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N . Defining the stabilized term ms as
+(20)
+ms
+�
+yN , qN ; µ
+�
+=
+�
+yN , qN �
+L2(Ω) +
+�
+K∈Th
+δK
+�
+yN , hK
+|η| TSSqN
+�
+K
+,
+then the backward Euler approach reads as: for each j ∈ {1, 2, · · · , Nt}, find yN
+j ∈ Y N s.t.
+(21)
+1
+∆tms
+�
+yN
+j (µ) − yN
+j−1(µ), qN ; µ
+�
++ as
+�
+yN
+j (µ), qN ; µ
+�
++ cs
+�
+uN
+j , qN ; µ
+�
+= Fs
+�
+qN ; µ
+�
+,
+given the initial condition yN
+0
+which satisfies
+�
+yN
+0 , qN �
+L2(Ω) =
+�
+y0, qN �
+L2(Ω) , ∀qN ∈ Y N .
+The matrix state equation to be solved becomes
+(22)
+Msyj + ∆tKsyj + ∆tCsuj = Msyj−1 + fsj∆t
+for j ∈ {1, 2, . . . , Nt} ,
+where the stabilized mass matrix of ms is Ms. Thus, we have
+�
+����
+Ms + ∆tKs
+0
+−Ms
+Ms + ∆tKs
+0
+...
+...
+0
+0
+−Ms
+Ms + ∆tKs
+�
+����
+�
+��
+�
+As
+y+∆t
+�
+��
+Cs
+0
+0
+...
+0
+0
+Cs
+�
+��
+�
+��
+�
+Cs
+u = Msy0 + ∆tf s,
+where Ms ∈ RN ·Nt×RN ·Nt is a block diagonal matrix which diagonal entries are [Ms, . . . , Ms].
+In a more compact notation, we have Asy+∆tCsu = Msy0 + ∆tf s.
+The final system considered and solved through an one shot approach is the following:
+(23)
+�
+�
+∆tMT
+s
+0
+AT
+s
+0
+α∆tM
+∆tCT
+As
+∆tCs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+∆tMT
+s yd
+0
+Msy0 + ∆tf s
+�
+� .
+
+8
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+4. Weighted ROMs for random inputs advection-dominated OCP(µ)s
+Numerical simulations for OCP(µ)s can be very expensive in relation to computational time and
+storage. To overcome this problem, in this work we will consider ROMs [6, 24, 41, 40, 39]. We will
+study the case when the parameter µ can be affected by randomness, i.e. it can follow a particular
+probability distribution. That is the case of random inputs OCP(µ)s. In this scenario, a suitable
+modification of the ROMs, the wROMs [11, 15, 13, 16, 17, 18, 49, 59, 61, 60], takes into account
+the uncertainty quantification (UQ) of the problems and shows efficient results concerning errors
+and computational time. For the sake of notation, from now on we denote a generic probability
+distribution with the symbol ρ. ROM procedure is divided in two stages: an offline phase and an
+online phase.
+To exploit the potential of the ROMs setting, we assume an affine decomposition of the forms in
+(15) [24]. Therefore, Assumption 4.0.1 is required here.
+Assumption 4.0.1. We request that all the forms in (15) are affine in µ = (µ1, . . . , µp) ∈ P. More
+precisely, we request that [15, 13]:
+(1) the random diffusivity γ : Ω × P → R is of the form
+(24)
+γ(µ, x) = γ0(x) +
+p
+�
+k=1
+θγ
+k(µk)γk(x),
+with γk ∈ L∞(Ω), for k = 0, . . . , p and θγ
+k depending only on µk;
+(2) the random advection field η : Ω × P → R2 is of the form
+(25)
+η(µ, x) = η0(x) +
+p
+�
+k=1
+θη
+k(µk)ηk(x),
+with ηk ∈ (L∞(Ω))2, for k = 0, . . . , p and θη
+k depending only on µk;
+(3) the random forcing term f : Ω × P → R is of the form
+(26)
+f(µ, x) = f0(x) +
+p
+�
+k=1
+θf
+k(µk)fk(x),
+with fk ∈ L2(Ω), for k = 0, . . . , p and θf
+k depending only on µk.
+For example, Assumption 4.0.1 can be satisfied by truncating a Karhunen–Lo`eve expansion [47].
+4.1. Offline phase. The offline phase is the most expensive stage of the wROMs, which usually
+depends on N. However, this should be done only once. The aim of this procedure is to build reduced
+spaces Y N, U N and (QN)∗ that are good approximations of the high-fidelity ones and to compute
+all block matrix components that are µ-independent. Then, everything is memorized in order to be
+ready to be used in the online phase. The construction of the reduced basis is achieved through a
+modified version of the POD algorithm: the wPOD [11, 60, 61], described in Section 4.1.1. Here,
+we firstly compute high-fidelity evaluation of optimal solutions
+�
+yN (µ), uN (µ), pN (µ)
+�
+for different
+parameters µ, the so-called snapshots, to build the bases. Because of good performance presented in
+literature [30, 34, 53], this process will go through a partitioned approach, i.e. the wPOD is executed
+separately for all three variables. After this step, the three reduced spaces for state, control and
+adjoint are constructed as, respectively,
+(27)
+Y N = span {ξy
+n, n = 1, . . . , N},
+U N = span {ξu
+n, n = 1, . . . , N} ,
+(QN)∗ = span {ξp
+n, n = 1, . . . , N} .
+In order to ensure well-posedness for the reduced space approximation, we need to implement an
+enriched space for state and adjoint variables. This means to impose GN ≡ Y N ≡ (QN)∗, where
+GN = span {σn, n = 1, . . . , 2N} and {σn}2N
+n=1 = {ξy
+n}N
+n=1 ∪ {ξp
+n}N
+n=1 [20, 23, 30, 31, 35, 34]. This
+whole discussion holds true for parabolic problems in a space-time context, too. As a matter of fact,
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+9
+when dealing with time-dependent OCP(µ)s in a space-time approach, the time instances are not
+separated in the wPOD algorithm. Therefore, each snapshot carries all the time instances.
+4.1.1. Weighted Proper Orthogonal Decomposition. The peculiarity of wPOD is to take into account
+the probability distribution that characterizes µ to create reduced spaces with less number of basis
+with respect to the deterministic case without losing in accuracy [11, 60, 61]. We will notice that
+there will be different ways to consider randomness in the wPOD: the general idea is to suitably
+attribute a larger weight to those samples that are more significant according to the distribution of µ.
+From now we will refer to the POD algorithm based on the Monte-Carlo procedure in a deterministic
+context, i.e. when the distribution ρ is the uniform one, as Standard POD to distinguish it from
+the wPOD. As we will consider a partitioned approach, we show the procedure for the state space:
+adjoint and control variables will follow the same process.
+To consider stochasticity, wPOD needs to find the N-dimensional space Y N, with N ≪ N, such
+that it minimizes the following estimate:
+(28)
+E =
+��
+P
+inf
+ζy∈Y N ∥yN (µ) − ζy∥2
+Y ρ(µ)dµ.
+Let us consider a set of Ntrain ordered parameters µ1, . . . , µNtrain ∈ PNtrain, where PNtrain ∈ P is a
+discretization of P called the training set and its cardinality is |PNtrain| = Ntrain. One can choose
+Ntrain so that PNtrain is a good approximation of P. We can relate µ1, . . . , µNtrain to the ordered
+snapshots yN (µ1) , . . . , yN �
+µNtrain
+�
+. Considering w : P → R+ a weight function, a discretization
+of problem (28) is meant to find the N-dimensional space Y N which minimize the quantity
+(29)
+1
+Ntrain
+Ntrain
+�
+k=1
+w (µk)
+��yN (µk) − yN (µk)
+��2
+Y .
+One could think that the natural choice can be w(µ) = ρ(µ) and in an UQ context this means
+to just discretize the expectation of the square error
+(30)
+E
+���yN − yN��2
+Y
+�
+:=
+�
+P
+��yN (µ) − yN(µ)
+��2 ρ(µ)dµ,
+which is the argument of the square root in (28). However, this is not the unique choice in this
+scenario: therefore it will be interesting to understand which method is better to approximate (30).
+Here we illustrate different techniques that we use in the numerical tests in Section 5 to approximate
+(30). Considering the training set PNtrain =
+�
+µ1, . . . , µNtrain
+�
+, which can be composed by the nodes
+of the chosen quadrature formula that approximates (30), we indicate with ω = (ω1, . . . , ωNtrain) the
+standard weights of a chosen quadrature rule, with ρ1, . . . , ρNtrain the values of the density ρ in the
+nodes in PNtrain, and with w = (w1, . . . , wNtrain) the definitive weights used in wPOD algorithm. For
+a node µk, we have the correspondent quantities ωk, ρk, and wk. As a final result of this first step,
+the wPOD furnished the following sum to minimize
+(31)
+1
+Ntrain
+Ntrain
+�
+k=1
+wk
+��yN (µk) − yN (µk)
+��2
+Y ,
+which is achieved here through the following algorithms:
+• Weighted Monte-Carlo method, where µ1, . . . , µNtrain are Ntrain parameters extracted from
+the random variable µ according to its distribution ρ and ρi are the values of the density ρ in
+these points. For this approximation, we have PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρ(µk),
+for all k = 1, . . . , Ntrain;
+• Pseudo-Random method based on a Halton Sequence, where µ1, . . . , µNtrain are the nodes
+extracted by a sampling completely based on the Halton sequence [57] and ρk = ρ(µk). Also
+in this case, PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρk, for all k = 1, . . . , Ntrain;
+
+10
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+• Tensor product Gauss-Jacobi rule, where µ1, . . . , µNtrain are the nodes of the tensor product
+Gauss-Jacobi quadrature rule and ω1, . . . , ωNtrain are the correspondent quadrature weights.
+We can use this formula when the distribution is a Beta(αk, βk), as suitable Jacobi polyno-
+mials are orthogonal to this distribution [43]. As a matter of fact, simulations in Section 5
+will consider different Beta distributions for all components of µ. Therefore, we implement
+a Gauss-Jacobi formula using (αk, βk) as its parameters in each dimension [43], accordingly
+to the distribution of µ. For this approximation, we have PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and
+wk = ωk, for all k = 1, . . . , Ntrain;
+• Tensor product Clenshaw-Curtis rule, where µ1, . . . , µNtrain are the nodes of the tensor prod-
+uct Clenshaw-Curtis quadrature rule and ω1, . . . , ωNtrain are the correspondent quadrature
+weights [57]. In this case we obtain PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρkωk, for all
+k = 1, . . . , Ntrain.
+In numerical tests of Section 5, we will respectively call as Weighted Monte-Carlo, Pseudo-
+Random, Gauss-Jacobi, and Clenshaw-Curtis wPOD algorithms the rules just specified. As it is
+know, tensor rule can be efficient, but their structure implies huge computational costs for elevate
+cardinality of the training set Ptrain or high-dimensional parameter space P. For this purpose, when
+we will use Clenshaw-Curtis or Gauss-Jacobi methods, we will consider sparse grid techniques based
+on a Smolyak algorithm, too [48, 62]: we will implement isotropic ones [36].
+Once chosen the rule (29) to approximate (30), the procedure to minimize (29) is described as
+follows. Let us define the deterministic correlation matrix of the snapshots of the state variable
+Dy ∈ RNtrain×Ntrain in the following way:
+(32)
+Dy
+kl :=
+1
+Ntrain
+�
+yN (µk) , yN (µl)
+�
+Y ,
+1 ≤ k, l ≤ Ntrain.
+Firstly, we define the weighted correlation matrix as
+(33)
+W y := W · Dy,
+where W := diag(w1, · · · , wNtrain) is the diagonal matrix whose elements are the weights of (29).
+The matrix W y is not symmetric in the usual matrix sense, but with respect to the scalar product
+induced by W y, hence W y is diagonalizable anyway [60]. Therefore, we seek the solution of the
+eigenvalue problem W ygy
+n = λy
+ngy
+n, 1 ≤ n ≤ Ntrain, where ∥gy
+n∥Y = 1, i.e. we pursue to find an
+eigenvalue λy
+n with the relative eigenvector of norm equal to one. We will indicate with (gy
+n)t the t-
+th component of the eigenvector gy
+n ∈ RNtrain. For the sake of simplicity, we rearrange the eigenvalues
+λy
+1, . . . , λy
+Ntrain in a decreasing layout. Then, let us look at the first N eigenvalue-eigenvector pairs
+(λy
+1, gy
+1), . . . , (gy
+N, λy
+N). The basis functions χy
+n for the state equation are constructed through the
+following relation:
+(34)
+ζy
+n =
+1
+√
+λy
+n
+Ntrain
+�
+t=1
+(gy
+n)t yN (µk) ,
+1 ≤ n ≤ N.
+In order to choose N, one can refer to same study of eigenvalues of W y [24, 39, 61]. At the end, our
+reduced spaces are built as (27) and, then, enriched spaces are constructed.
+We summarise all the wPOD procedure for OCP(µ)s in Algorithm 1.
+4.2. Online phase. In this stage, all operations have usually a N-independent cost. This process
+reflects to be computationally cheap and, therefore, it can be recalled multiple times using small
+machine resources. Firstly, we choose a parameter µ. We get all the µ-independent quantities and
+reduced spaces back from the storage. Immediately, we combined parameter independent part with
+the µ-dependent ones, that are rapidly calculated here. Then a Galerkin projector onto Y N, U N
+and (QN)∗ is performed, computing the reduced solution yN, uN and pN through a reduced block
+matrix system. As previously seen in Section 3, a stabilization is needed in the truth approximation.
+However, it could also not be the case for the online stage. This scenario lead to two possibilities:
+we do not use SUPG in the online phase, Offline-Only stabilization, or, on the contrary, stabilization
+occurs also here Offline-Online stabilization.
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+11
+Algorithm 1 Weighted POD algorithm for OCP(µ) problems through a partitioned approach
+Input: FEM spaces Y N , U N , and (QN )∗ parameter domain P, and Ntrain.
+Output: reduced spaces Y N, U N and (QN)∗.
+Considering the high-fidelity spaces Y N , U N and (QN )∗:
+1: Choose a quadrature rule (29) to approximate (30). This step defines a sample Ptrain ⊂ P and
+the respective weights w1, . . . , wNtrain. Define the matrix W := diag(w1, · · · , wNtrain) ;
+2: for all µ ∈ Ptrain do
+3:
+Solve the high-fidelity SUPG OCP(µ) system (15);
+4: end for
+5: Calculate the matrices Dy
+kl :=
+1
+Ntrain
+�
+yN (µk) , yN (µl)
+�
+Y , 1 ≤ k, l ≤ Ntrain and W y := W · Dy.
+Do the same for the control u and the adjoint p;
+6: Compute eigenvalues λy
+1, . . . , λy
+Ntrain
+and the corresponding orthonormalized eigenvectors
+gy
+1, . . . , gy
+Ntrain of W y. Do the same procedure for u and p variables;
+7: Fix N according to a certain criterion and construct Y N = span {ξy
+n, n = 1, . . . , N}, where
+ξy
+n =
+1
+√
+λy
+n
+�Ntrain
+t=1
+(gy
+n)t yN (µk). Do the same for u and p variables.
+8: Build the aggregated space GN = span
+�
+{ξy
+n}N
+n=1 ∪ {ξp
+n}N
+n=1
+�
+and set GN ≡ Y N ≡ (QN)∗.
+5. Numerical Results
+In this last part we illustrate numerical simulations concerning two Advection-Dominated OCP(µ)s
+under random inputs: the Graetz-Poiseuille Problem and the Propagating Front in a Square Prob-
+lem. In both experiments, the parameter µ will be a random vector and it will follow a prescribed
+probability density function that will be specified. The deterministic version of both experiments
+can be founded in [63].
+The Offline approximation will be always based on a P1−FEM, which means to consider a finite
+element method characterized by polynomials of degree less or equal than 1. In steady and unsteady
+simulations, the same stabilization parameter δK will be employed for both stabilization in the high-
+fidelity approximation and in the Online phase: namely in Offline-Online stabilization, δK is the
+same for both phases.
+For each simulation, relative errors between the FEM and the reduced solutions, i.e.
+(35)
+ey,N(µ) :=
+��yN (µ) − yN(µ)
+��
+Y
+∥yN (µ)∥Y
+, eu,N(µ) :=
+��uN (µ) − uN(µ)
+��
+U
+∥uN (µ)∥U
+, ep,N(µ) :=
+��pN (µ) − pN(µ)
+��
+Q∗
+∥pN (µ)∥Q∗
+,
+for the state, the control and the adjoint, respectively, will be shown. Due to the parametric nature
+of the problems, for each quantity in (35) a simple average is computed for µ distributed according
+to its probability density in a testing set Ptest ⊆ P of size Ntest, for every dimension N = 1, . . . , Nmax
+of the reduced space built through a chosen wPOD procedure. In every graph, the base-10 logarithm
+of these averages will be shown. When we will specify to use a POD procedure based on a Monte-
+Carlo sampling [57] of a uniform density distribution, we will talk about Standard POD. In order
+to compare the different wPOD possibilities, we use the same testing set for all of them: it will be
+taken using a Monte-Carlo method according to the distribution of µ. Obviously, the performance
+of the Standard POD will be based on a testing set of uniform density. The sum of the errors with
+respect to each discretized instant of time t will be taken into account in the unsteady versions.
+In order to compare the computational cost between the FEM solution with that of the reduced
+one for any possible dimension N, we use the speedup-index, i.e.
+(36)
+speedup-index = computational time of the high-fidelity solution
+computational time of the reduced solution
+,
+which will be calculated for any µ in the testing set. Again, we will shown its sample average
+for any dimension N. For each test case, we will use the same Ptest to compute relative errors and
+
+12
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+the speedup-index. The steady experiments are run using a machine with 16GB of RAM and Intel
+Core i7-7500U Dual Core, 2.7GHz for the CPU; whereas all parabolic simulations are computed
+considering 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.60GHz for the CPU.
+The code concerning steady experiments is implemented using the RBniCS library [2]; instead,
+the unsteady ones are provided using both RBniCS and multiphenics [1] libraries. These are python-
+based libraries, built on FEniCS [32].
+5.1. Numerical Tests for the Graetz-Poiseuille Problem. The Graetz-Poiseuille problem is
+an Advection-Diffusion problem that represents the heat conduction in a rectilinear pipe. Here the
+transfer of heat can be regulated through the walls of the duct, which can be held at certain fixed
+temperature [22, 37, 46, 59].
+Firstly, we present simulation concerning the stationary case, where a distributed control is em-
+ployed all over the whole domain. The parameter µ = (µ1, µ2) is composed by the diffusion compo-
+nent µ1 and the geometrical one µ2, which characterizes the length of the plate.
+Ωobs
+Ωobs
+Ωo
+Γo,1
+Γo,2
+Γo,3
+Γo,4
+Γo,5
+Γo,6
+(0,0)
+(1,0)
+(1+µ2,0)
+(1+µ2,0.2)
+(1+µ2,0.8)
+(1+µ2,1)
+(1,1)
+(0,1)
+Figure 1. Geometry of the Graetz-Poiseuille Problem.
+The problem is studied using (x0, x1) as spatial coordinates. Ωo is the domain observed for a
+certain value µ2 with boundary Γo.
+We deal with homogeneous Neumann boundary conditions
+(BC) on Γo,3 := {1 + µ2} × [0, 1] considering Figure 1. Instead, Dirichlet conditions are set on sides
+Γo,1 := [0, 1]×{0}, Γo,5 := [0, 1]×{1}, Γo,6 := {0}×[0, 1] by imposing y = 0 and Γo,2 := [1, 1+µ2]×{0}
+and Γo,4 := [1, 1 + µ2] × {1} by imposing y = 1.
+The formulation of the problem is the following: given µ ∈ P, find (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs(µ)
+(y(µ) − yd)2 dΩo(µ) + α
+2
+�
+Ωo(µ)
+u(µ)2 dΩo(µ),
+such that
+(37)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),
+in Ωo(µ),
+y(µ) = 0,
+on Γo,1(µ) ∪ Γo,5(µ) ∪ Γo,6(µ),
+y(µ) = 1,
+on Γo,2(µ) ∪ Γo,4(µ),
+∂y(µ)
+∂ν
+= 0,
+on Γo,3(µ),
+where ˜Y :=
+�
+v ∈ H1�
+Ωo
+�
+s.t. it satisfies the BC in (37)
+�
+and U = L2(Ωo). For the sake of clarity,
+a lifting function Ry ∈ H1(Ω) that fulfills the BC in (37) is used.
+Consequently, the variable
+¯y := y − Ry, with ¯y ∈ Y , is used, where
+Y :=
+�
+v ∈ H1
+0
+�
+Ω
+�
+s.t. ∂¯y
+∂ν = 0, on Γ3 and ¯y = 0 on Γ \ Γ3
+�
+.
+Furthermore, we settle Q := Y ∗ without any loss of generality. Therefore, the adjoint variable p
+is null on Γ \ Γ3. The observation domain is Ωobs := [1, 1 + µ2] × [0.8, 1] ∪ [1, 1 + µ2] × [0, 0.2] as
+illustrated in Figure 1. The value µ2 can change the domain under study. Having that the domain
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+13
+Ωo is µ-dependent itself, in the Offline Phase snapshots are based on different domains due to the
+sampling of the geometrical parameter components [24, 41, 45, 44]. To deal with the geometrical
+parametrization of the problem, we set a reference domain Ω and we build affine maps that transform
+Ω in Ωo for a defined µ. This procedure implies an automatic modification of some bilinear and
+linear forms involved in the weak formulation of Problem (37).
+We choose Ω = (0, 2) × (0, 1) as reference domain, that is the original one Ωo(µ) corresponding
+to µ2 = 1. We assume that µ2 is positive for the sake of simplicity. Considering Figure 1, we divide
+this into 2 subdomains, which are defined as Ω1 = (0, 1) × (0, 1) and Ω2 = (1, 2) × (0, 1). Then, we
+build two affine transformations:
+(38)
+V1(µ) : Ω1 → Ωo,1(µ) ⊂ R2,
+such that V1
+�� x
+y
+�
+; µ
+�
+:=
+� x
+y
+�
+,
+which is the identity map defined on the first subdomain Ω1 and V2(µ) : Ω2 → Ωo,2(µ) ⊂ R2 as
+(39)
+V2
+�� x
+y
+�
+; µ
+�
+=
+� µ2x
+y
+�
++
+� 1 − µ2
+0
+�
+= R2
+� x
+y
+�
++
+� 1 − µ2
+0
+�
+,
+where we have
+(40)
+R2 :=
+�
+µ2
+0
+0
+1
+�
+.
+Glueing together V1 and V2 for each µ ∈ P, we manage to build a one-to-one transformation
+V (µ) defined all over Ω. We denote the restrictions of Th to Ω1 and Ω2 with T 1
+h and T 2
+h , respec-
+tively. Therefore, we can express all the forms of the weak formulation under the effect of this
+transformation. For instance, after possible lifting, we have as = a + s and a∗
+s = a∗ + s∗ as
+(41)
+a
+�
+yN , qN ; µ
+�
+: =
+�
+Ω1
+1
+µ1
+∇yN · ∇qN + 4x1(1 − x1)∂x0yN qN
++
+�
+Ω2
+1
+µ1µ2
+∂x0yN ∂x0qN + µ2
+µ1
+∂x1yN ∂x1qN + 4x1(1 − x1)∂x0yN qN ,
+s
+�
+yN , qN ; µ
+�
+: =
+�
+K∈T 1
+h
+δKhK
+�
+K
+�
+4x1(1 − x1)∂x0yN �
+∂x0qN
++
+�
+K∈T 2
+h
+δK
+hK
+õ2
+�
+K
+�
+4x1(1 − x1)∂x0yN �
+∂x0qN ,
+a∗ �
+zN , pN ; µ
+�
+: =
+�
+Ω1
+1
+µ1
+∇pN · ∇zN − 4x1(1 − x1)∂x0pN zN
+−
+�
+Ω2
+1
+µ1µ2
+∂x0pN ∂x0zN − µ2
+µ1
+∂x1pN ∂x1zN − 4x1(1 − x1)∂x0pN zN ,
+s∗ �
+zN , pN ; µ
+�
+: =
+�
+K∈T 1
+h
+δKhK
+�
+K
+�
+4x1(1 − x1)∂x0pN �
+∂x0zN
++
+�
+K∈T 2
+h
+δK
+hK
+õ2
+�
+K
+�
+4x1(1 − x1)∂x0pN �
+∂x0zN ,
+for all yN , qN , zN , pN , ∈ Y N . In order to take into account the possible bad effect on stabilized
+forms due to a extension or shortening of our domain Ωo, we choose the stabilization parameter for
+K ∈ T 2
+h as δK
+hK
+õ2 , where õ2 =
+�
+| det(R2)| [35, 37, 58].
+For the FEM discretization, a quite coarse mesh of size h = 0.034 is used and the total dimension
+of the numerical problem is 13146. We take δK = 1.0 for all K ∈ Th. The parameter space is set
+as P :=
+�
+1, 105�
+×
+�
+0.5, 1.5
+�
+, from which we want to extract a training set Ptrain with cardinality
+Ntrain = 100. For the n bilinear form, we consider a penalization α = 0.01. Our aim is to minimize
+the L2-error between the state and the desired solution profile yd(x) = 1.0, function defined on Ωobs
+
+14
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 2. (top) FEM not stabilized and (bottom) FEM stabilized solution, y (right) and
+u (left), µ = (105, 1.5), h = 0.034, α = 0.01, δK = 1.0.
+of Figure 1. Each wPOD procedure is computed until a Nmax = 20 in a partitioned approach and
+then all algorithms are compared using a testing set Ptest of 100 elements in P.
+We suppose that µ follows a Beta(5, 3) distribution for both parameter µ1 and µ2, i.e.
+(42)
+µ1 ∼ 1 +
+�
+105 − 1
+�
+X1, where X1 ∼ Beta(5, 3),
+µ2 ∼ 0.5 +
+�
+1.5 − 0.5
+�
+X2, where X2 ∼ Beta(5, 3),
+where µ1 and µ2 are independent random variables. This implies that we consider more probable
+the parameters for which the Graetz-Poiseuille Problem has high values of the P´eclet number. In
+Figure 2, we highlight how the FEM solutions of the state and the control are for µ = (105, 1.5).
+The adjoint solution is not shown here because it is proportional to the control due to the gradient
+equation [19].
+From Figure 2, one can see that a stabilization is necessarily needed.
+We firstly exploit the
+Offline-Only stabilization procedure, which results regarding errors are shown in Figure 3. The
+performance is not good for any kind of wPOD. Moreover, the Standard POD does not perform
+good, either. Relative errors never drop under 10−2 for any variables, hence more stabilization is
+necessary in this case.
+In Figure 4 relative errors of the Offline-Online stabilization procedure are presented.
+Here
+the trend seems better than the Offline-Only one, because these quantities decay faster along the
+value of N. The wPOD Monte-Carlo is the best performer for all y, u, p variables, as a matter of
+fact, it reaches ey,16 = 2.13 · 10−7 for the state, for the adjoint ep,16 = 3.95 · 10−7 and the control
+eu,16 = 3.80·10−7. This procedure has a better performance of the Standard POD, which its accuracy
+is at least 100 times inferior of the wPOD Monte-Carlo after N > 11. Concerning other rules, it can
+be noticed that Smolyak grid techniques perform better than their tensor-rule counterparts, despite
+having a training set whose cardinality is similar, but less of 100: 93 and 91 for the Clenshaw-Curtis
+and Gauss-Jacobi sparse grids, respectively.
+In Figure 5 we visually compare the two possibilities of stabilization for the geometrical parametriza-
+tion of the Graetz-Pouiseuille problem for the wPOD Monte-Carlo.
+In Table 1 we compare the speedup-index for all wPOD algorithms. We see that computational
+values are all of the same order of magnitude. For the wPOD Monte-Carlo we calculate 87 reduced
+solutions in the time of a FEM one.
+Now we want to present the parabolic version of Problem (37). This unsteady problem has been
+studied without optimization in [37, 59] in a deterministic context and in [59] in a UQ one. Instead,
+the deterministic OCP(µ) Graetz Problem under boundary control without Advection-dominancy
+is studied in [54, 52] and the deterministicdistributed control configuration is analyzed in [63].
+
+1.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+0.5
+0
+-0.5
+-9.3e-011.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+-0.5
+- 0
+-0.5
+-9.3e-01STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+15
+Figure 3. Relative Errors for the Graetz-Poiseuille Problem - Offline-Only Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+Speedup-index Graetz-Poiseuille Problem: Offline-Online Stabilization - µ1, µ2 ∼ Beta(5,3)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+4
+113.0
+108.9
+110.1
+110.1
+106.5
+109.4
+112.0
+8
+108.4
+105.1
+104.9
+106.1
+102.1
+105.9
+107.4
+12
+103.3
+100.2
+99.9
+99.8
+99.1
+96.9
+101.7
+16
+97.2
+92.5
+95.1
+94.5
+92.6
+94.2
+96.9
+20
+90.5
+87.3
+87.0
+88.0
+85.8
+86.3
+89.7
+Table 1. Average Speedup-index of Offline-Online Stabilization for the Graetz-Poiseuille
+Problem under geometrical parametrization. From left to right: Standard POD, wPOD
+Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor,
+Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+Recalling Figure 1, for a fixed T > 0 the unsteady Graetz-Poiseuille Problem is posed as follows:
+find (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs(µ)×(0,T )
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω(µ)×(0,T )
+u(µ)2 dΩ,
+such that
+
+FEM vs ROM averaged relative error - y (state)
+101
+Log-Error
+100
+Relative L
+StandardPOD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gausslacobi-Smolyak
+ClenshawCurtis  tensor
+10-1
+ClenshawCurtis+Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - u (control)
+101
+100
+Standard POD
+Weighted POD Monte-Carlo
+Gaussjacobi-tensor
+10-1
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis  Smolyak
+PseudoRandom Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - p (adjoint)
+102
+Log-Error
+101
+Relative
+StandardPOD
+WeightedPoDMonte-Carlo
+Gaussjacobi-tensor
+100
+Gaussjacobi--Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis Smolyak
+PseudoRandom Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+N16
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 4. Relative Errors for the Graetz-Poiseuille Problem - Offline-Online Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+Figure 5. (top) wPOD Monte-Carlo Offline-Only stabilized and (bottom) Offline-Online
+stabilized solution, y (right) and u (left), µ = (105, 1.5), h = 0.034, α = 0.01, Ntrain = 100,
+δK = 1.0, N = 20.
+(43)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ty(µ) − 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),
+in Ω(µ) × (0, T),
+y(µ) = 0,
+on Γ1 ∪ Γ5 ∪ Γ6 × (0, T),
+y(µ) = 1,
+on Γ2(µ) ∪ Γ4(µ) × (0, T),
+∂y(µ)
+∂ν
+= 0,
+on Γ3(µ) × (0, T),
+y(µ)(0) = y0(x),
+in Ω(µ).
+
+FEM vs ROM averaged relative error - p (adjoint)
+101.4
+100
+10-1
+10-2
+10-3
+10
+Standard POD
+10-5
+Weighted POD Monte-Carlo
+Gaussjacobi+tensor
+GaussjacobiSmolyak
+ClenshawCurtis-tensor
+10-6
+ClenshawCurtis - Smolyak
+PseudoRandom - Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+N1.2e+00
+1
+0.8
+0.6
+0.4
+ 0.2
+-1.5e-021.0e+00
+0.5
+一
+0
+-0.5
+-9.3e-011.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+0.5
+0
+-0.5
+-9.3e-01FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10
+10-5
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi +tensor
+10-6
+Gaussjacobi↓. Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - u (control)
+10-1
+10-3
+10
+StandardPOD
+10-5
+Weighted PODMonte-Carlo
+Gaussjacobi+tensor
+Gaussjacobi + Smolyak
+10-6
+ClenshawCurtis.-.tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+17
+Figure 6. Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Only Sta-
+bilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue),
+wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak
+grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),
+Pseudo-Random based on Halton numbers (pink).
+As made in the steady version, we firstly consider a lifting procedure. Simulations are run following
+the space-time setting proposed in Section 3.2 for a prearranged number of time-steps Nt.
+The initial condition is y0(x) = 0 for all x ∈ Ω referring to Figure 1 and we set T = 3.0. The
+penalization parameter is α = 0.01 and we want the state solution to be similar in the L2-norm to
+a desired solution profile yd(x, t) = 1.0, function defined for all t ∈ (0, 3.0) and for all x in Ωobs in
+Figure 1. Choosing Nt = 30, the time step is ∆t = 0.1. For the spatial discretization a quite coarse
+mesh of h = 0.038 is implemented: consequently the total high-fidelity dimension is Ntot = 314820
+and a single FEM space is characterized by N = 3498 for a fixed instant t . Again, δK = 1.0 for all
+K ∈ Th. We take P :=
+�
+1, 105�
+×
+�
+1, 3.0
+�
+and µ is determined by the probability distribution
+(44)
+µ1 ∼ 1 +
+�
+105 − 1
+�
+X1, where X1 ∼ Beta(5, 3),
+µ2 ∼ 1.0 +
+�
+3.0 − 1.0
+�
+X2, where X2 ∼ Beta(5, 3).
+We choose a training set Ptrain of cardinality Ntrain = 100 (with exception of sparse grids, which
+have similar cardinality) and we performed the wPOD algorithms with Nmax = 15.
+In Figure 6 we present relative errors related to Offline-Only stabilization. Also in the parabolic
+case this procedure does not perform well. Therefore an online stabilization is needed.
+As a matter of fact, one can see in Figure 7 that the trends for Offline-Online stabilization seems
+a lot better than the previous strategy. Besides the Clenshaw-Curtis quadrature rule, errors decrease
+along the dimension N. Again, the best performance is given by the wPOD Monte-Carlo, where the
+following values are reached for N = 14: ey,14 = 9.71·10−7,ep,14 = 9.21·10−7, and eu,14 = 2.64·10−7.
+
+FEM vs ROM averaged relative error - y (state)
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis -tensor
+101
+ClenshawCurtis--Smolyak
+PseudoRandom-Halton
+Relative Log-Error
+100
+10-1
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - u (control)
+StandardPOD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+101
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis -Smolyak
+Relative Log-Error
+PseudoRandom - Halton
+100
+10-
+10-2
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - p (adjoint)
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi-Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+Relative Log-Error
+101
+100
+2
+4
+6
+8
+10
+12
+14
+N18
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 7. Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Online
+Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD
+(blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+Finally, in Table 2 we illustrate the performance of the speedup-index. All weighted algorithms
+performs similar: we compute an order of magnitude of 104 reduced solution in the time of a FEM
+one. This efficiency is given by the nature of the space-time procedure, where each snapshot carries
+all the time instances, and the reduction is very effective.
+Speedup-index Parabolic Graetz-Poiseuille Problem: Offline-Online Stab. - µ1, µ2 ∼ Beta(5,3)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+3
+14299.7
+14571.0
+13970.7
+14013.4
+14524.6
+14578.1
+14106.2
+6
+14666.3
+15393.5
+14621.8
+14952.8
+15302.6
+15117.8
+14482.0
+9
+14245.6
+14803.1
+14125.6
+14546.5
+14756.7
+14608.5
+13986.9
+12
+13693.6
+14206.2
+13554.3
+13935.7
+14050.5
+14075.4
+13453.0
+15
+13090.9
+13606.4
+13055.8
+13455.1
+13548.6
+13544.0
+12875.2
+Table 2. Average Speedup-index of Offline-Online Stabilization for the Parabolic Graetz-
+Poiseuille Problem under geometrical parametrization. From left to right: Standard POD,
+wPOD Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis
+tensor, Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+5.2. Numerical Tests for Propagating Front in a Square Problem. Here we analyze an
+Advection-Dominated PDE problem illustrated without control in a deterministic and in a stochastic
+context in [37, 59] and in [59], respectively. A distributed control is applied all over the domain Ω,
+
+FEM vs ROM averaged relative error - y (state)
+Standard POD
+Weighted POD Monte-Carlo
+101
+Gaussjacobi- tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+100
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10-
+10-2
+10-3
+10
+10-5
+10-6
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - u (control)
+101
+100
+10
+10-2
+10-3
+10-
+.4
+Standard POD
+10-5
+Weighted POD Monte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi-Smolyak
+10-6
+ClenshawCurtis.-.tensor
+ClenshawCurtis - Smolyak
+PseudoRandom -Halton
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - p (adioint)
+101
+10
+10-3
+Standard POD
+WeightedPODMonte-Cairlo
+Gaussjacobi-tensor
+10-5
+Gaussjacobi- Smolyak
+ClenshawCurtis - tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+2
+4
+6
+8
+10
+12
+14
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+19
+which is the square (0, 1) × (0, 1), as shown under Cartesian coordinates (x0, x1) in Figure 8. The
+boundary is composed as follows: Γ1 := {0} × [0, 0.25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1],
+Γ4 := [0, 1] × {1}, Γ5 := {0} × [0.25, 1]; instead Ωobs := [0.25, 1] × [0.75, 1].
+Γ1
+Γ2
+Γ3
+Γ4
+Γ5
+Ω
+Ωobs
+(0,0.25)
+(0,1)
+(1,0.75)
+(1,1)
+(0.25,1)
+(1,0)
+(0,0)
+Figure 8. Geometry of the Propagating Front in a Square Problem
+Given µ = (µ1, µ2), our aim is to solve the following OCP(µ) problem: find (y, u) ∈ ˜Y × U which
+solves
+min
+(y,u)
+1
+2
+�
+Ωobs
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω
+u(µ)2 dΩ,
+such that
+(45)
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + [cos µ2, sin µ2] · ∇y(µ) = u(µ),
+in Ω,
+y(µ) = 1,
+on Γ1 ∪ Γ2,
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5.
+In this case, we have that the domain of definition of our state y is
+˜Y :=
+�
+v ∈ H1�
+Ω
+�
+s.t. BC in (45)
+�
+.
+Again, we define a lifting function Ry ∈ H1�
+Ω
+�
+such that satisfies BC in (45), applying a lifting
+procedure before the Lagrangian approach. We define ¯y := y − Ry, with ¯y ∈ Y and Y := H1
+0(Ω),
+U = L2(Ω) and Q := Y ∗, with p = 0 on ∂Ω.
+The mesh size h is equal to 0.025, which entails an overall dimension of the truth approximation
+of 12087.
+Consequently, we have N = 4029 for state, control and adjoint spaces.
+Concerning
+stabilization, δK = 1.0 for all K ∈ Th. The penalization parameter is α = 0.01 and we pursue the
+state solution to be similar in the L2-norm to yd(x) = 0.5, defined for all x in Ωobs of Figure 8. In
+our test cases, P :=
+�
+1, 4 · 104�
+×
+�
+0.9, 1.5
+�
+and µ follow the subsequent probability distribution:
+(46)
+µ1 ∼ 1 +
+�
+4 · 104 − 1
+�
+X1, where X1 ∼ Beta(10, 10),
+µ2 ∼ 0.9 +
+�
+1.5 − 0.9
+�
+X2, where X2 ∼ Beta(10, 10),
+where µ1 and µ2 are independent random variables. The training set Ptrain and the testing set
+Ptest have both cardinality equal to ntrain = 100, with exception of sparse grid samplings, whose
+cardinality is similar to 100. We apply a wPOD procedure for a Nmax = 50 dimension. In Figure 9,
+we show the performance of relative errors for the Offline-Only stabilization procedure. As in the
+Graetz-Poiseuille Problem, these trends are not acceptable, as no quantity drops under 10−1 for all
+state, control and adjoint variables. Therefore, a stabilization applied in the Online Phase is needed,
+too.
+In Figure 10 relative errors for Offline-Online Stabilization procedure are shown. Again, wPOD
+Monte-Carlo presents the best behaviour: in this case it reaches ey,50 = 5.03 · 10−7 for the state, for
+the adjoint ep,50 = 1.07·10−6, and the control eu,50 = 4.21·10−6. Moreover, the wPOD Monte-Carlo
+has an accuracy of nearly a factor of 100 better than a Standard POD in a deterministic context for
+N > 20. Also here, Smolyak grids perform better than their tensor counterpart: for instance, we
+obtain in this case it reaches ey,50 = 2.77 · 10−6 for the state, for the adjoint ep,50 = 5.80 · 10−6, and
+
+20
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 9. Relative Errors for the Propagating Front in a Problem - Offline-Only Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+the control eu,50 = 1.02 · 10−5 for Gauss-Jacobi. Concerning the training set, we have Ntrain = 89
+and Ntrain = 93 for the Gauss-Jacobi and the Clenshaw-Curtis ones, respectively. In Figure 11
+we see a comparison between the FEM solution for the state and the adjoint without stabilization
+and the Offline-Online Stabilized wPOD Monte-Carlo reduced solution for these variables with
+µ = (2 · 104, 1.2).
+The values of the speedup-index for the Offline-Online stabilization for each type of wPOD are
+reported in Table 3. For N = 50 the wPOD Monte-Carlo is the best choice again with a computation
+of 50 reduced solutions in the time of a FEM one. All the other possibilities perform a little bit lower
+for N = 50; however, all weighted algorithms have similar performances concerning the speedup-
+index: an order of magnitude of 102 for the first 50 reduced basis.
+Numerical tests of the parabolic version of the Propagating Front in a Square Problem are here
+illustrated. For a fix T > 0 and a given µ ∈ P we have to find the pair (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs×(0,T )
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω×(0,T )
+u(µ)2 dΩ,
+such that
+(47)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ty(µ) − 1
+µ1
+∆y(µ) + [cos µ2, sin µ2] · ∇y(µ) = u(µ),
+in Ω × (0, T),
+y(µ) = 1,
+on Γ1 ∪ Γ2 × (0, T),
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5 × (0, T),
+y(µ)(0) = y0(x),
+in Ω,
+
+FEM vs ROM averaged relative error - y (state)
+Relative Log-Error
+StandardPOD
+100
+WeightedPoDMonte-Carlo
+GaussJacobi-tensor
+Gaussjacobi - Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - u (control)
+Relative Log-Error
+100
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi-tensor
+Gausslacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom -Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - p (adioint)
+Relative Log-Error
+101
+StandardPOD
+Weighted POD Monte-Carlo
+Gausslacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+21
+Figure 10. Relative Errors for the Propagating Front in a Problem - Offline-Online Sta-
+bilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue),
+wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak
+grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),
+Pseudo-Random based on Halton numbers (pink).
+Figure 11. FEM not stabilized and wPOD Monte-Carlo Offline-Online stabilized solu-
+tion for y (left) and for p (right), µ = (2 · 104, 1.2), h = 0.025 α = 0.01, Ntrain = 100,
+δK = 1.0, N = 50.
+where y0(x) = 0 for all x ∈ Ω in Figure 8. A final time T = 3.0 is set. Considering the time
+discretization, we chose a number of time steps equal to Nt = 30, then we have ∆t = 0.1. Instead,
+for the spatial approximation, the mesh size is set to h = 0.036, that implies an overall dimension
+of the space-time setting equal to Ntot = 174780.
+For a fixed instant t, a single FEM space is
+characterized by N = 1942.
+For the SUPG procedure, we impose δK = 1.0 for all K ∈ Th.
+Setting a penalization parameter α = 0.01, we try to achieve in a L2-mean a desired solution profile
+yd(x, t) = 0.5, defined for all t ∈ (0, 3) and x in Ωobs of Figure 8.
+P :=
+�
+1, 4 · 104�
+×
+�
+0.9, 1.5
+�
+, as in the steady version. We suppose that µ follows the probability
+distribution (46). Our training set has cardinality Ntrain = 100, with exception for Gauss-Jacobi and
+Clenshaw-Curtis Smolyak grids with Ntrain = 89 and Ntrain = 93, respectively, which are the number
+of nodes nearest to 100 for this kind of procedure. In Figure 12 and 13, we show a representative
+
+FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10
+4
+StandardPOD
+10-5
+WeightedPODMonte-Carlo
+GaussJacobi-tensor
+Gaussjaciobi-Smolyak
+ClenshawCurtis-tensor
+10-6
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - u (control)
+100
+Standard.POD
+Weighted POD Monte-Carlo
+GaussJacobi-tensor
+GaussJacobi - Smolyak
+ClenshawCurtis-tensor
+10-1
+ClenshawCurtis-Smolyak
+PseudoRandom -Halton
+10-2
+10-3
+10-5
+10
+20
+30
+40
+50
+NFEM ys ROM averaged relative error - p (adioint)
+100
+10-1
+10-2
+10-3
+StandardPOD
+WeightedPODMonte-Carlo
+10-5
+Gaussjacobi--tensor
+Gaussjacobi - Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis - Smolyak
+10-6
+PseudoRandom---Halton
+10
+20
+30
+40
+50
+N1.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-0322
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Speedup-index Propagating front in a Square Problem: Offline-Online Stab. - µ1, µ2 ∼ Beta(10,10)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+10
+151.3
+179.2
+175.0
+178.7
+181.5
+176.4
+173.9
+20
+123.3
+140.4
+139.8
+141.0
+140.9
+140.5
+143.6
+30
+88.5
+103.3
+102.6
+102.8
+100.6
+102.6
+104.3
+40
+61.6
+73.7
+73.2
+69.9
+68.6
+70.4
+70.2
+50
+43.4
+50.2
+49.0
+47.6
+46.8
+49.2
+48.2
+Table 3. Average Speedup-index of Offline-Online Stabilization for the Propagating
+Front in a Square Problem.
+From left to right: Standard POD, wPOD Monte-Carlo,
+Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-
+Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+Figure 12. wPOD Monte-Carlo Offline-Online stabilized reduced solution of y, for t =
+0.1, t = 1.5, t = 3.0, µ = (2 · 104, 1.2), h = 0.036, α = 0.01, Ntrain = 100, δK = 1.0,
+N = 30.
+Figure 13. wPOD Monte-Carlo Offline-Online stabilized reduced solution of p, for t =
+0.1, t = 1.5, t = 3.0, µ = (2 · 104, 1.2), h = 0.036, α = 0.01, Ntrain = 100, δK = 1.0,
+N = 30.
+stabilized FEM solution for µ = (2 · 104, 1.2) for some instants of time of the state y and the adjoint
+p, respectively. We choose to perform all wPOD procedure with Nmax = 30.
+Let us move to the error analysis. In Figure 14 we illustrate the relative errors for the Offline-Only
+stabilization. The performance are not satisfactory here, too, where no quantity drops below the
+accuracy of 10−1 for all N.
+Instead, Offline-Online stabilization procedure performs well, as one can notice from Figure 15.
+Again, wPOD Monte-Carlo has the best behaviour, it reaches ey,30 = 1.12 · 10−7 for the state, for
+the adjoint ep,30 = 4.55 · 10−7 and the control eu,30 = 1.36 · 10−7. Also in this case, isotropic sparse
+grid techniques is a better choice than tensor rules, both for Gauss-Jacobi and Clenshaw-Curtis
+approximations.
+In Table 4 we compare the speedup-index for all the weighted algorithms: performance are similar
+for all N, for N = 30 we computed nearly 4000 Offline-Online stabilized reduced solutions in the
+time of a FEM one.
+
+1.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-021.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-021.2e+00
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-039.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-039.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-03STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+23
+Figure 14. Relative Errors for the Parabolic Propagating Front in a Problem - Offline-
+Only Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD
+(blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+Speedup-index Parabolic Propagating front in a Square Problem: Offline-Online Stabilization
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+5
+6601.9
+6503.5
+6702.6
+6629.1
+6566.6
+6605.6
+6575.8
+10
+6275.9
+6208.0
+6336.0
+6277.9
+6204.4
+6293.4
+6212.4
+15
+5814.3
+5702.4
+5838.7
+5794.3
+5699.6
+5752.1
+5723.7
+20
+5327.9
+5190.4
+5329.8
+5270.3
+5277.2
+5235.9
+5197.6
+25
+4465.2
+4303.3
+4562.6
+4422.2
+4541.3
+4433.1
+4479.9
+30
+4061.5
+3959.5
+4140.3
+4026.3
+4100.3
+4035.6
+4043.8
+Table 4. Average Speedup-index of Offline-Online Stabilization for the Parabolic Propa-
+gating Front in a Square Problem. From left to right: Standard POD, wPOD Monte-Carlo,
+Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-
+Curtis Smolyak grid, Pseudo-Random based on Halton numbers. µ1, µ2 ∼ Beta(10,10)
+6. Conclusions and Perspectives
+In this work, we illustrated some numerical tests concerning stabilized Parametrized Advection-
+Dominated OCPs with random parametric inputs in a ROM context. We deal with both steady and
+unsteady cases and we took advantage of the SUPG stabilization to overcome numerical issues due
+to high values of the P´eclet number. Two possibilities of stabilization were analyzed: when SUPG
+is only used occurs in the offline phase, Offline-Only stabilization, or when it is provided in both
+online and offline phases, Offline-Online stabilization.
+
+FEM vs ROM averaged relative error - y (state)
+StandardPOD
+Weighted POD Monte-Carlo
+100
+Gaussjacobi- tensor
+Gaussjacobi -Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - u (control)
+Standard POD
+WeightedPODMonte-Carlo
+2 × 100
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis -Smolyak
+PseudoRandom-Halton
+100
+6×10-1
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - p (adjoint)
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+Log-Error
+101
+Relative
+100
+5
+10
+15
+20
+25
+30
+N24
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 15. Relative Errors for the Parabolic Propagating Front in a Problem - Offline-
+Online Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard
+POD (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+In order to deal with the uncertainty quantification caused by random inputs, we consider wROM.
+More precisely, we built our reduced bases using a wPOD in a partitioned approach, using different
+quadrature rules. We implemented wPOD Monte-Carlo, Gaussian quadrature formulae based on
+Jacobi polynomials in a tensor rule, approximation related to Clenshaw-Curtis tensor rule, Smolyak
+isotropic sparse grid approximation of the last two methods, quasi Monte-Carlo method as a Pseudo-
+Random rule defined on Halton numbers.
+We analyzed relative errors between the reduced and the high fidelity solutions and the speedup-
+index concerning the Graetz-Poiseuille and Propagating Front in a Square Problems, always under a
+distributed control. For the state, control and adjoint spaces we implemented a P1-FEM approxima-
+tion in a optimize-then-discretize framework as the truth solution. Concerning parabolic problems,
+a space-time approach is followed applying SUPG in a suitable way. In order to established which
+wPOD performs better, we compare them through the same testing set sampled by a Monte-Carlo
+method according to the probability distribution of the parameter.
+Offline-Only stabilization technique performed very poorly in terms of errors, this happened for
+all wROMs considered. Instead, in all the steady and unsteady experiments, the wROM technique
+performed excellently in an Offline-Online stabilization framework. For parabolic problems, the
+speedup-index features large values thanks to the space-time formulation. More precisely, wPOD
+Monte-Carlo technique was always the best performer for relative errors, instead, concerning compu-
+tational efficiency all methods seem equivalent. In addition, the efficiency of the wPOD Monte-Carlo
+
+FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10-5
+Standard POD
+WeightedPODMonte-Carlo
+10-6
+GaussJacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis - Smolyak
+10-7
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - u (control)
+10-1
+10-2
+10-3
+10-
+4
+10-5
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+10-6
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+10-7
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - p (adjoint)
+100
+10-
+10
+10-3
+10
+Standard POD
+10-5
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis -tensor
+10-6
+ClenshawCurtis -Smolyak
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+25
+is supported by the fact that after a small number of reduced basis it is nearly 100 times more accu-
+rate than a Standard POD in a deterministic context. Moreover, we notice that sparse grids perform
+better than relative tensor ones, although having a bit less number of quadrature nodes.
+Furthermore, in the Graetz-Poiseuille Problem we illustrate that under geometrical parametriza-
+tion affected by randomness, wROMs still have good performance, despite small fluctuations in the
+graph of relative errors.
+As a first perspective, it might be interesting to create a strongly-consistent stabilization technique
+that flattens all the fluctuations of geometrical parametrization in a UQ context. Moreover, we want
+to extend the study to boundary control. Finally, it might be interesting to study the performance
+of other stabilization techniques for the online phases, for instance, of the Online Vanishing Viscosity
+and the Online Rectification methods [4, 12, 33] combined with the SUPG technique in the offline
+phase or with the stabilization strategy used in [59].
+Acknowledgements
+We acknowledge the support by European Union Funding for Research and Innovation – Horizon
+2020 Program – in the framework of European Research Council Executive Agency: Consolidator
+Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods
+with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Nu-
+merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of
+complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-
+GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”. The computations in
+this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is
+an implementation in FEniCS [32] of several reduced order modelling techniques; we acknowledge
+developers and contributors to both libraries.
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+cation, 6(4):1475–1502, 2018.
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+inputs. Journal of Scientific Computing, 81(1):136–153, 2019.
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+Method for Advection-Dominated Partial Differential Equations under Optimal Control. preprint, 2022.
+
diff --git a/FtA0T4oBgHgl3EQfBf8H/content/tmp_files/load_file.txt b/FtA0T4oBgHgl3EQfBf8H/content/tmp_files/load_file.txt
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@@ -0,0 +1,1719 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf,len=1718
+page_content='STABILIZED WEIGHTED REDUCED ORDER METHODS FOR  PARAMETRIZED ADVECTION-DOMINATED OPTIMAL CONTROL  PROBLEMS GOVERNED BY PARTIAL DIFFERENTIAL EQUATIONS WITH  RANDOM INPUTS  FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this work, we analyze Parametrized Advection-Dominated distributed Optimal  Control Problems with random inputs in a Reduced Order Model (ROM) context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' All the simula-  tions are initially based on a finite element method (FEM) discretization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' moreover, a space-time  approach is considered when dealing with unsteady cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' To overcome numerical instabilities that  can occur in the optimality system for high values of the P´eclet number, we consider a Streamline  Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We com-  bine this method with the ROM framework in order to consider two possibilities of stabilization:  Offline-Only stabilization and Offline-Online stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Moreover we consider random parame-  ters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach  to deal with the issue of uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Several quadrature techniques are used to  derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We compare all the approaches analyzing relative errors between the FEM and ROM  solutions and the computational efficiency based on the speedup-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Introduction  Here we present a numerical study concerning stabilized Parametrized Advection-Dominated Op-  timal Control Problems (OCP(µ)s) with random inputs in a Reduced Order Methods (ROMs)  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As a matter of fact, engineering and scientific applications often need very fast evalu-  ations of the numerical solutions for many parameters that characterize the problem, for instance  in real-time scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A solution to these many-query situations can be to exploit the parameter  dependence of the OCP(µ)s using ROMs [6, 24, 41, 40, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This process is divided in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The former is the offline phase, when many numerical solutions for different values of parameters  are collected considering a first discretization of the OCP(µ), such a finite element method (FEM)  one, called the high-fidelity or truth approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then all parameter-independent components  are calculated and stored, and reduced spaces are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The latter is the online phase, when all  parameter-dependent parts and, then, the reduced solutions are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' More precisely, to deal  with the randomness which is hidden in the parameters, we consider a modification of the Proper  Orthogonal Decomposition (POD) that takes into account the probability distribution of the random  inputs: the weighted POD (wPOD) [61, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We apply this procedure in a partitioned approach,  following good results shown in literature [30, 34, 53, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As this algorithm aims to minimize the  expectation of the square error between the truth and the ROM solutions, we can identify different  types of weighted ROMs (wROMs) [11, 15, 13, 16, 17, 18, 49, 59, 61, 60] based on the chosen quad-  rature rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this work, we will exploit Monte-Carlo and Quasi Monte-Carlo procedures, tensor  rules based on Gauss-Jacobi and Clenshaw-Curtis quadrature techniques, and Smolyak isotropic  sparse grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland,  email: fabio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='zoccolan@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ch  2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  email: maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='strazzullo@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='it  3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  email: gianluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='rozza@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='it  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01975v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NA]  5 Jan 2023  2  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  The optimization problem will always concern a linear-quadratic cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We use FEM  as the truth approximation, both for steady and unsteady problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' At a first level, FEM approx-  imations of stochastic steady OCP(µ)s have been already presented, for example, in [26] consid-  ering stochastic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In the parabolic case, we discretize time-dependency via a space-time ap-  proach [25, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Concerning stabilization, we considered the Streamline Upwind Petrov–Galerkin  (SUPG) [9, 28, 38] suitably combined with the ROM setting in an optimize-then-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We exploit two possibilities: when stabilization only occurs in the offline phase, Offline-Only stabi-  lization or when SUPG is applied in both phases, Offline-Online stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Stabilized Advection-Dominated problems in a ROM framework without control are studied,  for instance, in [37, 59], both for steady and unsteady cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead, in [11] wROMs for generic  OCP(µ)s are applied to experiments concerning environmental sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead, SUPG Advection-  Dominated distributed OCP(µ)s are analyzed in a deterministic context in [63], both for elliptic and  parabolic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' To the best of our knowledge, this is the first time that stabilized Advection-  Dominated OCP(µ)s with random inputs are analyzed in a ROM context, both for elliptic and  parabolic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  This work is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In Section 2, we introduce linear-quadratic optimal control  theory for PDEs and its FEM discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Section 3 firstly concerns the basic theory of SUPG  stabilization for Advection-Dominated PDEs in an optimize-then-discretize approach [19], then the  space-time procedure that will be used is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' wROMs features will be illustrated in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Section 5 will concern numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Two Advection-Dominated problems under  distributed control and random inputs will be analyzed: the Graetz-Poiseuille Problem under ge-  ometrical parametrization and the Propagating Front in a Square Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will compare the  wPOD procedures through relative errors between the FEM and the ROM solutions and computa-  tional time considering the speedup-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Finally, conclusions follow in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Problem Formulation for Random Input Optimal Control Problems  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Mathematical Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let Ω be an open and bounded regular domain in R2, where ΓN and  ΓD will indicate the portions of the boundary ∂Ω where Neumann and Dirichlet boundary conditions  are imposed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' With the symbol Ωobs ⊆ Ω the observation domain will be indicated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  the subset of the domain where we seek the state variable to be similar to a desired solution profile  yd ∈ Y , with Y Hilbert space, in a sense that will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For time-dependent problems we  will also take into account the time interval (0, T) ⊂ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let us consider a compact set P ⊆ Rp, for  natural number p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will call P and as the parameter space and a p-vector µ ∈ P is the parameter  of our Parametric OCP(µ)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As the setting is completely general, for instance µ can characterize  our yd or geometrical and physical properties of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Furthermore, we denote with B(Q, R)  the space of linear continuous operators between Banach spaces Q and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The triplet (A, F, P) will denote a complete probability space, composed by A, which is the set  of outcomes ω ∈ A, F, that is a σ-algebra of events, and P : F → [0, 1] with P(A) = 1, which is  the chosen probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As dealing with random input OCP(µ)s, the parameter µ will be  a real-valued random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In detail, µ : (A, F) → (Rp, B) is a measurable function, where B is  the Borel σ-algebra on Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The distribution function of µ : A → P ⊂ Rp, being P the image of µ,  is defined as Pµ : P → [0, 1] such that  (1)  ∀µ ∈ P,  Pµ(µ) = P(ω ∈ A : µ(ω) ≤ µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Let dPµ(µ) denote the distribution measure of µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=',  (2)  ∀H ⊂ P,  P(µ ∈ H) =  �  H  dPµ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We assume that µ admits a Lebesgue density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' dPµ(µ) is absolutely continuous with respect  to the Lebesgue measure dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This practically means that there exists a probability density func-  tion ρµ : P → R+ such that ρµ(µ)dµ = dPµ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' It is worth to notice that the measure space  (P, B(P), ρµ(µ)dµ) is isometric to (A, F, P) under the random vector µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The aim of this work is to analyze random input OCP(µ)s from the numerical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  3  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 (Random Input Parametric Optimal Control Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Consider the state equation  E : Y × U → Q, with Y, U, and Q real Banach spaces, satisfying a set of boundary and/or initial  conditions, and a real functional J : Y ×U → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then for Pµ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' find the pair  �  y(µ), u(µ)  �  ∈ X :=  Y × U that minimizes cost functional J (y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) under the constraint E(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Let Xad be the set of all couples (y, u) solutions of E: we will only consider the case of full  admissibility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' when Xad = Y × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 looks for minimizers among all state-control  pairs such that:  min  (y(µ),u(µ))∈Y ×U J (y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' E(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  This can be achieved through the research of the critical points of the Lagrangian operator  L : Y × U × Q∗ → R defined as:  (3)  L(y(µ), u(µ), p(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) = J (y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) + ⟨p(µ), E(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ)⟩Q∗Q,  where p(µ) is a Lagrange multiplier belonging to the adjoint space Q∗, the dual space of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For  the sake of notation we write y := y(µ), u := u(µ) and p := p(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In case that Pµ is the uniform  distribution with support in P, then Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 is called to be deterministic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this  work linear-quadratic problems will be involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2 (Linear-Quadratic OCP(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let us consider the bilinear forms m : Z × Z → R  and n : U × U → R, which are symmetric and continuous, where Z is a Banach space called the the  observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Fix α > 0, a constant called the penalization parameter and consider a quadratic  objective functional J of the form  (4)  J (y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) = 1  2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α  2 n(u(µ), u(µ)),  where O : Y → Z is a linear and bounded operator called the observation map and zd(µ) ∈ Z is  the observed desired solution profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Consider an affine map E defined as  (5)  E(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) = A(µ)y + C(µ)u − f(µ),  ∀  �  y(µ), u(µ)  �  ∈ Y × U,  where A(µ) ∈ B(Y, Q), C(µ) ∈ B(U, Q) and f(µ) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For Linear-Quadratic OCP(µ)s well-posedness of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2 yields [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' More precisely, the  reader can refer to [10] to a comparison between the Lagrangian approach for the full-admissibility  case and the adjoint one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Via the functional derivative of L, we obtain a optimality system to be  solved to find the unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this case, this reads as finding (y, u, p) ∈ Y × U × Q∗ that  satisfies [10],  (6)  �  �  �  �  �  DyL(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ)(¯y) = 0 =⇒ m(Oy, O¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) ,  ∀¯y ∈ Y,  DuL(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ)(¯u) = 0 =⇒ αn(u, ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) + ⟨C∗(µ)p, ¯u⟩U ∗U = 0,  ∀¯u ∈ U,  DpL(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ)(¯p) = 0 =⇒ ⟨¯p, A(µ)y + C(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q,  ∀¯p ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In system (6), the first equation is called the adjoint equation, the second one is the gradient  equation and the last one is state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3 (Notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the sake of notation, when Hilbert spaces will be taken into account,  bilinear forms A, B and their adjoint counterparts will be indicate uniquely as:  ⟨A(µ)y, p⟩QQ∗ := a(y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ)  ⟨C(µ)u, p⟩QQ∗ := c(u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4 (Parabolic Problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Concerning unsteady problems, one must add more hypotheses  to the mathematical setting of Linear-Quadratic OCP(µ)ss to reach well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will consider  the following Hilbert spaces Y = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Y ), Y∗ = L2 (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Y ∗), Z = L2 (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Z), U = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' U)  with respective norms given by  (7) ∥y∥2  Y :=  T  �  0  ∥y∥2  Y dt,  ∥y∥2  Y∗ :=  T  �  0  ∥y∥2  Y ∗dt,  ∥z∥2  Z :=  T  �  0  ∥z∥2  Zdt,  and  ∥u∥2  U :=  T  �  0  ∥u∥2  Udt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  4  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Then we define the Hilbert space Yt := {y ∈ Y  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ∂ty ∈ Y∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For parabolic problems we will  also consider the case of full-admissibility as Xad = Yt × U [5, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' High-Fidelity Discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this work, the discretization of the optimality sistem (6)  follows an one shot or all-at-once approach [25, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' When we will consider Advection-Dominated  OCP(µ)s, a stabilization technique will be also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, a SUPG method will be applied  in a optimize-then-discretize approach, as we will see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  This means that firstly the  optimality conditions are computed obtaining system (6) and then we discretize and stabilize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Concerning numerical implementation, we use a FEM discretization for all three variables, where  Th is a regular triangularization on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Its elements K are triangles and the parameter h denotes the  mesh size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' the maximum diameter of an element of the chosen grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In addition, we define  Ωh := int  � �  K∈Th  K  �  ,  as a quasi-uniform mesh for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Considering Pr(K) as the space of polynomials of degree at most  equal to r defined on K and defining  XN ,r =  �  qN ∈ C(¯Ω) : qN  |K ∈ Pr(K), ∀K ∈ Th  �  we set Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and  �  QN �∗ = Q∗ ∩ XN ,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this work, the numerical  implementation will always made by a P1-FEM approximation and the same triangulation Th for  Y N , U N , and  �  QN �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A similar discussion can be made for time-dependent problem, as we will  see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This first discretization procedure will be indicated as the truth or high-fidelity  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  From now on, Y, U, Q will be always Hilbert spaces and we will consider the Identity restricted to  our observation domain Ωobs as the Observation function O for both steady and unsteady problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Therefore, Z = Y for steady problems and Z = Y for unsteady ones are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Our desired state  will be denoted by yd and with the same symbol will also indicate its FEM discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' SUPG stabilization for Advection-Dominated OCP(µ)s  In this work we only deal with Advection-Diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 (Advection-Diffusion Equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let us take into account the following problem:  (8)  T(µ)y := −γ(µ)∆y + η(µ) · ∇y = f(µ) in Ω ⊂ R2,  where suitable boundary conditions are applied on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In addition, we require that:  the diffusion coefficient γ : Ω → R is uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' there exists γmax, γmin > 0 such  that  (9)  P  �  ω ∈ A : γmin < γ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ) < γmax ∀x ∈ Ω  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  the advection field η : Ω → R2 belongs to (L∞(Ω))2 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' More precisely, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  µ the following inequality holds: 0 ≥ div η(x) ≥ −ϑ, ∀x ∈ Ω, with ϑ ∈ R+  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  f : Ω → R is an L2(Ω)-function for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' in addition, f has bounded second moments with  respect to the integral along A and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  With these hypotheses, Problem (8) is called Advection-Diffusion problem and the operator T(µ)y :=  −γ(µ)∆y + η(µ) · ∇y is said the Advection-Diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For more details regarding the well-posedness and theoretical results of Stochastic Advection-  Diffusion OCP(µ)s, we refer to [14, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2 (P´eclet number and Advection-Dominated problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let us consider the FEM  discretization related to an Advection-Diffusion problem and its regular triangulation Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For any  element K ∈ Th, the local P´eclet number is defined as [42, 38]:  (10)  PeK(x) := |η(x)|hK  2γ(x)  ∀x ∈ K,  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  5  where hK is the diameter of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' If PeK(x) > 1 ∀x ∈ K, ∀K ∈ Th, we say to study an Advection-  Dominated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Setting for Stabilized Advection-Dominated OCP(µ)s - Steady case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Numerical in-  stabilities can appear, when dealing with Advection-Dominated OCP(µ)s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' when |η(µ)| ≫ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In  order to adjust this unpleasant behaviour without modifying the mesh size, we use the Streamline  upwind/Petrov Galerkin (SUPG) method [9, 27, 28, 42] in a optimize-then-discretize approach [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  This assures the strongly consistency of the optimality system [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the sake of simplicity, we  define our Advection-Dominated problem on H1  0(Ω) and we do not indicate the parameter depen-  dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We denote with T ∗ the adjoint operator related to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This last operator can be split into its  symmetric and skew-symmetric parts as T = TS + TSS [42], where:  (11)  symmetric part: TSy = −γ∆y − 1  2(div η)y,  ∀y ∈ H1  0(Ω),  skew-symmetric part: TSSy = η · ∇y + 1  2(div η)y,  ∀y ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  This two parts can be immediately recovered using the formulae:  (12)  TS = T + T ∗  2  ,  TSS = T − T ∗  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  After having considered FEM spaces, the stabilization occurs in the bilinear and linear terms involved  in the state and the adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead, the gradient equation is left unstabilized [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We  recall that we use distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We defined the stabilized bilinear form as and cs, and the stabilized forcing term Fs as  (13)  as  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  := a  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  +  �  K∈Th  δK  �  TyN , hK  |η| TSSqN  �  K  ,  yN , qN ∈ Y N ,  cs  �  uN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  := −  �  Ω  uN qN −  �  K∈Th  δK  �  uN , hK  |η| TSSqN  �  K  ,  uN ∈ U N , qN ∈ Y N ,  Fs  �  qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  := F  �  qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  +  �  K∈Th  δK  �  f(µ), hK  |η| TSSqN  �  K  ,  ∀qN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  where δK is a local positive dimensionless parameter related to the element K ∈ Th, consequently  it can be different for each triangle, and (·, ·)K is the inner scalar product in L2(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In (13)  a  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  TyN , qN �  L2(Ω) and F  �  qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  f, qN �  L2(Ω), where f collects all forcing and  lifting terms of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For the remaining conditions of the optimality system, we will always consider m and n form as  the L2(Ωobs) and the L2(Ω) products for steady problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The adjoint equation is an Advection-  Dominated equation, too, where the advective term has opposite sign with respect to the state one:  indeed, T ∗ = TS − TSS from (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We use the next SUPG forms for zN ∈ Y N :  (14)  a∗  s  �  zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  := a∗ �  zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  +  �  K∈Th  δa  K  �  (TS − TSS)pN , hK  |η| (−TSS) zN  �  K  ,  �  yN − yd, zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  s :=  �  Ωobs  (yN − yd)zN dx +  �  K∈Th|Ωobs  δa  K  �  yN − yd, hK  |η| (−TSS) zN  �  K  ,  where a∗ is the adjoint form of a, δa  K is the positive stabilization parameter of the stabilized adjoint  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In our numerical experiments, we will always consider δK = δa  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Finally, the SUPG  optimality system for a steady OCP(µ) reads as:  (15)  discretized adjoint equation:  a∗  s  �  zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  +  �  yN − yd, zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  s = 0, ∀zN ∈ Y N ,  discretized gradient equation:  c∗�  vN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  + αn  �  uN , vN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  = 0, ∀vN ∈ U N ,  discretized state equation:  as  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  + cs  �  uN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  = Fs(qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ), ∀qN ∈  �  QN �∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  6  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  We denote with Ks and KT  s the stiffness matrices related to the stabilized forms as and a∗  s,  respectively, M is the not-stabilized mass matrix related to n, instead, Ms is the stabilized mass  matrix related to m after stabilization, Cs is the matrix linked to stable form cs, the block CT refers  to c, and fs is the vector that contains the coefficients of the stabilized force term as components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Moreover, we consider with the symbol y, u and p as the vectors of coefficients of yN , uN and pN ,  expressed in terms of the nodal basis of Y N , U N , (QN )∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Finally, the discretized block  system related to (15) is:  (16)  �  �  Ms  0  KT  s  0  αM  CT  Ks  Cs  0  �  �  �  �  y  u  p  �  � =  �  �  Msyd  0  fs  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Setting for Stabilized Advection-Dominated OCP(µ)s - Unsteady case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We show the  SUPG approach for time-dependent OCP(µ)s proposed in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A classical implicit Euler discretiza-  tion is applied to all forms including time-derivatives [3, 25, 50, 54, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We divide the time  interval (0, T) in Nt sub-intervals of equal length ∆t := tj − tj−1, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Nt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Starting from  this framework, a discretization along time is done, where each discrete instant of time is considered  as a steady-state Advection-Dominated equation in a space-time approach [25, 50, 51, 54, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In addition, the SUPG stabilization occurs for time-dependent forms, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The general scheme is  described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Let us firstly define the discrete vectors y =  �  yT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , yT  Nt  �T , u =  �  uT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , uT  Nt  �T and p =  �  pT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , pT  Nt  �T , where yi ∈ Y N , ui ∈ U N and pi ∈ (QN )∗ for 1 ≤ j ≤ Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Also here, yj, uj  and pj indicate the column vectors containing the coefficients of the FEM discretization for state,  control and adjoint, respectively (unlike the steady case, there are not denoted in bold style).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This  implies Ntot = 3 × Nt × N as the global dimension of the block system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We express all other  terms in based of the respective nodal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The vector representing the initial condition for the  state variable is y0 =  �  yT  0 , 0T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , 0T �T , where 0 is the zero vector in RN , yd =  �  yT  d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , yT  dNt  �T  is the vector including discrete time components of the discretized desired solution profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' instead,  f s =  �  f T  s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , f T  sNt  �T  corresponds to the stabilized forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We recall that Y, U, Q are Hilbert  Spaces and, for the sake of simplicity, we assume Y N ≡ (QN )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' So now we can see locally the time  block discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Adjoint equation: this equation is discretized backward in time using the forward Euler  method, which is equal to a backward Euler with respect to time T − t, for t ∈ (0, T) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Firstly, we add a stabilized term to the form related to ∂tp and a∗ defined as:  s∗ �  zN , pN (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  K∈Th  δK  �  −∂tpN (t) + T ∗pN (t), −hK  |η| TSSzN  �  K  ,  where we define the form  (17)  m∗  s  �  pN , zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  pN , zN �  L2(Ω) −  �  K∈Th  δK  �  pN , hK  |η| TSSzN  �  K  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Then, the time discretization is: for each j ∈ {Nt − 1, Nt − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=', 1}, find pN  j ∈ Y N  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (18)  1  ∆tm∗  s  �  pN  j (µ) − pN  j+1(µ), zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  + a∗  s  �  zN , pN  j (µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  = −  �  yN  j − ydj, zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  s  ∀zN ∈ Y N ,  Considering M T  s as the matrix inherent to m∗  s, the block subsystem reads  M T  s pj = M T  s pj+1 + ∆t  �  −M T  s yj − KT  s pj + M T  s ydj  �  for j ∈ {Nt − 1, Nt − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  7  Finally, we derive the following block system:  �  ����  M T  s + ∆tKT  s  −M T  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  M T  s + ∆tKT  s  −M T  s  M T  s + ∆tKT  s  �  ����  �  ��  �  AT  s  p +  �  �����  ∆tM T  s y1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ∆tM T  s yNt  �  �����  =  �  �����  ∆tM T  s yd1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ∆tM T  s ydNt  �  �����  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Setting the diagonal block matrix MT  s ∈ RN ·Nt×RN ·Nt with diagonal entries [M T  s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , M T  s ],  the adjoint system to be solved is: ∆tMT  s y + AT  s p = ∆tMT  s yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Gradient equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We seek the solution of α∆tMuj+∆tCT pj = 0, ∀j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Nt} ,  which is equal to the following block system:  (19)  α∆t  �  ����  M  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  M  �  ����  �  ��  �  M  �  ����  u1  u2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  uNt  �  ���� +∆t  �  ����  CT  0  · ·  CT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  CT  �  ����  �  ��  �  CT  �  ����  p1  p2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  pNt  �  ���� =  �  ����  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  0  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  More compactly, we solve α∆tMu+∆tCT p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  State equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A backward Euler method is used for a discretization forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The  stabilized term related to ∂ty and the bilinear form a is [29, 37, 58]:  s  �  yN (t), qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  K∈Th  δK  �  ∂tyN (t) + TyN (t), hK  |η| TSSqN  �  K  ,  where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Defining the stabilized term ms as  (20)  ms  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  �  yN , qN �  L2(Ω) +  �  K∈Th  δK  �  yN , hK  |η| TSSqN  �  K  ,  then the backward Euler approach reads as: for each j ∈ {1, 2, · · · , Nt}, find yN  j ∈ Y N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (21)  1  ∆tms  �  yN  j (µ) − yN  j−1(µ), qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  + as  �  yN  j (µ), qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  + cs  �  uN  j , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  = Fs  �  qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  ,  given the initial condition yN  0  which satisfies  �  yN  0 , qN �  L2(Ω) =  �  y0, qN �  L2(Ω) , ∀qN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The matrix state equation to be solved becomes  (22)  Msyj + ∆tKsyj + ∆tCsuj = Msyj−1 + fsj∆t  for j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Nt} ,  where the stabilized mass matrix of ms is Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Thus, we have  �  ����  Ms + ∆tKs  0  −Ms  Ms + ∆tKs  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  0  0  −Ms  Ms + ∆tKs  �  ����  �  ��  �  As  y+∆t  �  ��  Cs  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  0  0  Cs  �  ��  �  ��  �  Cs  u = Msy0 + ∆tf s,  where Ms ∈ RN ·Nt×RN ·Nt is a block diagonal matrix which diagonal entries are [Ms, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Ms].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In a more compact notation, we have Asy+∆tCsu = Msy0 + ∆tf s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The final system considered and solved through an one shot approach is the following:  (23)  �  �  ∆tMT  s  0  AT  s  0  α∆tM  ∆tCT  As  ∆tCs  0  �  �  �  �  y  u  p  �  � =  �  �  ∆tMT  s yd  0  Msy0 + ∆tf s  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  8  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Weighted ROMs for random inputs advection-dominated OCP(µ)s  Numerical simulations for OCP(µ)s can be very expensive in relation to computational time and  storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' To overcome this problem, in this work we will consider ROMs [6, 24, 41, 40, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will  study the case when the parameter µ can be affected by randomness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' it can follow a particular  probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' That is the case of random inputs OCP(µ)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this scenario, a suitable  modification of the ROMs, the wROMs [11, 15, 13, 16, 17, 18, 49, 59, 61, 60], takes into account  the uncertainty quantification (UQ) of the problems and shows efficient results concerning errors  and computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the sake of notation, from now on we denote a generic probability  distribution with the symbol ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' ROM procedure is divided in two stages: an offline phase and an  online phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  To exploit the potential of the ROMs setting, we assume an affine decomposition of the forms in  (15) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 is required here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We request that all the forms in (15) are affine in µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µp) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' More  precisely, we request that [15, 13]:  (1) the random diffusivity γ : Ω × P → R is of the form  (24)  γ(µ, x) = γ0(x) +  p  �  k=1  θγ  k(µk)γk(x),  with γk ∈ L∞(Ω), for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , p and θγ  k depending only on µk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (2) the random advection field η : Ω × P → R2 is of the form  (25)  η(µ, x) = η0(x) +  p  �  k=1  θη  k(µk)ηk(x),  with ηk ∈ (L∞(Ω))2, for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , p and θη  k depending only on µk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (3) the random forcing term f : Ω × P → R is of the form  (26)  f(µ, x) = f0(x) +  p  �  k=1  θf  k(µk)fk(x),  with fk ∈ L2(Ω), for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , p and θf  k depending only on µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For example, Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1 can be satisfied by truncating a Karhunen–Lo`eve expansion [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The offline phase is the most expensive stage of the wROMs, which usually  depends on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' However, this should be done only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The aim of this procedure is to build reduced  spaces Y N, U N and (QN)∗ that are good approximations of the high-fidelity ones and to compute  all block matrix components that are µ-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then, everything is memorized in order to be  ready to be used in the online phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The construction of the reduced basis is achieved through a  modified version of the POD algorithm: the wPOD [11, 60, 61], described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Here,  we firstly compute high-fidelity evaluation of optimal solutions  �  yN (µ), uN (µ), pN (µ)  �  for different  parameters µ, the so-called snapshots, to build the bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Because of good performance presented in  literature [30, 34, 53], this process will go through a partitioned approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' the wPOD is executed  separately for all three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' After this step, the three reduced spaces for state, control and  adjoint are constructed as, respectively,  (27)  Y N = span {ξy  n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , N},  U N = span {ξu  n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , N} ,  (QN)∗ = span {ξp  n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , N} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In order to ensure well-posedness for the reduced space approximation, we need to implement an  enriched space for state and adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This means to impose GN ≡ Y N ≡ (QN)∗, where  GN = span {σn, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , 2N} and {σn}2N  n=1 = {ξy  n}N  n=1 ∪ {ξp  n}N  n=1 [20, 23, 30, 31, 35, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This  whole discussion holds true for parabolic problems in a space-time context, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As a matter of fact,  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  9  when dealing with time-dependent OCP(µ)s in a space-time approach, the time instances are not  separated in the wPOD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, each snapshot carries all the time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Weighted Proper Orthogonal Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The peculiarity of wPOD is to take into account  the probability distribution that characterizes µ to create reduced spaces with less number of basis  with respect to the deterministic case without losing in accuracy [11, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will notice that  there will be different ways to consider randomness in the wPOD: the general idea is to suitably  attribute a larger weight to those samples that are more significant according to the distribution of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  From now we will refer to the POD algorithm based on the Monte-Carlo procedure in a deterministic  context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' when the distribution ρ is the uniform one, as Standard POD to distinguish it from  the wPOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As we will consider a partitioned approach, we show the procedure for the state space:  adjoint and control variables will follow the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  To consider stochasticity, wPOD needs to find the N-dimensional space Y N, with N ≪ N, such  that it minimizes the following estimate:  (28)  E =  ��  P  inf  ζy∈Y N ∥yN (µ) − ζy∥2  Y ρ(µ)dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Let us consider a set of Ntrain ordered parameters µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain ∈ PNtrain, where PNtrain ∈ P is a  discretization of P called the training set and its cardinality is |PNtrain| = Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' One can choose  Ntrain so that PNtrain is a good approximation of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We can relate µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain to the ordered  snapshots yN (µ1) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , yN �  µNtrain  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Considering w : P → R+ a weight function, a discretization  of problem (28) is meant to find the N-dimensional space Y N which minimize the quantity  (29)  1  Ntrain  Ntrain  �  k=1  w (µk)  ��yN (µk) − yN (µk)  ��2  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  One could think that the natural choice can be w(µ) = ρ(µ) and in an UQ context this means  to just discretize the expectation of the square error  (30)  E  ���yN − yN��2  Y  �  :=  �  P  ��yN (µ) − yN(µ)  ��2 ρ(µ)dµ,  which is the argument of the square root in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' However, this is not the unique choice in this  scenario: therefore it will be interesting to understand which method is better to approximate (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Here we illustrate different techniques that we use in the numerical tests in Section 5 to approximate  (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Considering the training set PNtrain =  �  µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain  �  , which can be composed by the nodes  of the chosen quadrature formula that approximates (30), we indicate with ω = (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , ωNtrain) the  standard weights of a chosen quadrature rule, with ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , ρNtrain the values of the density ρ in the  nodes in PNtrain, and with w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , wNtrain) the definitive weights used in wPOD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For  a node µk, we have the correspondent quantities ωk, ρk, and wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As a final result of this first step,  the wPOD furnished the following sum to minimize  (31)  1  Ntrain  Ntrain  �  k=1  wk  ��yN (µk) − yN (µk)  ��2  Y ,  which is achieved here through the following algorithms:  Weighted Monte-Carlo method, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain are Ntrain parameters extracted from  the random variable µ according to its distribution ρ and ρi are the values of the density ρ in  these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For this approximation, we have PNtrain =  �  µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain  �  and wk = ρ(µk),  for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Ntrain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Pseudo-Random method based on a Halton Sequence, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain are the nodes  extracted by a sampling completely based on the Halton sequence [57] and ρk = ρ(µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Also  in this case, PNtrain =  �  µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain  �  and wk = ρk, for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Ntrain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  10  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Tensor product Gauss-Jacobi rule, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain are the nodes of the tensor product  Gauss-Jacobi quadrature rule and ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , ωNtrain are the correspondent quadrature weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We can use this formula when the distribution is a Beta(αk, βk), as suitable Jacobi polyno-  mials are orthogonal to this distribution [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As a matter of fact, simulations in Section 5  will consider different Beta distributions for all components of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, we implement  a Gauss-Jacobi formula using (αk, βk) as its parameters in each dimension [43], accordingly  to the distribution of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For this approximation, we have PNtrain =  �  µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain  �  and  wk = ωk, for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Ntrain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Tensor product Clenshaw-Curtis rule, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain are the nodes of the tensor prod-  uct Clenshaw-Curtis quadrature rule and ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , ωNtrain are the correspondent quadrature  weights [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this case we obtain PNtrain =  �  µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , µNtrain  �  and wk = ρkωk, for all  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In numerical tests of Section 5, we will respectively call as Weighted Monte-Carlo, Pseudo-  Random, Gauss-Jacobi, and Clenshaw-Curtis wPOD algorithms the rules just specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As it is  know, tensor rule can be efficient, but their structure implies huge computational costs for elevate  cardinality of the training set Ptrain or high-dimensional parameter space P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For this purpose, when  we will use Clenshaw-Curtis or Gauss-Jacobi methods, we will consider sparse grid techniques based  on a Smolyak algorithm, too [48, 62]: we will implement isotropic ones [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Once chosen the rule (29) to approximate (30), the procedure to minimize (29) is described as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Let us define the deterministic correlation matrix of the snapshots of the state variable  Dy ∈ RNtrain×Ntrain in the following way:  (32)  Dy  kl :=  1  Ntrain  �  yN (µk) , yN (µl)  �  Y ,  1 ≤ k, l ≤ Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Firstly, we define the weighted correlation matrix as  (33)  W y := W · Dy,  where W := diag(w1, · · · , wNtrain) is the diagonal matrix whose elements are the weights of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The matrix W y is not symmetric in the usual matrix sense, but with respect to the scalar product  induced by W y, hence W y is diagonalizable anyway [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, we seek the solution of the  eigenvalue problem W ygy  n = λy  ngy  n, 1 ≤ n ≤ Ntrain, where ∥gy  n∥Y = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' we pursue to find an  eigenvalue λy  n with the relative eigenvector of norm equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We will indicate with (gy  n)t the t-  th component of the eigenvector gy  n ∈ RNtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the sake of simplicity, we rearrange the eigenvalues  λy  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , λy  Ntrain in a decreasing layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then, let us look at the first N eigenvalue-eigenvector pairs  (λy  1, gy  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , (gy  N, λy  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The basis functions χy  n for the state equation are constructed through the  following relation:  (34)  ζy  n =  1  √  λy  n  Ntrain  �  t=1  (gy  n)t yN (µk) ,  1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In order to choose N, one can refer to same study of eigenvalues of W y [24, 39, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' At the end, our  reduced spaces are built as (27) and, then, enriched spaces are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We summarise all the wPOD procedure for OCP(µ)s in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Online phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In this stage, all operations have usually a N-independent cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This process  reflects to be computationally cheap and, therefore, it can be recalled multiple times using small  machine resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Firstly, we choose a parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We get all the µ-independent quantities and  reduced spaces back from the storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Immediately, we combined parameter independent part with  the µ-dependent ones, that are rapidly calculated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then a Galerkin projector onto Y N, U N  and (QN)∗ is performed, computing the reduced solution yN, uN and pN through a reduced block  matrix system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As previously seen in Section 3, a stabilization is needed in the truth approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  However, it could also not be the case for the online stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This scenario lead to two possibilities:  we do not use SUPG in the online phase, Offline-Only stabilization, or, on the contrary, stabilization  occurs also here Offline-Online stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  11  Algorithm 1 Weighted POD algorithm for OCP(µ) problems through a partitioned approach  Input: FEM spaces Y N , U N , and (QN )∗ parameter domain P, and Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Output: reduced spaces Y N, U N and (QN)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Considering the high-fidelity spaces Y N , U N and (QN )∗:  1: Choose a quadrature rule (29) to approximate (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This step defines a sample Ptrain ⊂ P and  the respective weights w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , wNtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Define the matrix W := diag(w1, · · · , wNtrain) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  2: for all µ ∈ Ptrain do  3:  Solve the high-fidelity SUPG OCP(µ) system (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  4: end for  5: Calculate the matrices Dy  kl :=  1  Ntrain  �  yN (µk) , yN (µl)  �  Y , 1 ≤ k, l ≤ Ntrain and W y := W · Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Do the same for the control u and the adjoint p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  6: Compute eigenvalues λy  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , λy  Ntrain  and the corresponding orthonormalized eigenvectors  gy  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , gy  Ntrain of W y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Do the same procedure for u and p variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  7: Fix N according to a certain criterion and construct Y N = span {ξy  n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , N}, where  ξy  n =  1  √  λy  n  �Ntrain  t=1  (gy  n)t yN (µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Do the same for u and p variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  8: Build the aggregated space GN = span  �  {ξy  n}N  n=1 ∪ {ξp  n}N  n=1  �  and set GN ≡ Y N ≡ (QN)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Numerical Results  In this last part we illustrate numerical simulations concerning two Advection-Dominated OCP(µ)s  under random inputs: the Graetz-Poiseuille Problem and the Propagating Front in a Square Prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In both experiments, the parameter µ will be a random vector and it will follow a prescribed  probability density function that will be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The deterministic version of both experiments  can be founded in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The Offline approximation will be always based on a P1−FEM, which means to consider a finite  element method characterized by polynomials of degree less or equal than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In steady and unsteady  simulations, the same stabilization parameter δK will be employed for both stabilization in the high-  fidelity approximation and in the Online phase: namely in Offline-Online stabilization, δK is the  same for both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For each simulation, relative errors between the FEM and the reduced solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (35)  ey,N(µ) :=  ��yN (µ) − yN(µ)  ��  Y  ∥yN (µ)∥Y  , eu,N(µ) :=  ��uN (µ) − uN(µ)  ��  U  ∥uN (µ)∥U  , ep,N(µ) :=  ��pN (µ) − pN(µ)  ��  Q∗  ∥pN (µ)∥Q∗  ,  for the state, the control and the adjoint, respectively, will be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Due to the parametric nature  of the problems, for each quantity in (35) a simple average is computed for µ distributed according  to its probability density in a testing set Ptest ⊆ P of size Ntest, for every dimension N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' , Nmax  of the reduced space built through a chosen wPOD procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In every graph, the base-10 logarithm  of these averages will be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' When we will specify to use a POD procedure based on a Monte-  Carlo sampling [57] of a uniform density distribution, we will talk about Standard POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In order  to compare the different wPOD possibilities, we use the same testing set for all of them: it will be  taken using a Monte-Carlo method according to the distribution of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Obviously, the performance  of the Standard POD will be based on a testing set of uniform density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The sum of the errors with  respect to each discretized instant of time t will be taken into account in the unsteady versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In order to compare the computational cost between the FEM solution with that of the reduced  one for any possible dimension N, we use the speedup-index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (36)  speedup-index = computational time of the high-fidelity solution  computational time of the reduced solution  ,  which will be calculated for any µ in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Again, we will shown its sample average  for any dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For each test case, we will use the same Ptest to compute relative errors and  12  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  the speedup-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The steady experiments are run using a machine with 16GB of RAM and Intel  Core i7-7500U Dual Core, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='7GHz for the CPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' whereas all parabolic simulations are computed  considering 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='60GHz for the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The code concerning steady experiments is implemented using the RBniCS library [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' instead,  the unsteady ones are provided using both RBniCS and multiphenics [1] libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' These are python-  based libraries, built on FEniCS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Numerical Tests for the Graetz-Poiseuille Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The Graetz-Poiseuille problem is  an Advection-Diffusion problem that represents the heat conduction in a rectilinear pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Here the  transfer of heat can be regulated through the walls of the duct, which can be held at certain fixed  temperature [22, 37, 46, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Firstly, we present simulation concerning the stationary case, where a distributed control is em-  ployed all over the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The parameter µ = (µ1, µ2) is composed by the diffusion compo-  nent µ1 and the geometrical one µ2, which characterizes the length of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Ωobs  Ωobs  Ωo  Γo,1  Γo,2  Γo,3  Γo,4  Γo,5  Γo,6  (0,0)  (1,0)  (1+µ2,0)  (1+µ2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2)  (1+µ2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8)  (1+µ2,1)  (1,1)  (0,1)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Geometry of the Graetz-Poiseuille Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The problem is studied using (x0, x1) as spatial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Ωo is the domain observed for a  certain value µ2 with boundary Γo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We deal with homogeneous Neumann boundary conditions  (BC) on Γo,3 := {1 + µ2} × [0, 1] considering Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead, Dirichlet conditions are set on sides  Γo,1 := [0, 1]×{0}, Γo,5 := [0, 1]×{1}, Γo,6 := {0}×[0, 1] by imposing y = 0 and Γo,2 := [1, 1+µ2]×{0}  and Γo,4 := [1, 1 + µ2] × {1} by imposing y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The formulation of the problem is the following: given µ ∈ P, find (y, u) ∈ ˜Y × U which solves  min  (y,u)  1  2  �  Ωobs(µ)  (y(µ) − yd)2 dΩo(µ) + α  2  �  Ωo(µ)  u(µ)2 dΩo(µ),  such that  (37)  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  − 1  µ1  ∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),  in Ωo(µ),  y(µ) = 0,  on Γo,1(µ) ∪ Γo,5(µ) ∪ Γo,6(µ),  y(µ) = 1,  on Γo,2(µ) ∪ Γo,4(µ),  ∂y(µ)  ∂ν  = 0,  on Γo,3(µ),  where ˜Y :=  �  v ∈ H1�  Ωo  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' it satisfies the BC in (37)  �  and U = L2(Ωo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the sake of clarity,  a lifting function Ry ∈ H1(Ω) that fulfills the BC in (37) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Consequently, the variable  ¯y := y − Ry, with ¯y ∈ Y , is used, where  Y :=  �  v ∈ H1  0  �  Ω  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' ∂¯y  ∂ν = 0, on Γ3 and ¯y = 0 on Γ \\ Γ3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Furthermore, we settle Q := Y ∗ without any loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, the adjoint variable p  is null on Γ \\ Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The observation domain is Ωobs := [1, 1 + µ2] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8, 1] ∪ [1, 1 + µ2] × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2] as  illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The value µ2 can change the domain under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Having that the domain  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  13  Ωo is µ-dependent itself, in the Offline Phase snapshots are based on different domains due to the  sampling of the geometrical parameter components [24, 41, 45, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' To deal with the geometrical  parametrization of the problem, we set a reference domain Ω and we build affine maps that transform  Ω in Ωo for a defined µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This procedure implies an automatic modification of some bilinear and  linear forms involved in the weak formulation of Problem (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We choose Ω = (0, 2) × (0, 1) as reference domain, that is the original one Ωo(µ) corresponding  to µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We assume that µ2 is positive for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Considering Figure 1, we divide  this into 2 subdomains, which are defined as Ω1 = (0, 1) × (0, 1) and Ω2 = (1, 2) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Then, we  build two affine transformations:  (38)  V1(µ) : Ω1 → Ωo,1(µ) ⊂ R2,  such that V1  �� x  y  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  :=  � x  y  �  ,  which is the identity map defined on the first subdomain Ω1 and V2(µ) : Ω2 → Ωo,2(µ) ⊂ R2 as  (39)  V2  �� x  y  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  =  � µ2x  y  �  +  � 1 − µ2  0  �  = R2  � x  y  �  +  � 1 − µ2  0  �  ,  where we have  (40)  R2 :=  �  µ2  0  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Glueing together V1 and V2 for each µ ∈ P, we manage to build a one-to-one transformation  V (µ) defined all over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We denote the restrictions of Th to Ω1 and Ω2 with T 1  h and T 2  h , respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, we can express all the forms of the weak formulation under the effect of this  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For instance, after possible lifting, we have as = a + s and a∗  s = a∗ + s∗ as  (41)  a  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  : =  �  Ω1  1  µ1  ∇yN · ∇qN + 4x1(1 − x1)∂x0yN qN  +  �  Ω2  1  µ1µ2  ∂x0yN ∂x0qN + µ2  µ1  ∂x1yN ∂x1qN + 4x1(1 − x1)∂x0yN qN ,  s  �  yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  : =  �  K∈T 1  h  δKhK  �  K  �  4x1(1 − x1)∂x0yN �  ∂x0qN  +  �  K∈T 2  h  δK  hK  √µ2  �  K  �  4x1(1 − x1)∂x0yN �  ∂x0qN ,  a∗ �  zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  : =  �  Ω1  1  µ1  ∇pN · ∇zN − 4x1(1 − x1)∂x0pN zN  −  �  Ω2  1  µ1µ2  ∂x0pN ∂x0zN − µ2  µ1  ∂x1pN ∂x1zN − 4x1(1 − x1)∂x0pN zN ,  s∗ �  zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ  �  : =  �  K∈T 1  h  δKhK  �  K  �  4x1(1 − x1)∂x0pN �  ∂x0zN  +  �  K∈T 2  h  δK  hK  √µ2  �  K  �  4x1(1 − x1)∂x0pN �  ∂x0zN ,  for all yN , qN , zN , pN , ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In order to take into account the possible bad effect on stabilized  forms due to a extension or shortening of our domain Ωo, we choose the stabilization parameter for  K ∈ T 2  h as δK  hK  √µ2 , where √µ2 =  �  | det(R2)| [35, 37, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For the FEM discretization, a quite coarse mesh of size h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='034 is used and the total dimension  of the numerical problem is 13146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We take δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The parameter space is set  as P :=  �  1, 105�  ×  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  �  , from which we want to extract a training set Ptrain with cardinality  Ntrain = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the n bilinear form, we consider a penalization α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Our aim is to minimize  the L2-error between the state and the desired solution profile yd(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, function defined on Ωobs  14  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' (top) FEM not stabilized and (bottom) FEM stabilized solution, y (right) and  u (left), µ = (105, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='034, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Each wPOD procedure is computed until a Nmax = 20 in a partitioned approach and  then all algorithms are compared using a testing set Ptest of 100 elements in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We suppose that µ follows a Beta(5, 3) distribution for both parameter µ1 and µ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (42)  µ1 ∼ 1 +  �  105 − 1  �  X1, where X1 ∼ Beta(5, 3),  µ2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5 +  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  �  X2, where X2 ∼ Beta(5, 3),  where µ1 and µ2 are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This implies that we consider more probable  the parameters for which the Graetz-Poiseuille Problem has high values of the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In  Figure 2, we highlight how the FEM solutions of the state and the control are for µ = (105, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The adjoint solution is not shown here because it is proportional to the control due to the gradient  equation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  From Figure 2, one can see that a stabilization is necessarily needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We firstly exploit the  Offline-Only stabilization procedure, which results regarding errors are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The  performance is not good for any kind of wPOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Moreover, the Standard POD does not perform  good, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative errors never drop under 10−2 for any variables, hence more stabilization is  necessary in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Figure 4 relative errors of the Offline-Online stabilization procedure are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Here  the trend seems better than the Offline-Only one, because these quantities decay faster along the  value of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The wPOD Monte-Carlo is the best performer for all y, u, p variables, as a matter of  fact, it reaches ey,16 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='13 · 10−7 for the state, for the adjoint ep,16 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='95 · 10−7 and the control  eu,16 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='80·10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This procedure has a better performance of the Standard POD, which its accuracy  is at least 100 times inferior of the wPOD Monte-Carlo after N > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Concerning other rules, it can  be noticed that Smolyak grid techniques perform better than their tensor-rule counterparts, despite  having a training set whose cardinality is similar, but less of 100: 93 and 91 for the Clenshaw-Curtis  and Gauss-Jacobi sparse grids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Figure 5 we visually compare the two possibilities of stabilization for the geometrical parametriza-  tion of the Graetz-Pouiseuille problem for the wPOD Monte-Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Table 1 we compare the speedup-index for all wPOD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We see that computational  values are all of the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the wPOD Monte-Carlo we calculate 87 reduced  solutions in the time of a FEM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Now we want to present the parabolic version of Problem (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This unsteady problem has been  studied without optimization in [37, 59] in a deterministic context and in [59] in a UQ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead,  the deterministic OCP(µ) Graetz Problem under boundary control without Advection-dominancy  is studied in [54, 52] and the deterministicdistributed control configuration is analyzed in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5e-021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3e-011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5e-021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3e-01STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  15  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Graetz-Poiseuille Problem - Offline-Only Stabiliza-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD (blue), wPOD  Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),  Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-  Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Speedup-index Graetz-Poiseuille Problem: Offline-Online Stabilization - µ1, µ2 ∼ Beta(5,3)  N  POD  wPOD  Gauss tensor  Gauss Smolyak  CC tensor  CC Smolyak  Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Random  4  113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  8  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  12  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='7  16  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='8  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='7  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Average Speedup-index of Offline-Online Stabilization for the Graetz-Poiseuille  Problem under geometrical parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' From left to right: Standard POD, wPOD  Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor,  Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Recalling Figure 1, for a fixed T > 0 the unsteady Graetz-Poiseuille Problem is posed as follows:  find (y, u) ∈ ˜Y × U which solves  min  (y,u)  1  2  �  Ωobs(µ)×(0,T )  (y(µ) − yd)2 dΩ + α  2  �  Ω(µ)×(0,T )  u(µ)2 dΩ,  such that  FEM vs ROM averaged relative error - y (state)  101  Log-Error  100  Relative L  StandardPOD  WeightedPODMonte-Carlo  Gaussjacobi-tensor  Gausslacobi-Smolyak  ClenshawCurtis  tensor  10-1  ClenshawCurtis+Smolyak  PseudoRandom-Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  NFEM vs ROM averaged relative error - u (control)  101  100  Standard POD  Weighted POD Monte-Carlo  Gaussjacobi-tensor  10-1  Gaussjacobi- Smolyak  ClenshawCurtis-tensor  ClenshawCurtis  Smolyak  PseudoRandom Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  NFEM vs ROM averaged relative error - p (adjoint)  102  Log-Error  101  Relative  StandardPOD  WeightedPoDMonte-Carlo  Gaussjacobi-tensor  100  Gaussjacobi--Smolyak  ClenshawCurtis-tensor  ClenshawCurtis Smolyak  PseudoRandom Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  N16  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Graetz-Poiseuille Problem - Offline-Online Stabiliza-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD (blue), wPOD  Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),  Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-  Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' (top) wPOD Monte-Carlo Offline-Only stabilized and (bottom) Offline-Online  stabilized solution, y (right) and u (left), µ = (105, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='034, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, Ntrain = 100,  δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  (43)  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ∂ty(µ) − 1  µ1  ∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),  in Ω(µ) × (0, T),  y(µ) = 0,  on Γ1 ∪ Γ5 ∪ Γ6 × (0, T),  y(µ) = 1,  on Γ2(µ) ∪ Γ4(µ) × (0, T),  ∂y(µ)  ∂ν  = 0,  on Γ3(µ) × (0, T),  y(µ)(0) = y0(x),  in Ω(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  FEM vs ROM averaged relative error - p (adjoint)  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  100  10-1  10-2  10-3  10  Standard POD  10-5  Weighted POD Monte-Carlo  Gaussjacobi+tensor  GaussjacobiSmolyak  ClenshawCurtis-tensor  10-6  ClenshawCurtis - Smolyak  PseudoRandom - Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='0  N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2e+00  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='5e-021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  一  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='3e-011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='5e-021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='3e-01FEM vs ROM averaged relative error - y (state)  10-1  10-2  10-3  10  10-5  Standard POD  Weighted PODMonte-Carlo  Gaussjacobi +tensor  10-6  Gaussjacobi↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Smolyak  ClenshawCurtis -tensor  ClenshawCurtis-Smolyak  PseudoRandom-Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='0  NFEM vs ROM averaged relative error - u (control)  10-1  10-3  10  StandardPOD  10-5  Weighted PODMonte-Carlo  Gaussjacobi+tensor  Gaussjacobi + Smolyak  10-6  ClenshawCurtis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='tensor  ClenshawCurtis - Smolyak  PseudoRandom-Halton  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  17  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Only Sta-  bilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD (blue),  wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak  grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),  Pseudo-Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  As made in the steady version, we firstly consider a lifting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Simulations are run following  the space-time setting proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2 for a prearranged number of time-steps Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The initial condition is y0(x) = 0 for all x ∈ Ω referring to Figure 1 and we set T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The  penalization parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01 and we want the state solution to be similar in the L2-norm to  a desired solution profile yd(x, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, function defined for all t ∈ (0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0) and for all x in Ωobs in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Choosing Nt = 30, the time step is ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the spatial discretization a quite coarse  mesh of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='038 is implemented: consequently the total high-fidelity dimension is Ntot = 314820  and a single FEM space is characterized by N = 3498 for a fixed instant t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Again, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 for all  K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We take P :=  �  1, 105�  ×  �  1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  �  and µ is determined by the probability distribution  (44)  µ1 ∼ 1 +  �  105 − 1  �  X1, where X1 ∼ Beta(5, 3),  µ2 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 +  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  �  X2, where X2 ∼ Beta(5, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We choose a training set Ptrain of cardinality Ntrain = 100 (with exception of sparse grids, which  have similar cardinality) and we performed the wPOD algorithms with Nmax = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Figure 6 we present relative errors related to Offline-Only stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Also in the parabolic  case this procedure does not perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore an online stabilization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  As a matter of fact, one can see in Figure 7 that the trends for Offline-Online stabilization seems  a lot better than the previous strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Besides the Clenshaw-Curtis quadrature rule, errors decrease  along the dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Again, the best performance is given by the wPOD Monte-Carlo, where the  following values are reached for N = 14: ey,14 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='71·10−7,ep,14 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='21·10−7, and eu,14 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='64·10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='FEM vs ROM averaged relative error - y (state)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis--Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - u (control)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='StandardPOD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi- Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom - Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - p (adjoint)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis - Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='N18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Online  Stabilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD  (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi  Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid  (dark green), Pseudo-Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Finally, in Table 2 we illustrate the performance of the speedup-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' All weighted algorithms  performs similar: we compute an order of magnitude of 104 reduced solution in the time of a FEM  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' This efficiency is given by the nature of the space-time procedure, where each snapshot carries  all the time instances, and the reduction is very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Speedup-index Parabolic Graetz-Poiseuille Problem: Offline-Online Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' - µ1, µ2 ∼ Beta(5,3)  N  POD  wPOD  Gauss tensor  Gauss Smolyak  CC tensor  CC Smolyak  Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Random  3  14299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='7  14571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  13970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='7  14013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  14524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='9  12  13693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='4  13453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  15  13090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  13606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  13055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='8  13455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  13548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6  13544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0  12875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Average Speedup-index of Offline-Online Stabilization for the Parabolic Graetz-  Poiseuille Problem under geometrical parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' From left to right: Standard POD,  wPOD Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis  tensor, Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Numerical Tests for Propagating Front in a Square Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Here we analyze an  Advection-Dominated PDE problem illustrated without control in a deterministic and in a stochastic  context in [37, 59] and in [59], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A distributed control is applied all over the domain Ω,  FEM vs ROM averaged relative error - y (state)  Standard POD  Weighted POD Monte-Carlo  101  Gaussjacobi- tensor  GaussJacobi-Smolyak  ClenshawCurtis-tensor  100  ClenshawCurtis-Smolyak  PseudoRandom-Halton  10-  10-2  10-3  10  10-5  10-6  2  4  6  8  10  12  14  NFEM vs ROM averaged relative error - u (control)  101  100  10  10-2  10-3  10-  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  Standard POD  10-5  Weighted POD Monte-Carlo  Gaussjacobi-tensor  Gaussjacobi-Smolyak  10-6  ClenshawCurtis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='tensor  ClenshawCurtis - Smolyak  PseudoRandom -Halton  2  4  6  8  10  12  14  NFEM vs ROM averaged relative error - p (adioint)  101  10  10-3  Standard POD  WeightedPODMonte-Cairlo  Gaussjacobi-tensor  10-5  Gaussjacobi- Smolyak  ClenshawCurtis - tensor  ClenshawCurtis - Smolyak  PseudoRandom-Halton  2  4  6  8  10  12  14  NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  19  which is the square (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' 1) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' as shown under Cartesian coordinates (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' x1) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The  boundary is composed as follows: Γ1 := {0} × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1],  Γ4 := [0, 1] × {1}, Γ5 := {0} × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' instead Ωobs := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25, 1] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='75, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Γ1  Γ2  Γ3  Γ4  Γ5  Ω  Ωobs  (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25)  (0,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='75)  (1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25,1)  (1,0)  (0,0)  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Geometry of the Propagating Front in a Square Problem  Given µ = (µ1, µ2), our aim is to solve the following OCP(µ) problem: find (y, u) ∈ ˜Y × U which  solves  min  (y,u)  1  2  �  Ωobs  (y(µ) − yd)2 dΩ + α  2  �  Ω  u(µ)2 dΩ,  such that  (45)  �  �  �  �  �  �  �  − 1  µ1  ∆y(µ) + [cos µ2, sin µ2] · ∇y(µ) = u(µ),  in Ω,  y(µ) = 1,  on Γ1 ∪ Γ2,  y(µ) = 0,  on Γ3 ∪ Γ4 ∪ Γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In this case, we have that the domain of definition of our state y is  ˜Y :=  �  v ∈ H1�  Ω  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' BC in (45)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Again, we define a lifting function Ry ∈ H1�  Ω  �  such that satisfies BC in (45), applying a lifting  procedure before the Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We define ¯y := y − Ry, with ¯y ∈ Y and Y := H1  0(Ω),  U = L2(Ω) and Q := Y ∗, with p = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The mesh size h is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='025, which entails an overall dimension of the truth approximation  of 12087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Consequently, we have N = 4029 for state, control and adjoint spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Concerning  stabilization, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The penalization parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01 and we pursue the  state solution to be similar in the L2-norm to yd(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5, defined for all x in Ωobs of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In  our test cases, P :=  �  1, 4 · 104�  ×  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  �  and µ follow the subsequent probability distribution:  (46)  µ1 ∼ 1 +  �  4 · 104 − 1  �  X1, where X1 ∼ Beta(10, 10),  µ2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9 +  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  �  X2, where X2 ∼ Beta(10, 10),  where µ1 and µ2 are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The training set Ptrain and the testing set  Ptest have both cardinality equal to ntrain = 100, with exception of sparse grid samplings, whose  cardinality is similar to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We apply a wPOD procedure for a Nmax = 50 dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In Figure 9,  we show the performance of relative errors for the Offline-Only stabilization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' As in the  Graetz-Poiseuille Problem, these trends are not acceptable, as no quantity drops under 10−1 for all  state, control and adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Therefore, a stabilization applied in the Online Phase is needed,  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Figure 10 relative errors for Offline-Online Stabilization procedure are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Again, wPOD  Monte-Carlo presents the best behaviour: in this case it reaches ey,50 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='03 · 10−7 for the state, for  the adjoint ep,50 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='07·10−6, and the control eu,50 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='21·10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Moreover, the wPOD Monte-Carlo  has an accuracy of nearly a factor of 100 better than a Standard POD in a deterministic context for  N > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Also here, Smolyak grids perform better than their tensor counterpart: for instance, we  obtain in this case it reaches ey,50 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='77 · 10−6 for the state, for the adjoint ep,50 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='80 · 10−6, and  20  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Propagating Front in a Problem - Offline-Only Stabiliza-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD (blue), wPOD  Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),  Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-  Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  the control eu,50 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='02 · 10−5 for Gauss-Jacobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Concerning the training set, we have Ntrain = 89  and Ntrain = 93 for the Gauss-Jacobi and the Clenshaw-Curtis ones, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In Figure 11  we see a comparison between the FEM solution for the state and the adjoint without stabilization  and the Offline-Online Stabilized wPOD Monte-Carlo reduced solution for these variables with  µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  The values of the speedup-index for the Offline-Online stabilization for each type of wPOD are  reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For N = 50 the wPOD Monte-Carlo is the best choice again with a computation  of 50 reduced solutions in the time of a FEM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' All the other possibilities perform a little bit lower  for N = 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' however, all weighted algorithms have similar performances concerning the speedup-  index: an order of magnitude of 102 for the first 50 reduced basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Numerical tests of the parabolic version of the Propagating Front in a Square Problem are here  illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For a fix T > 0 and a given µ ∈ P we have to find the pair (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' u) ∈ ˜Y × U which solves  min  (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='u)  1  2  �  Ωobs×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='T )  (y(µ) − yd)2 dΩ + α  2  �  Ω×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='T )  u(µ)2 dΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  such that  (47)  �  �  �  �  �  �  �  �  �  �  �  �  �  ∂ty(µ) − 1  µ1  ∆y(µ) + [cos µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' sin µ2] · ∇y(µ) = u(µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  in Ω × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  y(µ) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  on Γ1 ∪ Γ2 × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  y(µ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  on Γ3 ∪ Γ4 ∪ Γ5 × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  y(µ)(0) = y0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='FEM vs ROM averaged relative error - y (state)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='StandardPOD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPoDMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi - Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - u (control)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Weighted PODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gausslacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom -Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - p (adioint)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='StandardPOD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Weighted POD Monte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gausslacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi- Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='50STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Propagating Front in a Problem - Offline-Online Sta-  bilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD (blue),  wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak  grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),  Pseudo-Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' FEM not stabilized and wPOD Monte-Carlo Offline-Online stabilized solu-  tion for y (left) and for p (right), µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='025 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, Ntrain = 100,  δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, N = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  where y0(x) = 0 for all x ∈ Ω in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A final time T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Considering the time  discretization, we chose a number of time steps equal to Nt = 30, then we have ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead,  for the spatial approximation, the mesh size is set to h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='036, that implies an overall dimension  of the space-time setting equal to Ntot = 174780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For a fixed instant t, a single FEM space is  characterized by N = 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  For the SUPG procedure, we impose δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Setting a penalization parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, we try to achieve in a L2-mean a desired solution profile  yd(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5, defined for all t ∈ (0, 3) and x in Ωobs of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  P :=  �  1, 4 · 104�  ×  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  �  , as in the steady version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We suppose that µ follows the probability  distribution (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Our training set has cardinality Ntrain = 100, with exception for Gauss-Jacobi and  Clenshaw-Curtis Smolyak grids with Ntrain = 89 and Ntrain = 93, respectively, which are the number  of nodes nearest to 100 for this kind of procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In Figure 12 and 13, we show a representative  FEM vs ROM averaged relative error - y (state)  10-1  10-2  10-3  10  4  StandardPOD  10-5  WeightedPODMonte-Carlo  GaussJacobi-tensor  Gaussjaciobi-Smolyak  ClenshawCurtis-tensor  10-6  ClenshawCurtis-Smolyak  PseudoRandom-Halton  10  20  30  40  50  NFEM vs ROM averaged relative error - u (control)  100  Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='POD  Weighted POD Monte-Carlo  GaussJacobi-tensor  GaussJacobi - Smolyak  ClenshawCurtis-tensor  10-1  ClenshawCurtis-Smolyak  PseudoRandom -Halton  10-2  10-3  10-5  10  20  30  40  50  NFEM ys ROM averaged relative error - p (adioint)  100  10-1  10-2  10-3  StandardPOD  WeightedPODMonte-Carlo  10-5  Gaussjacobi--tensor  Gaussjacobi - Smolyak  ClenshawCurtis-tensor  ClenshawCurtis - Smolyak  10-6  PseudoRandom---Halton  10  20  30  40  50  N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2e+00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='4e-0322  STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  Speedup-index Propagating front in a Square Problem: Offline-Online Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' - µ1, µ2 ∼ Beta(10,10)  N  POD  wPOD  Gauss tensor  Gauss Smolyak  CC tensor  CC Smolyak  Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='2  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Average Speedup-index of Offline-Online Stabilization for the Propagating  Front in a Square Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  From left to right: Standard POD, wPOD Monte-Carlo,  Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-  Curtis Smolyak grid, Pseudo-Random based on Halton numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' wPOD Monte-Carlo Offline-Online stabilized reduced solution of y, for t =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='036, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, Ntrain = 100, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0,  N = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' wPOD Monte-Carlo Offline-Online stabilized reduced solution of p, for t =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='1, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0, µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='036, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='01, Ntrain = 100, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='0,  N = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  stabilized FEM solution for µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2) for some instants of time of the state y and the adjoint  p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We choose to perform all wPOD procedure with Nmax = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Let us move to the error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In Figure 14 we illustrate the relative errors for the Offline-Only  stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The performance are not satisfactory here, too, where no quantity drops below the  accuracy of 10−1 for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Instead, Offline-Online stabilization procedure performs well, as one can notice from Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Again, wPOD Monte-Carlo has the best behaviour, it reaches ey,30 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='12 · 10−7 for the state, for  the adjoint ep,30 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='55 · 10−7 and the control eu,30 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='36 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Also in this case, isotropic sparse  grid techniques is a better choice than tensor rules, both for Gauss-Jacobi and Clenshaw-Curtis  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In Table 4 we compare the speedup-index for all the weighted algorithms: performance are similar  for all N, for N = 30 we computed nearly 4000 Offline-Online stabilized reduced solutions in the  time of a FEM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='4e-03STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  23  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Parabolic Propagating Front in a Problem - Offline-  Only Stabilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard POD  (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi  Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid  (dark green), Pseudo-Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Speedup-index Parabolic Propagating front in a Square Problem: Offline-Online Stabilization  N  POD  wPOD  Gauss tensor  Gauss Smolyak  CC tensor  CC Smolyak  Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content='8  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Average Speedup-index of Offline-Online Stabilization for the Parabolic Propa-  gating Front in a Square Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' From left to right: Standard POD, wPOD Monte-Carlo,  Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-  Curtis Smolyak grid, Pseudo-Random based on Halton numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' µ1, µ2 ∼ Beta(10,10)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Conclusions and Perspectives  In this work, we illustrated some numerical tests concerning stabilized Parametrized Advection-  Dominated OCPs with random parametric inputs in a ROM context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We deal with both steady and  unsteady cases and we took advantage of the SUPG stabilization to overcome numerical issues due  to high values of the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Two possibilities of stabilization were analyzed: when SUPG  is only used occurs in the offline phase, Offline-Only stabilization, or when it is provided in both  online and offline phases, Offline-Online stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='FEM vs ROM averaged relative error - y (state)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='StandardPOD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Weighted POD Monte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi- tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi -Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - u (control)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='2 × 100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='6×10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - p (adjoint)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Weighted PODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis - Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Log-Error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Relative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='N24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Relative Errors for the Parabolic Propagating Front in a Problem - Offline-  Online Stabilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' State (top-left), Control (top-right), Adjoint (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Standard  POD (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi  Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid  (dark green), Pseudo-Random based on Halton numbers (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  In order to deal with the uncertainty quantification caused by random inputs, we consider wROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  More precisely, we built our reduced bases using a wPOD in a partitioned approach, using different  quadrature rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We implemented wPOD Monte-Carlo, Gaussian quadrature formulae based on  Jacobi polynomials in a tensor rule, approximation related to Clenshaw-Curtis tensor rule, Smolyak  isotropic sparse grid approximation of the last two methods, quasi Monte-Carlo method as a Pseudo-  Random rule defined on Halton numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  We analyzed relative errors between the reduced and the high fidelity solutions and the speedup-  index concerning the Graetz-Poiseuille and Propagating Front in a Square Problems, always under a  distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For the state, control and adjoint spaces we implemented a P1-FEM approxima-  tion in a optimize-then-discretize framework as the truth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Concerning parabolic problems,  a space-time approach is followed applying SUPG in a suitable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In order to established which  wPOD performs better, we compare them through the same testing set sampled by a Monte-Carlo  method according to the probability distribution of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Offline-Only stabilization technique performed very poorly in terms of errors, this happened for  all wROMs considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Instead, in all the steady and unsteady experiments, the wROM technique  performed excellently in an Offline-Online stabilization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' For parabolic problems, the  speedup-index features large values thanks to the space-time formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' More precisely, wPOD  Monte-Carlo technique was always the best performer for relative errors, instead, concerning compu-  tational efficiency all methods seem equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' the efficiency of the wPOD Monte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='FEM vs ROM averaged relative error - y (state)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi- Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis - Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - u (control)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='GaussJacobi-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis-Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NFEM vs ROM averaged relative error - p (adjoint)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Standard POD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='WeightedPODMonte-Carlo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi-tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='Gaussjacobi- Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -tensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10-6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='ClenshawCurtis -Smolyak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='PseudoRandom-Halton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='is supported by the fact that after a small number of reduced basis it is nearly 100 times more accu-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='rate than a Standard POD in a deterministic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Moreover, we notice that sparse grids perform  better than relative tensor ones, although having a bit less number of quadrature nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Furthermore, in the Graetz-Poiseuille Problem we illustrate that under geometrical parametriza-  tion affected by randomness, wROMs still have good performance, despite small fluctuations in the  graph of relative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  As a first perspective, it might be interesting to create a strongly-consistent stabilization technique  that flattens all the fluctuations of geometrical parametrization in a UQ context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Moreover, we want  to extend the study to boundary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Finally, it might be interesting to study the performance  of other stabilization techniques for the online phases, for instance, of the Online Vanishing Viscosity  and the Online Rectification methods [4, 12, 33] combined with the SUPG technique in the offline  phase or with the stabilization strategy used in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  Acknowledgements  We acknowledge the support by European Union Funding for Research and Innovation – Horizon  2020 Program – in the framework of European Research Council Executive Agency: Consolidator  Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods  with Applications in Computational Fluid Dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' We also acknowledge the PRIN 2017 “Nu-  merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of  complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-  GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' The computations in  this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is  an implementation in FEniCS [32] of several reduced order modelling techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' we acknowledge  developers and contributors to both libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  References  [1] multiphenics - easy prototyping of multiphysics problems in FEniCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content=' POD-Galerkin model order reduction for parametrized  nonlinear time-dependent optimal flow control: an application to shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Journal of Numerical  Mathematics, 30(1):63–84, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  [57] Timothy John Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content=' Stabilized reduced basis method for transport PDEs with random inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' Master’s thesis, University  of Trieste and SISSA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+page_content=' Hesthaven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' High-order collocation methods for differential equations with random inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  SIAM Journal on Scientific Computing, 27(3):1118–1139, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content='  [63] Fabio Zoccolan, Maria Strazzullo, and Gianluigi Rozza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' A Streamline Upwind Petrov-Galerkin Reduced Order  Method for Advection-Dominated Partial Differential Equations under Optimal Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
+page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtA0T4oBgHgl3EQfBf8H/content/2301.01975v1.pdf'}
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+1
+Joint Beamforming Design for Dual-Functional
+MIMO Radar and Communication Systems
+Guaranteeing Physical Layer Security
+Fuwang Dong, Wei Wang, Senior Member, IEEE, Xin Li, Fan Liu, Sheng Chen, Fellow, IEEE, and Lajos
+Hanzo, Life Fellow, IEEE
+Abstract—The
+dual-functional
+radar
+and
+communication
+(DFRC) technique constitutes a promising next-generation wire-
+less solution, due to its benefits in terms of power consumption,
+physical hardware, and spectrum exploitation. In this paper,
+we propose sophisticated beamforming designs for multi-user
+DFRC systems by additionally taking the physical layer security
+(PLS) into account. We show that appropriately designed radar
+waveforms can also act as the traditional artificial noise conceived
+for drowning out the eavesdropping channel and for attaining
+increased design degrees of freedom (DoF). The joint beamform-
+ing design is formulated as a non-convex optimization problem
+for striking a compelling trade-off amongst the conflicting design
+objectives of radar transmit beampattern, communication quality
+of service (QoS), and the PLS level. Then, we propose a
+semidefinite relaxation (SDR)-based algorithm and a reduced-
+complexity version to tackle the non-convexity, where the globally
+optimal solutions are found. Moreover, a robust beamforming
+method is also developed for considering realistic imperfect
+channel state information (CSI) knowledge. Finally, simulation
+results are provided for corroborating our theoretical results and
+show the proposed methods’ superiority.
+Index Terms—Dual-functional radar and communication sys-
+tem, joint beamforming design, physical layer security, multi-user
+MIMO.
+I. INTRODUCTION
+The proliferation of wireless mobile services exhibits an
+exponential trend, leading to a scarcity of spectral resources
+and to escalating spectrum prices. For example, it has been
+reported that the number of connected devices is expected
+to be 80 billion by 2030 with an annual growth rate of
+around 25%, and that of the active Internet of Things (IoT)
+This work is supported in part by the National Natural Science Foundation
+udner Grant 62271163, in part by the Fundamental Research Funds for the
+Central Universities (3072022QBZ0401, 3072021CFT0404). F. Liu would
+like to acknowledge the financial support of the National Natural Science
+Foundation of China under Grant 62101234, as well as of the Young Elite
+Scientist Sponsorship Program by the China Association for Science and
+Technology (CAST) under Grant No. YESS20210055. L. Hanzo would like to
+acknowledge the financial support of the Engineering and Physical Sciences
+Research Council projects EP/W016605/1 and EP/P003990/1 (COALESCE)
+as well as of the European Research Council’s Advanced Fellow Grant
+QuantCom (Grant No. 789028). (Corresponding author: Wei Wang.)
+Fuwang Dong, and Fan Liu are with the Department of Electronic and Elec-
+trical Engineering, Southern University of Science and Technology, Shenzhen
+518055, China (email: dongfw@sustech.edu.cn; liuf6@sustech.edu.cn)
+Wei Wang, and Xin Li are with the College of Intelligent System Science
+and Engineering, Harbin Engineering University, Harbin, 150001, China
+(email: wangwei407@hrbeu.edu.cn; xinxin forever@126.com ).
+Sheng Chen, and Lajos Hanzo are with the School of Electronic and
+Computer Science, University of Southampton, Southampton SO17 1BJ, U.K.
+(email: sqc@ecs.soton.ac.uk; lh@ecs.soton.ac.uk).
+devices will reach 24.1 billion by 2030 [1], [2]. Recently, the
+concept and scope of Integrated Sensing and Communication
+(ISAC) technology have been formally defined in [3], [4],
+enabling sensing and communication simultaneously in the
+same frequency band or/and hardware platform, which can
+significantly improve the resource utilization. Due to the
+numerous advantages offered by ISAC, it is envisioned to
+be a promising technique in terms of supporting autonomous
+vehicles [5], [6] and the IoT in 6G wireless networks [7].
+There are two main ISAC categories in terms of transmitted
+signal: radar and communication spectrum coexistence and
+dual functional radar-communication (DFRC) [8], [9]. In this
+paper, we consider a DFRC system, which transmits dual-
+functional signals/waveforms from a single hardware platform,
+to gain benefits from joint sensing and signaling operations
+via real-time cooperation. The main motivation of transmit
+beamforming is to synthesize multiple beams towards both
+the communication users and the radar targets by exploiting
+the associated spatial degrees of freedom (DoF). In [10], the
+authors considered the radar targets as virtual downlink users
+encountering a line of sight (LoS) channel. Therefore, the
+beamforming matrix was designed for closely matching the
+desired radar beampattern, while simultaneously guaranteeing
+the signal to interference and noise ratio (SINR) attained by the
+downlink users. Furthermore, the authors of [11], [12] studied
+the associated symbol/waveform level probing signal design
+issues, where the multi-user interference energy was mini-
+mized under the similarity and constant modulus constraints of
+the radar waveform. However, the above-mentioned schemes
+only utilize the communication waveform as the DFRC wave-
+form to implement target detection, hence leading to a DoF
+reduction, thereby to a radar performance degradation. To this
+end, the authors of [13] firstly proposed a jointly precoded
+individual communication and radar waveforms based scheme,
+where the communication signal can be regarded as a special
+case relying on nullifying the dedicated radar waveforms.
+Therefore, by exploiting the inherent advantages of the radar
+waveform, the DoF erosion can be efficiently compensated,
+hence resulting in target detection performance improvements,
+especially for a small number of downlink users.
+Another critical problem in the DFRC system, which has
+been largely overlooked in the relevant literature, is how to
+guarantee the privacy and security of the desired informa-
+tion [14]. The DFRC base station (BS) transmits the dual-
+functional probing waveform for detection purposes, but also
+arXiv:2301.00340v1  [eess.SP]  1 Jan 2023
+
+2
+TABLE I
+OUR CONTRIBUTIONS IN CONTRAST TO THE STATE-OF-THE-ART.
+[10]
+[13]
+[17]
+[18]
+[19]
+Our work
+Secure Transmission
+
+
+
+
+Jointly precoded communication and radar waveforms
+
+
+Precoder design rather than covariance matrix
+
+
+
+
+Radar beampattern optimization
+
+
+
+Multiple users
+
+
+
+
+Imperfect CSI estimations
+
+
+Multiple eavesdroppers
+
+Using radar signal as artificial noise
+
+Tight solution for PLS design
+
+sends confidential information to the targets. Evidently, private
+information might be leaked to the targets, which may act as
+potential eavesdroppers (Eves). Recently, several schemes have
+been proposed for guaranteeing secure data transmission by
+exploiting constructive interference [15], frequency hopping
+[16], and additional artificial noise (AN) [17]–[19], etc. As a
+low complexity yet powerful technique, the AN method has
+been widely harnessed in the communication community for
+enhancing the physical layer security (PLS). The basic princi-
+ple of AN-aided secure transmission is that of contaminating
+the transmit signal by well-designed AN to degrade Eve’s
+reception without affecting the legitimate users (LUs) [20].
+In [17], several optimization problems, including secrecy
+rate maximization, target return SINR maximization, and
+transmit power minimization were formulated for a DFRC
+system in the presence of a single target and a single communi-
+cation receiver. To tackle the non-convexity of the secrecy rate
+expression, an approximate algorithm based on the first-order
+Taylor expansion was proposed, which however resulted in
+a performance gap between the original non-convex problem
+and the approximated one. The authors of [18] considered a
+unified joint passive radar and communication system, where
+the SNR at the passive radar receiver was maximized, while
+keeping the secrecy rate above a certain target. Moreover,
+several practical constraints, such as realistic target direction
+estimation and imperfect channel state information (CSI) were
+taken into account in the associated robust beamforming
+proposal of [19]. However, at the time of writing, most of
+the contributions on secure DFRC systems have the following
+two drawbacks: (1) They only design the covariance matrix of
+the AN, yet no further analysis of the DFRC system’s radar
+detection is offered; (2) Several relaxation algorithms are used
+such as Taylor expansions or semidefinite relaxation (SDR)
+techniques, but the performance loss compared to the original
+non-convex problem is overlooked.
+Motivated by filling the above-mentioned knowledge gap
+in the literature, we develop jointly precoded communication
+and radar waveforms for secure transmission in a multiple-
+input multiple-output (MIMO) DFRC system inspired by [13],
+serving multiple LUs and detecting the targets simultaneously.
+On one hand, the DFRC platform relying on the ISAC tech-
+nique eliminates duplication in the system’s hardware. On the
+other hand, the bespoke transmit signals can simultaneously
+meet the requirements of radar, communications, and PLS,
+circumventing redundancy in the resource consumption for
+each functionality, hence also the power dissipation. Compared
+to the current DFRC schemes such as those in [10]–[12], [19],
+our method achieves superior radar detection performance
+thanks to the increased DoFs attained by the additional radar
+waveforms. In contrast to [13], the PLS level is also considered
+in our work, where the targets may act as potential Eves. The
+radar waveforms conveying no confidential information may
+also be exploited as the AN imposed on the communication
+signals for contaminating the eavesdropping channels. The
+main contributions of this paper are summarized as follows,
+and they are also boldly and explicitly contrasted to the
+literature at a glance in Table 1.
+• We develop jointly precoded communication and radar
+waveforms for secure transmission. Specifically, the AN
+of traditional PLS designs can be replaced by bespoke
+radar signals specifically designed for inflicting interfer-
+ence upon the Eves, whilst additionally increasing the
+DoF available for target detection.
+• We formulate the joint beamforming design as a non-
+convex optimization problem under the consideration of
+both radar, communication and security performance. An
+SDR-based and the associated low complexity algorithms
+are also conceived for tackling the non-convexity of the
+problem, where we prove that the relaxation used in our
+scheme is tight.
+• We propose a robust beamforming design for the more
+practical scenarios of imperfect estimations, including the
+uncertain target directions and the imperfect CSI acquired
+for the LUs. We also show that the globally optimal
+reconstruction method proposed for ideal scenarios still
+applicable to our robust beamforming scheme.
+• We analyze the performance trade-offs among radar, com-
+munication and PLS both theoretically and by simulation
+for providing new insights into flexible beamforming.
+The rest of this paper is organized as follows. In Section II,
+we establish the mathematical model of joint communication
+and radar signal transmission and introduce the performance
+metrics of radar detection, multiuser communication, and
+system security, respectively. The proposed SDR-based beam-
+forming and the low complexity ZF-based algorithms are
+characterized in Section III. Furthermore, Section IV provides
+our robust beamforming method relying on imperfect CSI
+knowledge, while the performance vs. the complexity of the
+proposed algorithms is analyzed in Section V. Finally, our
+simulation results and conclusions are provided in Section VI
+
+3
+TABLE II
+FREQUENTLY USED SYMBOLS
+Notation
+Description
+R
+Covariance matrix of the transmitted signals
+H
+Communication CSI matrix
+Wr (Wc)
+Radar (Communication) beamforming matrix
+Γe (Γc)
+SINR threshold at Eves (LUs)
+K
+Number of the LUs
+Q
+Number of the targets (Eves)
+M
+Number of antennas
+β
+Path-loss coefficient for radar channel
+Lr(R, α)
+Least square function for MIMO radar beampattern
+γk (˜γq)
+SINR of the k-th LU (the q-th Eve)
+and VII, respectively.
+The notations used in this paper are as follows. Upper-
+case A (lower-case a) bold characters denote matrices (column
+vectors), and lower case normal letters a are scalars; (·)T ,
+(·)∗ and (·)H represent the transpose, conjugate and complex
+conjugate transpose operations respectively; |a| and ∥a∥2 stand
+for the magnitude of a scalar a and the ℓ2-norm of the
+vector a; E{·} is the statistical expectation; diag{a} stands
+for a diagonal matrix using the elements of a as its diagonal
+elements; for a matrix A, [A][i,j] denotes the (i, j)th element;
+A[:,1:k] and A[1:k,:] represent the sub-matrices containing the
+first k columns and rows of A respectively; IM is the n-
+dimensional identity matrix and 0M×N is the M × N matrix
+having all-zero entries. Frequently used symbols in this paper
+are summarized in Table II.
+II. SYSTEM MODEL AND PERFORMANCE METRICS
+A. Transmission and Reception Signal Model
+As shown in Fig. 1, a colocated MIMO BS transmits DFRC
+signals to detect Q targets and K LUs simultaneously. For
+the consideration of our PLS design, all the targets considered
+are non-cooperative, such as unmanned aerial vehicle (UAV)
+which are regarded as the potential Eves at the same time. We
+assume that the BS is equipped with M antennas arranged in
+a uniform linear array (ULA), and all the Eves and LUs have
+a single antenna. The proposed beamforming design can be
+readily extended to multi-antenna scenarios.
+Following [13], the discrete-time transmitted signal at time
+slot n, which is the weighted sum of the communication
+signals and radar waveforms, can be expressed as
+x[n] = Wrs[n] + Wcc[n], n = 0, 1, · · · , N − 1,
+(1)
+where s[n] = [s1[n], · · · , sM[n]]T represents the individual
+radar signals and c[n] = [c1[n], · · · , cK[n]]T stands for the K
+parallel communication symbol streams intended for the LUs.
+N is the total number of symbols. Furthermore, Wr ∈ CM×M
+and Wc ∈ CM×K denote the beamforming matrices (or pre-
+coders) designed for the radar waveforms and communication
+waveforms. The conventional transmit signal strategy which
+only exploits the communication signals for detection in [10]–
+[12], [19], can be regarded as the special case associated with
+Wr = 0. In line with the literature, the following assumptions
+are stipulated for the transmitted signals (1).
+Fig. 1.
+The DFRC system detects the targets (Eves) and serves downlink
+users by transmitting mixture waveform.
+• Both the radar and communication signals have zero
+mean, and they are temporally white wide-sense station-
+ary stochastic processes;
+• The radar and the communication waveforms are statis-
+tically independent, hence we have E{scH} = 0M×K;
+• The M radar waveforms are orthogonal to each other,
+then we have E{ssH} = IM;
+• The communication symbols transmitted to different LUs
+are uncorrelated, i.e., E{ccH} = IK;
+Here, the signal power is normalized to unity. Thus, the
+covariance matrix of the transmitted signal can be written as
+R = E{xxH} = WrWH
+r + WcWH
+c .
+(2)
+Let y = [y1, y2, · · · , yK]T denote the received signal vector
+of all the LUs, which can be expressed by
+y = Hx + nc,
+(3)
+where H = [h∗
+1, · · · , h∗
+K]T ∈ CK×M is the channel matrix
+and hk represents the channel vector spanning from the BS to
+the kth LU, and nc ∼ CN(0, σ2
+cIK) denotes the additive white
+Gaussian noise (AWGN). Moreover, the targets of interest can
+be viewed as virtual downlink users located in the LoS channel
+of DFRC systems [10]. Therefore, the signal received by the
+qth target (Eve) can be modeled as [19]
+rq = βqaH(θq)x + ne,
+(4)
+where βq is the complex path-loss coefficient, ne is the AWGN
+with covariance σ2
+e, and a(θ) represents the ULA arrays’
+steering vector, which can be expressed as
+a(θ) =
+1
+√
+M
+�
+1, eȷ2π d
+λ sin(θ), · · · , eȷ2π(M−1) d
+λ sin(θ)�T
+.
+(5)
+Here, d is the antenna spacing, λ is the carrier wavelength,
+and θ is the azimuth of the target.
+The BS has to acquire the CSI for both LUs and Eves
+before the beamforming design. In general, the CSI marix H of
+LUs can be obtained through channel estimation and feedback
+techniques [21]. By contrast, the CSI from the BS to the Eve
+is challenging to acquire, since the Eves tend to be passive
+in general. Fortunately, the sensing functionality of the DFRC
+
+4
+Fig. 2. Flow of the mathematical analysis.
+signal can be exploited for estimating the azimuth and path-
+loss coefficient through radar parameter estimation techniques
+[22], [23]. Since we only focus on the beamforming design,
+the processes of radar parameter estimation and information
+demodulation are ignored in this paper. The elaborate details
+can be found in [1], [24]. Before proceeding to our mathe-
+matical analysis, we have depicted in Fig. 2 the flow of the
+analysis described in the sequel, which allows readers to grasp
+the overall structure of this paper at a glance.
+B. Performance Metrics
+In our proposed physical layer beamformer designed for
+secure transmission, some important properties related to the
+symbol-level waveform design [11], [12] are not considered,
+such as the radar’s ambiguity function, the peak-to-average
+power ratio (PAPR), etc. Next, we introduce our performance
+metrics used for the target detection, for the communication
+quality of service (QoS), and for the PLS level, respectively.
+(1) Performance metric for MIMO radar: In general, there
+are two primary MIMO radar functions, namely detection and
+tracking. MIMO radar tends to create both spatially orthogonal
+waveforms and omni-directional beampatterns (i.e., R = I)
+for detecting the potential targets in the detection stage, since
+there is no prior information concerning the targets. Then, in
+the tracking stage, MIMO radar steers the beam to the target
+directions of interest acquired during the previous observa-
+tions. Instead of maximizing the SINR at radar receiver [25],
+we focus on the radar transmit beampattern performance. The
+synthesized radar beampattern at azimuth θ can be formulated
+as
+P(θ; R) = E{aH(θ)xxHa(θ)} = aH(θ)Ra(θ).
+(6)
+Additionally, the cross-correlation pattern between direction
+θ1 and θ2 can be written as
+Pc(θ1, θ2; R) = aH(θ2)Ra(θ1).
+(7)
+The objectives of beamformer design for MIMO radar include
+the following [26]
+• Optimize the beampattern over the sectors of interest to
+concentrate the signal power while maintaining a low
+sidelobe level;
+• Reduce the cross-correlation pattern over the set of target
+angles to achieve an excellent adaptive performance;
+To this end, we adopt the loss function defined in terms of
+the least squares as our performance metric for MIMO radar,
+which is formulated as
+Lr(R, α) = Lb(R, α) + ηLc(R),
+(8)
+where η is the weighting factor representing the relative
+importance of the two terms based on the associated practical
+requirements. The first term represents the mean squared error
+between the designed and desired beampatterns, which can be
+formulated as
+Lb(R, α) = 1
+L
+L
+�
+l=1
+|αΦ(θl) − P(θl; R)|2.
+(9)
+Here, α is a scaling factor, Φ(θ) denotes the desired transmit
+beampattern, and {θl}L
+l=1 represents the fine grid of points that
+cover the targets of interest. Let ∆ denote the beam-width,
+then the desired beampattern at azimuth θ⋆ is given by
+Φ(θ) =
+�
+�
+�
+1, θ⋆ − ∆
+2 ≤ θ ≤ θ⋆ + ∆
+2
+0, otherwise.
+(10)
+Moreover, the second term is the mean-squared cross-
+correlation pattern, given by
+Lc(R) =
+2
+P 2 − P
+P −1
+�
+p=1
+P
+�
+q=p+1
+|Pc(¯θp, ¯θq; R)|2,
+(11)
+where {θp}P
+p=1 are the given directions of the targets. We refer
+the reader to [13], [26] for more details.
+(2) Performance metric for multi-user communication:
+The achievable transmission rate related to the SINR of the sig-
+nal received by the downlink users is a standard performance
+measure in multiuser communication systems. For notation
+convenience, we introduce W = [Wc, Wr], where wi is the
+ith column of W for i = 1, · · · , K + M. Then, the signal
+covariance matrix can be rewritten as
+R = WWH =
+K+M
+�
+i=1
+wiwH
+i =
+K+M
+�
+i=1
+Ri,
+(12)
+where Ri = wiwH
+i
+is the rank 1 covariance matrix. Specif-
+ically, R1, · · · , RK are the covariance matrices of communi-
+cation symbols, where the last M ones are those of the radar
+
+ertormance metrics ol MiMJ, sccultransmission (15).SteOntim1zin
+undertheStep 2.
+beamformer:rom tne covaliance matrix h telms of tneeamformer interms of the
+ Section Ill-B.★5
+waveforms. Thus, the SINR at the kth LU can be formulated
+as
+γk =
+E{|hH
+k wkck|2}
+K
+�
+i=1,i̸=k
+E{|hH
+k wici|2} +
+M
+�
+j=1
+E{|hH
+k wj+Ksj|2} + σ2c
+=
+hH
+k Rkhk
+K+M
+�
+i=1,i̸=k
+hH
+k Rihk + σ2c
+.
+(13)
+There are two popular design criteria for multiuser beamform-
+ing [27]. One of them is the throughput criterion to maximize
+the system’s sum-rate. The other is the fairness criterion used
+for maximizing the minimal SINR at each user, which can be
+expressed as
+max min{γ1, · · · , γK}.
+(14)
+In this work, the SINR-fairness is adopted as the performance
+metric for multiuser communication. On the one hand, the
+fairness metric guarantees that each LU can obtain satisfactory
+QoS. On the other hand, the fairness metric based optimization
+is more tractable than the NP-hard optimal throughput beam-
+forming problem. Given a minimal level of communication
+QoS Γc, the SNR-fairness metric can be transformed to forcing
+the minimal SINR of the users to be higher than the target
+threshold, i.e., γk ≥ Γc, k = 1, · · · , K.
+(3) Performance metric for PLS level: When the targets
+become Eves, the achievable data rates at Eves are non-
+negligible. A straightforward method is to increase the propor-
+tion of interference signal power to the detriment of the useful
+signal. According to the previous analysis, the radar waveform
+conveying no desired information can be regarded as the
+interference contaminating the reception of Eves. Accordingly,
+by recalling the received signal model (4), the SINR for the
+qth Eve can be formulated as
+˜γq =
+|βq|2a(θq)H �K
+k=1 Rka(θq)
+|βq|2a(θq)H �K+M
+i=K+1 Ria(θq) + σ2e
+.
+(15)
+Following [19], we consider the worst-case SINR in (15),
+assuming that all the information intended for the K LUs is
+the desired signal for Eves. As stated in [28], there will exist
+modulation and coding schemes that allow the LUs rather than
+the Eves to reliably decode the transmit information, as long
+as γk > ˜γq, for ∀k, q. Therefore, we restrict the maximal
+SINR at Eves to be less than a given threshold Γe, instead
+of optimizing the secrecy rate [log(1 + γk) − log(1 + ˜γq)]+
+defined in [17], to achieve a satisfactory PLS level. On the one
+hand, the system’s secrecy rate is difficult to determine due to
+its non-convexity with respect to Ri. On the other hand, since
+SINR-fairness based schemes are still capable of maintaining
+a minimal communication rate due to the monotonicity of the
+log function, we can equivalently achieve a desired secrecy
+rate [log(1 + Γc) − log(1 + Γe)]+ by appropriately choosing
+the thresholds Γc and Γe.
+III. THE BEAMFORMING DESIGN FOR IDEAL SCENARIOS
+In this section, we aim for designing the transmit beamform-
+ing matrices Wr and Wc under the consideration of the perfor-
+mance metrics for the radar beampattern, the communication
+QoS and the PLS levels given in the previous section. We first
+consider the ideal conditions, where the BS perfectly knows
+the CSI both for the LUs and Eves, and leave the beamformer
+design under the more practical imperfect CSI scenario for the
+next section.
+A. The proposed SDR-based beamforming algorithm
+Our beamforming design objective is to minimize the dif-
+ference between the desired transmit beampattern and that
+generated by the BS to achieve good target detection and
+tracking performance. Meanwhile, the beamforming design
+also guarantees that the downlink SINR at the LUs remains
+higher than the given threshold, while that of the Eve is lower.
+Recalling the definition (12), instead of directly optimizing
+the precoding matrix W, the SDR based optimization problem
+with respect to the variables Ri can be formulated as
+minimize
+R,{Ri},α
+Lr(R, α)
+(P0)
+subject to
+R =
+K+M
+�
+i=1
+Ri ∈ S+
+M, α > 0,
+(16a)
+Ri ∈ S+
+M, i = 1, · · · , K + M,
+(16b)
+rank(Ri) = 1, i = 1, · · · , K + M,
+(16c)
+[R][m,m] = Pt/M, m = 1, · · · , M,
+(16d)
+γk ≥ Γc, k = 1, · · · , K,
+(16e)
+˜γq ≤ Γe, q = 1, · · · , Q,
+(16f)
+where S+
+M represents the set consisting of all M-dimensional
+complex positive semidefinite matrices, i.e., S+
+M = {A|A ∈
+CM×M, A = AH, A ⪰ 0}. The rank-1 constraint in (16c) is
+equivalent to Ri = wiwH
+i . (16d) represents the per-antenna
+power constraints, and Pt is the total transmit power of the
+BS. Furthermore, the objective function and the constraints
+(16e), (16f) are the performance metrics introduced in Section
+II-B, where Γc and Γe are the predefined SINR thresholds at
+the LUs and Eve, respectively.
+Upon substituting the SINR expressions (13) as well as (15)
+into the constraints and applying some simple mathematical
+manipulations, (16e) and (16f) can be recast as
+(1 + Γ−1
+c )hH
+k Rkhk ≥ hH
+k Rhk + σ2
+c, ∀k
+(17a)
+(1 + Γ−1
+e )aH
+q
+K
+�
+k=1
+Rkaq ≤ aH
+q Raq +
+σ2
+e
+|βq|2 , ∀q
+(17b)
+where aq is the abbreviated form of a(θq). It can be observed
+that the individual matrices {Ri}i≥K+1 have no effect on the
+SINR constraints, which motivates us to remove these matrix
+variables from the original problem P0 of (16). As a result,
+the number of matrix variables is reduced from K + M + 1
+to K + 1, leading to much reduced memory requirements.
+
+6
+By reformulating the constraint (16a), problem P0 can be
+transformed to
+minimize
+R,R1,··· ,RK,α
+Lr(R, α)
+(P1)
+subject to
+R ∈ S+
+M, R −
+K
+�
+k=1
+Rk ∈ S+
+M,
+(18a)
+α > 0, Rk ∈ S+
+M, k = 1, · · · , K,
+(18b)
+rank(Rk) = 1, k = 1, · · · , K,
+(18c)
+[R][m,m] = Pt/M, m = 1, · · · , M,
+(18d)
+(17a), (17b).
+However, problem P1 is non-convex due to the rank-1 con-
+straints. Thus, the SDR relaxation based version of problem
+P1 can be obtained by omitting the rank-1 constraints (18c),
+which is denoted by problem P2. Thus, the problem P2 has
+become a standard quadratic semidefinite program (QSDP),
+since the objective function is a positive-semidefinite quadratic
+form and all the constraints are either linear or semidefinite.
+Hence, the global optimum can be obtained in polynomial time
+with the aid of standard convex optimization toolboxes [29],
+[30]. Note that the optimal solutions of the relaxed problem
+P2 are not necessarily rank-1 matrices, hence either the classic
+eigenvalue decomposition or Gaussian randomization methods
+[31] can be leveraged to obtain the solutions of the origi-
+nal problem P1. Unfortunately, these kinds of approximate
+algorithms usually only provide suboptimal solutions of the
+original problem, hence resulting in a loss of performance.
+To circumvent this deficiency, we set out to find a global
+optimum for problem P1, which means that the SDR relax-
+ation is tight. Inspired by the result in [13], we propose the
+following proposition.
+Proposition 1: Let ˆR, ˆR1, · · · , ˆRK be the optimal solution
+of the QSDP problem P2. There also exists a global optimum
+˜R, ˜R1, · · · , ˜RK for problem P1, where we have
+˜R = ˆR, ˜wk = (hH
+k ˆRkhk)−1/2 ˆRkhk, ˜Rk = ˜wk ˜wH
+k ,
+(19)
+for k = 1, · · · , K.
+Proof: The proof is relegated to Appendix A.
+■
+According to Proposition 1, we can get the global rank-
+1 optimal solution for problem P1 from its QSDP relaxation
+based version P2, where the relaxation is tight. The remaining
+step is to find the optimal solution for the original problem
+P0, i.e. obtaining the precoding matrix Wr for the radar
+waveforms. To meet the constrains of (16a) and (16b), the
+M precoding vectors {wi}i≥K+1 can be obtained by the
+following decomposition
+WrWH
+r = Rrad = ˜R −
+K
+�
+k=1
+˜Rk,
+(20)
+where Wr = [wK+1, · · · , wK+M]. Actually, since the associ-
+ated waveform level design is not considered in this work, the
+decomposition (20) is not unique, but it is trivial thanks to the
+positive semi-definite nature of the radar signal’s covariance
+matrix. Several decomposition methods such as the square root
+matrix (Wr = R
+1
+2
+radU, U is an arbitrary unitary matrix) based
+one [32] and the Cholesky decomposition based one may be
+applied [33].
+B. The ZF-based low complexity algorithm
+The main computational complexity burden in the proposed
+SDR-based algorithm is imposed by that of solving the QSDP
+problem P2, which motivates us to seek a low-complexity
+solution. Inspired by the zero forcing (ZF) based method of
+[13], we develop a reduced-complexity sub-optimal algorithm
+by incorporating ZF constraints into problem P2. The ZF
+method is widely used in low-complexity linear precoders,
+because its performance tends to that of the optimal non-
+linear precoder, especially for a large number of antennas [34],
+[35]. Its main appeal is that of eliminating the inter-user and
+radar interferences, hence achieving a high SINR at each user.
+Mathematically, the ZF constraints can be expressed as
+HWc = diag(√ρ1, · · · , √ρK), HWr = 0K×M,
+(21)
+where ρk represents the signal power at the kth user, for 1 ≤
+k ≤ K. Upon recalling the definition W = [Wc, Wr] and R =
+WWH, (21) can be equivalently transformed to the following
+form (Theorem 2, [13])
+HRHH = diag(ρ),
+(22)
+where ρ = [ρ1, · · · , ρK]. Moreover, substituting (21) or (22)
+into the SINR expression (13), the associated SINR constraints
+(17a) can be simplified by
+ρk ≥ Γcσ2
+c, ∀k.
+(23)
+It can be observed that the individual matrix variable Rk
+has been removed from the SINR constraints for the LUs by
+imposing the ZF constraints. Following the same methodology
+for further reducing the number of matrix variables, and by
+introducing the auxiliary matrix variable Rcom = �K
+k=1 Rk,
+the PLS constraint (17b) can be rewritten as follows
+(1 + Γ−1
+e )aH
+q Rcomaq ≤ aH
+q Raq +
+σ2
+e
+|βq|2 , ∀q
+(24)
+Furthermore, we can immediately infer the ZF constraint for
+Rcom as
+HRcomHH = HWcWH
+c HH = diag(ρ).
+(25)
+As a consequence, either the communication SINR constraint
+or the PLS constraint no longer contains the individual matrix
+variable Rk. Accordingly, problem P2 can be converted to
+minimize
+R,Rcom,ρ,α
+Lr(R, α)
+(P3)
+subject to
+R ∈ S+
+M, R − Rcom ∈ S+
+M, Rcom ∈ S+
+M,
+(26a)
+[R][m,m] = Pt/M, m = 1, 2, · · · , M,
+(26b)
+HRHH = diag(ρ),
+(26c)
+HRcomHH = diag(ρ),
+(26d)
+α > 0, (23), (24).
+Problem P3 is also a standard QSDP problem, because the
+objective function has a positive-semidefinite quadratic form
+
+7
+Algorithm 1 The proposed SDR(ZF)-based beamforming
+algorithm designed for secure DFRC.
+Input:
+Total transmit power of base station Pt;
+Radar desired beampattern Φ(θ);
+Instantaneous downlink channel H;
+SINR threshold at LUs Γc and at Eves Γe;
+The directions of Eves θq, q = 1, · · · , Q;
+Output
+The overall precoding matrix W = [w1, · · · , wK+M].
+Steps
+1. Compute the optimal solution of P2 (or P3) via convex
+optimization solver;
+2. Compute w1, · · · , wK by (19) (or by (28));
+3. Compute wK+1, · · · , wK+M by (20) (or by (29));
+and all the constraints are either linear or semidefinite. Simi-
+larly, the optimal solutions ˆR and ˆRcom can be obtained by a
+standard convex optimization toolbox in polynomial time.
+The next step is to recover the precoding matrix W from
+the optimal solutions ˆR and ˆRcom. Inspired by Theorem 2
+of [13], we conceive the following procedure of constructing
+the radar and communication precoding matrices, respectively.
+First, either the classic Cholesky decomposition or square
+root method is used by exploiting the positive-semidefinite
+property for ˆRcom = LcLH
+c . Then, we employ the row QR
+decomposition of HLc, yielding
+HLc = [Lh, 0K×(M−K)]Q,
+(27)
+where Lh is a K × K lower triangular matrix and Q is a
+M × M unitary matrix. Thus, the communication precoder
+can be formulated as
+Wc = Lc[QH][:,1:K],
+(28)
+while the radar precoding matrix Wr can be expressed as
+WrWH
+r = ˆRrad = ˆR − WcWH
+c .
+(29)
+Subsequently, we analyze the feasibility of the proposed pre-
+coder design method by introducing the following proposition.
+Proposition 2: Given the optimal solution ˆR and ˆRcom of
+problem P3, the matrices Wc in (28) and Wr in (29) are
+also the optimal precoders of problem P3 and satisfy the ZF
+constraint (21) at the same time.
+Proof: The proof is divided into three parts, and it is
+relegated to Appendix B.
+■
+Proposition 2 illustrates the feasibility and efficiency of
+the proposed precoding matrices recovered from the optimal
+solution of problem P3. In summary, we can obtain the
+optimal beamforming for DFRC secure transmission with the
+perfectly known CSI by the proposed SDR-based and the low
+complexity ZF-based algorithms. The detailed procedure of
+the proposed algorithms are summarized in Algorithm 1.
+IV. ROBUST BEAMFORMING DESIGN WITH IMPERFECT
+CSI KNOWLEDGE
+In practice, it is challenging to obtain the exact CSI due
+to the estimation errors, feedback quantization, hardware defi-
+ciencies, etc., resulting in imperfect CSI knowledge at the BS.
+Specifically, for the radar targets, we assume that the direction
+of the q-th target is roughly known by the BS within an angular
+interval of [θq − ∆θq, θq + ∆θq], where ∆θq represents the
+associated angle uncertainty. Moreover, for the communication
+LUs, the additive error model of the CSI matrix for the k-th
+LU is considered as hk = ˆhk + ϵk, where ˆhk is the estimated
+CSI matrix and ϵk denotes the channel uncertainty. To this
+end, we aim for designing the robust beamforming scheme
+for secure transmission in this section.
+A. Wide main-lobe beampattern design
+The uncertainties of the target directions have an impact
+on both the objective function and the PLS constraints in
+problem P0. On one hand, the BS should form a wide main-
+lobe to avoid missing the target. Thus, the beam-width ∆ in
+(10) should be appropriately chosen according to the angular
+uncertainty ∆θq, in order to cover all the possible locations
+of the target.
+On the other hand, since Eve may be located in an arbitrary
+direction within the angular interval, we should guarantee a
+satisfactory secrecy rate for every possible direction. Con-
+sequently, the SINR constraints (17b) should be modified
+according to
+(1 + Γ−1
+e )aH
+qi
+K
+�
+k=1
+Rkaqi ≤ aH
+qiRaqi + σ2
+e
+|βq|2 , ∀θqi ∈ ¯Ωq, (30)
+where ¯Ωq is a discrete set that covers the potential directions
+of the q-th Eve, and aqi represents the compact form of a(θqi).
+It can be observed that the angular uncertainty introduces
+more constraints similar to (17b) over the associated angular
+interval. Evidently, the proposed Algorithm 1 is also capable
+of handling the modified constraints (30). In other words,
+the number of targets and the uncertainty of target directions
+determine the number of PLS constraints. Naturally, imposing
+a large number of constraints for securing certain PLS levels
+results in degraded radar beampattern and communication
+QoS. We will illustrate this phenomenon in Section VI.
+B. Robust beamforming for mitigating CSI error of LUs
+Similar to [19], [37], we assume that the CSI uncertainty is
+bounded by a spherical region as
+Sk := {ˆhk + ϵk | ||ϵk|| ≤ uk}, ∀k.
+(31)
+In this case, the SINR expression for the k-th LU in (13)
+should be replaced by the worst-case SINR over the set Sk,
+namely
+¯γk = min
+hk∈Sk γk, ∀k.
+(32)
+Thus, based on the definitions (31) and (32), the SINR
+constraint in (17a) can be reformulated as
+(ˆhk+ϵk)H �
+(1 + Γ−1
+c )Rk − R
+�
+(ˆhk+ϵk)−σ2
+c ≥ 0, ∀k. (33)
+Then, we adopt the popular S-procedure of robust optimization
+to tackle the SINR constraints mentioned above. By introduc-
+ing an auxiliary vector t = [t1, · · · , tK], the original problem
+
+8
+P1 can be reformulated as the following robust beamforming
+version [19], [37]
+minimize
+R,R1,··· ,RK,t,α
+Lr(R, α)
+(P4)
+subject to
+(18a) − (18d), (17b) or (30),
+�
+Sk + tkIM
+Skˆhk
+ˆh
+H
+k Sk
+hH
+k Skhk − σ2
+c − tku2
+k
+�
+⪰ 0, ∀k
+Sk := (1 + Γ−1
+c )Rk − R, tk ≥ 0.
+(34)
+Again, by dropping the rank-1 constraints (18c), problem
+P4 becomes a QSDP, which can be efficiently solved in
+polynomial time. Then, we will show that the optimal solution
+of the QSDP reconstruction method in (19) also holds for the
+proposed robust beamforming.
+Proposition 3: Let ˆR, ˆR1, · · · , ˆRK be the optimal solution
+of the relaxed version of problem P4. Then the ˜R, ˜R1, · · · , ˜RK
+associated with the expression of (19) is also the optimal
+solution of the original problem P4.
+Proof: By employing the result in Proposition 1, the proof
+becomes straightforward upon substituting (19) into the con-
+straints (33).
+■
+V. PERFORMANCE AND COMPLEXITY ANALYSIS
+A. Complexity Analysis
+The complexity of the proposed algorithms is dominated by
+the QSDP problem. For a given solution accuracy ϵ, the worst-
+case complexity order of solving problem P2 using the primal-
+dual interior-point algorithm is O[(K + Q)6.5M 6.5log(1/ϵ)]
+[13], [38], where K + Q and M refer to the number of
+semidefinite constraints and the dimension of matrix variables,
+respectively. Compared to the SDR algorithm, the low com-
+plexity ZF beamforming problem P3 includes 5 = O(1) such
+constraints, hence the worst-case complexity order becomes
+O[Q6.5M 6.5log(1/ϵ)]. Furthermore, for the robust beamform-
+ing algorithm with imperfect CSI knowledge, the complexity
+also depends on the number of elements in the set ¯Ωq of
+(30). Specifically, upon denoting the cardinality of the set
+¯Ωq as P, the worst-case complexity is on the order of
+O[K6.5P 6.5M 6.5log(1/ϵ)].
+B. Performance Analysis
+In this subsection, we provide the performance analysis of
+the proposed algorithms.
+(I) We can immediately spot the performance trade-off
+among the radar beampattern, the communication QoS, and
+the PLS level in problem P1. The constraints (17a) and (17b)
+always hold, when we have Γc = 0 and Γe → ∞. In this case,
+problem P1 is reduced to the conventional radar-only beam-
+forming design, determining the optimal beampattern for radar
+detection. Explicitly, any improvements of the communication
+QoS and PLS level are attained at the cost of sacrificing the
+radar performance, since the radar loss function will increase
+upon increasing Γc or decreasing Γe.
+(II) Compared to the SDR-based algorithm, the low com-
+plexity ZF-based algorithm forces the radar and inter-user in-
+terference to zero, potentially raising the SINR at the LUs to a
+certain threshold (denoted by ˆΓ). Thus, for the communication
+constraints, we have
+�
+γZF
+k = ˆΓ > γSDR
+k
+≥ Γc, when Γc < ˆΓ,
+γZF
+k = γSDR
+k
+≥ Γc ≥ ˆΓ, when Γc ≥ ˆΓ.
+(35)
+For a relatively low threshold Γc, the interference encountered
+by the users do not have to be as low as zero to satisfy the
+SINR constraint, resulting in γZF
+k > γSDR
+k
+. By contrast, the in-
+terference in γSDR
+k
+has to be eliminated to meet the high SINR
+requirements, resulting in γZF
+k = γSDR
+k
+. According to (35), we
+can immediately conclude the following properties of the ZF-
+based algorithm. (1) It results in worse radar beampattern than
+the SDR-based algorithm because more severe restrictions are
+imposed by the ZF constraint when Γc < ˆΓ. (2) The radar
+loss function and the users’ SINR remains constant, as long
+as the threshold Γc is lower than a positive value ˆΓ. (3) The
+performance of ZF-based beamforming tends to be similar to
+that of SDR-based beamforming at high SINRs, i.e., Γc ≥ ˆΓ.
+(III) For the SDR-based algorithm, the system’s secrecy rate
+is always approximated by log2(1+Γc)−log2(1+Γe) given the
+thresholds Γc and Γe, because the optimal solution generally
+reaches the boundary of the feasible region. By contrast, the
+secrecy rate of the ZF-based algorithm may become higher
+than the above value for small Γc values due to the potentially
+high SINR achieved under the ZF constraint. The proposed
+algorithms guarantee to have a secrecy rate above a certain
+lower bound.
+(IV) Upon considering the extreme case that the channels of
+the users and Eves have the same quality, i.e., βka(θk) = hk,
+the communication QoS constraint (17a) and the PLS level
+constraint (17b) are contradictory to each other, hence lead-
+ing to the infeasibility of problem P1. This means that the
+feasibility probability of problem P1 significantly depends on
+the values of Γc as well as Γe, and on the distances between
+the targets and Eves. The proposed joint beamforming design
+method will become invalid, when the Eves are at the same
+directions as the users. The symbol-level range sidelobe design
+[36] may be a promising remedy, which we will leave for
+future research.
+(V) It should be pointed out that the joint PLS beamforming
+design of [19] minimized the SINR at Eve, which is different
+from the proposed method optimizing the radar transmit
+beampattern. Even though it cannot be compared directly due
+to the different functional requirements, the proposed method
+has the following advantages over [19]. (1) The fractional
+programming approach is adopted in [19], where a sequence
+of SDPs has to be solved by iteration, imposing a heavy
+computational burden. By contrast, the proposed methods
+only have to solve a SDP or QSDP problem with the same
+number of matrix variables. (2) The eigenvalue decomposition
+or Gaussian randomization techniques of [19] result in a sub-
+optimal solution, when the ranks of the optimal matrices
+obtained by the SDP solver are not equal to 1. By contrast, the
+proposed SDR relaxation is tight. (3) When using the SINR
+instead of the secrecy rate as the objective, the difference
+between the achievable rate of users and that of Eves may
+become negative, leading to a secrecy rate of SR = 0. By
+
+9
+contrast, the proposed algorithms can always guarantee a
+satisfactory secrecy rate.
+VI. SIMULATION RESULTS
+In this section, we evaluate the proposed joint PLS beam-
+forming algorithm by numerical simulations. The system pa-
+rameters are set as follows, unless specified otherwise. The
+BS is equipped with a ULA having half-wavelength spacing
+between adjacent antennas, i.e. d/λ = 1/2. The number of
+antennas is set to M = 10, and the total transmit power is
+normalized as Pt = 1. The angular directions are obtained by
+uniform sampling with resolution of 0.1◦, including {θl}L
+l=1
+in (9) with the range of [−90◦, 90◦], and Ωq in (30). Without
+loss of generality, we adopt the Rayleigh fading model for
+the multi-user communication channel so that each entry of
+H obeys the standard complex Gaussian distribution with
+hi,j ∼ CN(0, 1). Additionally, we assume the noise levels
+at the Eves and LUs to be the same, i.e., σ2
+c = σ2
+e = 0.01 for
+convenience. The individual radar waveforms and communica-
+tion symbols are generated as random quadrature-phase-shift-
+keying (QPSK) modulated sequences, with the total number
+of symbols being N = 1024.
+For comparison, we choose the joint beamforming de-
+sign method and its low-complexity counterpart proposed in
+[13] termed as Benchmark 1 and Benchmark 2, respectively.
+Compared to [10], where only the communication signal is
+exploited by the DFRC system, the superiority of the combined
+radar waveforms and communication signals in terms of
+increasing the DoFs has been shown in [13]. Therefore, we
+refer to [13] for circumventing repetition.
+First, we numerically characterize the MIMO radar transmit
+beampattern, where the proposed SDR-based algorithm and
+its low-complexity version are referred to as SDR and ZF,
+respectively. We set the direction of a single target to θ0 = 0◦,
+the threshold for the LUs’ SINRs to Γc = 10dB, and the
+threshold for the Eve’s SINR to Γe = 0dB. Fig. 3 illustrates
+the trade-off among the radar beampattern, the communication
+QoS and the PLS level. Although the proposed algorithms
+impose a performance degradation on the transmit beampattern
+compared to their counterparts, the target secrecy rate (SR) can
+still be guaranteed. By contrast, Benchmark 1 and 2 form better
+beampatterns, but their SR becomes zero. Then, we evaluate
+the system performance versus the predefined SINR thresholds
+Γc and Γe, respectively.
+A. System Performance Evaluation vs. the Threshold Γc
+In this subsection, we keep the SINR threshold of Eves
+Γe = 0dB as a constant, and sweep Γc of LUs from 10dB to
+18dB to test its impact. All of the simulation results represent
+averaged values over 500 Monte Carlo trials. In each trial,
+the target direction θq is chosen randomly in the range of
+[−60◦, 60◦], and the CSI of the link spanning from the BS
+and the LUs obey the standard Complex Gaussian distribution.
+The radar performance is evaluated as the difference between
+the DFRC transmit beampattern and the optimal radar-only
+-100
+-50
+0
+50
+100
+Spatial Direction (degree)
+-40
+-30
+-20
+-10
+0
+10
+20
+Transmit Beampattern (dB) 
+Radar Only
+Benchmark 1,SR=0
+Benchmark 2,SR=0
+SDR,SR=3.46
+ZF,SR=5.20
+30
+40
+50
+60
+-30
+-28
+-26
+-24
+-22
+Fig. 3. Radar transmit beampattern for the direction θ0 = 0◦, with K = 2,
+Γc = 10dB, and Γe = 0dB.
+beampattern by defining the mean square error (MSE) metric
+as
+MSE = 1
+L
+L
+�
+l=1
+|P(θl; ˆR) − P(θl; R⋆)|2,
+(35)
+where R⋆ is the optimal radar-only variance matrix by the 3dB
+low sidelobe beampattern design scheme of [26] .
+Fig. 4 shows the beampattern MSE versus the SINR thresh-
+old Γc of the LUs. We can observe the following three
+phenomena from Fig. 4. (1) The beampattern MSEs of all
+algorithms increase upon increasing Γc, which is consistent
+with the previous analysis. As expected, the MSE of the ZF-
+based algorithms remains constant in the scenarios of K = 2
+and for SINRs below 16dB at K = 4. This is because the
+ZF-based methods force the interference to zero, leading to
+a potentially high SINR. Thus, the performance will remain
+constant until the SINR thresholds become higher than the
+potential SINR achieved by the ZF constraint. The perfor-
+mance gaps between the SDR-based and ZF-based methods
+become quite small for high enough values of Γc. (2) The
+benchmark algorithms formulate better beampattern, since the
+PLS aspects of confidential information protection is not taken
+into account in these methods. (3) The more users have to be
+supported, the higher the beampattern MSE becomes. Notably,
+the impact of the number of users K on the beampattern MSE
+is more significant than that of the SINR threshold Γc, which
+implies that serving more downlink users is more restrictive
+than improving the SINR level of the users.
+In Fig. 5, we quantify the achievable sum-rate versus the
+SINR threshold Γc, where the system sum-rate is defined by
+�K
+k=1 log2(1 + γk). The SDR and Benchmark 1 curves are
+fairly similar and increase linearly with the SINR constraint
+Γc. This is because the optimal solution should reach the
+SINR boundary related to the given threshold. Conversely, as
+seen in the analysis of Section V-B, the ZF-based beamformer
+achieves a higher communication sum rate to the detriment of
+the radar performance. Meanwhile, the performances of the
+SDR-based and ZF-based beamformer tend to become similar
+at high SINR thresholds for both K = 2 and 4. Furthermore,
+the curves of the Benchmark 2 are slightly higher than those
+
+10
+10
+11
+12
+13
+14
+15
+16
+17
+18
+SINR Threshold for LUs (dB)
+10-3
+10-2
+10-1
+100
+Beampattern MSE
+Benchmark1
+Benchmark2
+SDR
+ZF
+K=4
+K=2
+Fig. 4. Beampattern MSE versus SINR thresh-
+old Γc for LUs, Γe = 0dB.
+10
+11
+12
+13
+14
+15
+16
+17
+18
+SINR Threshold for Downlink Users (dB)
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+System Sum Rate (bit/(s·Hz))
+Benchmark1
+Benchmark2
+SDR
+ZF
+K=2
+K=4
+Fig. 5.
+The achievable sum rate versus SINR
+threshold Γc for LUs, Γe = 0dB.
+10
+11
+12
+13
+14
+15
+16
+17
+18
+SINR Threshold for Downlink Users (dB)
+0
+1
+2
+3
+4
+5
+Secrecy Rate (bit/(s·Hz))
+Benchmark1,K=2
+Benchmark2,K=2
+SDR,K=2
+ZF,K=2
+Benchmark1,K=4
+Benchmark2,K=4
+SDR,K=4
+ZF,K=4
+Fig. 6. The secrecy rate versus SINR threshold
+Γc for LUs, Γe = 0dB.
+-20
+-15
+-10
+-5
+0
+SINR Threshold for Eavesdropper (dB)
+10-4
+10-3
+10-2
+10-1
+100
+Beampattern MSE
+Benchmark1
+Benchmark2
+SDR
+ZF
+K=4
+K=2
+Fig. 7. Beampattern MSE versus SINR thresh-
+old Γe for Eves, Γc = 10dB.
+-20
+-15
+-10
+-5
+0
+SINR Threshold for Eavesdropper (dB)
+6
+8
+10
+12
+14
+16
+18
+20
+22
+System Sum Rate (bit/(s·Hz))
+Benchmark1
+Benchmark2
+SDR
+ZF
+K=2
+K=4
+Fig. 8.
+The achievable sum rate versus SINR
+threshold Γe for Eves, Γc = 10dB.
+-20
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+-2
+SINR Threshold for Eavesdropper (dB)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+Secrecy Rate (bit/(s·Hz))
+SDR,K=2
+ZF,K=2
+SDR,K=4
+ZF,K=4
+Benchmark1,K=2
+Benchmark2,K=2
+Benchmark1,K=4
+Benchmark2,K=4
+Fig. 9. The secrecy rate versus SINR threshold
+Γe for Eves, Γc = 10dB.
+of the proposed ZF algorithms, since there is an additional
+minimum PLS constraint imposed on the ZF algorithm.
+Fig. 6 illustrates the system’s secrecy rate versus the SINR
+threshold Γc. Observe that the curves of SDR associated with
+K = 2 and K = 4 are coincident and increase linearly
+upon increasing Γc. Recall from Section V-B that the system’s
+secrecy rate will only reach the value of log2(1 + Γc) −
+log2(1+Γe), if the optimization problem is feasible, regardless
+of how the other parameters change. Additionally, the ZF-
+based beamformer associated with K = 2 achieves a higher
+secrecy rate than that of the SDR-based algorithm at small
+values of Γc, since it can reach a higher SINR level than
+the given threshold. However, the secrecy rate of these two
+algorithms becomes similar for K = 4. Actually, supporting
+more communication users imposes more restrictions on the
+optimization problem P3, hence forcing the minimal SINR
+level to approximate the threshold Γc. Moreover, the proposed
+PLS-protected beamforming design guarantees a satisfactory
+PLS level by appropriately choosing the thresholds. By con-
+trast, the benchmark 1 and 2 are not capable of secrecy
+protection, especially not for numerous legitimate users K.
+B. System Performance Evaluation vs. the Threshold Γe
+In this subsection, we evaluate the system performance
+versus the SINR threshold Γe of the Eves. Accordingly, we set
+Γc = 10dB as a constant, while all other system parameters
+remain unchanged. The SINR threshold Γe is varied from
+−20dB to 0dB with intervals of 2dB. It should be highlighted
+that the benchmark curves of [13] remain constant in all the
+figures of this subsection. This is because these algorithms do
+not take the PLS into account, hence the change of threshold
+Γe does not affect these performances.
+Fig. 7 shows that the radar beampattern MSE decreases
+upon increasing Γe both for the proposed SDR and ZF
+algorithms. Specifically, we can see that the curves of Fig.
+7 remain near-constant, when Γe is less than −12dB, while
+decreasing noticeably, when Γe is higher than −10dB. Similar
+trends may also be observed in Fig. 8 and Fig. 9, which implies
+that the performance is not sensitive to the choice of Γe, when
+Γe is less than −12dB for this set of parameters. Having
+excessively low Γe increases the infeasibility probability of
+the optimization problem considered.
+In Fig. 8, we can see that the system’s sum-rate also remains
+unchanged for the SDR algorithm as a result of the constant
+threshold Γc being close to the optimal solution. By contrast,
+the curves of ZF show an increasing trend in Fig. 8 upon
+increasing Γe, since a higher Γe implies that less severe
+restrictions are imposed on the ZF-based beamforming.
+In Fig. 9, the SDR and the ZF for K = 4 reach the
+boundary of the secrecy rate log2(1 + Γc) − log2(1 + Γe).
+Meanwhile, the ZF for K = 2 attains a higher secrecy rate
+
+11
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+Angular Direction (degree)
+-15
+-10
+-5
+0
+5
+10
+Transmit Beampattern (dB) 
+Radar Only
+ZF,SR=3.08
+SDR,SR=3.05
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+SINR Level
+ZF
+SDR
+Fig. 10.
+Transmit beampattern for multiple
+targets with uncertain directions.
+10
+11
+12
+13
+14
+15
+16
+17
+18
+SINR Threshold for Downlink Users (dB)
+10-3
+10-2
+10-1
+Beampattern MSE
+SDR,
+=0
+ZF,
+=0
+SDR,
+=5
+ZF,
+=5
+SDR,
+=10
+ZF,
+=10
+Fig. 11.
+Beampattern MSE comparison with
+different angular uncertainties of the Eves.
+0
+0.2
+0.4
+0.6
+0.8
+1
+CSI error bound for LUs (uk
+2)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Estimated Secrecy Rate (bit/(s·Hz))
+Perfect CSI, e= 0dB
+Imperfect CSI, 
+e= 0dB
+c = 10dB
+c = 5dB
+Fig. 12.
+Estimated secrecy rate calculated by
+the known imperfect CSI versus error bound.
+than its counterparts, since supporting less LUs imposes less
+restrictions on the beamforming design. Furthermore, we can
+infer from Fig. 8 and Fig. 9 that although a low Γe reduces the
+achievable data rate of the Eve, it also results in a low data rate
+for the LUs. Therefore, no obvious secrecy rate improvement
+is attained upon reducing Γe.
+C. System Performance Evaluation for imperfect CSI
+First, we evaluate the impact of angular uncertainties of
+the Eves on the system performance. We set Q = 3 targets
+having the directions of θ1 = −40◦, θ2 = 0◦, and θ3 = 40◦,
+respectively. Each target has the same direction uncertainty
+of ∆θ = 5◦. The BS detects and tracks these targets, while
+serving K = 3 LUs. The SINR thresholds for the LUs and
+the Eves are set to Γc = 10dB and Γe = 0dB, respectively.
+Fig. 10 illustrates the radar transmit beampattern synthe-
+sized by the proposed algorithms. The SINR level defined by
+(15) is calculated over the set of [−90◦, 90◦] angular direction.
+It can be observed that although the BS forms multi-beams
+pointing to the directions of the Eves, the SINR levels in
+each interval covering the Eves are controlled by the threshold
+Γe. This is because the signal power of radar waveforms is
+higher than that of the communication symbols, which have
+to be protected. Moreover, although the beampattern of the
+ZF algorithm is less beneficial than that of the SDR (higher
+side-lobe), the average spatial SINR level is lower than that
+of the SDR algorithm. In Fig. 11, we evaluate the impact of
+the direction uncertainties on the optimization performance
+upon varying Γc from 10dB to 18dB. As expected, further
+constraints are introduced by the uncertainty of the target
+directions, hence leading to an eroded radar performance.
+Fig. 12 shows the estimated secrecy rate calculated by the
+known imperfect CSI versus the error bound for the scenario
+of K = 2. It can be observed that the estimated secrecy
+rates remain constant and are equal to the secrecy rates in
+the case of perfect CSI. By contrast, the curves obtained in
+the case of imperfect CSI exhibit an increasing trend. This is
+because the worst-case secrecy rate is forced to be larger than
+a given threshold in our robust beamforming algorithm, while
+the statistical difference between the worst-case and estimated
+secrecy rate becomes larger upon increasing the error bound.
+VII. CONCLUSION
+A DFRC multi-user communication system was pro-
+posed, while taking the physical layer security into account.
+The weighted sum of the communication signal and radar
+waveform was adopted for dual-functional transmission. We
+demonstrated that the additional radar waveform conveying no
+confidential information improves the DoF in target detection
+and simultaneously contaminates the eavesdropping channel.
+Subsequently, the SDR and the low complexity ZF algorithms
+were proposed for finding the global optimal solution of the
+formulated non-convex beamforming design problem. Further-
+more, we also designed the robust beamforming for the more
+practical scenarios of imperfect CSI knowledge. Finally, we
+evaluated the impact of the parameters on the attainable sys-
+tem performance by numerical simulations, which showed an
+excellent consistency with the theoretical analysis. Designing
+PLS systems operating in the face of other types of legitimate
+and eavesdropping channels as well as hardware impairments
+is left for our future research. Another promising area of
+research is the design of Pareto-optimal multi-component
+systems relying on the full set of optimal operating points
+in terms of throughput, bit error rate (BER), package loss,
+latency, etc.
+APPENDIX A
+THE PROOF OF PROPOSITION 1
+By applying the Theorem 1 in [13], we only have to prove
+that the PLS constraint (17b) holds for ˜R, ˜R1, · · · , ˜RK, if it
+holds for ˆR, ˆR1, · · · , ˆRK. First, we show that
+aH(θ)ˆRka(θ) ≥ aH(θ)˜Rka(θ),
+(36)
+for arbitrary θ. Upon substituting the expression of ˜Rk into
+(19), the right-hand side term of the inequality can be ex-
+panded as
+aH ˜Rka = aH ˜wk ˜wH
+k a
+= (hH
+k ˆRkhk)−1aH ˆRkhkhH
+k ˆRka
+= (hH
+k ˆRkhk)−1|aH ˆRkhk|2.
+(37)
+Additionally, according to the Cauchy-Schwarz inequality, we
+have
+(hH
+k ˆRkhk)(aH ˆRka) ≥ |aH ˆRkhk|2.
+(38)
+
+12
+Therefore, it can be readily seen from (37) and (38) that (36)
+holds. Thus, we can expound as follows
+aH
+q ˜Raq + σ2
+e
+|β|2
+(a)
+= aH
+q ˆRaq + σ2
+e
+|β|2
+≥ (1 + Γ−1
+e )aH
+q
+K
+�
+k=1
+ˆRkaq
+(b)
+≥(1 + Γ−1
+e )aH
+q
+K
+�
+k=1
+˜Rkaq,
+(39)
+where (a) and (b) follow the first equation in (19) and the
+inequality (36), respectively. Thus, the PLS constraint (17b)
+holds for ˜R, ˜R1, · · · , ˜RK, hence completing the proof.
+APPENDIX B
+THE PROOF OF PROPOSITION 2
+The proof is divided into the following three parts:
+(1) We show that the radar covariance matrix ˆRrad in (29)
+is a positive semidefinite matrix, hence it can be decomposed
+by either the Cholesky decomposition or by the square root
+method. Actually, we have
+ˆR − WcWH
+c
+= ˆR − ˆRcom + ˆRcom − WcWH
+c
+= ˆR − ˆRcom + Lc(I − [QH][:,1:K][Q][1:K,:])LH
+c .
+(40)
+Here, ˆR − ˆRcom is positive semidefinite due to the constraint
+(26a). Since [QH][:,1:K] is the sub-matrix containing the first
+K columns of unitary matrix, (I − [QH][:,1:K][Q][1:K,:]) is
+a positive semidefinite matrix, thereby the last term is also
+positive semidefinite.
+(2)
+We
+show
+that
+the
+proposed
+precoding
+matri-
+ces satisfy the ZF constraint (21). Upon letting F
+=
+diag(√ρ1, · · · , √ρK), we have
+HRcomHH = HLcLH
+c HH = LhLH
+h = FFH.
+(41)
+Note that LhLH
+h and FFH are the Cholesky decompositions
+of the matrix diag(ρ), therefore we have Lh = F according
+to the uniqueness of the Cholesky decomposition of a positive
+definite matrix. Thus, we have
+HWc = HLc[QH][:,1:K]
+= [Lh, 0K×(M−K)]Q[QH][:,1:K]
+= Lh = F.
+(42)
+Moreover, for the radar precoding matrix, we arrive at
+HWrWH
+r HH = H(ˆR − WcWH
+c )HH
+= FFH − FFH = 0.
+(43)
+Thus we can readily obtain HWr = 0 from (43).
+(3) We show that the proposed precoding matrices meet the
+PLS constraint (24). According to the positive semidefinite
+property, we can show that
+yH(I − [QH][:,1:K][Q][1:K,:])y ≥ 0,
+(44)
+for an arbitrary non-zero vector y. Upon letting y = LH
+c aq,
+we have
+aH
+q Lc(I − [QH][:,1:K][Q][1:K,:])LH
+c aq
+= aH
+0 ˆRcomaq − aH
+q WcWH
+c aq ≥ 0.
+(45)
+By applying the inequality (45), we can see that
+aH
+q ˆRaq + σ2
+e
+|β|2
+(a)
+≥(1 + Γ−1
+e )aH
+q ˆRcomaq
+≥ (1 + Γ−1
+e )aH
+q WcWH
+c aq,
+(46)
+where (a) is valid, because ˆR and ˆRcom are the feasible
+solution of problem P3 and ˆR follows the relationship (29).
+Consequently, it can be observed that the precodering matrix
+constructs also satisfy the PLS constraint, hence completing
+the proof.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf,len=968
+page_content='1  Joint Beamforming Design for Dual-Functional  MIMO Radar and Communication Systems  Guaranteeing Physical Layer Security  Fuwang Dong, Wei Wang, Senior Member, IEEE, Xin Li, Fan Liu, Sheng Chen, Fellow, IEEE, and Lajos  Hanzo, Life Fellow, IEEE  Abstract—The  dual-functional  radar  and  communication  (DFRC) technique constitutes a promising next-generation wire-  less solution, due to its benefits in terms of power consumption,  physical hardware, and spectrum exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In this paper,  we propose sophisticated beamforming designs for multi-user  DFRC systems by additionally taking the physical layer security  (PLS) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We show that appropriately designed radar  waveforms can also act as the traditional artificial noise conceived  for drowning out the eavesdropping channel and for attaining  increased design degrees of freedom (DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The joint beamform-  ing design is formulated as a non-convex optimization problem  for striking a compelling trade-off amongst the conflicting design  objectives of radar transmit beampattern, communication quality  of service (QoS), and the PLS level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, we propose a  semidefinite relaxation (SDR)-based algorithm and a reduced-  complexity version to tackle the non-convexity, where the globally  optimal solutions are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, a robust beamforming  method is also developed for considering realistic imperfect  channel state information (CSI) knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Finally, simulation  results are provided for corroborating our theoretical results and  show the proposed methods’ superiority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Index Terms—Dual-functional radar and communication sys-  tem, joint beamforming design, physical layer security, multi-user  MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' INTRODUCTION  The proliferation of wireless mobile services exhibits an  exponential trend, leading to a scarcity of spectral resources  and to escalating spectrum prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' For example, it has been  reported that the number of connected devices is expected  to be 80 billion by 2030 with an annual growth rate of  around 25%, and that of the active Internet of Things (IoT)  This work is supported in part by the National Natural Science Foundation  udner Grant 62271163, in part by the Fundamental Research Funds for the  Central Universities (3072022QBZ0401, 3072021CFT0404).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Liu would  like to acknowledge the financial support of the National Natural Science  Foundation of China under Grant 62101234, as well as of the Young Elite  Scientist Sponsorship Program by the China Association for Science and  Technology (CAST) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' YESS20210055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Hanzo would like to  acknowledge the financial support of the Engineering and Physical Sciences  Research Council projects EP/W016605/1 and EP/P003990/1 (COALESCE)  as well as of the European Research Council’s Advanced Fellow Grant  QuantCom (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 789028).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (Corresponding author: Wei Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=')  Fuwang Dong, and Fan Liu are with the Department of Electronic and Elec-  trical Engineering, Southern University of Science and Technology, Shenzhen  518055, China (email: dongfw@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' liuf6@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='cn)  Wei Wang, and Xin Li are with the College of Intelligent System Science  and Engineering, Harbin Engineering University, Harbin, 150001, China  (email: wangwei407@hrbeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' xinxin forever@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='com ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Sheng Chen, and Lajos Hanzo are with the School of Electronic and  Computer Science, University of Southampton, Southampton SO17 1BJ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (email: sqc@ecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='soton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' lh@ecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='soton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  devices will reach 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='1 billion by 2030 [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Recently, the  concept and scope of Integrated Sensing and Communication  (ISAC) technology have been formally defined in [3], [4],  enabling sensing and communication simultaneously in the  same frequency band or/and hardware platform, which can  significantly improve the resource utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Due to the  numerous advantages offered by ISAC, it is envisioned to  be a promising technique in terms of supporting autonomous  vehicles [5], [6] and the IoT in 6G wireless networks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  There are two main ISAC categories in terms of transmitted  signal: radar and communication spectrum coexistence and  dual functional radar-communication (DFRC) [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In this  paper, we consider a DFRC system, which transmits dual-  functional signals/waveforms from a single hardware platform,  to gain benefits from joint sensing and signaling operations  via real-time cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The main motivation of transmit  beamforming is to synthesize multiple beams towards both  the communication users and the radar targets by exploiting  the associated spatial degrees of freedom (DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In [10], the  authors considered the radar targets as virtual downlink users  encountering a line of sight (LoS) channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Therefore, the  beamforming matrix was designed for closely matching the  desired radar beampattern, while simultaneously guaranteeing  the signal to interference and noise ratio (SINR) attained by the  downlink users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, the authors of [11], [12] studied  the associated symbol/waveform level probing signal design  issues, where the multi-user interference energy was mini-  mized under the similarity and constant modulus constraints of  the radar waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' However, the above-mentioned schemes  only utilize the communication waveform as the DFRC wave-  form to implement target detection, hence leading to a DoF  reduction, thereby to a radar performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' To this  end, the authors of [13] firstly proposed a jointly precoded  individual communication and radar waveforms based scheme,  where the communication signal can be regarded as a special  case relying on nullifying the dedicated radar waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Therefore, by exploiting the inherent advantages of the radar  waveform, the DoF erosion can be efficiently compensated,  hence resulting in target detection performance improvements,  especially for a small number of downlink users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Another critical problem in the DFRC system, which has  been largely overlooked in the relevant literature, is how to  guarantee the privacy and security of the desired informa-  tion [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The DFRC base station (BS) transmits the dual-  functional probing waveform for detection purposes, but also  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='00340v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='SP]  1 Jan 2023  2  TABLE I  OUR CONTRIBUTIONS IN CONTRAST TO THE STATE-OF-THE-ART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  [10]  [13]  [17]  [18]  [19]  Our work  Secure Transmission  \x13  \x13  \x13  \x13  Jointly precoded communication and radar waveforms  \x13  \x13  Precoder design rather than covariance matrix  \x13  \x13  \x13  \x13  Radar beampattern optimization  \x13  \x13  \x13  Multiple users  \x13  \x13  \x13  \x13  Imperfect CSI estimations  \x13  \x13  Multiple eavesdroppers  \x13  Using radar signal as artificial noise  \x13  Tight solution for PLS design  \x13  sends confidential information to the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Evidently, private  information might be leaked to the targets, which may act as  potential eavesdroppers (Eves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Recently, several schemes have  been proposed for guaranteeing secure data transmission by  exploiting constructive interference [15], frequency hopping  [16], and additional artificial noise (AN) [17]–[19], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' As a  low complexity yet powerful technique, the AN method has  been widely harnessed in the communication community for  enhancing the physical layer security (PLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The basic princi-  ple of AN-aided secure transmission is that of contaminating  the transmit signal by well-designed AN to degrade Eve’s  reception without affecting the legitimate users (LUs) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  In [17], several optimization problems, including secrecy  rate maximization, target return SINR maximization, and  transmit power minimization were formulated for a DFRC  system in the presence of a single target and a single communi-  cation receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' To tackle the non-convexity of the secrecy rate  expression, an approximate algorithm based on the first-order  Taylor expansion was proposed, which however resulted in  a performance gap between the original non-convex problem  and the approximated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The authors of [18] considered a  unified joint passive radar and communication system, where  the SNR at the passive radar receiver was maximized, while  keeping the secrecy rate above a certain target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover,  several practical constraints, such as realistic target direction  estimation and imperfect channel state information (CSI) were  taken into account in the associated robust beamforming  proposal of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' However, at the time of writing, most of  the contributions on secure DFRC systems have the following  two drawbacks: (1) They only design the covariance matrix of  the AN, yet no further analysis of the DFRC system’s radar  detection is offered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (2) Several relaxation algorithms are used  such as Taylor expansions or semidefinite relaxation (SDR)  techniques, but the performance loss compared to the original  non-convex problem is overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Motivated by filling the above-mentioned knowledge gap  in the literature, we develop jointly precoded communication  and radar waveforms for secure transmission in a multiple-  input multiple-output (MIMO) DFRC system inspired by [13],  serving multiple LUs and detecting the targets simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  On one hand, the DFRC platform relying on the ISAC tech-  nique eliminates duplication in the system’s hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On the  other hand, the bespoke transmit signals can simultaneously  meet the requirements of radar, communications, and PLS,  circumventing redundancy in the resource consumption for  each functionality, hence also the power dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Compared  to the current DFRC schemes such as those in [10]–[12], [19],  our method achieves superior radar detection performance  thanks to the increased DoFs attained by the additional radar  waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In contrast to [13], the PLS level is also considered  in our work, where the targets may act as potential Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The  radar waveforms conveying no confidential information may  also be exploited as the AN imposed on the communication  signals for contaminating the eavesdropping channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The  main contributions of this paper are summarized as follows,  and they are also boldly and explicitly contrasted to the  literature at a glance in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  We develop jointly precoded communication and radar  waveforms for secure transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Specifically, the AN  of traditional PLS designs can be replaced by bespoke  radar signals specifically designed for inflicting interfer-  ence upon the Eves, whilst additionally increasing the  DoF available for target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  We formulate the joint beamforming design as a non-  convex optimization problem under the consideration of  both radar, communication and security performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' An  SDR-based and the associated low complexity algorithms  are also conceived for tackling the non-convexity of the  problem, where we prove that the relaxation used in our  scheme is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  We propose a robust beamforming design for the more  practical scenarios of imperfect estimations, including the  uncertain target directions and the imperfect CSI acquired  for the LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We also show that the globally optimal  reconstruction method proposed for ideal scenarios still  applicable to our robust beamforming scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  We analyze the performance trade-offs among radar, com-  munication and PLS both theoretically and by simulation  for providing new insights into flexible beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In Section II,  we establish the mathematical model of joint communication  and radar signal transmission and introduce the performance  metrics of radar detection, multiuser communication, and  system security, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The proposed SDR-based beam-  forming and the low complexity ZF-based algorithms are  characterized in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, Section IV provides  our robust beamforming method relying on imperfect CSI  knowledge, while the performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' the complexity of the  proposed algorithms is analyzed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' our  simulation results and conclusions are provided in Section VI  3  TABLE II  FREQUENTLY USED SYMBOLS  Notation  Description  R  Covariance matrix of the transmitted signals  H  Communication CSI matrix  Wr (Wc)  Radar (Communication) beamforming matrix  Γe (Γc)  SINR threshold at Eves (LUs)  K  Number of the LUs  Q  Number of the targets (Eves)  M  Number of antennas  β  Path-loss coefficient for radar channel  Lr(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' α)  Least square function for MIMO radar beampattern  γk (˜γq)  SINR of the k-th LU (the q-th Eve)  and VII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The notations used in this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Upper-  case A (lower-case a) bold characters denote matrices (column  vectors), and lower case normal letters a are scalars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (·)T ,  (·)∗ and (·)H represent the transpose, conjugate and complex  conjugate transpose operations respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' |a| and ∥a∥2 stand  for the magnitude of a scalar a and the ℓ2-norm of the  vector a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' E{·} is the statistical expectation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' diag{a} stands  for a diagonal matrix using the elements of a as its diagonal  elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' for a matrix A, [A][i,j] denotes the (i, j)th element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  A[:,1:k] and A[1:k,:] represent the sub-matrices containing the  first k columns and rows of A respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' IM is the n-  dimensional identity matrix and 0M×N is the M × N matrix  having all-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Frequently used symbols in this paper  are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' SYSTEM MODEL AND PERFORMANCE METRICS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Transmission and Reception Signal Model  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 1, a colocated MIMO BS transmits DFRC  signals to detect Q targets and K LUs simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' For  the consideration of our PLS design, all the targets considered  are non-cooperative, such as unmanned aerial vehicle (UAV)  which are regarded as the potential Eves at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We  assume that the BS is equipped with M antennas arranged in  a uniform linear array (ULA), and all the Eves and LUs have  a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The proposed beamforming design can be  readily extended to multi-antenna scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Following [13], the discrete-time transmitted signal at time  slot n, which is the weighted sum of the communication  signals and radar waveforms, can be expressed as  x[n] = Wrs[n] + Wcc[n], n = 0, 1, · · · , N − 1,  (1)  where s[n] = [s1[n], · · · , sM[n]]T represents the individual  radar signals and c[n] = [c1[n], · · · , cK[n]]T stands for the K  parallel communication symbol streams intended for the LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  N is the total number of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, Wr ∈ CM×M  and Wc ∈ CM×K denote the beamforming matrices (or pre-  coders) designed for the radar waveforms and communication  waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The conventional transmit signal strategy which  only exploits the communication signals for detection in [10]–  [12], [19], can be regarded as the special case associated with  Wr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In line with the literature, the following assumptions  are stipulated for the transmitted signals (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The DFRC system detects the targets (Eves) and serves downlink  users by transmitting mixture waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Both the radar and communication signals have zero  mean, and they are temporally white wide-sense station-  ary stochastic processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The radar and the communication waveforms are statis-  tically independent, hence we have E{scH} = 0M×K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The M radar waveforms are orthogonal to each other,  then we have E{ssH} = IM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The communication symbols transmitted to different LUs  are uncorrelated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', E{ccH} = IK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Here, the signal power is normalized to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the  covariance matrix of the transmitted signal can be written as  R = E{xxH} = WrWH  r + WcWH  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (2)  Let y = [y1, y2, · · · , yK]T denote the received signal vector  of all the LUs, which can be expressed by  y = Hx + nc,  (3)  where H = [h∗  1, · · · , h∗  K]T ∈ CK×M is the channel matrix  and hk represents the channel vector spanning from the BS to  the kth LU, and nc ∼ CN(0, σ2  cIK) denotes the additive white  Gaussian noise (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, the targets of interest can  be viewed as virtual downlink users located in the LoS channel  of DFRC systems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Therefore, the signal received by the  qth target (Eve) can be modeled as [19]  rq = βqaH(θq)x + ne,  (4)  where βq is the complex path-loss coefficient, ne is the AWGN  with covariance σ2  e, and a(θ) represents the ULA arrays’  steering vector, which can be expressed as  a(θ) =  1  √  M  �  1, eȷ2π d  λ sin(θ), · · · , eȷ2π(M−1) d  λ sin(θ)�T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (5)  Here, d is the antenna spacing, λ is the carrier wavelength,  and θ is the azimuth of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The BS has to acquire the CSI for both LUs and Eves  before the beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In general, the CSI marix H of  LUs can be obtained through channel estimation and feedback  techniques [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the CSI from the BS to the Eve  is challenging to acquire, since the Eves tend to be passive  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Fortunately, the sensing functionality of the DFRC  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Flow of the mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  signal can be exploited for estimating the azimuth and path-  loss coefficient through radar parameter estimation techniques  [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Since we only focus on the beamforming design,  the processes of radar parameter estimation and information  demodulation are ignored in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The elaborate details  can be found in [1], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Before proceeding to our mathe-  matical analysis, we have depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 2 the flow of the  analysis described in the sequel, which allows readers to grasp  the overall structure of this paper at a glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Performance Metrics  In our proposed physical layer beamformer designed for  secure transmission, some important properties related to the  symbol-level waveform design [11], [12] are not considered,  such as the radar’s ambiguity function, the peak-to-average  power ratio (PAPR), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Next, we introduce our performance  metrics used for the target detection, for the communication  quality of service (QoS), and for the PLS level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (1) Performance metric for MIMO radar: In general, there  are two primary MIMO radar functions, namely detection and  tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' MIMO radar tends to create both spatially orthogonal  waveforms and omni-directional beampatterns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', R = I)  for detecting the potential targets in the detection stage, since  there is no prior information concerning the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, in  the tracking stage, MIMO radar steers the beam to the target  directions of interest acquired during the previous observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Instead of maximizing the SINR at radar receiver [25],  we focus on the radar transmit beampattern performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The  synthesized radar beampattern at azimuth θ can be formulated  as  P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' R) = E{aH(θ)xxHa(θ)} = aH(θ)Ra(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (6)  Additionally, the cross-correlation pattern between direction  θ1 and θ2 can be written as  Pc(θ1, θ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' R) = aH(θ2)Ra(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (7)  The objectives of beamformer design for MIMO radar include  the following [26]  Optimize the beampattern over the sectors of interest to  concentrate the signal power while maintaining a low  sidelobe level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Reduce the cross-correlation pattern over the set of target  angles to achieve an excellent adaptive performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  To this end, we adopt the loss function defined in terms of  the least squares as our performance metric for MIMO radar,  which is formulated as  Lr(R, α) = Lb(R, α) + ηLc(R),  (8)  where η is the weighting factor representing the relative  importance of the two terms based on the associated practical  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The first term represents the mean squared error  between the designed and desired beampatterns, which can be  formulated as  Lb(R, α) = 1  L  L  �  l=1  |αΦ(θl) − P(θl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (9)  Here, α is a scaling factor, Φ(θ) denotes the desired transmit  beampattern, and {θl}L  l=1 represents the fine grid of points that  cover the targets of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Let ∆ denote the beam-width,  then the desired beampattern at azimuth θ⋆ is given by  Φ(θ) =  �  �  �  1, θ⋆ − ∆  2 ≤ θ ≤ θ⋆ + ∆  2  0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (10)  Moreover, the second term is the mean-squared cross-  correlation pattern, given by  Lc(R) =  2  P 2 − P  P −1  �  p=1  P  �  q=p+1  |Pc(¯θp, ¯θq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' R)|2,  (11)  where {θp}P  p=1 are the given directions of the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We refer  the reader to [13], [26] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (2) Performance metric for multi-user communication:  The achievable transmission rate related to the SINR of the sig-  nal received by the downlink users is a standard performance  measure in multiuser communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' For notation  convenience, we introduce W = [Wc, Wr], where wi is the  ith column of W for i = 1, · · · , K + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, the signal  covariance matrix can be rewritten as  R = WWH =  K+M  �  i=1  wiwH  i =  K+M  �  i=1  Ri,  (12)  where Ri = wiwH  i  is the rank 1 covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Specif-  ically, R1, · · · , RK are the covariance matrices of communi-  cation symbols, where the last M ones are those of the radar  ertormance metrics ol MiMJ, sccultransmission (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='SteOntim1zin  undertheStep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  beamformer:rom tne covaliance matrix h telms of tneeamformer interms of the  Section Ill-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='★5  waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the SINR at the kth LU can be formulated  as  γk =  E{|hH  k wkck|2}  K  �  i=1,i̸=k  E{|hH  k wici|2} +  M  �  j=1  E{|hH  k wj+Ksj|2} + σ2c  =  hH  k Rkhk  K+M  �  i=1,i̸=k  hH  k Rihk + σ2c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (13)  There are two popular design criteria for multiuser beamform-  ing [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' One of them is the throughput criterion to maximize  the system’s sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The other is the fairness criterion used  for maximizing the minimal SINR at each user, which can be  expressed as  max min{γ1, · · · , γK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (14)  In this work, the SINR-fairness is adopted as the performance  metric for multiuser communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On the one hand, the  fairness metric guarantees that each LU can obtain satisfactory  QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On the other hand, the fairness metric based optimization  is more tractable than the NP-hard optimal throughput beam-  forming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Given a minimal level of communication  QoS Γc, the SNR-fairness metric can be transformed to forcing  the minimal SINR of the users to be higher than the target  threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', γk ≥ Γc, k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (3) Performance metric for PLS level: When the targets  become Eves, the achievable data rates at Eves are non-  negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' A straightforward method is to increase the propor-  tion of interference signal power to the detriment of the useful  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' According to the previous analysis, the radar waveform  conveying no desired information can be regarded as the  interference contaminating the reception of Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Accordingly,  by recalling the received signal model (4), the SINR for the  qth Eve can be formulated as  ˜γq =  |βq|2a(θq)H �K  k=1 Rka(θq)  |βq|2a(θq)H �K+M  i=K+1 Ria(θq) + σ2e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (15)  Following [19], we consider the worst-case SINR in (15),  assuming that all the information intended for the K LUs is  the desired signal for Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' As stated in [28], there will exist  modulation and coding schemes that allow the LUs rather than  the Eves to reliably decode the transmit information, as long  as γk > ˜γq, for ∀k, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Therefore, we restrict the maximal  SINR at Eves to be less than a given threshold Γe, instead  of optimizing the secrecy rate [log(1 + γk) − log(1 + ˜γq)]+  defined in [17], to achieve a satisfactory PLS level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On the one  hand, the system’s secrecy rate is difficult to determine due to  its non-convexity with respect to Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On the other hand, since  SINR-fairness based schemes are still capable of maintaining  a minimal communication rate due to the monotonicity of the  log function, we can equivalently achieve a desired secrecy  rate [log(1 + Γc) − log(1 + Γe)]+ by appropriately choosing  the thresholds Γc and Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' THE BEAMFORMING DESIGN FOR IDEAL SCENARIOS  In this section, we aim for designing the transmit beamform-  ing matrices Wr and Wc under the consideration of the perfor-  mance metrics for the radar beampattern, the communication  QoS and the PLS levels given in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We first  consider the ideal conditions, where the BS perfectly knows  the CSI both for the LUs and Eves, and leave the beamformer  design under the more practical imperfect CSI scenario for the  next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The proposed SDR-based beamforming algorithm  Our beamforming design objective is to minimize the dif-  ference between the desired transmit beampattern and that  generated by the BS to achieve good target detection and  tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Meanwhile, the beamforming design  also guarantees that the downlink SINR at the LUs remains  higher than the given threshold, while that of the Eve is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Recalling the definition (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' instead of directly optimizing  the precoding matrix W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' the SDR based optimization problem  with respect to the variables Ri can be formulated as  minimize  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='{Ri},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='α  Lr(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' α)  (P0)  subject to  R =  K+M  �  i=1  Ri ∈ S+  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' α > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16a)  Ri ∈ S+  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' K + M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16b)  rank(Ri) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' K + M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16c)  [R][m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='m] = Pt/M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16d)  γk ≥ Γc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16e)  ˜γq ≤ Γe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' q = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (16f)  where S+  M represents the set consisting of all M-dimensional  complex positive semidefinite matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', S+  M = {A|A ∈  CM×M, A = AH, A ⪰ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The rank-1 constraint in (16c) is  equivalent to Ri = wiwH  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (16d) represents the per-antenna  power constraints, and Pt is the total transmit power of the  BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, the objective function and the constraints  (16e), (16f) are the performance metrics introduced in Section  II-B, where Γc and Γe are the predefined SINR thresholds at  the LUs and Eve, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Upon substituting the SINR expressions (13) as well as (15)  into the constraints and applying some simple mathematical  manipulations, (16e) and (16f) can be recast as  (1 + Γ−1  c )hH  k Rkhk ≥ hH  k Rhk + σ2  c, ∀k  (17a)  (1 + Γ−1  e )aH  q  K  �  k=1  Rkaq ≤ aH  q Raq +  σ2  e  |βq|2 , ∀q  (17b)  where aq is the abbreviated form of a(θq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' It can be observed  that the individual matrices {Ri}i≥K+1 have no effect on the  SINR constraints, which motivates us to remove these matrix  variables from the original problem P0 of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' As a result,  the number of matrix variables is reduced from K + M + 1  to K + 1, leading to much reduced memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  6  By reformulating the constraint (16a), problem P0 can be  transformed to  minimize  R,R1,··· ,RK,α  Lr(R, α)  (P1)  subject to  R ∈ S+  M, R −  K  �  k=1  Rk ∈ S+  M,  (18a)  α > 0, Rk ∈ S+  M, k = 1, · · · , K,  (18b)  rank(Rk) = 1, k = 1, · · · , K,  (18c)  [R][m,m] = Pt/M, m = 1, · · · , M,  (18d)  (17a), (17b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  However, problem P1 is non-convex due to the rank-1 con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the SDR relaxation based version of problem  P1 can be obtained by omitting the rank-1 constraints (18c),  which is denoted by problem P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the problem P2 has  become a standard quadratic semidefinite program (QSDP),  since the objective function is a positive-semidefinite quadratic  form and all the constraints are either linear or semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Hence, the global optimum can be obtained in polynomial time  with the aid of standard convex optimization toolboxes [29],  [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Note that the optimal solutions of the relaxed problem  P2 are not necessarily rank-1 matrices, hence either the classic  eigenvalue decomposition or Gaussian randomization methods  [31] can be leveraged to obtain the solutions of the origi-  nal problem P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Unfortunately, these kinds of approximate  algorithms usually only provide suboptimal solutions of the  original problem, hence resulting in a loss of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  To circumvent this deficiency, we set out to find a global  optimum for problem P1, which means that the SDR relax-  ation is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Inspired by the result in [13], we propose the  following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proposition 1: Let ˆR, ˆR1, · · · , ˆRK be the optimal solution  of the QSDP problem P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' There also exists a global optimum  ˜R, ˜R1, · · · , ˜RK for problem P1, where we have  ˜R = ˆR, ˜wk = (hH  k ˆRkhk)−1/2 ˆRkhk, ˜Rk = ˜wk ˜wH  k ,  (19)  for k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proof: The proof is relegated to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  ■  According to Proposition 1, we can get the global rank-  1 optimal solution for problem P1 from its QSDP relaxation  based version P2, where the relaxation is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The remaining  step is to find the optimal solution for the original problem  P0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' obtaining the precoding matrix Wr for the radar  waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' To meet the constrains of (16a) and (16b), the  M precoding vectors {wi}i≥K+1 can be obtained by the  following decomposition  WrWH  r = Rrad = ˜R −  K  �  k=1  ˜Rk,  (20)  where Wr = [wK+1, · · · , wK+M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Actually, since the associ-  ated waveform level design is not considered in this work, the  decomposition (20) is not unique, but it is trivial thanks to the  positive semi-definite nature of the radar signal’s covariance  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Several decomposition methods such as the square root  matrix (Wr = R  1  2  radU, U is an arbitrary unitary matrix) based  one [32] and the Cholesky decomposition based one may be  applied [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The ZF-based low complexity algorithm  The main computational complexity burden in the proposed  SDR-based algorithm is imposed by that of solving the QSDP  problem P2, which motivates us to seek a low-complexity  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Inspired by the zero forcing (ZF) based method of  [13], we develop a reduced-complexity sub-optimal algorithm  by incorporating ZF constraints into problem P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The ZF  method is widely used in low-complexity linear precoders,  because its performance tends to that of the optimal non-  linear precoder, especially for a large number of antennas [34],  [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Its main appeal is that of eliminating the inter-user and  radar interferences, hence achieving a high SINR at each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Mathematically, the ZF constraints can be expressed as  HWc = diag(√ρ1, · · · , √ρK), HWr = 0K×M,  (21)  where ρk represents the signal power at the kth user, for 1 ≤  k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Upon recalling the definition W = [Wc, Wr] and R =  WWH, (21) can be equivalently transformed to the following  form (Theorem 2, [13])  HRHH = diag(ρ),  (22)  where ρ = [ρ1, · · · , ρK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, substituting (21) or (22)  into the SINR expression (13), the associated SINR constraints  (17a) can be simplified by  ρk ≥ Γcσ2  c, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (23)  It can be observed that the individual matrix variable Rk  has been removed from the SINR constraints for the LUs by  imposing the ZF constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Following the same methodology  for further reducing the number of matrix variables, and by  introducing the auxiliary matrix variable Rcom = �K  k=1 Rk,  the PLS constraint (17b) can be rewritten as follows  (1 + Γ−1  e )aH  q Rcomaq ≤ aH  q Raq +  σ2  e  |βq|2 , ∀q  (24)  Furthermore, we can immediately infer the ZF constraint for  Rcom as  HRcomHH = HWcWH  c HH = diag(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (25)  As a consequence, either the communication SINR constraint  or the PLS constraint no longer contains the individual matrix  variable Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Accordingly, problem P2 can be converted to  minimize  R,Rcom,ρ,α  Lr(R, α)  (P3)  subject to  R ∈ S+  M, R − Rcom ∈ S+  M, Rcom ∈ S+  M,  (26a)  [R][m,m] = Pt/M, m = 1, 2, · · · , M,  (26b)  HRHH = diag(ρ),  (26c)  HRcomHH = diag(ρ),  (26d)  α > 0, (23), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Problem P3 is also a standard QSDP problem, because the  objective function has a positive-semidefinite quadratic form  7  Algorithm 1 The proposed SDR(ZF)-based beamforming  algorithm designed for secure DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Input:  Total transmit power of base station Pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Radar desired beampattern Φ(θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Instantaneous downlink channel H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  SINR threshold at LUs Γc and at Eves Γe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The directions of Eves θq, q = 1, · · · , Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Output  The overall precoding matrix W = [w1, · · · , wK+M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Steps  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Compute the optimal solution of P2 (or P3) via convex  optimization solver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Compute w1, · · · , wK by (19) (or by (28));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Compute wK+1, · · · , wK+M by (20) (or by (29));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  and all the constraints are either linear or semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Simi-  larly, the optimal solutions ˆR and ˆRcom can be obtained by a  standard convex optimization toolbox in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The next step is to recover the precoding matrix W from  the optimal solutions ˆR and ˆRcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Inspired by Theorem 2  of [13], we conceive the following procedure of constructing  the radar and communication precoding matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  First, either the classic Cholesky decomposition or square  root method is used by exploiting the positive-semidefinite  property for ˆRcom = LcLH  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, we employ the row QR  decomposition of HLc, yielding  HLc = [Lh, 0K×(M−K)]Q,  (27)  where Lh is a K × K lower triangular matrix and Q is a  M × M unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the communication precoder  can be formulated as  Wc = Lc[QH][:,1:K],  (28)  while the radar precoding matrix Wr can be expressed as  WrWH  r = ˆRrad = ˆR − WcWH  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (29)  Subsequently, we analyze the feasibility of the proposed pre-  coder design method by introducing the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proposition 2: Given the optimal solution ˆR and ˆRcom of  problem P3, the matrices Wc in (28) and Wr in (29) are  also the optimal precoders of problem P3 and satisfy the ZF  constraint (21) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proof: The proof is divided into three parts, and it is  relegated to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  ■  Proposition 2 illustrates the feasibility and efficiency of  the proposed precoding matrices recovered from the optimal  solution of problem P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In summary, we can obtain the  optimal beamforming for DFRC secure transmission with the  perfectly known CSI by the proposed SDR-based and the low  complexity ZF-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The detailed procedure of  the proposed algorithms are summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' ROBUST BEAMFORMING DESIGN WITH IMPERFECT  CSI KNOWLEDGE  In practice, it is challenging to obtain the exact CSI due  to the estimation errors, feedback quantization, hardware defi-  ciencies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', resulting in imperfect CSI knowledge at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Specifically, for the radar targets, we assume that the direction  of the q-th target is roughly known by the BS within an angular  interval of [θq − ∆θq, θq + ∆θq], where ∆θq represents the  associated angle uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, for the communication  LUs, the additive error model of the CSI matrix for the k-th  LU is considered as hk = ˆhk + ϵk, where ˆhk is the estimated  CSI matrix and ϵk denotes the channel uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' To this  end, we aim for designing the robust beamforming scheme  for secure transmission in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Wide main-lobe beampattern design  The uncertainties of the target directions have an impact  on both the objective function and the PLS constraints in  problem P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' On one hand, the BS should form a wide main-  lobe to avoid missing the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the beam-width ∆ in  (10) should be appropriately chosen according to the angular  uncertainty ∆θq, in order to cover all the possible locations  of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  On the other hand, since Eve may be located in an arbitrary  direction within the angular interval, we should guarantee a  satisfactory secrecy rate for every possible direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Con-  sequently, the SINR constraints (17b) should be modified  according to  (1 + Γ−1  e )aH  qi  K  �  k=1  Rkaqi ≤ aH  qiRaqi + σ2  e  |βq|2 , ∀θqi ∈ ¯Ωq, (30)  where ¯Ωq is a discrete set that covers the potential directions  of the q-th Eve, and aqi represents the compact form of a(θqi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  It can be observed that the angular uncertainty introduces  more constraints similar to (17b) over the associated angular  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Evidently, the proposed Algorithm 1 is also capable  of handling the modified constraints (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In other words,  the number of targets and the uncertainty of target directions  determine the number of PLS constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Naturally, imposing  a large number of constraints for securing certain PLS levels  results in degraded radar beampattern and communication  QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We will illustrate this phenomenon in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Robust beamforming for mitigating CSI error of LUs  Similar to [19], [37], we assume that the CSI uncertainty is  bounded by a spherical region as  Sk := {ˆhk + ϵk | ||ϵk|| ≤ uk}, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (31)  In this case, the SINR expression for the k-th LU in (13)  should be replaced by the worst-case SINR over the set Sk,  namely  ¯γk = min  hk∈Sk γk, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (32)  Thus, based on the definitions (31) and (32), the SINR  constraint in (17a) can be reformulated as  (ˆhk+ϵk)H �  (1 + Γ−1  c )Rk − R  �  (ˆhk+ϵk)−σ2  c ≥ 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (33)  Then, we adopt the popular S-procedure of robust optimization  to tackle the SINR constraints mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By introduc-  ing an auxiliary vector t = [t1, · · · , tK], the original problem  8  P1 can be reformulated as the following robust beamforming  version [19], [37]  minimize  R,R1,··· ,RK,t,α  Lr(R, α)  (P4)  subject to  (18a) − (18d), (17b) or (30),  �  Sk + tkIM  Skˆhk  ˆh  H  k Sk  hH  k Skhk − σ2  c − tku2  k  �  ⪰ 0, ∀k  Sk := (1 + Γ−1  c )Rk − R, tk ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (34)  Again, by dropping the rank-1 constraints (18c), problem  P4 becomes a QSDP, which can be efficiently solved in  polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, we will show that the optimal solution  of the QSDP reconstruction method in (19) also holds for the  proposed robust beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proposition 3: Let ˆR, ˆR1, · · · , ˆRK be the optimal solution  of the relaxed version of problem P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then the ˜R, ˜R1, · · · , ˜RK  associated with the expression of (19) is also the optimal  solution of the original problem P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Proof: By employing the result in Proposition 1, the proof  becomes straightforward upon substituting (19) into the con-  straints (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  ■  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' PERFORMANCE AND COMPLEXITY ANALYSIS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Complexity Analysis  The complexity of the proposed algorithms is dominated by  the QSDP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' For a given solution accuracy ϵ, the worst-  case complexity order of solving problem P2 using the primal-  dual interior-point algorithm is O[(K + Q)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5log(1/ϵ)]  [13], [38], where K + Q and M refer to the number of  semidefinite constraints and the dimension of matrix variables,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Compared to the SDR algorithm, the low com-  plexity ZF beamforming problem P3 includes 5 = O(1) such  constraints, hence the worst-case complexity order becomes  O[Q6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5log(1/ϵ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, for the robust beamform-  ing algorithm with imperfect CSI knowledge, the complexity  also depends on the number of elements in the set ¯Ωq of  (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Specifically, upon denoting the cardinality of the set  ¯Ωq as P, the worst-case complexity is on the order of  O[K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5P 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5log(1/ϵ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Performance Analysis  In this subsection, we provide the performance analysis of  the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (I) We can immediately spot the performance trade-off  among the radar beampattern, the communication QoS, and  the PLS level in problem P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The constraints (17a) and (17b)  always hold, when we have Γc = 0 and Γe → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In this case,  problem P1 is reduced to the conventional radar-only beam-  forming design, determining the optimal beampattern for radar  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Explicitly, any improvements of the communication  QoS and PLS level are attained at the cost of sacrificing the  radar performance, since the radar loss function will increase  upon increasing Γc or decreasing Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (II) Compared to the SDR-based algorithm, the low com-  plexity ZF-based algorithm forces the radar and inter-user in-  terference to zero, potentially raising the SINR at the LUs to a  certain threshold (denoted by ˆΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, for the communication  constraints, we have  �  γZF  k = ˆΓ > γSDR  k  ≥ Γc, when Γc < ˆΓ,  γZF  k = γSDR  k  ≥ Γc ≥ ˆΓ, when Γc ≥ ˆΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (35)  For a relatively low threshold Γc, the interference encountered  by the users do not have to be as low as zero to satisfy the  SINR constraint, resulting in γZF  k > γSDR  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the in-  terference in γSDR  k  has to be eliminated to meet the high SINR  requirements, resulting in γZF  k = γSDR  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' According to (35), we  can immediately conclude the following properties of the ZF-  based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (1) It results in worse radar beampattern than  the SDR-based algorithm because more severe restrictions are  imposed by the ZF constraint when Γc < ˆΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (2) The radar  loss function and the users’ SINR remains constant, as long  as the threshold Γc is lower than a positive value ˆΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (3) The  performance of ZF-based beamforming tends to be similar to  that of SDR-based beamforming at high SINRs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', Γc ≥ ˆΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (III) For the SDR-based algorithm, the system’s secrecy rate  is always approximated by log2(1+Γc)−log2(1+Γe) given the  thresholds Γc and Γe, because the optimal solution generally  reaches the boundary of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the  secrecy rate of the ZF-based algorithm may become higher  than the above value for small Γc values due to the potentially  high SINR achieved under the ZF constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The proposed  algorithms guarantee to have a secrecy rate above a certain  lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (IV) Upon considering the extreme case that the channels of  the users and Eves have the same quality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', βka(θk) = hk,  the communication QoS constraint (17a) and the PLS level  constraint (17b) are contradictory to each other, hence lead-  ing to the infeasibility of problem P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This means that the  feasibility probability of problem P1 significantly depends on  the values of Γc as well as Γe, and on the distances between  the targets and Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The proposed joint beamforming design  method will become invalid, when the Eves are at the same  directions as the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The symbol-level range sidelobe design  [36] may be a promising remedy, which we will leave for  future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (V) It should be pointed out that the joint PLS beamforming  design of [19] minimized the SINR at Eve, which is different  from the proposed method optimizing the radar transmit  beampattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Even though it cannot be compared directly due  to the different functional requirements, the proposed method  has the following advantages over [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (1) The fractional  programming approach is adopted in [19], where a sequence  of SDPs has to be solved by iteration, imposing a heavy  computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the proposed methods  only have to solve a SDP or QSDP problem with the same  number of matrix variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (2) The eigenvalue decomposition  or Gaussian randomization techniques of [19] result in a sub-  optimal solution, when the ranks of the optimal matrices  obtained by the SDP solver are not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the  proposed SDR relaxation is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (3) When using the SINR  instead of the secrecy rate as the objective, the difference  between the achievable rate of users and that of Eves may  become negative, leading to a secrecy rate of SR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By  9  contrast, the proposed algorithms can always guarantee a  satisfactory secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we evaluate the proposed joint PLS beam-  forming algorithm by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The system pa-  rameters are set as follows, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The  BS is equipped with a ULA having half-wavelength spacing  between adjacent antennas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' d/λ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The number of  antennas is set to M = 10, and the total transmit power is  normalized as Pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The angular directions are obtained by  uniform sampling with resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='1◦, including {θl}L  l=1  in (9) with the range of [−90◦, 90◦], and Ωq in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Without  loss of generality, we adopt the Rayleigh fading model for  the multi-user communication channel so that each entry of  H obeys the standard complex Gaussian distribution with  hi,j ∼ CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Additionally, we assume the noise levels  at the Eves and LUs to be the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=', σ2  c = σ2  e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='01 for  convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The individual radar waveforms and communica-  tion symbols are generated as random quadrature-phase-shift-  keying (QPSK) modulated sequences, with the total number  of symbols being N = 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  For comparison, we choose the joint beamforming de-  sign method and its low-complexity counterpart proposed in  [13] termed as Benchmark 1 and Benchmark 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Compared to [10], where only the communication signal is  exploited by the DFRC system, the superiority of the combined  radar waveforms and communication signals in terms of  increasing the DoFs has been shown in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Therefore, we  refer to [13] for circumventing repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  First, we numerically characterize the MIMO radar transmit  beampattern, where the proposed SDR-based algorithm and  its low-complexity version are referred to as SDR and ZF,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We set the direction of a single target to θ0 = 0◦,  the threshold for the LUs’ SINRs to Γc = 10dB, and the  threshold for the Eve’s SINR to Γe = 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 3 illustrates  the trade-off among the radar beampattern, the communication  QoS and the PLS level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Although the proposed algorithms  impose a performance degradation on the transmit beampattern  compared to their counterparts, the target secrecy rate (SR) can  still be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, Benchmark 1 and 2 form better  beampatterns, but their SR becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Then, we evaluate  the system performance versus the predefined SINR thresholds  Γc and Γe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' System Performance Evaluation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' the Threshold Γc  In this subsection, we keep the SINR threshold of Eves  Γe = 0dB as a constant, and sweep Γc of LUs from 10dB to  18dB to test its impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' All of the simulation results represent  averaged values over 500 Monte Carlo trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In each trial,  the target direction θq is chosen randomly in the range of  [−60◦, 60◦], and the CSI of the link spanning from the BS  and the LUs obey the standard Complex Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The radar performance is evaluated as the difference between  the DFRC transmit beampattern and the optimal radar-only  100  50  0  50  100  Spatial Direction (degree)  40  30  20  10  0  10  20  Transmit Beampattern (dB)  Radar Only  Benchmark 1,SR=0  Benchmark 2,SR=0  SDR,SR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='46  ZF,SR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='20  30  40  50  60  30  28  26  24  22  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Radar transmit beampattern for the direction θ0 = 0◦, with K = 2,  Γc = 10dB, and Γe = 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  beampattern by defining the mean square error (MSE) metric  as  MSE = 1  L  L  �  l=1  |P(θl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' ˆR) − P(θl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' R⋆)|2,  (35)  where R⋆ is the optimal radar-only variance matrix by the 3dB  low sidelobe beampattern design scheme of [26] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 4 shows the beampattern MSE versus the SINR thresh-  old Γc of the LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We can observe the following three  phenomena from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (1) The beampattern MSEs of all  algorithms increase upon increasing Γc, which is consistent  with the previous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' As expected, the MSE of the ZF-  based algorithms remains constant in the scenarios of K = 2  and for SINRs below 16dB at K = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This is because the  ZF-based methods force the interference to zero, leading to  a potentially high SINR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the performance will remain  constant until the SINR thresholds become higher than the  potential SINR achieved by the ZF constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The perfor-  mance gaps between the SDR-based and ZF-based methods  become quite small for high enough values of Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (2) The  benchmark algorithms formulate better beampattern, since the  PLS aspects of confidential information protection is not taken  into account in these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' (3) The more users have to be  supported, the higher the beampattern MSE becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Notably,  the impact of the number of users K on the beampattern MSE  is more significant than that of the SINR threshold Γc, which  implies that serving more downlink users is more restrictive  than improving the SINR level of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 5, we quantify the achievable sum-rate versus the  SINR threshold Γc, where the system sum-rate is defined by  �K  k=1 log2(1 + γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The SDR and Benchmark 1 curves are  fairly similar and increase linearly with the SINR constraint  Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This is because the optimal solution should reach the  SINR boundary related to the given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Conversely, as  seen in the analysis of Section V-B, the ZF-based beamformer  achieves a higher communication sum rate to the detriment of  the radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Meanwhile, the performances of the  SDR-based and ZF-based beamformer tend to become similar  at high SINR thresholds for both K = 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore,  the curves of the Benchmark 2 are slightly higher than those  10  10  11  12  13  14  15  16  17  18  SINR Threshold for LUs (dB)  10-3  10-2  10-1  100  Beampattern MSE  Benchmark1  Benchmark2  SDR  ZF  K=4  K=2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Beampattern MSE versus SINR thresh-  old Γc for LUs, Γe = 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  10  11  12  13  14  15  16  17  18  SINR Threshold for Downlink Users (dB)  6  8  10  12  14  16  18  20  22  24  26  System Sum Rate (bit/(s·Hz))  Benchmark1  Benchmark2  SDR  ZF  K=2  K=4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The achievable sum rate versus SINR  threshold Γc for LUs, Γe = 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  10  11  12  13  14  15  16  17  18  SINR Threshold for Downlink Users (dB)  0  1  2  3  4  5  Secrecy Rate (bit/(s·Hz))  Benchmark1,K=2  Benchmark2,K=2  SDR,K=2  ZF,K=2  Benchmark1,K=4  Benchmark2,K=4  SDR,K=4  ZF,K=4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The secrecy rate versus SINR threshold  Γc for LUs, Γe = 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  20  15  10  5  0  SINR Threshold for Eavesdropper (dB)  10-4  10-3  10-2  10-1  100  Beampattern MSE  Benchmark1  Benchmark2  SDR  ZF  K=4  K=2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Beampattern MSE versus SINR thresh-  old Γe for Eves, Γc = 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  20  15  10  5  0  SINR Threshold for Eavesdropper (dB)  6  8  10  12  14  16  18  20  22  System Sum Rate (bit/(s·Hz))  Benchmark1  Benchmark2  SDR  ZF  K=2  K=4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The achievable sum rate versus SINR  threshold Γe for Eves, Γc = 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  20  18  16  14  12  10  8  6  4  2  SINR Threshold for Eavesdropper (dB)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='5  Secrecy Rate (bit/(s·Hz))  SDR,K=2  ZF,K=2  SDR,K=4  ZF,K=4  Benchmark1,K=2  Benchmark2,K=2  Benchmark1,K=4  Benchmark2,K=4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The secrecy rate versus SINR threshold  Γe for Eves, Γc = 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  of the proposed ZF algorithms, since there is an additional  minimum PLS constraint imposed on the ZF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 6 illustrates the system’s secrecy rate versus the SINR  threshold Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Observe that the curves of SDR associated with  K = 2 and K = 4 are coincident and increase linearly  upon increasing Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Recall from Section V-B that the system’s  secrecy rate will only reach the value of log2(1 + Γc) −  log2(1+Γe), if the optimization problem is feasible, regardless  of how the other parameters change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Additionally, the ZF-  based beamformer associated with K = 2 achieves a higher  secrecy rate than that of the SDR-based algorithm at small  values of Γc, since it can reach a higher SINR level than  the given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' However, the secrecy rate of these two  algorithms becomes similar for K = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Actually, supporting  more communication users imposes more restrictions on the  optimization problem P3, hence forcing the minimal SINR  level to approximate the threshold Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, the proposed  PLS-protected beamforming design guarantees a satisfactory  PLS level by appropriately choosing the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By con-  trast, the benchmark 1 and 2 are not capable of secrecy  protection, especially not for numerous legitimate users K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' System Performance Evaluation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' the Threshold Γe  In this subsection, we evaluate the system performance  versus the SINR threshold Γe of the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Accordingly, we set  Γc = 10dB as a constant, while all other system parameters  remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The SINR threshold Γe is varied from  −20dB to 0dB with intervals of 2dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' It should be highlighted  that the benchmark curves of [13] remain constant in all the  figures of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This is because these algorithms do  not take the PLS into account, hence the change of threshold  Γe does not affect these performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 7 shows that the radar beampattern MSE decreases  upon increasing Γe both for the proposed SDR and ZF  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Specifically, we can see that the curves of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  7 remain near-constant, when Γe is less than −12dB, while  decreasing noticeably, when Γe is higher than −10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Similar  trends may also be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 9, which implies  that the performance is not sensitive to the choice of Γe, when  Γe is less than −12dB for this set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Having  excessively low Γe increases the infeasibility probability of  the optimization problem considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 8, we can see that the system’s sum-rate also remains  unchanged for the SDR algorithm as a result of the constant  threshold Γc being close to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast,  the curves of ZF show an increasing trend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 8 upon  increasing Γe, since a higher Γe implies that less severe  restrictions are imposed on the ZF-based beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 9, the SDR and the ZF for K = 4 reach the  boundary of the secrecy rate log2(1 + Γc) − log2(1 + Γe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Meanwhile, the ZF for K = 2 attains a higher secrecy rate  11  80  60  40  20  0  20  40  60  80  Angular Direction (degree)  15  10  5  0  5  10  Transmit Beampattern (dB)  Radar Only  ZF,SR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='08  SDR,SR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='8  2  SINR Level  ZF  SDR  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Transmit beampattern for multiple  targets with uncertain directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  10  11  12  13  14  15  16  17  18  SINR Threshold for Downlink Users (dB)  10-3  10-2  10-1  Beampattern MSE  SDR,  =0  ZF,  =0  SDR,  =5  ZF,  =5  SDR,  =10  ZF,  =10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Beampattern MSE comparison with  different angular uncertainties of the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='8  1  CSI error bound for LUs (uk  2)  0  1  2  3  4  5  6  7  8  Estimated Secrecy Rate (bit/(s·Hz))  Perfect CSI, e= 0dB  Imperfect CSI,  e= 0dB  c = 10dB  c = 5dB  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Estimated secrecy rate calculated by  the known imperfect CSI versus error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  than its counterparts, since supporting less LUs imposes less  restrictions on the beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Furthermore, we can  infer from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 9 that although a low Γe reduces the  achievable data rate of the Eve, it also results in a low data rate  for the LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Therefore, no obvious secrecy rate improvement  is attained upon reducing Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' System Performance Evaluation for imperfect CSI  First, we evaluate the impact of angular uncertainties of  the Eves on the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We set Q = 3 targets  having the directions of θ1 = −40◦, θ2 = 0◦, and θ3 = 40◦,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Each target has the same direction uncertainty  of ∆θ = 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The BS detects and tracks these targets, while  serving K = 3 LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The SINR thresholds for the LUs and  the Eves are set to Γc = 10dB and Γe = 0dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 10 illustrates the radar transmit beampattern synthe-  sized by the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' The SINR level defined by  (15) is calculated over the set of [−90◦, 90◦] angular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  It can be observed that although the BS forms multi-beams  pointing to the directions of the Eves, the SINR levels in  each interval covering the Eves are controlled by the threshold  Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This is because the signal power of radar waveforms is  higher than that of the communication symbols, which have  to be protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Moreover, although the beampattern of the  ZF algorithm is less beneficial than that of the SDR (higher  side-lobe), the average spatial SINR level is lower than that  of the SDR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 11, we evaluate the impact of  the direction uncertainties on the optimization performance  upon varying Γc from 10dB to 18dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' As expected, further  constraints are introduced by the uncertainty of the target  directions, hence leading to an eroded radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' 12 shows the estimated secrecy rate calculated by the  known imperfect CSI versus the error bound for the scenario  of K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' It can be observed that the estimated secrecy  rates remain constant and are equal to the secrecy rates in  the case of perfect CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' By contrast, the curves obtained in  the case of imperfect CSI exhibit an increasing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' This is  because the worst-case secrecy rate is forced to be larger than  a given threshold in our robust beamforming algorithm, while  the statistical difference between the worst-case and estimated  secrecy rate becomes larger upon increasing the error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' CONCLUSION  A DFRC multi-user communication system was pro-  posed, while taking the physical layer security into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  The weighted sum of the communication signal and radar  waveform was adopted for dual-functional transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' We  demonstrated that the additional radar waveform conveying no  confidential information improves the DoF in target detection  and simultaneously contaminates the eavesdropping channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Subsequently, the SDR and the low complexity ZF algorithms  were proposed for finding the global optimal solution of the  formulated non-convex beamforming design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Further-  more, we also designed the robust beamforming for the more  practical scenarios of imperfect CSI knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Finally, we  evaluated the impact of the parameters on the attainable sys-  tem performance by numerical simulations, which showed an  excellent consistency with the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Designing  PLS systems operating in the face of other types of legitimate  and eavesdropping channels as well as hardware impairments  is left for our future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Another promising area of  research is the design of Pareto-optimal multi-component  systems relying on the full set of optimal operating points  in terms of throughput, bit error rate (BER), package loss,  latency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  APPENDIX A  THE PROOF OF PROPOSITION 1  By applying the Theorem 1 in [13], we only have to prove  that the PLS constraint (17b) holds for ˜R, ˜R1, · · · , ˜RK, if it  holds for ˆR, ˆR1, · · · , ˆRK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' First, we show that  aH(θ)ˆRka(θ) ≥ aH(θ)˜Rka(θ),  (36)  for arbitrary θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Upon substituting the expression of ˜Rk into  (19), the right-hand side term of the inequality can be ex-  panded as  aH ˜Rka = aH ˜wk ˜wH  k a  = (hH  k ˆRkhk)−1aH ˆRkhkhH  k ˆRka  = (hH  k ˆRkhk)−1|aH ˆRkhk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (37)  Additionally, according to the Cauchy-Schwarz inequality, we  have  (hH  k ˆRkhk)(aH ˆRka) ≥ |aH ˆRkhk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (38)  12  Therefore, it can be readily seen from (37) and (38) that (36)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, we can expound as follows  aH  q ˜Raq + σ2  e  |β|2  (a)  = aH  q ˆRaq + σ2  e  |β|2  ≥ (1 + Γ−1  e )aH  q  K  �  k=1  ˆRkaq  (b)  ≥(1 + Γ−1  e )aH  q  K  �  k=1  ˜Rkaq,  (39)  where (a) and (b) follow the first equation in (19) and the  inequality (36), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, the PLS constraint (17b)  holds for ˜R, ˜R1, · · · , ˜RK, hence completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  APPENDIX B  THE PROOF OF PROPOSITION 2  The proof is divided into the following three parts:  (1) We show that the radar covariance matrix ˆRrad in (29)  is a positive semidefinite matrix, hence it can be decomposed  by either the Cholesky decomposition or by the square root  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Actually, we have  ˆR − WcWH  c  = ˆR − ˆRcom + ˆRcom − WcWH  c  = ˆR − ˆRcom + Lc(I − [QH][:,1:K][Q][1:K,:])LH  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (40)  Here, ˆR − ˆRcom is positive semidefinite due to the constraint  (26a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Since [QH][:,1:K] is the sub-matrix containing the first  K columns of unitary matrix, (I − [QH][:,1:K][Q][1:K,:]) is  a positive semidefinite matrix, thereby the last term is also  positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (2)  We  show  that  the  proposed  precoding  matri-  ces satisfy the ZF constraint (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Upon letting F  =  diag(√ρ1, · · · , √ρK), we have  HRcomHH = HLcLH  c HH = LhLH  h = FFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (41)  Note that LhLH  h and FFH are the Cholesky decompositions  of the matrix diag(ρ), therefore we have Lh = F according  to the uniqueness of the Cholesky decomposition of a positive  definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Thus, we have  HWc = HLc[QH][:,1:K]  = [Lh, 0K×(M−K)]Q[QH][:,1:K]  = Lh = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (42)  Moreover, for the radar precoding matrix, we arrive at  HWrWH  r HH = H(ˆR − WcWH  c )HH  = FFH − FFH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (43)  Thus we can readily obtain HWr = 0 from (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (3) We show that the proposed precoding matrices meet the  PLS constraint (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' According to the positive semidefinite  property, we can show that  yH(I − [QH][:,1:K][Q][1:K,:])y ≥ 0,  (44)  for an arbitrary non-zero vector y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content=' Upon letting y = LH  c aq,  we have  aH  q Lc(I − [QH][:,1:K][Q][1:K,:])LH  c aq  = aH  0 ˆRcomaq − aH  q WcWH  c aq ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  (45)  By applying the inequality (45), we can see that  aH  q ˆRaq + σ2  e  |β|2  (a)  ≥(1 + Γ−1  e )aH  q ˆRcomaq  ≥ (1 + Γ−1  e )aH  q WcWH  c aq,  (46)  where (a) is valid, because ˆR and ˆRcom are the feasible  solution of problem P3 and ˆR follows the relationship (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
+page_content='  Consequently, it can be observed that the precodering matrix  constructs also satisfy the PLS constraint, hence completing  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
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+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfffhM/content/2301.00340v1.pdf'}
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+An investigation of the flow structure beneath solitary waves with
+constant vorticity on a conducting fluid under normal electric fields
+M. V. Flamarion1, T. Gao2 & R. Ribeiro-Jr3
+1 Unidade Acadêmica do Cabo de Santo Agostinho, UFRPE/Rural Federal University of Pernambuco, BR 101 Sul, Cabo de
+Santo Agostinho-PE, Brazil, 54503-900
+marcelo.flamarion@ufrpe.br
+2 Department of Mathematical Sciences, University of Essex, Colchester CO4 3SQ, UK.
+t.gao@essex.ac.uk
+3 UFPR/Federal University of Paraná, Departamento de Matemática, Centro Politécnico, Jardim das Américas, Caixa Postal
+19081, Curitiba, PR, 81531-980, Brazil
+robertoribeiro@ufpr.br
+Abstract
+The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked
+the attention of researchers from different areas. While there is an abundance of studies investigating the
+free surface wave properties, very few works have examined the associated velocity field within the bulk of
+the fluid. Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with
+constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric
+fields in the framework of a weakly nonlinear theory. Elevation and depression solitary waves with constant
+vorticity are computed by a pseudo-spectral method and a parameter sweep on the intensity of the electric
+field is carried out in order to study its role in the appearance of stagnation points. We find that for elevation
+solitary waves the location of stagnation points does not change significantly with variations of the electric
+field. For depression solitary waves, on the other hand, the electric field acts as a catalyser that makes
+possible the appearance of stagnation points – in the sense that in its absence there is no stagnation point.
+1
+Introduction
+Electrohydrodynamics (EHD) is an interdisciplinary subject that studies the coupling of fluid dynamics and
+electromagnetism. The motivation comes from the engineering applications of manipulating fluid motion by
+electric fields. The readers may refer to [1] for more details.
+An EHD problem is usually concerned with an interface between two fluids, and therefore the fluid motion
+under the effect of electric fields is governed by the Navier-Stokes equations (or the Euler equations in the
+inviscid case) coupled with Maxwell’s equations. A complete review has been produced by Papageorgiou [2].
+The motion of a free surface wave in an EHD flow has been widely studied by different frameworks. Of
+note, many reduced models have been derived for different configurations under certain assumptions, such as
+long-wave approximations in order to understand the mechanism of fluid-electric coupling. The readers may
+refer to [3, 4] for a comprehensive review of the linear theory and the weakly nonlinear theory respectively.
+However, very few works have focused on the features of a velocity field associated with free surface waves in
+EHD flows. To our knowledge, the only study in this direction is the one carried out by Flamarion et al. [5] in
+which the authors show that normal electric field acts as a mechanism that helps the appearance of stagnation
+points beneath periodic waves with constant vorticity. Stagnation points can be understood as points in the
+fluid domain that travels at the same speed as the wave.
+No work has been achieved to study the flow structure beneath solitary waves under electric fields to our
+best knowledge. To fill such a gap, we consider the same configuration as in Gleeson et al. [6] who derived
+a Korteweg-de Vries Benjamin-Ono equation to describe the fluid interface. Then we compute numerically
+solitary waves with constant vorticity and investigate the electrical effect on the streamlines. We shall focus on
+comprehending the role of the electrical field in the appearance of stagnation points.
+We will proceed using Korteweg-de Vries Benjamin-Ono equation to approximate the velocity field in the
+bulk of the fluid and then extract information about the flow structure beneath the wave. This methodology of
+approximating the velocity field through reduced models has been adopted in other studies such as in irrotational
+gravity flows [7, 8, 9], in gravity flows with constant vorticity [10, 11, 12, 13], in capillary-gravity flows with
+constant vorticity [14, 15] and in gravity flows with variable vorticity [16]. There is no doubt that solving the
+1
+arXiv:2301.05573v1  [physics.flu-dyn]  13 Jan 2023
+
+full Euler equations provides a more complete description of the flow. However, reduced models can reproduce
+the main features of the flow with little computational effort.
+The paper is structured as follows. We recall the formulation in section 2. The numerical methods are
+introduced in section 3 and 4. The results are presented in section 5.
+2
+Mathematical Formulation
+We consider an incompressible flow of an inviscid conducting fluid of constant density ρ and depth h0 bounded
+by a solid boundary below and an infinitely long layer of perfectly dielectric gas with permittivity ϵd above in
+a two-dimensional Cartesian x-y coordinate system. The gravity acts in the negative y-direction. The interface
+between the fluid and the gas is free to move and is usually called a free surface. Without losing generality, we
+set the undisturbed free surface at y = 0 and the bottom boundary at y = −h0. Electric fields E are active in
+the vertical direction. In the upper layer occupied by the dielectric gas, the induced magnetic field is negligible
+so that the electric fields admit a potential function V (x, y, t), i.e. E = ∇V , and satisfies V ∼ E0y as y → ∞
+where E0 is a constant. Its irrotational nature also implies that V satisfies the Laplace equation in the gas layer.
+Meanwhile, there is no variation in the electric potential within the fluid bulk so V is constant, which is assumed
+to be zero without losing generality in the lower layer due to the conducting nature of the fluid. A schematic is
+displayed in Figure 1. We consider a travelling wave, whose profile is described by ζ(x, t), propagating in the
+positive x-direction. The velocity field in the bulk of fluid is denoted by (u (x, y, t) , v (x, y, t)). We denote l by
+a typical horizontal length scale and a by a typical wave amplitude. Following [6], the dimensionless variables
+are defined by
+x = lx′ ,
+t = lt′
+c0
+,
+V = E0lV ′ ,
+y− = h0y′ ,
+y+ = λ˜y′ ,
+η = aη′ ,
+(1)
+in which c0 = √gh0 is the long-wave speed, y+ and y− are the ordinates in the upper and lower layer respectively.
+The primes are dropped to ease the notations. We follow to introduce two parameters as follows
+α = a
+h0
+,
+β = h2
+0
+λ2
+(2)
+to measure amplitude and depth. In the dimensionless variables, the bottom boundary is at y = −1 and the
+free surface is at y = αη(x, t) or ˜y = α√β η(x, t). It follows that the dimensionless governing equations are
+written by
+Vxx + V˜y˜y = 0, for y > αη(x, t),
+Vx + α
+�
+βηxV˜y = 0, at y = αη(x, t),
+(3)
+and
+ut + uux + vuy = −px for
+− 1 < y < αη(x, t),
+β
+�
+vt + uvx + vuy
+�
+= −py at y = αη(x, t),
+βvx − uy = Ω, for
+− 1 < y < αη(x, t),
+ux + vy = 0, for
+− 1 < y < αη(x, t),
+ηt + uηx = v
+α, at y = αη(x, t),
+(4)
+where Ω = hω/c0 is the dimensionless vorticity. The Young-Laplace equation at the free surface reads
+p − αη − F 2
+E
+2
+= −
+F 2
+E
+1 + α2β(ηx)2
+�
+α2β(ηx)2T11 − 2α
+�
+βηxT12 + T22
+�
+− Bαβ
+η2
+x
+(1 + α2β(ηx)2)3/2 ,
+(5)
+where T is the Maxwell stress tensor given by
+T11 = V 2
+x − V 2
+˜y
+2
+= −T22 ,
+T12 = VxV˜y ,
+(6)
+and
+F 2
+E = ϵdE2
+0
+ρgh ,
+B =
+σ
+ρgh2 ,
+(7)
+are called the electric Froude number and the Bond number respectively. The former parameter measures the
+ratio of the strength of the electric field over gravity, and the latter measures the ratio of the capillary force over
+2
+
+gravity. It is assumed that the flow is in the presence of a depth-dependent imposed current (U(y), 0), which
+dominates the velocity field. This work concerns investigating the appearance of stagnation points in the bulk
+of the fluid beneath a solitary wave. To this end, we first derive an asymptotic model for the velocity field and
+the free surface in the long-wave limits which is the so-called Korteweg-de Vries Benjamin-Ono Equation [6].
+As the derivation is well acknowledged, we only present the main results. The readers may refer to Hunt and
+Dutykh [17] for more details.
+y = −1
+E
+y = 0
+v = 0
+Conductor
+Permittivity ϵd
+Dielectric
+y = ζ(x, t)
+g
+y
+x
+Figure 1: Schematic of the problem.
+In the KdV scaling, we select α = β = ϵ ≪ 1. For a travelling-wave solution, the variables become ξ = x−ct
+and τ = ϵt in which c is the linear phase speed. We seek an asymptotic solutions of (3)-(5) in the form of
+u(ξ, y, τ) = −Ωy + ϵu0(ξ, y, τ) + ϵ2u1(ξ, y, τ) + o(ϵ2)
+v(ξ, y, τ) = ϵv0(ξ, y, τ) + ϵ2u1(ξ, y, τ) + o(ϵ2)
+p(ξ, y, τ) = ϵp0(ξ, y, τ) + ϵ2p1(ξ, y, τ) + o(ϵ2)
+V (ξ, y, τ) = −y + ϵ3/2V1(ξ, y, τ) + o(ϵ3/2)
+η(ξ, τ) = η0(ξ, τ) + ϵη1(ξ, τ) + o(ϵ)
+(8)
+Substituting (8) in equations (3)-(5), it is discovered at the leading order that
+c2 − Ωc = 1.
+(9)
+Here, we choose the solution with a positive sign, i.e. c = Ω
+2 +
+√
+Ω2+4
+2
+. We note that (9) is in fact the linear
+dispersion relation in the long-wave limit, i.e. when k → 0, and therefore the surface tension and the electric
+fields do not contribute. At the quadratic order, a KdV-Benjamin-Ono equation that incorporates the surface
+tension, vorticity effects and electric forces is obtained to be
+η0τ + µη0η0ξ + νη0ξξξ + γH[η0ξξ] = 0
+(10)
+where the coefficients are given by
+µ = Ω2 + 3
+2c − Ω, ν =
+1
+2c − Ω
+�c2
+3 − B
+�
+, γ = −
+F 2
+E
+2c − Ω.
+(11)
+and H is the Hilbert operator which is defined as
+�
+H[f(ξ)] = −i sign(k) ˆf(k),
+(12)
+where �[·] represents the Fourier transform.
+In the absence of electric forces (γ = 0), equation (10) reduces to a standard KdV equation which admits
+solitary wave solutions described by the formula [18]
+η0(ξ, τ) = A sech2
+��
+µA
+12ν
+�
+ξ − µA
+3 τ
+��
+.
+(13)
+It is noted that the KdV equation collapses when the nonlinearity disappears at ν = 0, or at
+B = Bc ≡ c2
+3 .
+(14)
+3
+
+Under such circumstances, a different scaling is required to derive a fifth-order KdV equation. As it is irrelevant
+to the main aim of this work, it will not be further discussed. When ν is non-zero (and γ = 0), equation (10)
+admits elevation solitary wave solutions when 0 ≤ B < Bc and depression solitary wave solutions when B > Bc.
+When electric forces are present, solitary waves of equation (10) do not have a closed form. We consider a
+solitary wave solution of (10) denoted by Θ = Θ(ξ −Cτ) propagating with speed C. It immediately follows that
+η0(ξ, τ) = Θ(ξ −Cτ). As we are interested in investigating particle trajectories for the Euler equations using the
+KdV-Benjamin-Ono model as an approximation, we have to express the free surface and the horizontal velocity
+at the bottom of the channel using the Euler coordinates. The solitary wave solution and the approximation of
+the velocity field in the Euler coordinates are written respectively by
+η0(x, t) = Θ(x − (c + ϵC)t),
+(15)
+and
+u0(x, y, t) = cΘ(x − (c + ϵC)t)) and v0(x, y, t) = −cΘx(x − (c + ϵC)t))(y + 1).
+(16)
+In the next section, we present the numerical methods to compute depression solitary waves of (10) to investigate
+particle trajectories in the bulk of the fluid.
+3
+Numerical methods
+Solitary waves Θ with speed C, amplitude A and crest located at x = 0 for the KdV-Benjamin-Ono equation
+(10) are computed through Newton’s method by solving the equations
+−CΘξ + αΘΘξ + βΘξξξ + γH[Θξξ] = 0,
+Θ(0) − A = 0,
+(17)
+in a periodic computational domain [−L, L) with a uniform grid with even points N. The spatial points are
+discretised as
+ξj = −L + (j − 1)∆ξ , for j = 1, 2, . . . , N, where ∆ξ = 2L/N,
+(18)
+and the frequencies as
+(k1, k2, · · · , kN) = π
+L(0, 1, · · · , N/2 − 1, 0, −N/2 + 1, · · · , −1).
+(19)
+On the grid points defined in equation (18), we denote by Θj = Θ(ξj), Θξ,j = Θξ(ξj), Θξξ,j = Θξξ(ξj) and
+Θξξξ,j = Θξξξ(ξj). The discretised version of equations (17) gives rise to a system of (N + 1) equations with
+(N + 1) unknowns
+Gj(Θ1, Θ2, ..., ΘN, C) := −CΘξ,j + αΘjΘξ,j + βΘξξξ,j + γH[Θξξ,j] = 0, for j = 1, 2, . . . , N.
+GN+1(Θ1, Θ2, ..., ΘN, C) := ΘN/2+1 − A = 0.
+(20)
+The discretisation chosen allows us to compute all spatial derivatives and the nonlocal operator H in equations
+(17) with spectral accuracy in Fourier space through the FFT [19]. The system’s Jacobian for the Newton
+iteration is found by finite variations in the unknowns and the stopping criterion considered is
+�N+1
+j=1 |Gj(Θ1, Θ2, ..., ΘN, C)|
+N + 1
+< δ,
+(21)
+where δ is the tolerance value set to be 10−10. For a fixed value of A, Ω and FE, we choose the solitary wave
+solution of equation (10) in the absence of electric forces
+Θ0(ξ) = A sech2(kξ) ,
+C0 = −αA
+3 .
+(22)
+as the initial guess. The solution is then computed by a continuation method in the parameter FE by using the
+prior converged solution of the Newton method as the initial guess.
+Typical numerical solitary waves are displayed in Figure 2. We recall that elevation solitary waves occur
+when B < Bc and depression ones when B > Bc. It is noted that the elevation and depression solitary waves
+have more ripples appearing on the side of the main pulse when the third-order dispersive term is weak and the
+electric term in the Hilbert transform is strong, i.e. ν small and γ big in (10). These solutions with decaying
+oscillatory tails have been previously reported by [20] and [21]. For the purpose of this work, we only focus on
+the waves shown in Figure 2.
+4
+
+-40
+-20
+0
+20
+40
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+FE
+2=0
+FE
+2=0.2
+FE
+2=0.4
+FE
+2=0.6
+FE
+2=0.8
+-20
+-10
+0
+10
+20
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+FE
+2=0
+FE
+2=0.1
+FE
+2=0.2
+FE
+2=0.25
+FE
+2=0.3
+-40
+-20
+0
+20
+40
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+FE
+2=0
+FE
+2=0.2
+FE
+2=0.4
+FE
+2=0.6
+FE
+2=0.8
+-20
+-10
+0
+10
+20
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+FE
+2=0
+FE
+2=4
+FE
+2=6
+FE
+2=8
+FE
+2=10
+Figure 2: Top: Solitary wave solutions of equation (10) in the absence of vorticity (Ω = 0) and different values
+of FE. Parameters: B = 0 (left) and B = 0.4 (right). Bottom: Solitary wave solutions of equation (10) with
+Ω = 5 and different values of FE. Parameters: B = 0 (left) and B = 17 (right).
+4
+Particle trajectories
+Particle trajectories beneath the solitary wave (15) can be computed approximately by solving the dynamical
+system
+dx
+dt = −Ωy + ϵu(x, y, t) ≈ −Ωy + ϵcΘ(x − (c + ϵC)t)),
+dy
+dt = ϵv(x, y, t) ≈ −ϵcΘx(x − (c + ϵC)t))(y + 1).
+(23)
+In order to compute stagnation points, it is convenient to solve equations (23) in the frame that moves with
+the wave speed, for this purpose we consider the new variables X = x − (c + ϵC)t and Y = y. In this new
+reference frame, the streamlines are solutions of the autonomous dynamical system
+dX
+dt = −ΩY + ϵcΘ(X) − (c + ϵC),
+dY
+dt = −ϵcΘX(X)(Y + 1),
+(24)
+which can be seen as the level curves of the Hamiltonian Ψ(X, Y ) given by
+Ψ(X, Y ) = ϵcΘ(X)(Y + 1) − Ω
+2 Y 2 − (c + ϵC)Y.
+(25)
+Notice that once the solitary wave Θ is computed numerically through the method proposed in the previous
+section, the level curves can be easily computed using the function contour that is implemented in MATLAB.
+In the absence of surface tension and electric fields, Guan [10] investigated particle trajectories beneath
+solitary waves in the presence of a linear sheared current through the Korteweg-de Vries equation. He showed
+that the orbits obtained from the asymptotic approximation agree well with the ones computed through the
+full Euler equations when the solitary waves have small amplitudes. Based on his results, in all simulations
+presented in this article, we fix ϵ = 0.1.
+5
+
+5
+Results and discussion
+5.1
+Elevation solitary waves
+In the absence of an electric field, the increase of the vorticity could cause the appearance of stagnation points
+(see [22]). It first appears at the bottom and below the crest. As the vorticity increases further, other stagnation
+points appear in the bulk of the fluid creating a recirculation zone [22]. Therefore, in order to discuss the influence
+of the electric field in the flow structure beneath solitary waves for 0 ≤ B < Bc, we first find the smallest value
+of the vorticity such that a stagnation point appears at the bottom and below the solitary wave crest in the
+absence of the electric field then follow to study the case where the electric fields are switched on.
+The value of the vorticity for which we have a single stagnation point located at the bottom and below the
+solitary wave crest is obtained by solving for Ω equation (24) evaluated at X = 0 and Y = −1 which yields the
+equation
+0 = Ω + ϵcA − (c + ϵC).
+(26)
+The solution to equation (26) for FE = 0 and B = 0 is Ω∗ ≈ 5.2962 and this value does not vary considerably
+with B because as pointed out by Flamarion [15] surface tension does not create stagnation points. Moreover,
+it barely changes the position of the stagnation point below the crest (when it does exist).
+0
+0.2
+0.4
+0.6
+0.8
+1
+5.25
+5.26
+5.27
+5.28
+5.29
+5.3
+Figure 3: The graph represents the vorticity as a function of the parameter FE in which the first stagnation
+point gives rise at the bottom of the channel.
+Figure 3 displays the solution of equation (26) for different values of the Bond number.
+These curves
+correspond to flows with a single stagnation point on the bottom and beneath the crest. Firstly, it is noticed
+that the solution does not vary much for different values of the Bond number and small values of the parameter
+FE. Besides, we observe that the appearance of the stagnation point on the bottom can occur at a tinnier
+vorticity with the increase of intensity of the electric field. Secondly, we can regard these curves as bifurcation
+points that separate the parameter space in two regions according to the number of stagnation points beneath
+the solitary wave. For those (F 2
+E, Ω) below these curves, there is no stagnation point in the fluid domain. On
+the other side, for those (F 2
+E, Ω) above these curves, there exist three stagnation points, namely, two saddles at
+the bottom of the channel and a centre in the bulk of the fluid aligned with the crest of the solitary wave. And
+there is only one stagnation point at the bottom for those (F 2
+E, Ω) right on the curves. A typical example of
+this bifurcation is depicted in Figure 4.
+We follow to analyse how the strength of the electric field affects the location of the stagnation points. To
+this end, we fix the vorticity and the surface tension and let FE vary. The left panel of Figure 5 shows the
+vertical position (Y ∗) of the stagnation point located below the wave crest and the right panel of the same
+figure presents the horizontal coordinate (X∗) of the saddle point as a function of the parameter FE for Ω = Ω∗
+and B = 0. Of note, the intensity of FE barely impacts the position of the centre point, however, it does affect
+the position of the saddle points.
+Flamarion et al. [5] showed that the appearance of stagnation points beneath periodic travelling waves
+can occur at small vorticity with the help of electric fields.
+Besides, it was shown that the position of all
+the stagnation points changed significantly with variations in the electric field. The features differ from the
+discussion presented above for elevation solitary waves where the electric field does not act as a mechanism to
+help the generation of stagnation points.
+6
+
+-10
+-5
+0
+5
+10
+-0.1
+-0.05
+0
+0.05
+-1.01
+-0.99
+-0.96
+-10
+-5
+0
+5
+10
+-0.1
+-0.05
+0
+0.05
+-1.01
+-0.99
+-0.96
+-10
+-5
+0
+5
+10
+-0.1
+-0.05
+0
+0.05
+-1.01
+-0.99
+-0.96
+-10
+-5
+0
+5
+10
+-0.1
+-0.05
+0
+0.05
+-1.01
+-0.99
+-0.96
+Figure 4: Phase portrait for different values of the vorticity with a solitary wave with amplitude A = 0.5,
+F 2
+E = 0.5 and B = 0. The critical value of the vorticity in which the first stagnation point appears at the
+bottom is Ω∗ ≈ 5.2780.
+0
+2
+4
+6
+8
+10
+-1
+-0.9995
+-0.999
+-0.9985
+-0.998
+-0.9975
+0
+2
+4
+6
+8
+10
+0
+0.2
+0.4
+0.6
+0.8
+1
+Figure 5: The effect of the electric field in the position of the centre bellow the crest of the solitary wave for
+B = 0 and Ω = Ω∗.
+5.2
+Depression solitary waves
+It is known that in the absence of an electric field typical depression solitary wave solutions of equation (10)
+are sech2 −like. For such waves, it is well established that stagnation points never give rise to the bulk of the
+fluid. As shown by Flamarion [14] stagnation points beneath depression solitary waves can occur only in the
+presence of decaying or oscillatory tails, which cannot be captured by a third-order KdV equation. As can be
+seen from Figure 2, under the electrical effect, equation (10) admits depression solitary wave solutions with two
+elevation dimples on the side of the wave trough. Consequently, an immediate interesting question is whether
+stagnation points can take place in the bulk of the fluid beneath such depression solitary waves, which will be
+examined in the rest of the paper.
+Other authors have studied the appearance of stagnation points beneath depression solitary waves [14, 15, 23].
+However, these works considered gravity-capillary waves in the absence of electric fields. Moreover, it has been
+shown that the location of the stagnation points does not change much for choices of B. Having said this, we
+focus on investigating the effect of the electric field as a mechanism to create stagnation points. To address this
+issue we fix the vorticity, the Bond number and vary the intensity of the electric field.
+7
+
+-10
+-5
+0
+5
+10
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.99
+-10
+-5
+0
+5
+10
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.98
+-10
+-5
+0
+5
+10
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.99
+-10
+-5
+0
+5
+10
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.95
+-10
+-5
+0
+5
+10
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.95
+-10
+-5
+0
+5
+10
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+-1.001
+-0.95
+Figure 6: Phase portrait for different values of the vorticity with a solitary wave with amplitude A = 0.5, Ω = 5
+and B = 17. The critical value of the vorticity in which the first stagnation point appears at the bottom is
+Ω∗ ≈ 5.2780.
+Figure 6 depicts a series of simulations from which we can see that in the presence of a strong electric field
+stagnation points can appear in the fluid domain. The location of the stagnation points is determined in two
+ways– (i) by finding the equilibrium points of the dynamical system (24), i.e., we find the zeros of the velocity
+field or (ii) by the contour function of MATLAB. The flow structure beneath the depression solitary wave can
+have (i) zero, (ii) two centres (at the bottom), (iii) two centres (in the bulk of the fluid) and four saddles (at the
+bottom) or (iv) two centres (in the bulk of the fluid) and four saddles (two at the bottom and two in the bulk
+of the fluid) as stagnation points depending on the intensity of the electric field. This features the bifurcation
+of flow according to the F 2
+E parameter. Similar descriptions of the arrangement of the stagnation point in the
+context of gravity-capillary waves were reported in the work of Flamarion [14].
+It is well acknowledged that the full Euler equations are the most realistic model to reproduce EHD scenarios
+in inviscid fluids. However, reduced models can reproduce qualitatively the same features of the flow with
+comparatively little effort. For instance, our results show that the weakly nonlinear weakly dispersive regime
+can capture rich flow structures, such as recirculation zones and stagnation points.
+6
+Conclusion
+In the presented study the flow structure beneath EHD flows with constant vorticity was investigated numerically
+in the Korteweg Benjamin-Ono equation framework. Solitary waves were computed numerically through the
+standard Newton’s method combined with Fourier spectral methods. This approach allowed us to approximate
+the velocity field beneath the free surface. As a consequence, the location of stagnation points and details of the
+8
+
+recirculation zones formed by them were determined. For elevation solitary waves, we showed that the location
+of the centre points does not change significantly by variations on the electric field. It is remarkable that for
+depression solitary waves the electric field acts as a mechanism for the creation of stagnation points. In the
+absence of an electric field even when the vorticity is strong there is no stagnation point in the bulk of the fluid.
+The results presented in this work are expected to agree well with the full nonlinear model. An attempt to
+compare the results predicted by both models is a natural path to be pursued in future.
+Acknowledgments
+M. V. F and R.R.-Jr are grateful to IMPA for hosting them as visitors during the 2023 Post-Doctoral Summer
+Program.
+Data Availability Statement
+Data sharing is not applicable to this article as the parameters used in the numerical experiments are informed
+in this paper.
+References
+[1] X. Chen, J. Cheng and X. Yin. Advances and applications of electrohydrodynamics, Chin. Sci. Bull. 48
+(2003) 1055–1063.
+[2] D. T. Papageorgiou, Film flows in the presence of electric fields, Ann. Rev. Fluid Mech. 51 (2019) 155–187.
+[3] A. Doak, T. Gao, J.-M. Vanden-Broeck & J. J. S. Kandola, Capillary-gravity waves on the interface of two
+dielectric fluid layers under normal electric fields, Q. J. Mech. Appl. Math. 73 (2020) 231–250.
+[4] Z. Wang, Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids,
+Proc. R. Soc. A 473 (2017) 20160817.
+[5] M. V. Flamarion, T. Gao, R. Ribeiro-Jr & A. Doak. Flow structure beneath periodic waves with constant
+vorticity under normal electric fields. Phys. Fluids 34, 127119 (2022).
+[6] H. Gleeson, P. Hammerton, D. Papageorgiou, J.-M. Vanden-Broeck, A new application of the Korteweg-de
+Vries Benjamin-Ono equation in interfacial electrohydrodynamics, Phys. Fluids 19 (2007) 031703.
+[7] H. Borluk, H. Kalisch, Particle dynamics in the KdV approximation, Wave Motion 49 (2012) 691-709.
+[8] L. Gagnon, Qualitative description of the particle trajectories for n-solitons solution of the korteweg-de Vries
+equation, Discrete Contin. Dyn. Syst. 37 (2017) 1489-1507.
+[9] Z. Khorsand, Particle trajectories in the Serre equations, Appl. Math. Comput. 230 (2014) 35-42.
+[10] X. Guan, Particle trajectories under interactions between solitary waves and a linear shear current. Theor.
+App. Mech. Lett. 10 (2020) 125-131.
+[11] A. Alfatih, H. Kalisch, Reconstruction of the pressure in long-wave models with constant vorticity, Eur. J.
+Mech. B Fluids 37 (2013) 187-194.
+[12] C. Cutis, J. Carter, H. Kalisch, Particle paths in nonlinear Schrödinger models in the presence of linear
+shear currents, J. Fluid Mech. 855 (2018)
+[13] J. Carter, C. Curtis, H. Kalisch, Particle trajectories in nonlinear Schr¨dinger models, Water Waves 2 (2020)
+31-57.
+[14] M.V. Flamarion, Complex flow structures beneath rotational depression solitary waves. Wave Motion. 117,
+(2023) 103108.
+[15] M.V. Flamarion, Stagnation points beneath rotational solitary waves in gravity-capillary flows. Trends in
+Computational and Applied Mathematics. (in press) (2023).
+[16] M.V. Flamarion, R. Ribeiro-Jr, Solitary Waves on Flows with an Exponentially Sheared Current and
+Stagnation Points. Quart. J. Mech. Appl. Math., (2023).
+9
+
+[17] M. J. Hunt & D. Dutykh, Free Surface Flows in Electrohydrodynamics with a Constant Vorticity Distri-
+bution. Water Waves 3, 297–317 (2021).
+[18] G.B. Whitham, Linear and Nonlinear Waves. Wiley. 1974.
+[19] Trefethen LN Spectral Methods in MATLAB. Philadelphia: SIAM; 2001.
+[20] J. P. Albert, J. L. Bona & J. M. Restrepo,
+Solitary-wave solutions of the Benjamin equation. SIAM J.
+Appl. Math. 59(6), 2139-2161 (2014).
+[21] V. Dougalis, A. Duran, & D. Mitsotakis, Numerical solution of the Benjamin equation. Wave Motion. 52,
+194-215 (2015).
+[22] R. Ribeiro-Jr, P.A. Milewski, A. Nachbin, Flow structure beneath rotational water waves with stagnation
+points. J. Fluid. Mech. 812 (2017) 792-814.
+[23] Z. Wang, X. Guan & J-M. Vanden-Broeck, Progressive flexural-gravity waves with constant vorticity. J.
+Fluid. Mech. 995 (2020) A12.
+10
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf,len=486
+page_content='An investigation of the flow structure beneath solitary waves with  constant vorticity on a conducting fluid under normal electric fields  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Flamarion1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Gao2 & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Ribeiro-Jr3  1 Unidade Acadêmica do Cabo de Santo Agostinho, UFRPE/Rural Federal University of Pernambuco, BR 101 Sul, Cabo de  Santo Agostinho-PE, Brazil, 54503-900  marcelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='flamarion@ufrpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='br  2 Department of Mathematical Sciences, University of Essex, Colchester CO4 3SQ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='gao@essex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='uk  3 UFPR/Federal University of Paraná, Departamento de Matemática, Centro Politécnico, Jardim das Américas, Caixa Postal  19081, Curitiba, PR, 81531-980, Brazil  robertoribeiro@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='br  Abstract  The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked  the attention of researchers from different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' While there is an abundance of studies investigating the  free surface wave properties, very few works have examined the associated velocity field within the bulk of  the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with  constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric  fields in the framework of a weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Elevation and depression solitary waves with constant  vorticity are computed by a pseudo-spectral method and a parameter sweep on the intensity of the electric  field is carried out in order to study its role in the appearance of stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We find that for elevation  solitary waves the location of stagnation points does not change significantly with variations of the electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For depression solitary waves, on the other hand, the electric field acts as a catalyser that makes  possible the appearance of stagnation points – in the sense that in its absence there is no stagnation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  1  Introduction  Electrohydrodynamics (EHD) is an interdisciplinary subject that studies the coupling of fluid dynamics and  electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The motivation comes from the engineering applications of manipulating fluid motion by  electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The readers may refer to [1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  An EHD problem is usually concerned with an interface between two fluids, and therefore the fluid motion  under the effect of electric fields is governed by the Navier-Stokes equations (or the Euler equations in the  inviscid case) coupled with Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' A complete review has been produced by Papageorgiou [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  The motion of a free surface wave in an EHD flow has been widely studied by different frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Of  note, many reduced models have been derived for different configurations under certain assumptions, such as  long-wave approximations in order to understand the mechanism of fluid-electric coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The readers may  refer to [3, 4] for a comprehensive review of the linear theory and the weakly nonlinear theory respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  However, very few works have focused on the features of a velocity field associated with free surface waves in  EHD flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' To our knowledge, the only study in this direction is the one carried out by Flamarion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' [5] in  which the authors show that normal electric field acts as a mechanism that helps the appearance of stagnation  points beneath periodic waves with constant vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Stagnation points can be understood as points in the  fluid domain that travels at the same speed as the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  No work has been achieved to study the flow structure beneath solitary waves under electric fields to our  best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' To fill such a gap, we consider the same configuration as in Gleeson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' [6] who derived  a Korteweg-de Vries Benjamin-Ono equation to describe the fluid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Then we compute numerically  solitary waves with constant vorticity and investigate the electrical effect on the streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We shall focus on  comprehending the role of the electrical field in the appearance of stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  We will proceed using Korteweg-de Vries Benjamin-Ono equation to approximate the velocity field in the  bulk of the fluid and then extract information about the flow structure beneath the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' This methodology of  approximating the velocity field through reduced models has been adopted in other studies such as in irrotational  gravity flows [7, 8, 9], in gravity flows with constant vorticity [10, 11, 12, 13], in capillary-gravity flows with  constant vorticity [14, 15] and in gravity flows with variable vorticity [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' There is no doubt that solving the  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05573v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='flu-dyn]  13 Jan 2023  full Euler equations provides a more complete description of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' However, reduced models can reproduce  the main features of the flow with little computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We recall the formulation in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The numerical methods are  introduced in section 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The results are presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  2  Mathematical Formulation  We consider an incompressible flow of an inviscid conducting fluid of constant density ρ and depth h0 bounded  by a solid boundary below and an infinitely long layer of perfectly dielectric gas with permittivity ϵd above in  a two-dimensional Cartesian x-y coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The gravity acts in the negative y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The interface  between the fluid and the gas is free to move and is usually called a free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Without losing generality, we  set the undisturbed free surface at y = 0 and the bottom boundary at y = −h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Electric fields E are active in  the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' In the upper layer occupied by the dielectric gas, the induced magnetic field is negligible  so that the electric fields admit a potential function V (x, y, t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' E = ∇V , and satisfies V ∼ E0y as y → ∞  where E0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Its irrotational nature also implies that V satisfies the Laplace equation in the gas layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Meanwhile, there is no variation in the electric potential within the fluid bulk so V is constant, which is assumed  to be zero without losing generality in the lower layer due to the conducting nature of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' A schematic is  displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We consider a travelling wave, whose profile is described by ζ(x, t), propagating in the  positive x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The velocity field in the bulk of fluid is denoted by (u (x, y, t) , v (x, y, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We denote l by  a typical horizontal length scale and a by a typical wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Following [6], the dimensionless variables  are defined by  x = lx′ ,  t = lt′  c0  ,  V = E0lV ′ ,  y− = h0y′ ,  y+ = λ˜y′ ,  η = aη′ ,  (1)  in which c0 = √gh0 is the long-wave speed, y+ and y− are the ordinates in the upper and lower layer respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  The primes are dropped to ease the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We follow to introduce two parameters as follows  α = a  h0  ,  β = h2  0  λ2  (2)  to measure amplitude and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' In the dimensionless variables, the bottom boundary is at y = −1 and the  free surface is at y = αη(x, t) or ˜y = α√β η(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It follows that the dimensionless governing equations are  written by  Vxx + V˜y˜y = 0, for y > αη(x, t),  Vx + α  �  βηxV˜y = 0, at y = αη(x, t),  (3)  and  ut + uux + vuy = −px for  − 1 < y < αη(x, t),  β  �  vt + uvx + vuy  �  = −py at y = αη(x, t),  βvx − uy = Ω, for  − 1 < y < αη(x, t),  ux + vy = 0, for  − 1 < y < αη(x, t),  ηt + uηx = v  α, at y = αη(x, t),  (4)  where Ω = hω/c0 is the dimensionless vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The Young-Laplace equation at the free surface reads  p − αη − F 2  E  2  = −  F 2  E  1 + α2β(ηx)2  �  α2β(ηx)2T11 − 2α  �  βηxT12 + T22  �  − Bαβ  η2  x  (1 + α2β(ηx)2)3/2 ,  (5)  where T is the Maxwell stress tensor given by  T11 = V 2  x − V 2  ˜y  2  = −T22 ,  T12 = VxV˜y ,  (6)  and  F 2  E = ϵdE2  0  ρgh ,  B =  σ  ρgh2 ,  (7)  are called the electric Froude number and the Bond number respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The former parameter measures the  ratio of the strength of the electric field over gravity, and the latter measures the ratio of the capillary force over  2  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It is assumed that the flow is in the presence of a depth-dependent imposed current (U(y), 0), which  dominates the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' This work concerns investigating the appearance of stagnation points in the bulk  of the fluid beneath a solitary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' To this end, we first derive an asymptotic model for the velocity field and  the free surface in the long-wave limits which is the so-called Korteweg-de Vries Benjamin-Ono Equation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  As the derivation is well acknowledged, we only present the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The readers may refer to Hunt and  Dutykh [17] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  y = −1  E  y = 0  v = 0  Conductor  Permittivity ϵd  Dielectric  y = ζ(x, t)  g  y  x  Figure 1: Schematic of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  In the KdV scaling, we select α = β = ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For a travelling-wave solution, the variables become ξ = x−ct  and τ = ϵt in which c is the linear phase speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We seek an asymptotic solutions of (3)-(5) in the form of  u(ξ, y, τ) = −Ωy + ϵu0(ξ, y, τ) + ϵ2u1(ξ, y, τ) + o(ϵ2)  v(ξ, y, τ) = ϵv0(ξ, y, τ) + ϵ2u1(ξ, y, τ) + o(ϵ2)  p(ξ, y, τ) = ϵp0(ξ, y, τ) + ϵ2p1(ξ, y, τ) + o(ϵ2)  V (ξ, y, τ) = −y + ϵ3/2V1(ξ, y, τ) + o(ϵ3/2)  η(ξ, τ) = η0(ξ, τ) + ϵη1(ξ, τ) + o(ϵ)  (8)  Substituting (8) in equations (3)-(5), it is discovered at the leading order that  c2 − Ωc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (9)  Here, we choose the solution with a positive sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' c = Ω  2 +  √  Ω2+4  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We note that (9) is in fact the linear  dispersion relation in the long-wave limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' when k → 0, and therefore the surface tension and the electric  fields do not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' At the quadratic order, a KdV-Benjamin-Ono equation that incorporates the surface  tension, vorticity effects and electric forces is obtained to be  η0τ + µη0η0ξ + νη0ξξξ + γH[η0ξξ] = 0  (10)  where the coefficients are given by  µ = Ω2 + 3  2c − Ω, ν =  1  2c − Ω  �c2  3 − B  �  , γ = −  F 2  E  2c − Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (11)  and H is the Hilbert operator which is defined as  �  H[f(ξ)] = −i sign(k) ˆf(k),  (12)  where �[·] represents the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  In the absence of electric forces (γ = 0), equation (10) reduces to a standard KdV equation which admits  solitary wave solutions described by the formula [18]  η0(ξ, τ) = A sech2  ��  µA  12ν  �  ξ − µA  3 τ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (13)  It is noted that the KdV equation collapses when the nonlinearity disappears at ν = 0, or at  B = Bc ≡ c2  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (14)  3  Under such circumstances, a different scaling is required to derive a fifth-order KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As it is irrelevant  to the main aim of this work, it will not be further discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' When ν is non-zero (and γ = 0), equation (10)  admits elevation solitary wave solutions when 0 ≤ B < Bc and depression solitary wave solutions when B > Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  When electric forces are present, solitary waves of equation (10) do not have a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We consider a  solitary wave solution of (10) denoted by Θ = Θ(ξ −Cτ) propagating with speed C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It immediately follows that  η0(ξ, τ) = Θ(ξ −Cτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As we are interested in investigating particle trajectories for the Euler equations using the  KdV-Benjamin-Ono model as an approximation, we have to express the free surface and the horizontal velocity  at the bottom of the channel using the Euler coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The solitary wave solution and the approximation of  the velocity field in the Euler coordinates are written respectively by  η0(x, t) = Θ(x − (c + ϵC)t),  (15)  and  u0(x, y, t) = cΘ(x − (c + ϵC)t)) and v0(x, y, t) = −cΘx(x − (c + ϵC)t))(y + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (16)  In the next section, we present the numerical methods to compute depression solitary waves of (10) to investigate  particle trajectories in the bulk of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  3  Numerical methods  Solitary waves Θ with speed C, amplitude A and crest located at x = 0 for the KdV-Benjamin-Ono equation  (10) are computed through Newton’s method by solving the equations  −CΘξ + αΘΘξ + βΘξξξ + γH[Θξξ] = 0,  Θ(0) − A = 0,  (17)  in a periodic computational domain [−L, L) with a uniform grid with even points N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The spatial points are  discretised as  ξj = −L + (j − 1)∆ξ , for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' , N, where ∆ξ = 2L/N,  (18)  and the frequencies as  (k1, k2, · · · , kN) = π  L(0, 1, · · · , N/2 − 1, 0, −N/2 + 1, · · · , −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (19)  On the grid points defined in equation (18), we denote by Θj = Θ(ξj), Θξ,j = Θξ(ξj), Θξξ,j = Θξξ(ξj) and  Θξξξ,j = Θξξξ(ξj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The discretised version of equations (17) gives rise to a system of (N + 1) equations with  (N + 1) unknowns  Gj(Θ1, Θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=', ΘN, C) := −CΘξ,j + αΘjΘξ,j + βΘξξξ,j + γH[Θξξ,j] = 0, for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  GN+1(Θ1, Θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=', ΘN, C) := ΘN/2+1 − A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (20)  The discretisation chosen allows us to compute all spatial derivatives and the nonlocal operator H in equations  (17) with spectral accuracy in Fourier space through the FFT [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The system’s Jacobian for the Newton  iteration is found by finite variations in the unknowns and the stopping criterion considered is  �N+1  j=1 |Gj(Θ1, Θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=', ΘN, C)|  N + 1  < δ,  (21)  where δ is the tolerance value set to be 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For a fixed value of A, Ω and FE, we choose the solitary wave  solution of equation (10) in the absence of electric forces  Θ0(ξ) = A sech2(kξ) ,  C0 = −αA  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (22)  as the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The solution is then computed by a continuation method in the parameter FE by using the  prior converged solution of the Newton method as the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Typical numerical solitary waves are displayed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' We recall that elevation solitary waves occur  when B < Bc and depression ones when B > Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It is noted that the elevation and depression solitary waves  have more ripples appearing on the side of the main pulse when the third-order dispersive term is weak and the  electric term in the Hilbert transform is strong, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' ν small and γ big in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' These solutions with decaying  oscillatory tails have been previously reported by [20] and [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For the purpose of this work, we only focus on  the waves shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  4  40  20  0  20  40  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5  FE  2=0  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='6  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='8  20  10  0  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0  FE  2=0  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='25  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  40  20  0  20  40  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5  FE  2=0  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='6  FE  2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='8  20  10  0  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0  FE  2=0  FE  2=4  FE  2=6  FE  2=8  FE  2=10  Figure 2: Top: Solitary wave solutions of equation (10) in the absence of vorticity (Ω = 0) and different values  of FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Parameters: B = 0 (left) and B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Bottom: Solitary wave solutions of equation (10) with  Ω = 5 and different values of FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Parameters: B = 0 (left) and B = 17 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  4  Particle trajectories  Particle trajectories beneath the solitary wave (15) can be computed approximately by solving the dynamical  system  dx  dt = −Ωy + ϵu(x, y, t) ≈ −Ωy + ϵcΘ(x − (c + ϵC)t)),  dy  dt = ϵv(x, y, t) ≈ −ϵcΘx(x − (c + ϵC)t))(y + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (23)  In order to compute stagnation points, it is convenient to solve equations (23) in the frame that moves with  the wave speed, for this purpose we consider the new variables X = x − (c + ϵC)t and Y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' In this new  reference frame, the streamlines are solutions of the autonomous dynamical system  dX  dt = −ΩY + ϵcΘ(X) − (c + ϵC),  dY  dt = −ϵcΘX(X)(Y + 1),  (24)  which can be seen as the level curves of the Hamiltonian Ψ(X, Y ) given by  Ψ(X, Y ) = ϵcΘ(X)(Y + 1) − Ω  2 Y 2 − (c + ϵC)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (25)  Notice that once the solitary wave Θ is computed numerically through the method proposed in the previous  section, the level curves can be easily computed using the function contour that is implemented in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  In the absence of surface tension and electric fields, Guan [10] investigated particle trajectories beneath  solitary waves in the presence of a linear sheared current through the Korteweg-de Vries equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' He showed  that the orbits obtained from the asymptotic approximation agree well with the ones computed through the  full Euler equations when the solitary waves have small amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Based on his results, in all simulations  presented in this article, we fix ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  5  5  Results and discussion  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  Elevation solitary waves  In the absence of an electric field, the increase of the vorticity could cause the appearance of stagnation points  (see [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It first appears at the bottom and below the crest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As the vorticity increases further, other stagnation  points appear in the bulk of the fluid creating a recirculation zone [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Therefore, in order to discuss the influence  of the electric field in the flow structure beneath solitary waves for 0 ≤ B < Bc, we first find the smallest value  of the vorticity such that a stagnation point appears at the bottom and below the solitary wave crest in the  absence of the electric field then follow to study the case where the electric fields are switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  The value of the vorticity for which we have a single stagnation point located at the bottom and below the  solitary wave crest is obtained by solving for Ω equation (24) evaluated at X = 0 and Y = −1 which yields the  equation  0 = Ω + ϵcA − (c + ϵC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  (26)  The solution to equation (26) for FE = 0 and B = 0 is Ω∗ ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2962 and this value does not vary considerably  with B because as pointed out by Flamarion [15] surface tension does not create stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Moreover,  it barely changes the position of the stagnation point below the crest (when it does exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='8  1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='28  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='3  Figure 3: The graph represents the vorticity as a function of the parameter FE in which the first stagnation  point gives rise at the bottom of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Figure 3 displays the solution of equation (26) for different values of the Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  These curves  correspond to flows with a single stagnation point on the bottom and beneath the crest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Firstly, it is noticed  that the solution does not vary much for different values of the Bond number and small values of the parameter  FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Besides, we observe that the appearance of the stagnation point on the bottom can occur at a tinnier  vorticity with the increase of intensity of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Secondly, we can regard these curves as bifurcation  points that separate the parameter space in two regions according to the number of stagnation points beneath  the solitary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For those (F 2  E, Ω) below these curves, there is no stagnation point in the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' On  the other side, for those (F 2  E, Ω) above these curves, there exist three stagnation points, namely, two saddles at  the bottom of the channel and a centre in the bulk of the fluid aligned with the crest of the solitary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' And  there is only one stagnation point at the bottom for those (F 2  E, Ω) right on the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' A typical example of  this bifurcation is depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  We follow to analyse how the strength of the electric field affects the location of the stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' To  this end, we fix the vorticity and the surface tension and let FE vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The left panel of Figure 5 shows the  vertical position (Y ∗) of the stagnation point located below the wave crest and the right panel of the same  figure presents the horizontal coordinate (X∗) of the saddle point as a function of the parameter FE for Ω = Ω∗  and B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Of note, the intensity of FE barely impacts the position of the centre point, however, it does affect  the position of the saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Flamarion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' [5] showed that the appearance of stagnation points beneath periodic travelling waves  can occur at small vorticity with the help of electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Besides, it was shown that the position of all  the stagnation points changed significantly with variations in the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The features differ from the  discussion presented above for elevation solitary waves where the electric field does not act as a mechanism to  help the generation of stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  6  10  5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='96  10  5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='96  10  5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='96  10  5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='96  Figure 4: Phase portrait for different values of the vorticity with a solitary wave with amplitude A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5,  F 2  E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5 and B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The critical value of the vorticity in which the first stagnation point appears at the  bottom is Ω∗ ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  0  2  4  6  8  10  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='9995  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='9985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='9975  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='8  1  Figure 5: The effect of the electric field in the position of the centre bellow the crest of the solitary wave for  B = 0 and Ω = Ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2  Depression solitary waves  It is known that in the absence of an electric field typical depression solitary wave solutions of equation (10)  are sech2 −like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For such waves, it is well established that stagnation points never give rise to the bulk of the  fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As shown by Flamarion [14] stagnation points beneath depression solitary waves can occur only in the  presence of decaying or oscillatory tails, which cannot be captured by a third-order KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As can be  seen from Figure 2, under the electrical effect, equation (10) admits depression solitary wave solutions with two  elevation dimples on the side of the wave trough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Consequently, an immediate interesting question is whether  stagnation points can take place in the bulk of the fluid beneath such depression solitary waves, which will be  examined in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Other authors have studied the appearance of stagnation points beneath depression solitary waves [14, 15, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  However, these works considered gravity-capillary waves in the absence of electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Moreover, it has been  shown that the location of the stagnation points does not change much for choices of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Having said this, we  focus on investigating the effect of the electric field as a mechanism to create stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' To address this  issue we fix the vorticity, the Bond number and vary the intensity of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
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+page_content='95  Figure 6: Phase portrait for different values of the vorticity with a solitary wave with amplitude A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='5, Ω = 5  and B = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The critical value of the vorticity in which the first stagnation point appears at the bottom is  Ω∗ ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='2780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Figure 6 depicts a series of simulations from which we can see that in the presence of a strong electric field  stagnation points can appear in the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The location of the stagnation points is determined in two  ways– (i) by finding the equilibrium points of the dynamical system (24), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=', we find the zeros of the velocity  field or (ii) by the contour function of MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' The flow structure beneath the depression solitary wave can  have (i) zero, (ii) two centres (at the bottom), (iii) two centres (in the bulk of the fluid) and four saddles (at the  bottom) or (iv) two centres (in the bulk of the fluid) and four saddles (two at the bottom and two in the bulk  of the fluid) as stagnation points depending on the intensity of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' This features the bifurcation  of flow according to the F 2  E parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Similar descriptions of the arrangement of the stagnation point in the  context of gravity-capillary waves were reported in the work of Flamarion [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  It is well acknowledged that the full Euler equations are the most realistic model to reproduce EHD scenarios  in inviscid fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' However, reduced models can reproduce qualitatively the same features of the flow with  comparatively little effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For instance, our results show that the weakly nonlinear weakly dispersive regime  can capture rich flow structures, such as recirculation zones and stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  6  Conclusion  In the presented study the flow structure beneath EHD flows with constant vorticity was investigated numerically  in the Korteweg Benjamin-Ono equation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Solitary waves were computed numerically through the  standard Newton’s method combined with Fourier spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' This approach allowed us to approximate  the velocity field beneath the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' As a consequence, the location of stagnation points and details of the  8  recirculation zones formed by them were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' For elevation solitary waves, we showed that the location  of the centre points does not change significantly by variations on the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' It is remarkable that for  depression solitary waves the electric field acts as a mechanism for the creation of stagnation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' In the  absence of an electric field even when the vorticity is strong there is no stagnation point in the bulk of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  The results presented in this work are expected to agree well with the full nonlinear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' An attempt to  compare the results predicted by both models is a natural path to be pursued in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Acknowledgments  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' F and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='-Jr are grateful to IMPA for hosting them as visitors during the 2023 Post-Doctoral Summer  Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Data Availability Statement  Data sharing is not applicable to this article as the parameters used in the numerical experiments are informed  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
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+page_content=' Vanden-Broeck, Progressive flexural-gravity waves with constant vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  Fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content=' 995 (2020) A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE5T4oBgHgl3EQfYQ_L/content/2301.05573v1.pdf'}
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+arXiv:2301.08716v1  [eess.SY]  20 Jan 2023
+Minimum Time Control of a Gantry Crane System with Rate Constraints
+Adrian Steina, Tarunraj Singha
+aDepartment of Mechanical and Aerospace Engineering, University at Buffalo (SUNY),
+Buffalo, NY 14260-4400, USA
+Abstract
+This paper focuses on the development of minimum time control profiles for point-to-point motion of a
+gantry crane system in the presence of uncertainties in modal parameters. Assuming that the velocity of
+the trolley of the crane can be commanded and is subject to limits, an optimal control problem is posed to
+determine the bang-off-bang control profile to transition the system from a point of rest to the terminal states
+with no residual vibrations. Both undamped and underdamped systems are considered and the variation
+of the structure of the optimal control profiles as a function of the final displacement is studied. As the
+magnitude of the rigid body displacement is increased, the collapse and birthing of switches in the optimal
+control profile are observed and explained. Robustness to uncertainties in modal parameters is accounted
+for by forcing the state sensitivities at the terminal time to zero. The observation that the time-optimal
+control profile merges with the robust time-optimal control is noted for specific terminal displacements and
+the migration of zeros of the time-delay filter parameterizing the optimal control profile are used to explain
+this counter intuitive result. A two degree of freedom gantry crane system is used to experimentally validate
+the observations of the numerical studies and the tradeoff of increase in maneuver time to the reduction of
+residual vibrations is experimentally illustrated.
+Keywords:
+Input Shaper, Gantry Crane, Time-Optimal Control, Rest-to-Rest Maneuvers.
+1. Introduction
+Control of cranes is a topic that has garnered increased interest over the past three decades coinciding
+with the growth in the use of prefiltering approaches to minimize residual vibrations of systems characterized
+by underdamped motion. A vast majority of crane controllers can be classified as open-loop or closed-loop,
+with a few combining feedforward and feedback controllers in a tracking framework. One open-loop approach
+is called input shaping [1] which consists of a time-delay filter which is designed to cancel the underdamped
+poles of the system [2].
+The domain of input shaping has matured and can account for uncertainties
+in model parameters.
+To account for uncertainties in the estimated damping or natural frequencies of
+the underdamped poles, multiple zeros of the time-delay filter are placed at the nominal locations of the
+underdamped poles, resulting in robustness to uncertainties in the modal parameters. Controllers which are
+robust around the nominal model [1] and those that account for interval domains of uncertainties [3] have
+been developed. Constraints on jerk [4] and deflection [5] have also been taken into account in the design.
+Recently distributed delay input shapers [6] have been studied which introduce a novel parameterization in
+the design of input shapers. Including input shapers within a feedback loop has also been considered [7, 8]
+as researchers explore techniques to exploit the strengths of input shapers.
+Noakes, Petterson, and Werner [9] proposed a switching control profile to generate oscillation-damped
+transport and swing-free stop. Their technique consists of bang-off-bang acceleration profiles in which the
+pulses are timed to minimize the cable sway during the maneuver and results in a swing-free stop. They
+∗Corresponding author
+Email address: tsingh@buffalo.edu (Tarunraj Singh)
+1
+
+experimentally demonstrated the results of the open-loop control design. Shah and Hong [10] applied input
+shaping for the underwater transport of nuclear power plant’s fuel. There have been numerous publications
+related to the use of input shapers [1, 2, 11] for sway control of cranes [12, 5, 13, 14, 15, 16, 17, 18, 19,
+20, 21, 22]. Maghsoudi et al. [23] applied a distributed time-delay filter on a gantry crane. They used the
+method proposed by Vyhlidal et al. [6] for uncertainty studies for gantry crane control and demonstrated
+that applying a distributed time-delay filter lead to an asymmetric robustness behaviour of the residual
+sway about the nominal stage. Ramli et al. [24] designed a neural network-based input shaper while Yavuz
+and Beller [25] used neural networks for a closed-loop controller when it is combined with an input shaper.
+Wahrburg et al. [26] and Ramli et al. [27] applied input shaping for maneuvers of an overhead crane with
+non-zero initial conditions, with an objective of zero residual oscillations of the payload. Additionally, work
+has been done in command shaping control for non-zero initial and final conditions [28]. Stein and Singh [29]
+presented simulation results of velocity constrained design of input shaped control profiles for a gantry crane
+system. Fliess et al. [30] used the concept of differential flatness to control the traversing and hoisting of an
+overhead crane. This differential flatness based design was extended to discrete time design by Diwold et
+al. [31].
+Alli and Singh [32] designed passive controllers for a distributed parameter representation of the crane
+cable for point-to-point maneuvers where the integral of the time absolute error is minimized. O’Connor [33]
+used a wave equation representation of the cable dynamics assuming that the velocity of the trolley could be
+commanded. The velocity of the trolley was assumed to be constrained and the damping was assumed to be
+zero. Researchers included particle swarm optimization into controller design for overhead cranes [34, 35].
+Golovin et al. [36] developed a H∞ robust controller for actively damping the structural vibrations of the
+gantry crane system. Various control methods have also been proposed where the overhead crane is modelled
+as a double pendulum system [37, 38, 39, 40]. A few papers proposed controllers which combine closed-loop
+controller used in conjunction with open-loop shaped profiles to track. Kolar et al. [41] proposed a hybrid
+solution combining an open-loop generated crane trajectory as a reference signal and closed-loop controller
+for handling external disturbances. Li et al. [42] introduced an online planning method for minimum-time
+control of overhead cranes. Furthermore, many papers considered the payload as a point mass whereas Stein
+and Singh [43] proposed an input shaper used in conjunction with a proportional-derivative controller for a
+crane with an inertial payload. Other work has considered sliding mode [44, 19, 45], adaptive control [46],
+discrepancy-based control [47] and compared different control strategies [48] while including various external
+disturbances on cranes. Apart from time-optimal control, Sun et al. [49] investigated an energy-optimal
+controller for an underactuated double pendulum crane with state and control constraints. Compared to
+active vibration suppression of a crane’s payload, Yurchenko et al. [50] used a passive method with an
+absorber.
+This paper considers a tabletop gantry crane system driven by stepper motors which permits commanding
+the position of the trolley. By imposing velocity limits on the trolley motion, this paper considers the design
+of velocity constrained time-optimal point-to-point control of a crane moving in two dimensions.
+Since
+the pendular motion is almost undamped, an undamped system model was first considered. Subsequently,
+the structure of switching function was used to parameterize the bang-off-bang profile and the resulting
+nonlinear programming problem was solved to determine the optimal solution for any arbitrary maneuver.
+The bang-off-bang control structure was then generalized to cater to multi-mode systems with underdamped
+modes.
+The main contributions of this work include: (1) Development of a velocity constrained time-optimal
+control profile for a gantry crane which is robust to uncertainties in modal parameters, (2) Illustration of
+the non-intuitive result that the robust and non-robust solutions are coincident for specific displacements,
+(3) Illustration that a rectangular pulse input can attenuate the dominant vibratory modes for specific
+displacements, (4) Illustration of the change in structure of the optimal control profile, and (5) Experimental
+validation of all the aforementioned observations.
+Section 2 presents the development of velocity constrained time-optimal control for an undamped gantry
+crane system followed by the development of controllers which are robust to uncertainties in the undamped
+frequencies in Section 3. Section 4 generalizes the optimal control formulation for a multi-mode system
+with damped or undamped modes. Section 5 presents a simple approach to determine the transition in the
+2
+
+structure of the control profile, followed by the validation of the design on a two degree of freedom gantry
+crane system in Section 6. The paper concludes with a brief summary of the results of the paper.
+2. Undamped System
+β
+m
+xi
+g
+L
+x = xi + Lβ
+xi
+x
+k
+m
+Figure 1: Equivalent spring-mass system for small angle displacements.
+The gantry crane setup includes a trolley driven by a stepper motor which permits the command of
+the trolley’s velocity by assuming that the acceleration is zero as the commanded velocity transitions. A
+schematic of the crane and an equivalent spring-mass system are shown in Fig. 1 where a small angle
+displacement is assumed [51] [52]. The spring-mass model can be written as:
+mL ¨β(t) + mg sin (β(t)) = 0
+(1)
+m¨x(t) − m✟✟✟
+✯0
+¨xi(t)
++ mg
+�x(t) − xi(t)
+L
+�
+= 0
+(2)
+↔ m¨x(t) + kx(t) − kxi(t) = 0
+(3)
+˙xi(t) = v(t)
+(4)
+where v the velocity of the trolley is considered as the input and is constrained 0 ≤ v ≤ Vm. The assumption
+that velocity of the trolley can be used as a control input is based on papers which demonstrate velocity
+input control of industrial cranes [53, 54, 55, 44]. Mass normalization leads to the state space equation:
+
+
+˙x1
+˙x2
+˙x3
+
+
+� �� �
+˙X
+=
+
+
+0
+1
+0
+−ω2
+n
+0
+ω2
+n
+0
+0
+0
+
+
+�
+��
+�
+A
+
+
+x1
+x2
+x3
+
+
+� �� �
+X
++
+
+
+0
+0
+1
+
+
+����
+B
+v
+(5)
+where ωn =
+�
+k
+m. The time-optimal control problem can be posed as:
+min J =
+� tf
+0
+dt
+(6a)
+subject to
+˙X = AX + Bv
+(6b)
+X(0) =
+�0
+0
+0�T
+(6c)
+X(tf) =
+�xf
+0
+xf
+�T
+(6d)
+0 ≤ v ≤ Vm
+∀t
+(6e)
+Defining the Hamiltonian as:
+H = 1 + λT (AX + Bv) ,
+(7)
+3
+
+the necessary conditions for optimality can be derived using calculus of variations, resulting in the equations
+˙X = ∂H
+∂λ = AX + Bv
+(8a)
+˙λ = −∂H
+∂X = −AT λ
+(8b)
+v = VmH(−BTλ)
+(8c)
+X(0) = 0 and X(tf) =
+�
+xf
+0
+xf
+�T
+(8d)
+H = 0 at t = 0
+(8e)
+where H is the Heaviside step function. Eq. (8c) is derived using Pontryagin’s minimum principle (PMP)
+which requires the optimal trolley velocity to be bang-off-bang.
+Eq. (8b) can be solved in closed form
+resulting in the equation:
+λ(t) = e−AT tλ(0)
+(9)
+which can also be written as:
+λ1(t) = cos(ωnt)λ1(0) + ωn sin(ωnt)λ2(0)
+(10)
+λ2(t) = −sin(ωnt)
+ωn
+λ1(0) + cos(ωnt)λ2(0)
+(11)
+λ3(t) = (1 − cos(ωnt))λ1(0) − ωn sin(ωnt)λ2(0) + λ3(0)
+(12)
+for undamped systems. Since the switching function is BTλ, it reduces to the third costate which reveals
+that the switching function is a non-zero mean harmonic. This structure will help comprehend the change in
+structure of the optimal control profile v(t) as a function of the final displacement xf. Since the Hamiltonian
+H at all times, including the initial time, is zero, we have:
+H(0) = 1 + λT (AX(0) + BVm) = 0
+(13)
+⇒ λ3(0) = − 1
+Vm
+.
+(14)
+Further, for a rest-to-rest maneuver, which is the focus of this paper, the Hamiltonian H at the final time
+is:
+H(tf) = 1 + λT (AX(tf) + BVm) = 0
+(15)
+⇒ λ3(tf) = − 1
+Vm
+.
+(16)
+Substituting Eq. (16) into Eq. (12), we can show that:
+λ2(0)
+λ1(0) = 1
+ωn
+tan
+�ωntf
+2
+�
+.
+(17)
+Evaluating the slope of λ3(t) at the initial time and final time, we can show that:
+˙λ3(0) = −ω2
+nλ2(0), and ˙λ3(tf) = ω2
+nλ2(0),
+(18)
+which permits us to conclude that the switching function is anti-symmetric about the mid-maneuver time,
+which implies there will always be an even number of switches and that pairs of switch times are equally
+distant from the mid-maneuver time. Using Eq. (17), the time derivative of λ3(t) evaluated at the mid-
+maneuver time is:
+˙λ3
+�tf
+2
+�
+= ωn sin
+�
+ωn
+tf
+2
+�
+λ1(0) − ω2
+n cos
+�
+ωn
+tf
+2
+�
+λ2(0) = 0
+(19)
+4
+
+which implies that the slope of the switching curve is always zero at the mid-maneuver time. This prompts
+parameterising the optimal control profile as:
+v(t) = Vm (1 − H(t − (T2 − T1)) + H(t − (T2 + T1)) − H(t − 2T2))
+(20)
+as illustrated in Fig. 2, which in the frequency domain is:
+Figure 2: Anti-symmetric time-optimal control profile.
+V (s) = Vm
+s
+�
+1 − e−s(T2−T1) + e−s(T2+T1) − e−2sT2�
+�
+��
+�
+Gc(s)
+(21)
+where Gc(s) is the transfer function of the time-delay filter which generates the bang-off-bang control profile
+when subjected to a step input. By requiring that a pair of zeros of Gc(s) cancel the undamped poles of
+the system and the pole at the origin, one can derive the constraint to formulate a parameter optimization
+problem. It can be seen that:
+Gc(s = 0) = 1 − 1 + 1 − 1 = 0
+(22)
+which implies that the transfer function has a zero at the origin. The rigid body boundary condition is
+determined by integrating the bang-off-bang velocity profile leading to the equation:
+xi(t) = Vm (t − (t − T2 + T1)H(t − T2 + T1) + (t − T2 − T1)H(t − T2 − T1) − (t − 2T2)H(t − 2T2))
+(23)
+which at the final time of t = tf = 2T2 leads to the equation:
+xi(2T2) =xf = Vm (2T2 − (T2 + T1) + (T2 − T1))
+(24)
+↔ xf =2Vm(T2 − T1) ⇒ (T2 − T1) = xf
+2Vm
+.
+(25)
+Furthermore, to cancel the undamped poles at s = ±jωn, we have:
+Gc(s = jωn) = 1 − e−jωn(T2−T1) + e−jωn(T2+T1) − e−2jωnT2 = 0
+(26)
+which reduces to:
+1 − cos(ωn(T2 − T1)) + cos(ωn(T2 + T1)) − cos(2ωnT2) = 0
+(27)
+sin(ωn(T2 − T1)) − sin(ωn(T2 + T1)) + sin(2ωnT2) = 0
+(28)
+which simplifies to:
+2 sin(ωnT2) (sin(ωnT2) − sin(ωnT1)) = 0
+(29)
+2 cos(ωnT2) (sin(ωnT2) − sin(ωnT1)) = 0
+(30)
+5
+
+which leads to the solution:
+T2 = π
+ωn
+− T1
+(31)
+which results in the closed form solution:
+2T2 = π
+ωn
++ xf
+2Vm
+and T1 =
+π
+2ωn
+− xf
+4Vm
+(32)
+where 2T2 is the maneuver time tf. Since both T1 and T2 are functions of the maneuver xf, the scenario of
+collapse of the switches requires T1 = 0 which results in the constraint:
+xf = 2πVm
+ωn
+(33)
+which is a bang profile or a constant velocity profile. It should be noted that the collapse of the switches is
+proportional to one period of the switching function 2π
+ωn . Any increase in time associated with a maneuver
+greater than the bang profile will introduce two additional switches. This results from the fact that there
+will be two peaks or troughs of the switching profile, which results in four switches. An observation which
+will be exploited later is the fact that since the switching function is a harmonic with a bias, the width of
+all the off zones in the bang-off-bang profiles will be the same. Consider a constant velocity profile with no
+zero zones which can be parameterized as:
+v(t) = Vm (1 − H(t − 2T2))
+(34)
+which can be represented in the frequency domain as:
+V (s) = Vm
+s
+�
+1 − e−2sT2�
+�
+��
+�
+Gc(s)
+.
+(35)
+The final displacement is given by the equation:
+xf = 2VmT2.
+(36)
+The requirement that the transfer function Gc(s) places a zero at the location of the undamped poles of the
+system s = ±jωn requires the constraints:
+1 − cos(2ωnT2) = 0
+(37)
+− sin(2ωnT2) = 0
+(38)
+which results in the solution:
+T2 = nπ
+ωn
+, where n = 1, 2, 3, . . ..
+(39)
+Substituting the solution T2 into Eq. (36), we have:
+xf = 2nπVm
+ωn
+.
+(40)
+The number of zero zones n depends on the desired terminal displacement and is given by the constraint:
+2(n − 1)πVm
+ωn
+≤ xf ≤ 2nπVm
+ωn
+.
+(41)
+For example, when 0 ≤ xf ≤
+2πVm
+ωn , one zero velocity zone exists in the optimal velocity profile. Since
+all the off zones of the bang-off-bang profiles are the same, we parameterize the width of each zero zone
+as 2T1 and only two parameters are needed to determine the velocity limited time-optimal control profile.
+Fig. 3 illustrates the variation of the switch time for each zone as the terminal displacement increases. The
+6
+
+Figure 3: Switching structure and switch time variation as a function of xf and τi = Ti/tf . Shaded areas illustrate the time
+span of T1 and the blue curve shows the switching function. Different background colors illustrate the three different picked
+Zones for (I) Zone 1, (II) Zone 2, (III) Zone 3.
+7
+
+change in structure of the time-optimal control profile with increasing terminal displacements is illustrated
+for three consecutive zones. Zone 1 is characterized by two switches which collapse for a specific terminal
+displacement. Any increase in terminal displacement results in the introduction of two additional switches
+in the optimal control profile. Fig. 3 also plots the switching function which illustrates that as the optimal
+control profile transitions from one zone to the next. The switching function includes a phase shift of π
+radians, which results in new switches being birthed at the initial and terminal times of the maneuver. For
+higher zones, switches can be introduced at the mid-maneuver time and 2π/ωn distance from each other.
+To determine the values for T1, which is half the width of each zero velocity zone, and T2, which is
+half of the maneuver time, a parameter optimization problem is solved which minimizes T2 subject to the
+constraints:
+T2 − nT1 = xf
+2Vm
+(42)
+(−1)n+1n sin(ωnT1) − sin(ωnT2) = 0
+(43)
+where 2T1 is the width of each of the zero pulse and 2T2 is the maneuver time. Eq. (42) is derived from
+the final displacement constraint and Eq.(43) is derived from the constraint that the time-delay filter needs
+to locate a pair of zeros at the location of the undamped poles of the system. Eqs. (42) and (43) can be
+transformed into polynomial equations which can be solved efficiently to identify the optimal parameters
+T1 and T2. The Appendix provides the details of the transformation to generate polynomial equations for
+zones 2 and 3 for illustrative purposes.
+It is interesting to note that as xf → 0, Eq. (42) requires T1 → T2. Eq. (43) where n = 1 requires
+ωnT2 = π − ωnT1
+(44)
+which results in the equation:
+2T2 = π
+ωn
+(45)
+which one can note is analogous to the two-impulse input shaper [1] which for an undamped system places
+the two impulses of equal magnitude half a period of oscillation apart. Note that 2T2 corresponds to the
+maneuver time.
+The upper graph of Fig. 4 illustrates the variation of the switch times and the maneuver time as a
+function of the terminal displacement xf. The solid colored zone corresponds to the time intervals when the
+input (velocity command) is at the maximum and the rest of the time intervals are when the input is zero.
+When xf = 240 mm, the two switches collapse resulting in a constant velocity profile. This is followed by
+the birth of four switches which collapse concurrently for a terminal displacement of xf = 480 mm following
+which a six switch optimal control profile is birthed. The lower half of Fig. 4 illustrates the optimal solution
+for three unique displacements in zones 1, 2 and 3, labelled “a”, “b”, and “c” respectively.
+3. Robust Control
+The challenge of dealing with model parameter uncertainties is ubiquitous and there have been numerous
+approaches proposed for the design of robust open-loop controllers including enforcing robustness around
+the nominal model of the system or a minimax problem formulation where the maximum residual energy is
+minimized over an interval of uncertainty. In this research we determined the sensitivity of the states of the
+system with respect to uncertainty in the spring stiffness which correspond to uncertainties in the natural
+frequency and force the state sensitivities with respect to the uncertain frequency to zero at the terminal
+8
+
+Figure 4: Switch and maneuver time variation of a non-robust time-optimal controller for an undamped 1 mode system. (a),
+(b) and (c) show the control profiles for xf = 100 mm, xf = 300 mm and xf = 550 mm in Zone 1, Zone 2 and Zone 3
+respectively.
+time. The resulting augmented state space model is:
+˙x1(t) = x2(t)
+(46)
+˙x2(t) = −ω2
+nx1(t) + ω2
+nx3(t)
+(47)
+d ˙x1(t)
+dωn
+= dx2(t)
+dωn
+(48)
+d ˙x2(t)
+dωn
+= −2ωnx1(t) − ω2
+n
+dx1(t)
+dωn
++ 2ωnx3(t)
+(49)
+˙x3(t) = v(t)
+(50)
+0 ≤ v ≤ Vm.
+(51)
+and is subject to the initial and final conditions:
+x1(0) = x2(0) = x3(0) = 0
+(52)
+dx1(0)
+dωn
+= dx2(0)
+dωn
+= 0
+(53)
+x1(tf) = x3(tf) = xf
+(54)
+x2(tf) = dx1(tf)
+dωn
+= dx2(tf)
+dωn
+= 0.
+(55)
+The robust velocity constrained time-optimal control problem for the undamped system is solved for various
+displacements. The variation in the optimal control profile parameterized by the switch times and maneuver
+time are illustrated in Fig. 5 as a function of the terminal displacement. Unlike the variation of the switch
+times as a function of terminal displacement and the concurrent collapse of the switches, this phenomena
+is not observed in the spectrum of switch times in Fig. 5. It nevertheless should be noted that the optimal
+control profile births and collapses switches as a function of terminal displacements, resulting in an optimal
+control profile where the number of switches is a function of the terminal displacement. Fig. 6 illustrates the
+variation of the residual energy at the terminal time for the non-robust and robust time-optimal controllers
+over a range of uncertain natural frequencies for the undamped system. It is clear that the red line, which
+represents the variation of residual energy of the robust control, outperforms the non-robust design illustrated
+by the blue line. These graphs are generated for a terminal displacement of xf = 50 mm and a natural
+frequency of ωn = 2π. The residual energy is given by:
+9
+
+Figure 5: Switch and maneuver time variation of a robust time-optimal controller for an undamped 1 mode system.
+4
+8
+12
+0
+16
+x103
+Residual Energy V
+4
+5
+4.5
+5.5
+6
+6.5
+7
+7.5
+8
+8.5
+non-robust
+robust
+ωn [rad/s]
+Figure 6: Residual energy at tf of a non-robust and robust time-optimal controller for a perturbation of ±30 % in ωn at
+xf = 50 mm.
+V (t) = 1
+2 ˙x(t)2 + 1
+2ω2
+n (x(t) − xf)2 .
+(56)
+The sensitivity of the residual energy with respect to the natural frequency is:
+dV (t)
+dωn
+= ˙x(t)d ˙x(t)
+dωn
++ ωn (x(t) − xf)2 + ω2
+n (x(t) − xf) dx(t)
+dωn
+.
+(57)
+Since the time-optimal control profile forces the terminal states to be x(tf) = xf and ˙x(tf) = 0, V (tf) and
+dV (tf)
+dωn
+= 0, irrespective of the magnitude of the sensitivity states, dV (t)
+dωn
+= 0 at the terminal time. The
+second derivative of the residual energy with respect to the natural frequency is:
+d2V (t)
+dω2n
+=
+�d ˙x(t)
+dωn
+�2
++ ˙xd2 ˙x(t)
+dω2n
++ (x(t) − xf)2 + 4ωn (x(t) − xf) dx(t)
+dωn
++ ω2
+n
+�dx(t)
+dωn
+�2
+...
+... + ω2
+n (x(t) − xf) d2x(t)
+dω2n
+(58)
+which is the curvature since the first derivative is zero at the nominal frequency ωn.
+The curvature is
+defined as the reciprocal of the radius of a circle which best approximates the curve V (ωn). The curvature
+can be used as a measure of the robustness of the control profile in the proximity of the nominal model
+with a smaller curvature (i.e. an osculating circle with large radius) indicating a smaller residual energy
+variation or greater robustness to uncertainties in the natural frequency. The solid and dashed lines in Fig. 7
+correspond to the respective non-robust and robust solutions, the blue curves represent the variation in the
+maneuver time as a function of terminal displacement, and the black lines represent the variation of the
+residual energy and its derivatives with terminal displacement. The third panel of Fig. 7 shows that there
+is a profound variation in the curvature of residual energy function for the non-robust solution as a function
+of terminal displacement in comparison to that of the robust control which is zero for all displacements. It
+10
+
+tf,nonrobust
+tf,robust 
+V, dV/d�n , d2V/d�n
+2 (non-robust)
+V, dV/d�n , d2V/d�n
+2 (robust)
+Residual Energy V
+0
+200
+400
+600
+0
+2
+4
+0
+2
+4
+1
+0
+�1
+2
+0
+4
+6 x105
+Final Displacement xf [mm]
+d2V/d
+�
+n
+2
+Final Time tf [s]
+dV/d
+�
+n
+0
+2
+4
+1
+0
+�1
+Figure 7: Variation of terminal time, residual energy, and its derivatives as a function of xf for a non-robust and robust
+time-optimal controller to illustrate the collapses of the switch times for an undamped 1 mode system.
+11
+
+is also intriguing that, for specific terminal displacements, the robust and non-robust solutions are identical,
+i.e., the maneuver times are the same and the curvatures are both zero. To comprehend this unique result
+the loci of the zeros of the time-delay filter which generates the bang-off-bang control profiles are generated
+and are illustrated in Fig. 8.
+Figure 8: Loci of zeros of the optimal time-delay filter for a non-robust and robust controller as a function of xf for a 1 mode
+system.
+All the graphs are generated for a nominal frequency of ωn = 2π. As illustrated by the upper graph of
+Fig. 8, the non-robust design requires that the time-delay filter place zeros at the location of the nominal
+undamped poles of the plant. Meanwhile, the lower graph of Fig. 8 shows that the robust design mandates
+that a pairs of zeros be located at the nominal poles of the plant. The horizontal loci of zeros for both the
+robust and non-robust designs illustrates that the time-delay filter places single and double sets of zeros at
+the nominal location of the poles for the non-robust and robust designs respectively. It can also be noted
+that the zero loci for the non-robust design intersects the horizontal loci for a terminal displacement of
+xf = 266.5 mm (shown by the red cross), resulting in a pair of coincident zeros which is what the robust-
+controller is mandated to do. This is the reason why, for specific terminal displacements, the robust and
+non-robust designs results in identical solutions. It can also be seen that the same phenomena occurs for a
+terminal displacement of xf = 513.3 mm.
+Fig. 9 illustrates the location of the zeros of the time-delay filter for the non-robust and robust designs
+for the terminal displacement xf = 500 mm. The blue circles correspond to the non-robust design and the
+red to the robust design. It should be noted that the robust design includes two pairs of zeros at ±2πj and
+the non-robust design includes one pair of zeros. The arrows associated with the blue and red circles are
+12
+
+Figure 9: Residual energy versus uncertain model frequency for a non-robust and robust controller at xf = 500 mm for a 1
+mode system. The arrows indicate the slopes of zero-loci for xf > 500 mm. ωn,design shows the frequency the time-delay
+filters were designed for.
+the slopes of the loci of the zeros as a function of terminal displacement xf. It is evident that a blue zero is
+transitioning from above 2πj with a negative slope and is shown to coincide with the nominal poles of the
+system for xf = 513.3 mm, resulting in identical robust and non-robust designs.
+The red and blue lines in Fig. 9 illustrate the variation in residual energy as the location of the nominal
+poles of the system are varied. The non-robust design outperforms the robust design for perturbation of
+the uncertain frequency above ωn = 2π, while the robust design outperforms the non-robust design for
+perturbations below the nominal frequency. This asymmetry is due to the fact that the non-robust design
+has a zero located immediately above the nominal frequency and ends up acting like the zero locations of a
+minimax design [3]. It can also be seen that, for large perturbations of the uncertain frequency, the residual
+energy goes to zero at locations where the time-delay filter has a zero.
+4. Generalization
+Section 2 dealt with an undamped system with one mode which permitted a reduced order param-
+eterization of the optimal control profile by exploiting the symmetrical nature of the control about the
+mid-maneuver time. This symmetry is attributed to the fact that the oscillatory motion excited by the
+input does not damp out and the symmetric input can, by virtue of linearity of the system, generate an out
+of phase motion of the undamped modes which cancels the existing oscillations. For a system with damping,
+the symmetric nature of the control profile is lost since the amplitude of the oscillatory mode decays over
+time. Consequently, the optimal control profiles need to explicitly parameterize every switch in addition to
+the maneuver time. Furthermore, there may be multiple modes which contribute to the output of interest
+and those modes need to be quiescent at the end of the maneuver as well. For a system with multiple modes,
+characterized by damped or undamped modes, the bang-off-bang control profile is parameterized as:
+v(s) = Vm
+s
+�
+1 +
+N+1
+�
+i=1
+(−1)ie−sTi
+�
+�
+��
+�
+=Gc(s)
+(59)
+where N is the number of switches which is an even number and TN+1 is the maneuver time.
+13
+
+The constraints to identify the optimal values of Ti are derived by requiring the zeros of the time-delay
+filter to cancel the pole at the origin and the underdamped poles of the system. This results in the constraint:
+Gc(s = 0) = 1 +
+N+1
+�
+i=1
+(−1)i = 0
+(60)
+which is automatically satisfied by the parameterization of the optimal control profile. To cancel the complex
+conjugate poles at s = −σk ± j ωd,k = −ζkωn,k ± j ωn,k
+�
+1 − ζ2
+k, where k = 1, 2, . . . , m are the m modes
+whose poles need to be canceled by the zeros of the time-delay filter. This results in the constraints:
+1+
+N+1
+�
+i=1
+(−1)ieσkTi cos(ωd,kTi) = 0
+(61)
+N+1
+�
+i=1
+(−1)ieσkTi sin(ωd,kTi) = 0, k = 1, 2, . . ., m.
+(62)
+To satisfy the terminal rigid body displacement, we require:
+x(TN+1) = xf =
+� tf
+0
+Vmdt
+(63)
+= Vm
+�
+TN+1 +
+N
+�
+i=1
+(−1)i(TN+1 − Ti)
+�
+(64)
+→ xf = Vm
+�
+TN+1 −
+N
+�
+i=1
+(−1)iTi
+�
+.
+(65)
+The nonlinear optimization problem that requires solving is:
+min J = TN+1
+(66a)
+subject to
+Vm
+�
+TN+1 −
+N
+�
+i=1
+(−1)iTi
+�
+= xf
+(66b)
+1 +
+N+1
+�
+i=1
+(−1)ieσkTi cos(ωd,kTi) = 0
+(66c)
+N+1
+�
+i=1
+(−1)ieσkTi sin(ωd,kTi) = 0, k = 1, 2, . . ., m
+(66d)
+0 ≤ (Ti+1 − Ti), for i = 0, 1, 2, . . ., N
+(66e)
+where T0 = 0. As in the case of the undamped system, the number of switches necessary to parameterize
+the optimal control profile changes with the terminal displacement. Fig. 10 illustrates the variation of the
+switch times as a function of the final displacement and the control input over the maneuver time for a single
+mode system with an underdamped system model (ζ = 0.01). For terminal displacements of 0 − 700 mm,
+three zones have been identified, separated by the vertical dashed lines. Three representative optimal control
+profiles are presented corresponding to three displacements highlighted by the dotted lines and labelled a,
+b, and c. It is interesting to note that in zone 3, which includes the dotted line c, the number of switches
+required for the optimal control profile starts with four switches, transitions to six switches, and returns to
+a four switch optimal control profile before finally transitioning to a two switch profile.
+14
+
+Figure 10: Switch and maneuver time variation of a non-robust time-optimal controller for an underdamped 1 mode system.
+5. Switching Profile Transition
+It is clear from Figs. 4 and 10 that the structure of bang-off-bang control profile changes as a function
+of the terminal displacement. In Fig. 4, for small displacements (xf ≤ 240 mm), the optimal control profile
+is first characterized by two switches and the switches, then, after collapsing, result in a pulse control
+profile which births a four switch control profile which subsequently transitions to a six switch control
+profile. The change in structure of the optimal control profile for underdamped systems is more involved
+as all switches do not collapse for the same value of the terminal displacement. To exactly determine the
+terminal displacement which corresponds to the birth or collapse of two or more switches, the constraint is
+that the switching function and its time derivative are simultaneously zero at some time instant. For the
+underdamped system the switching function λ3(t) can be represented as:
+λ3(tcr, xf) = ˙λ3(tcr, xf) = 0
+(67)
+where tcr is the switch time where two switches collapse. tcr and xf can be determined by solving the
+two nonlinear simultaneous equations in two unknowns while satisfying all the necessary conditions for
+optimality.
+6. Experimental Results
+A scaled model of a gantry crane was fabricated to test and validate the time-optimal control profiles
+presented in this paper. The workspace of the gantry crane includes a cuboid of dimensions 8’ × 4’ × 3’
+as shown in Fig. 11. Its main element is a trolley which slides on a rail and is able to move a payload over
+a 2-dimensional space (the winch motion of the crane is not active). The crane is equipped with 3 stepper
+motors of type NEMA 23 with a driver setting of 400 steps per revolution, where two of them are used for
+the y-axis and one for the x-axis. For this experiment just the motor of the x-axis is used. The drivers of the
+stepper motors are connected to an Arduino MEGA 2560 which receives the switch times and the desired
+position of the trolley as inputs. The maximum velocity of the trolley is set to 240 mm/s. The inset of
+Fig. 11 includes two images. The one on the left illustrates the 3D printed chassis which houses a cylindrical
+steel mass of 500g. A rope and a hook enable a connection between the chassis and the trolley making it a
+double pendulum system. The right panel illustrates the sensor integrated into the payload which includes
+two Arduino Nanos, a 3-axis gyroscope (MPU-6050), and two nRF24L01 single chip radio transceivers. One
+Arduino Nano is housed within the chassis to process the data provided by the gyroscope and to transmit
+the data to the other Arduino Nano, which is connected to a receiver. To permit a repeatable evaluation of
+the robustness of the controller, a cable deployment setup was designed which permits changing the length
+15
+
+Figure 11: Experimental setup of a gantry crane with an inset of the sensor part, which consists of a powerbank, Arduino
+Nano, 3-axis gyroscope (MPU-6050), and an nRF24L01 transceiver. Rotation around the y-axis is introduced as the swinging
+angle α. L1 is denoted as the rope length and L2 = 7.86 cm
+16
+
+OOKEREDOOKEREDL1 of the pendulum about a nominal length. The change in cable length changes the modal frequency of
+the system and was used to test the robustness of the controller to uncertainties in modal parameters of the
+gantry crane system.
+A series of experiments were conducted to illustrate some of the novel observations of the analytical
+study. The first of which included the collapse of the switches of the velocity constrained time-optimal
+control profile for a system with a single undamped mode. The simplest model of the gantry crane includes
+representing the suspended mass as a single undamped pendulum and a rectangular command, i.e., a bang
+control profile should cancel the undamped mode for a specific displacement. The natural frequency and
+damping ratio of the first mode of the pendulum with the nominal length were experimentally determined
+to be ωn = 0.6832 Hz and ζ1 = 0.001517 which was approximated to be zero in the development of the
+controller.
+Fig. 12 illustrates experimental results for terminal displacements resident in zone 1, which corresponds
+to the optimal control profile being characterized by a two switch bang-off-bang profile.
+A pulse with
+increasing width is progressively applied to illustrate the fact that for specific terminal displacements, a
+bang profile (pulse) results in zero terminal vibration. Starting from a terminal displacement of xf = 200
+mm and progressively increasing it until xf = 351.3 mm, the maneuver time increases, but at the same
+time, the maximum angular displacement of the pendulum α is decreasing until it is not present at the end
+of the maneuver for a displacement of 351.3 mm. It can be noted that a small high frequency oscillation is
+evident which corresponds to the second mode of the double pendulum system which is not considered in
+the controller design. For the next set of experiments, the double pendulum model of the crane is assumed
+Figure 12: Residual vibration variation when changing the final displacement from xf = 200 mm to xf = 351.3 mm for a pulse
+control profile. The blue curve illustrates the residual vibration at xf = 351.3 mm which validates the cancellation of the first
+mode of vibration.
+in the identification of the parameters of the two modes of oscillation. The second mode’s parameters are
+identified to be ωn,2 = 6.159 Hz and ζ2 = 0.026065 where the damping of the second mode is an order of
+magnitude greater than that of the first. For the two mode system, three controllers were tested. The first
+controller was designed to cancel the first mode only. The second controller was designed to cancel both of
+the modes at the end of the maneuver and the third controller was designed to be robust to uncertainties
+in both the modes’ natural frequencies. Fig. 13 illustrates the time variation of the angular displacement
+of the pendular payload where the green region corresponds to when the control input is active and the
+yellow region is the post-actuation domain. The first panel of Fig. 13 illustrates the impact of cancelling
+the first mode, the second panel corresponds to the controller cancelling both modes, and the third panels
+display the elimination of residual vibration when the robust time-optimal control profile is used to drive
+the system. These results were generated for a terminal displacement of 100 mm and the pendulum length
+L1 which corresponds to the nominal model which was used to identify the modal parameters. To illustrate
+the variation in residual energy as the pendular length is varied, five experiments were conducted for each
+17
+
+0
+1
+2
+3
+4
+Swinging Angle α [deg]
+1 mode filter (non-robust)
+2 mode filter (non-robust)
+2 mode filter (robust)
+1 �
+0
+�
+ 
+ 
+
+ 
+ 
+
+ 
+Time [s]
+operating time
+residual time
+tf
+Figure 13: Swinging angle α (around y-axis) for a 1 mode time-delay filter (non-robust), 2 mode time-delay filter (non-robust),
+and 2 mode time-delay filter (robust) when xf = 100 mm. Shaded regions help to distinguish between the operating and
+residual (off) time of the controller. The final time tf for each scenario is highlighted by a red dashed line.
+pendular length to account for uncertainties in initial conditions. With a total of seven perturbed lengths
+of the pendulum on either side of the nominal length, a total of 150 experiments were conducted. Box and
+whisker charts are used to illustrate uncertainties in the residual energy for each of the perturbed models.
+Fig. 14 illustrates the profound improvement in the use of the robust controller for which the residual
+vibration for the various pendulum lengths appear negligible compared to the residual energy resulting from
+the use of the non-robust control profiles. The final set of experiments illustrates a rather counter intuitive
+result which claims that the non-robust design and robust designs are coincident since the non-robust design
+for specific terminal displacements of the single-mode model places multiple zeros of the time-delay filter at
+the nominal location of the poles of the system. Fig. 15 illustrates the residual vibration box and whisker
+charts for a terminal displacement of xf = 390.1 mm, where it is evident that the residual energy curve is
+relatively flat. It should be pointed out that in this design, only the first mode of vibration was considered
+in the design, since observation of the coincidence of the robust and non-robust design has been observed
+for system with single undamped modes.
+7. Conclusions
+This paper presents an optimal control based development of a velocity limited minimum time control
+of a gantry crane system which is characterized by two modes of vibratory motion. The variation in the
+structure of the optimal control profile is presented for a single mode system where the vibratory modes
+are undamped or underdamped. It is noted that, as the final displacement increases, there is an increase
+in the number of switches in the optimal control profile with periodic terminal displacements requiring a
+pulse control profile with no switches. The optimal control framework is extended to account for multiple
+vibratory modes such as when the crane is modeled as a double pendulum. The state sensitivities are used to
+determine controllers which are robust to uncertainties in model parameters. Experimental results validate
+the counter intuitive observation that for specific terminal displacement the robust and non-robust optimal
+control profiles are coincident. The experimental results also clearly demonstrate the profound reduction in
+18
+
+Rope Length L1 [cm]
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+x10
+4
+Residual Energy Vy
+non-robust
+robust
+39.9
+41.5
+43.1
+44.7
+46.3
+47.9
+49.5
+51.1
+Figure 14: Residual energy variation at tf of time-optimal controllers for xf = 100 mm for different rope lengths L1 varying
+from 39.9 cm to 51.1 cm in 15 equally spaced intervals. The time-optimal controllers are designed for L1 = 45.5 cm. Blue and
+red shaded illustrate the residual energy statistics for the 2 mode non-robust and robust controller respectively. Furthermore,
+the time history of the swinging angle α for a non-robust vs robust scenario is illustrated by insets. The overall box and whisker
+chart includes 150 experiments.
+Rope Length L1 [cm]
+39.9
+41.5
+43.1
+44.7
+46.3
+47.9
+49.5
+51.1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+x10−4
+Residual Energy Vy
+n      
+ !
+"
+#$ % & '(
+Figure 15: Residual energy at tf of a time-optimal controller at xf = 390.1 mm for different rope length L1 varying from 39.9
+cm to 51.1 cm in 15 equally spaced intervals. The controller is designed for a system with L1 = 45.5 cm where the non-robust
+and robust solution collapse. The change of swinging angle α for the controller is illustrated by insets. The overall box and
+whisker chart includes 75 experiments.
+19
+
+residual vibrations of the double pendulum system when the length of the pendulum is changed to serve as
+a proxy for uncertainties in natural frequencies of the model.
+Acknowledgment
+The authors acknowledge the support of this work by the US National Science Foundation through
+CMMI Award number 2021710. The authors would like to thank Dr. Claude F. Leibovici for his help in
+transforming Eq. (42) and (43) to a polynomial equation.
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+
+Appendix
+The time-optimal control profile for zone 1 illustrated in Fig. 4 parameterized with two variables T1 and
+T2 has a closed form solution. For higher order zones, the transcendental constraint equations cannot be
+solved in closed form, but can be converted to a polynomial equation as illustrated for zones 2 and 3.
+Zone 2
+The parameterization of the time-optimal control for Zone 2 leads to the constraints:
+2 sin(ωnT1) + sin(ωnT2) = 0
+(A-1)
+2T2 − 4T1 = xf
+Vm
+.
+(A-2)
+Let xf/Vm = 2aωn, which gives 2T2 − 4T1 = 2aωn or T2 = 2T1 + aωn. Eq. (A-1) becomes 2 sin(ωnT1) +
+sin(2ωnT1 + aω2
+n) = 0. This can be further simplified to:
+2 sin (ωnT1) + sin (2ωnT1) cos
+�
+aω2
+n
+�
++ sin
+�
+aω2
+n
+�
+cos (2ωnT1) = 0.
+(A-3)
+Let α = sin
+�
+aω2
+n
+�
+, β = cos
+�
+aω2
+n
+�
+and t = ωnT1:
+2 sin(t) + β sin(2t) + α cos(2t) = 0.
+(A-4)
+Let t = cos−1(z), resulting in the simplified equation:
+2 sin(cos−1(z)) + β sin(2cos−1(z)) + α cos(2cos−1(z)) = 0
+(A-5)
+2
+�
+1 − z2 + β2z
+�
+1 − z2 + α
+�
+2z2 − 1
+�
+= 0
+(A-6)
+2
+�
+1 − z2 (βz + 1) + α
+�
+2z2 − 1
+�
+= 0
+(A-7)
+α
+�
+2z2 − 1
+�
+= −2
+�
+1 − z2 (βz + 1)
+(A-8)
+α2 �
+4z4 − 4z2 + 1
+�
+= 4
+�
+1 − z2� �
+β2z2 + 2βz + 1
+�
+(A-9)
+�
+4α2 + 4β2�
+z4 + 8βz3 +
+�
+−4α2 − 4β2 + 4
+�
+z2 − 8βz + α2 − 4 = 0.
+(A-10)
+Exploiting the knowledge that α2 + β2 = 1, we have:
+4z4 + 8βz3 − 8βz + α − 4 = 0.
+(A-11)
+From Eq. (A-11) the parameter T1 can be calculated by using t = cos−1(z) and T1 = t/ωn. Assuming
+that ωn is given and the user can choose any xf which lies in the bounds of the zone, the quartic equation
+provides a solution for the switch time T1. From there, the mid-maneuver time T2 can easily be calculated
+by Eq. (A-2). For illustrative purposes, assume ωn = 2π, xf = 400 mm and Vm = 240 mm/s. Since the
+discriminant of the quartic equation is negative, we have two real and two complex conjugate roots. We
+disregard the complex roots, obtain T1,1 = 0.0409 s and T1,2 = 0.4247 s, and from there T2,1 = 0.9151 s and
+T2,2 = 1.6827 s follows. The time-optimal problem requires the shorter time which is why T2,2 is discarded.
+Zone 3
+The switch parameterization for Zone 3 can be written as:
+3 sin(ωnT1) − sin(ωnT2) = 0
+(A-12)
+2T2 − 6T1 = xf
+Vm
+.
+(A-13)
+Let xf/Vm = 2aωn, which gives 2T2 − 6T1 = 2aωn or T2 = 3T1 + aωn. Eq. (A-13) becomes −3 sin (ωnT1) +
+sin
+�
+2ωnT1 + aω2
+n + ωnT1
+�
+= 0. This can be further simplified to:
+−3 sin(ωnT1) + sin (2ωnT1) cos
+�
+aω2
+n + ωnT1
+�
++ sin
+�
+aω2
+n + ωnT1
+�
+cos (2ωnT1) = 0.
+(A-14)
+23
+
+Let α = sin
+�
+aω2
+n
+�
+, β = cos
+�
+aω2
+n
+�
+and t = ωnT1, Eq. (A-14) can be rewritten as:
+−3 sin(t) + sin(2t) (β cos(t) − α sin(t)) + cos(2t) (α cos(t) + β sin(t)) = 0.
+(A-15)
+Let t = cos−1(z):
+−3 sin(cos−1(z)) + sin
+�
+2 cos−1(z)
+� �
+β cos(cos−1(z)) − α sin(cos−1(z))
+�
+...
+... + cos(2 cos−1(z))
+�
+α cos(cos−1(z)) + β sin(cos−1(z))
+�
+= 0
+(A-16)
+−3
+�
+1 − z2 + 2z
+�
+1 − z2
+�
+βz − α
+�
+1 − z2
+�
++
+�
+2z2 − 1
+� �
+αz + β
+�
+1 − z2
+�
+= 0
+(A-17)
+−3
+�
+1 − z2 + 2βz2�
+1 − z2 − 2αz(1 − z2) + (2z2 − 1)αz +
+�
+2z2 − 1
+�
+β
+�
+1 − z2 = 0
+(A-18)
+�
+1 − z2 �
+4βz2 − β − 3
+�
+= −4αz3 + 3αz
+(A-19)
+�
+1 − z2� �
+16β2z4 − 8β(β + 3)z2 + β2 + 6β + 9
+�
+= 16α2z6 − 24α2z4 + 9α2z2.
+(A-20)
+With the knowledge that α2 + β2 = 1, we have:
+−16z6 + (24 + 24β)z4 + (−30β − 18)z2 + β2 + 6β + 9 = 0
+(A-21)
+which can be reduced to a cubic equation by introducing z2 = w, resulting in the equation:
+−16w3 + (24 + 24β)w2 + (−30β − 18)w + β2 + 6β + 9 = 0.
+(A-22)
+If we pick ωn = 2π, xf = 600 mm and Vm = 240 mm/s, the only real solution which would satisfy Eq. (A-12)
+is T1 = 0.0395 s and therefore T2 = 1.3684 s. Note, t = cos−1(z) supports two real solutions and 4 complex
+conjugate solutions. From there, T1 can be calculated but in this case just one of the real solutions satisfies
+Eq. (A-12).
+24
+
diff --git a/T9FAT4oBgHgl3EQf2x4q/content/tmp_files/load_file.txt b/T9FAT4oBgHgl3EQf2x4q/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fdc90800ee529dab154620816a33ba6bc0b6a69b
--- /dev/null
+++ b/T9FAT4oBgHgl3EQf2x4q/content/tmp_files/load_file.txt
@@ -0,0 +1,1011 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf,len=1010
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='08716v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='SY]  20 Jan 2023  Minimum Time Control of a Gantry Crane System with Rate Constraints  Adrian Steina, Tarunraj Singha  aDepartment of Mechanical and Aerospace Engineering, University at Buffalo (SUNY),  Buffalo, NY 14260-4400, USA  Abstract  This paper focuses on the development of minimum time control profiles for point-to-point motion of a  gantry crane system in the presence of uncertainties in modal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Assuming that the velocity of  the trolley of the crane can be commanded and is subject to limits, an optimal control problem is posed to  determine the bang-off-bang control profile to transition the system from a point of rest to the terminal states  with no residual vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Both undamped and underdamped systems are considered and the variation  of the structure of the optimal control profiles as a function of the final displacement is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' As the  magnitude of the rigid body displacement is increased, the collapse and birthing of switches in the optimal  control profile are observed and explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Robustness to uncertainties in modal parameters is accounted  for by forcing the state sensitivities at the terminal time to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The observation that the time-optimal  control profile merges with the robust time-optimal control is noted for specific terminal displacements and  the migration of zeros of the time-delay filter parameterizing the optimal control profile are used to explain  this counter intuitive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' A two degree of freedom gantry crane system is used to experimentally validate  the observations of the numerical studies and the tradeoff of increase in maneuver time to the reduction of  residual vibrations is experimentally illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Keywords:  Input Shaper, Gantry Crane, Time-Optimal Control, Rest-to-Rest Maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Introduction  Control of cranes is a topic that has garnered increased interest over the past three decades coinciding  with the growth in the use of prefiltering approaches to minimize residual vibrations of systems characterized  by underdamped motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' A vast majority of crane controllers can be classified as open-loop or closed-loop,  with a few combining feedforward and feedback controllers in a tracking framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' One open-loop approach  is called input shaping [1] which consists of a time-delay filter which is designed to cancel the underdamped  poles of the system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The domain of input shaping has matured and can account for uncertainties  in model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  To account for uncertainties in the estimated damping or natural frequencies of  the underdamped poles, multiple zeros of the time-delay filter are placed at the nominal locations of the  underdamped poles, resulting in robustness to uncertainties in the modal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Controllers which are  robust around the nominal model [1] and those that account for interval domains of uncertainties [3] have  been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Constraints on jerk [4] and deflection [5] have also been taken into account in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Recently distributed delay input shapers [6] have been studied which introduce a novel parameterization in  the design of input shapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Including input shapers within a feedback loop has also been considered [7, 8]  as researchers explore techniques to exploit the strengths of input shapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Noakes, Petterson, and Werner [9] proposed a switching control profile to generate oscillation-damped  transport and swing-free stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Their technique consists of bang-off-bang acceleration profiles in which the  pulses are timed to minimize the cable sway during the maneuver and results in a swing-free stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' They  ∗Corresponding author  Email address: tsingh@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='edu (Tarunraj Singh)  1  experimentally demonstrated the results of the open-loop control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Shah and Hong [10] applied input  shaping for the underwater transport of nuclear power plant’s fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' There have been numerous publications  related to the use of input shapers [1, 2, 11] for sway control of cranes [12, 5, 13, 14, 15, 16, 17, 18, 19,  20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Maghsoudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [23] applied a distributed time-delay filter on a gantry crane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' They used the  method proposed by Vyhlidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [6] for uncertainty studies for gantry crane control and demonstrated  that applying a distributed time-delay filter lead to an asymmetric robustness behaviour of the residual  sway about the nominal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Ramli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [24] designed a neural network-based input shaper while Yavuz  and Beller [25] used neural networks for a closed-loop controller when it is combined with an input shaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Wahrburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [26] and Ramli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [27] applied input shaping for maneuvers of an overhead crane with  non-zero initial conditions, with an objective of zero residual oscillations of the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Additionally, work  has been done in command shaping control for non-zero initial and final conditions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Stein and Singh [29]  presented simulation results of velocity constrained design of input shaped control profiles for a gantry crane  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fliess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [30] used the concept of differential flatness to control the traversing and hoisting of an  overhead crane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This differential flatness based design was extended to discrete time design by Diwold et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Alli and Singh [32] designed passive controllers for a distributed parameter representation of the crane  cable for point-to-point maneuvers where the integral of the time absolute error is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' O’Connor [33]  used a wave equation representation of the cable dynamics assuming that the velocity of the trolley could be  commanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The velocity of the trolley was assumed to be constrained and the damping was assumed to be  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Researchers included particle swarm optimization into controller design for overhead cranes [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Golovin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [36] developed a H∞ robust controller for actively damping the structural vibrations of the  gantry crane system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Various control methods have also been proposed where the overhead crane is modelled  as a double pendulum system [37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' A few papers proposed controllers which combine closed-loop  controller used in conjunction with open-loop shaped profiles to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Kolar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [41] proposed a hybrid  solution combining an open-loop generated crane trajectory as a reference signal and closed-loop controller  for handling external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [42] introduced an online planning method for minimum-time  control of overhead cranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Furthermore, many papers considered the payload as a point mass whereas Stein  and Singh [43] proposed an input shaper used in conjunction with a proportional-derivative controller for a  crane with an inertial payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Other work has considered sliding mode [44, 19, 45], adaptive control [46],  discrepancy-based control [47] and compared different control strategies [48] while including various external  disturbances on cranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Apart from time-optimal control, Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [49] investigated an energy-optimal  controller for an underactuated double pendulum crane with state and control constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Compared to  active vibration suppression of a crane’s payload, Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' [50] used a passive method with an  absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  This paper considers a tabletop gantry crane system driven by stepper motors which permits commanding  the position of the trolley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' By imposing velocity limits on the trolley motion, this paper considers the design  of velocity constrained time-optimal point-to-point control of a crane moving in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Since  the pendular motion is almost undamped, an undamped system model was first considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Subsequently,  the structure of switching function was used to parameterize the bang-off-bang profile and the resulting  nonlinear programming problem was solved to determine the optimal solution for any arbitrary maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The bang-off-bang control structure was then generalized to cater to multi-mode systems with underdamped  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The main contributions of this work include: (1) Development of a velocity constrained time-optimal  control profile for a gantry crane which is robust to uncertainties in modal parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (2) Illustration of  the non-intuitive result that the robust and non-robust solutions are coincident for specific displacements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (3) Illustration that a rectangular pulse input can attenuate the dominant vibratory modes for specific  displacements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (4) Illustration of the change in structure of the optimal control profile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' and (5) Experimental  validation of all the aforementioned observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Section 2 presents the development of velocity constrained time-optimal control for an undamped gantry  crane system followed by the development of controllers which are robust to uncertainties in the undamped  frequencies in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Section 4 generalizes the optimal control formulation for a multi-mode system  with damped or undamped modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Section 5 presents a simple approach to determine the transition in the  2  structure of the control profile, followed by the validation of the design on a two degree of freedom gantry  crane system in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The paper concludes with a brief summary of the results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Undamped System  β  m  xi  g  L  x = xi + Lβ  xi  x  k  m  Figure 1: Equivalent spring-mass system for small angle displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The gantry crane setup includes a trolley driven by a stepper motor which permits the command of  the trolley’s velocity by assuming that the acceleration is zero as the commanded velocity transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' A  schematic of the crane and an equivalent spring-mass system are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 1 where a small angle  displacement is assumed [51] [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The spring-mass model can be written as:  mL ¨β(t) + mg sin (β(t)) = 0  (1)  m¨x(t) − m✟✟✟  ✯0  ¨xi(t)  + mg  �x(t) − xi(t)  L  �  = 0  (2)  ↔ m¨x(t) + kx(t) − kxi(t) = 0  (3)  ˙xi(t) = v(t)  (4)  where v the velocity of the trolley is considered as the input and is constrained 0 ≤ v ≤ Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The assumption  that velocity of the trolley can be used as a control input is based on papers which demonstrate velocity  input control of industrial cranes [53, 54, 55, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Mass normalization leads to the state space equation:  \uf8ee  \uf8f0  ˙x1  ˙x2  ˙x3  \uf8f9  \uf8fb  � �� �  ˙X  =  \uf8ee  \uf8f0  0  1  0  −ω2  n  0  ω2  n  0  0  0  \uf8f9  \uf8fb  �  ��  �  A  \uf8ee  \uf8f0  x1  x2  x3  \uf8f9  \uf8fb  � �� �  X  +  \uf8ee  \uf8f0  0  0  1  \uf8f9  \uf8fb  ����  B  v  (5)  where ωn =  �  k  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The time-optimal control problem can be posed as:  min J =  � tf  0  dt  (6a)  subject to  ˙X = AX + Bv  (6b)  X(0) =  �0  0  0�T  (6c)  X(tf) =  �xf  0  xf  �T  (6d)  0 ≤ v ≤ Vm  ∀t  (6e)  Defining the Hamiltonian as:  H = 1 + λT (AX + Bv) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (7)  3  the necessary conditions for optimality can be derived using calculus of variations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' resulting in the equations  ˙X = ∂H  ∂λ = AX + Bv  (8a)  ˙λ = −∂H  ∂X = −AT λ  (8b)  v = VmH(−BTλ)  (8c)  X(0) = 0 and X(tf) =  �  xf  0  xf  �T  (8d)  H = 0 at t = 0  (8e)  where H is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (8c) is derived using Pontryagin’s minimum principle (PMP)  which requires the optimal trolley velocity to be bang-off-bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (8b) can be solved in closed form  resulting in the equation:  λ(t) = e−AT tλ(0)  (9)  which can also be written as:  λ1(t) = cos(ωnt)λ1(0) + ωn sin(ωnt)λ2(0)  (10)  λ2(t) = −sin(ωnt)  ωn  λ1(0) + cos(ωnt)λ2(0)  (11)  λ3(t) = (1 − cos(ωnt))λ1(0) − ωn sin(ωnt)λ2(0) + λ3(0)  (12)  for undamped systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Since the switching function is BTλ, it reduces to the third costate which reveals  that the switching function is a non-zero mean harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This structure will help comprehend the change in  structure of the optimal control profile v(t) as a function of the final displacement xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Since the Hamiltonian  H at all times, including the initial time, is zero, we have:  H(0) = 1 + λT (AX(0) + BVm) = 0  (13)  ⇒ λ3(0) = − 1  Vm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (14)  Further, for a rest-to-rest maneuver, which is the focus of this paper, the Hamiltonian H at the final time  is:  H(tf) = 1 + λT (AX(tf) + BVm) = 0  (15)  ⇒ λ3(tf) = − 1  Vm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (16)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (16) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (12), we can show that:  λ2(0)  λ1(0) = 1  ωn  tan  �ωntf  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (17)  Evaluating the slope of λ3(t) at the initial time and final time, we can show that:  ˙λ3(0) = −ω2  nλ2(0), and ˙λ3(tf) = ω2  nλ2(0),  (18)  which permits us to conclude that the switching function is anti-symmetric about the mid-maneuver time,  which implies there will always be an even number of switches and that pairs of switch times are equally  distant from the mid-maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (17), the time derivative of λ3(t) evaluated at the mid-  maneuver time is:  ˙λ3  �tf  2  �  = ωn sin  �  ωn  tf  2  �  λ1(0) − ω2  n cos  �  ωn  tf  2  �  λ2(0) = 0  (19)  4  which implies that the slope of the switching curve is always zero at the mid-maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This prompts  parameterising the optimal control profile as:  v(t) = Vm (1 − H(t − (T2 − T1)) + H(t − (T2 + T1)) − H(t − 2T2))  (20)  as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 2, which in the frequency domain is:  Figure 2: Anti-symmetric time-optimal control profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  V (s) = Vm  s  �  1 − e−s(T2−T1) + e−s(T2+T1) − e−2sT2�  �  ��  �  Gc(s)  (21)  where Gc(s) is the transfer function of the time-delay filter which generates the bang-off-bang control profile  when subjected to a step input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' By requiring that a pair of zeros of Gc(s) cancel the undamped poles of  the system and the pole at the origin, one can derive the constraint to formulate a parameter optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It can be seen that:  Gc(s = 0) = 1 − 1 + 1 − 1 = 0  (22)  which implies that the transfer function has a zero at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The rigid body boundary condition is  determined by integrating the bang-off-bang velocity profile leading to the equation:  xi(t) = Vm (t − (t − T2 + T1)H(t − T2 + T1) + (t − T2 − T1)H(t − T2 − T1) − (t − 2T2)H(t − 2T2))  (23)  which at the final time of t = tf = 2T2 leads to the equation:  xi(2T2) =xf = Vm (2T2 − (T2 + T1) + (T2 − T1))  (24)  ↔ xf =2Vm(T2 − T1) ⇒ (T2 − T1) = xf  2Vm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (25)  Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' to cancel the undamped poles at s = ±jωn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' we have:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='Gc(s = jωn) = 1 − e−jωn(T2−T1) + e−jωn(T2+T1) − e−2jωnT2 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(26)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='which reduces to:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1 − cos(ωn(T2 − T1)) + cos(ωn(T2 + T1)) − cos(2ωnT2) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(27)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='sin(ωn(T2 − T1)) − sin(ωn(T2 + T1)) + sin(2ωnT2) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(28)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='which simplifies to:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2 sin(ωnT2) (sin(ωnT2) − sin(ωnT1)) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(29)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2 cos(ωnT2) (sin(ωnT2) − sin(ωnT1)) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(30)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='which leads to the solution:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='T2 = π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='ωn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='− T1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(31)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='which results in the closed form solution:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2T2 = π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='ωn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='+ xf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2Vm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='and T1 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2ωn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='− xf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='4Vm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='(32)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='where 2T2 is the maneuver time tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Since both T1 and T2 are functions of the maneuver xf, the scenario of  collapse of the switches requires T1 = 0 which results in the constraint:  xf = 2πVm  ωn  (33)  which is a bang profile or a constant velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It should be noted that the collapse of the switches is  proportional to one period of the switching function 2π  ωn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Any increase in time associated with a maneuver  greater than the bang profile will introduce two additional switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This results from the fact that there  will be two peaks or troughs of the switching profile, which results in four switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' An observation which  will be exploited later is the fact that since the switching function is a harmonic with a bias, the width of  all the off zones in the bang-off-bang profiles will be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Consider a constant velocity profile with no  zero zones which can be parameterized as:  v(t) = Vm (1 − H(t − 2T2))  (34)  which can be represented in the frequency domain as:  V (s) = Vm  s  �  1 − e−2sT2�  �  ��  �  Gc(s)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (35)  The final displacement is given by the equation:  xf = 2VmT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (36)  The requirement that the transfer function Gc(s) places a zero at the location of the undamped poles of the  system s = ±jωn requires the constraints:  1 − cos(2ωnT2) = 0  (37)  − sin(2ωnT2) = 0  (38)  which results in the solution:  T2 = nπ  ωn  , where n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='.  (39)  Substituting the solution T2 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (36), we have:  xf = 2nπVm  ωn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (40)  The number of zero zones n depends on the desired terminal displacement and is given by the constraint:  2(n − 1)πVm  ωn  ≤ xf ≤ 2nπVm  ωn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (41)  For example, when 0 ≤ xf ≤  2πVm  ωn , one zero velocity zone exists in the optimal velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Since  all the off zones of the bang-off-bang profiles are the same, we parameterize the width of each zero zone  as 2T1 and only two parameters are needed to determine the velocity limited time-optimal control profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 3 illustrates the variation of the switch time for each zone as the terminal displacement increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The  6  Figure 3: Switching structure and switch time variation as a function of xf and τi = Ti/tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Shaded areas illustrate the time  span of T1 and the blue curve shows the switching function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Different background colors illustrate the three different picked  Zones for (I) Zone 1, (II) Zone 2, (III) Zone 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  7  change in structure of the time-optimal control profile with increasing terminal displacements is illustrated  for three consecutive zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Zone 1 is characterized by two switches which collapse for a specific terminal  displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Any increase in terminal displacement results in the introduction of two additional switches  in the optimal control profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 3 also plots the switching function which illustrates that as the optimal  control profile transitions from one zone to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The switching function includes a phase shift of π  radians, which results in new switches being birthed at the initial and terminal times of the maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For  higher zones, switches can be introduced at the mid-maneuver time and 2π/ωn distance from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  To determine the values for T1, which is half the width of each zero velocity zone, and T2, which is  half of the maneuver time, a parameter optimization problem is solved which minimizes T2 subject to the  constraints:  T2 − nT1 = xf  2Vm  (42)  (−1)n+1n sin(ωnT1) − sin(ωnT2) = 0  (43)  where 2T1 is the width of each of the zero pulse and 2T2 is the maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (42) is derived from  the final displacement constraint and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (43) is derived from the constraint that the time-delay filter needs  to locate a pair of zeros at the location of the undamped poles of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (42) and (43) can be  transformed into polynomial equations which can be solved efficiently to identify the optimal parameters  T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The Appendix provides the details of the transformation to generate polynomial equations for  zones 2 and 3 for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  It is interesting to note that as xf → 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (42) requires T1 → T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (43) where n = 1 requires  ωnT2 = π − ωnT1  (44)  which results in the equation:  2T2 = π  ωn  (45)  which one can note is analogous to the two-impulse input shaper [1] which for an undamped system places  the two impulses of equal magnitude half a period of oscillation apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Note that 2T2 corresponds to the  maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The upper graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 4 illustrates the variation of the switch times and the maneuver time as a  function of the terminal displacement xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The solid colored zone corresponds to the time intervals when the  input (velocity command) is at the maximum and the rest of the time intervals are when the input is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  When xf = 240 mm, the two switches collapse resulting in a constant velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This is followed by  the birth of four switches which collapse concurrently for a terminal displacement of xf = 480 mm following  which a six switch optimal control profile is birthed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The lower half of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 4 illustrates the optimal solution  for three unique displacements in zones 1, 2 and 3, labelled “a”, “b”, and “c” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Robust Control  The challenge of dealing with model parameter uncertainties is ubiquitous and there have been numerous  approaches proposed for the design of robust open-loop controllers including enforcing robustness around  the nominal model of the system or a minimax problem formulation where the maximum residual energy is  minimized over an interval of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' In this research we determined the sensitivity of the states of the  system with respect to uncertainty in the spring stiffness which correspond to uncertainties in the natural  frequency and force the state sensitivities with respect to the uncertain frequency to zero at the terminal  8  Figure 4: Switch and maneuver time variation of a non-robust time-optimal controller for an undamped 1 mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (a),  (b) and (c) show the control profiles for xf = 100 mm, xf = 300 mm and xf = 550 mm in Zone 1, Zone 2 and Zone 3  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The resulting augmented state space model is:  ˙x1(t) = x2(t)  (46)  ˙x2(t) = −ω2  nx1(t) + ω2  nx3(t)  (47)  d ˙x1(t)  dωn  = dx2(t)  dωn  (48)  d ˙x2(t)  dωn  = −2ωnx1(t) − ω2  n  dx1(t)  dωn  + 2ωnx3(t)  (49)  ˙x3(t) = v(t)  (50)  0 ≤ v ≤ Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (51)  and is subject to the initial and final conditions:  x1(0) = x2(0) = x3(0) = 0  (52)  dx1(0)  dωn  = dx2(0)  dωn  = 0  (53)  x1(tf) = x3(tf) = xf  (54)  x2(tf) = dx1(tf)  dωn  = dx2(tf)  dωn  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (55)  The robust velocity constrained time-optimal control problem for the undamped system is solved for various  displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The variation in the optimal control profile parameterized by the switch times and maneuver  time are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 5 as a function of the terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Unlike the variation of the switch  times as a function of terminal displacement and the concurrent collapse of the switches, this phenomena  is not observed in the spectrum of switch times in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It nevertheless should be noted that the optimal  control profile births and collapses switches as a function of terminal displacements, resulting in an optimal  control profile where the number of switches is a function of the terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 6 illustrates the  variation of the residual energy at the terminal time for the non-robust and robust time-optimal controllers  over a range of uncertain natural frequencies for the undamped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It is clear that the red line, which  represents the variation of residual energy of the robust control, outperforms the non-robust design illustrated  by the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' These graphs are generated for a terminal displacement of xf = 50 mm and a natural  frequency of ωn = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The residual energy is given by:  9  Figure 5: Switch and maneuver time variation of a robust time-optimal controller for an undamped 1 mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  4  8  12  0  16  x103  Residual Energy V  4  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  non-robust  robust  ωn [rad/s]  Figure 6: Residual energy at tf of a non-robust and robust time-optimal controller for a perturbation of ±30 % in ωn at  xf = 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  V (t) = 1  2 ˙x(t)2 + 1  2ω2  n (x(t) − xf)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (56)  The sensitivity of the residual energy with respect to the natural frequency is:  dV (t)  dωn  = ˙x(t)d ˙x(t)  dωn  + ωn (x(t) − xf)2 + ω2  n (x(t) − xf) dx(t)  dωn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (57)  Since the time-optimal control profile forces the terminal states to be x(tf) = xf and ˙x(tf) = 0, V (tf) and  dV (tf)  dωn  = 0, irrespective of the magnitude of the sensitivity states, dV (t)  dωn  = 0 at the terminal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The  second derivative of the residual energy with respect to the natural frequency is:  d2V (t)  dω2n  =  �d ˙x(t)  dωn  �2  + ˙xd2 ˙x(t)  dω2n  + (x(t) − xf)2 + 4ωn (x(t) − xf) dx(t)  dωn  + ω2  n  �dx(t)  dωn  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' + ω2  n (x(t) − xf) d2x(t)  dω2n  (58)  which is the curvature since the first derivative is zero at the nominal frequency ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The curvature is  defined as the reciprocal of the radius of a circle which best approximates the curve V (ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The curvature  can be used as a measure of the robustness of the control profile in the proximity of the nominal model  with a smaller curvature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' an osculating circle with large radius) indicating a smaller residual energy  variation or greater robustness to uncertainties in the natural frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The solid and dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 7  correspond to the respective non-robust and robust solutions, the blue curves represent the variation in the  maneuver time as a function of terminal displacement, and the black lines represent the variation of the  residual energy and its derivatives with terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The third panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 7 shows that there  is a profound variation in the curvature of residual energy function for the non-robust solution as a function  of terminal displacement in comparison to that of the robust control which is zero for all displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It  10  tf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='nonrobust  tf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='robust  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' dV/d�n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' d2V/d�n  2 (non-robust)  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' dV/d�n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' d2V/d�n  2 (robust)  Residual Energy V  0  200  400  600  0  2  4  0  2  4  1  0  �1  2  0  4  6 x105  Final Displacement xf [mm]  d2V/d  �  n  2  Final Time tf [s]  dV/d  �  n  0  2  4  1  0  �1  Figure 7: Variation of terminal time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' residual energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' and its derivatives as a function of xf for a non-robust and robust  time-optimal controller to illustrate the collapses of the switch times for an undamped 1 mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  11  is also intriguing that, for specific terminal displacements, the robust and non-robust solutions are identical,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=', the maneuver times are the same and the curvatures are both zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' To comprehend this unique result  the loci of the zeros of the time-delay filter which generates the bang-off-bang control profiles are generated  and are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Figure 8: Loci of zeros of the optimal time-delay filter for a non-robust and robust controller as a function of xf for a 1 mode  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  All the graphs are generated for a nominal frequency of ωn = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' As illustrated by the upper graph of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 8, the non-robust design requires that the time-delay filter place zeros at the location of the nominal  undamped poles of the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Meanwhile, the lower graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 8 shows that the robust design mandates  that a pairs of zeros be located at the nominal poles of the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The horizontal loci of zeros for both the  robust and non-robust designs illustrates that the time-delay filter places single and double sets of zeros at  the nominal location of the poles for the non-robust and robust designs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It can also be noted  that the zero loci for the non-robust design intersects the horizontal loci for a terminal displacement of  xf = 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5 mm (shown by the red cross), resulting in a pair of coincident zeros which is what the robust-  controller is mandated to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This is the reason why, for specific terminal displacements, the robust and  non-robust designs results in identical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It can also be seen that the same phenomena occurs for a  terminal displacement of xf = 513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 9 illustrates the location of the zeros of the time-delay filter for the non-robust and robust designs  for the terminal displacement xf = 500 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The blue circles correspond to the non-robust design and the  red to the robust design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It should be noted that the robust design includes two pairs of zeros at ±2πj and  the non-robust design includes one pair of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The arrows associated with the blue and red circles are  12  Figure 9: Residual energy versus uncertain model frequency for a non-robust and robust controller at xf = 500 mm for a 1  mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The arrows indicate the slopes of zero-loci for xf > 500 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' ωn,design shows the frequency the time-delay  filters were designed for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  the slopes of the loci of the zeros as a function of terminal displacement xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It is evident that a blue zero is  transitioning from above 2πj with a negative slope and is shown to coincide with the nominal poles of the  system for xf = 513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm, resulting in identical robust and non-robust designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  The red and blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 9 illustrate the variation in residual energy as the location of the nominal  poles of the system are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The non-robust design outperforms the robust design for perturbation of  the uncertain frequency above ωn = 2π, while the robust design outperforms the non-robust design for  perturbations below the nominal frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This asymmetry is due to the fact that the non-robust design  has a zero located immediately above the nominal frequency and ends up acting like the zero locations of a  minimax design [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It can also be seen that, for large perturbations of the uncertain frequency, the residual  energy goes to zero at locations where the time-delay filter has a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Generalization  Section 2 dealt with an undamped system with one mode which permitted a reduced order param-  eterization of the optimal control profile by exploiting the symmetrical nature of the control about the  mid-maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This symmetry is attributed to the fact that the oscillatory motion excited by the  input does not damp out and the symmetric input can, by virtue of linearity of the system, generate an out  of phase motion of the undamped modes which cancels the existing oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For a system with damping,  the symmetric nature of the control profile is lost since the amplitude of the oscillatory mode decays over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Consequently, the optimal control profiles need to explicitly parameterize every switch in addition to  the maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Furthermore, there may be multiple modes which contribute to the output of interest  and those modes need to be quiescent at the end of the maneuver as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For a system with multiple modes,  characterized by damped or undamped modes, the bang-off-bang control profile is parameterized as:  v(s) = Vm  s  �  1 +  N+1  �  i=1  (−1)ie−sTi  �  �  ��  �  =Gc(s)  (59)  where N is the number of switches which is an even number and TN+1 is the maneuver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  13  The constraints to identify the optimal values of Ti are derived by requiring the zeros of the time-delay  filter to cancel the pole at the origin and the underdamped poles of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This results in the constraint:  Gc(s = 0) = 1 +  N+1  �  i=1  (−1)i = 0  (60)  which is automatically satisfied by the parameterization of the optimal control profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' To cancel the complex  conjugate poles at s = −σk ± j ωd,k = −ζkωn,k ± j ωn,k  �  1 − ζ2  k, where k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' , m are the m modes  whose poles need to be canceled by the zeros of the time-delay filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This results in the constraints:  1+  N+1  �  i=1  (−1)ieσkTi cos(ωd,kTi) = 0  (61)  N+1  �  i=1  (−1)ieσkTi sin(ωd,kTi) = 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (62)  To satisfy the terminal rigid body displacement, we require:  x(TN+1) = xf =  � tf  0  Vmdt  (63)  = Vm  �  TN+1 +  N  �  i=1  (−1)i(TN+1 − Ti)  �  (64)  → xf = Vm  �  TN+1 −  N  �  i=1  (−1)iTi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (65)  The nonlinear optimization problem that requires solving is:  min J = TN+1  (66a)  subject to  Vm  �  TN+1 −  N  �  i=1  (−1)iTi  �  = xf  (66b)  1 +  N+1  �  i=1  (−1)ieσkTi cos(ωd,kTi) = 0  (66c)  N+1  �  i=1  (−1)ieσkTi sin(ωd,kTi) = 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=', m  (66d)  0 ≤ (Ti+1 − Ti), for i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=', N  (66e)  where T0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' As in the case of the undamped system, the number of switches necessary to parameterize  the optimal control profile changes with the terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 10 illustrates the variation of the  switch times as a function of the final displacement and the control input over the maneuver time for a single  mode system with an underdamped system model (ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For terminal displacements of 0 − 700 mm,  three zones have been identified, separated by the vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Three representative optimal control  profiles are presented corresponding to three displacements highlighted by the dotted lines and labelled a,  b, and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It is interesting to note that in zone 3, which includes the dotted line c, the number of switches  required for the optimal control profile starts with four switches, transitions to six switches, and returns to  a four switch optimal control profile before finally transitioning to a two switch profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  14  Figure 10: Switch and maneuver time variation of a non-robust time-optimal controller for an underdamped 1 mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Switching Profile Transition  It is clear from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 4 and 10 that the structure of bang-off-bang control profile changes as a function  of the terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 4, for small displacements (xf ≤ 240 mm), the optimal control profile  is first characterized by two switches and the switches, then, after collapsing, result in a pulse control  profile which births a four switch control profile which subsequently transitions to a six switch control  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The change in structure of the optimal control profile for underdamped systems is more involved  as all switches do not collapse for the same value of the terminal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' To exactly determine the  terminal displacement which corresponds to the birth or collapse of two or more switches, the constraint is  that the switching function and its time derivative are simultaneously zero at some time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For the  underdamped system the switching function λ3(t) can be represented as:  λ3(tcr, xf) = ˙λ3(tcr, xf) = 0  (67)  where tcr is the switch time where two switches collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' tcr and xf can be determined by solving the  two nonlinear simultaneous equations in two unknowns while satisfying all the necessary conditions for  optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Experimental Results  A scaled model of a gantry crane was fabricated to test and validate the time-optimal control profiles  presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The workspace of the gantry crane includes a cuboid of dimensions 8’ × 4’ × 3’  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Its main element is a trolley which slides on a rail and is able to move a payload over  a 2-dimensional space (the winch motion of the crane is not active).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The crane is equipped with 3 stepper  motors of type NEMA 23 with a driver setting of 400 steps per revolution, where two of them are used for  the y-axis and one for the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For this experiment just the motor of the x-axis is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The drivers of the  stepper motors are connected to an Arduino MEGA 2560 which receives the switch times and the desired  position of the trolley as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The maximum velocity of the trolley is set to 240 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The inset of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 11 includes two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The one on the left illustrates the 3D printed chassis which houses a cylindrical  steel mass of 500g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' A rope and a hook enable a connection between the chassis and the trolley making it a  double pendulum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The right panel illustrates the sensor integrated into the payload which includes  two Arduino Nanos, a 3-axis gyroscope (MPU-6050), and two nRF24L01 single chip radio transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' One  Arduino Nano is housed within the chassis to process the data provided by the gyroscope and to transmit  the data to the other Arduino Nano, which is connected to a receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' To permit a repeatable evaluation of  the robustness of the controller, a cable deployment setup was designed which permits changing the length  15  Figure 11: Experimental setup of a gantry crane with an inset of the sensor part, which consists of a powerbank, Arduino  Nano, 3-axis gyroscope (MPU-6050), and an nRF24L01 transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Rotation around the y-axis is introduced as the swinging  angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' L1 is denoted as the rope length and L2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='86 cm  16  OOKEREDOOKEREDL1 of the pendulum about a nominal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The change in cable length changes the modal frequency of  the system and was used to test the robustness of the controller to uncertainties in modal parameters of the  gantry crane system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  A series of experiments were conducted to illustrate some of the novel observations of the analytical  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The first of which included the collapse of the switches of the velocity constrained time-optimal  control profile for a system with a single undamped mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The simplest model of the gantry crane includes  representing the suspended mass as a single undamped pendulum and a rectangular command, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=', a bang  control profile should cancel the undamped mode for a specific displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The natural frequency and  damping ratio of the first mode of the pendulum with the nominal length were experimentally determined  to be ωn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='6832 Hz and ζ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='001517 which was approximated to be zero in the development of the  controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 12 illustrates experimental results for terminal displacements resident in zone 1, which corresponds  to the optimal control profile being characterized by a two switch bang-off-bang profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  A pulse with  increasing width is progressively applied to illustrate the fact that for specific terminal displacements, a  bang profile (pulse) results in zero terminal vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Starting from a terminal displacement of xf = 200  mm and progressively increasing it until xf = 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm, the maneuver time increases, but at the same  time, the maximum angular displacement of the pendulum α is decreasing until it is not present at the end  of the maneuver for a displacement of 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It can be noted that a small high frequency oscillation is  evident which corresponds to the second mode of the double pendulum system which is not considered in  the controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For the next set of experiments, the double pendulum model of the crane is assumed  Figure 12: Residual vibration variation when changing the final displacement from xf = 200 mm to xf = 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm for a pulse  control profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The blue curve illustrates the residual vibration at xf = 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3 mm which validates the cancellation of the first  mode of vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  in the identification of the parameters of the two modes of oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The second mode’s parameters are  identified to be ωn,2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='159 Hz and ζ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='026065 where the damping of the second mode is an order of  magnitude greater than that of the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For the two mode system, three controllers were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The first  controller was designed to cancel the first mode only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The second controller was designed to cancel both of  the modes at the end of the maneuver and the third controller was designed to be robust to uncertainties  in both the modes’ natural frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 13 illustrates the time variation of the angular displacement  of the pendular payload where the green region corresponds to when the control input is active and the  yellow region is the post-actuation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The first panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 13 illustrates the impact of cancelling  the first mode, the second panel corresponds to the controller cancelling both modes, and the third panels  display the elimination of residual vibration when the robust time-optimal control profile is used to drive  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' These results were generated for a terminal displacement of 100 mm and the pendulum length  L1 which corresponds to the nominal model which was used to identify the modal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' To illustrate  the variation in residual energy as the pendular length is varied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' five experiments were conducted for each  17  0  1  2  3  4  Swinging Angle α [deg]  1 mode filter (non-robust)  2 mode filter (non-robust)  2 mode filter (robust)  1 �  0  �  \x0e  \x0f\x10 \x11  \x12 \x13  \x14  \x15\x16 \x17  Time [s]  operating time  residual time  tf  Figure 13: Swinging angle α (around y-axis) for a 1 mode time-delay filter (non-robust),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 2 mode time-delay filter (non-robust),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  and 2 mode time-delay filter (robust) when xf = 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Shaded regions help to distinguish between the operating and  residual (off) time of the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The final time tf for each scenario is highlighted by a red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  pendular length to account for uncertainties in initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' With a total of seven perturbed lengths  of the pendulum on either side of the nominal length, a total of 150 experiments were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Box and  whisker charts are used to illustrate uncertainties in the residual energy for each of the perturbed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 14 illustrates the profound improvement in the use of the robust controller for which the residual  vibration for the various pendulum lengths appear negligible compared to the residual energy resulting from  the use of the non-robust control profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The final set of experiments illustrates a rather counter intuitive  result which claims that the non-robust design and robust designs are coincident since the non-robust design  for specific terminal displacements of the single-mode model places multiple zeros of the time-delay filter at  the nominal location of the poles of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 15 illustrates the residual vibration box and whisker  charts for a terminal displacement of xf = 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1 mm, where it is evident that the residual energy curve is  relatively flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It should be pointed out that in this design, only the first mode of vibration was considered  in the design, since observation of the coincidence of the robust and non-robust design has been observed  for system with single undamped modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Conclusions  This paper presents an optimal control based development of a velocity limited minimum time control  of a gantry crane system which is characterized by two modes of vibratory motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The variation in the  structure of the optimal control profile is presented for a single mode system where the vibratory modes  are undamped or underdamped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' It is noted that, as the final displacement increases, there is an increase  in the number of switches in the optimal control profile with periodic terminal displacements requiring a  pulse control profile with no switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The optimal control framework is extended to account for multiple  vibratory modes such as when the crane is modeled as a double pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The state sensitivities are used to  determine controllers which are robust to uncertainties in model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Experimental results validate  the counter intuitive observation that for specific terminal displacement the robust and non-robust optimal  control profiles are coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The experimental results also clearly demonstrate the profound reduction in  18  Rope Length L1 [cm]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  5  x10  \x184  Residual Energy Vy  non-robust  robust  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='7  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1  Figure 14: Residual energy variation at tf of time-optimal controllers for xf = 100 mm for different rope lengths L1 varying  from 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9 cm to 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1 cm in 15 equally spaced intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The time-optimal controllers are designed for L1 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Blue and  red shaded illustrate the residual energy statistics for the 2 mode non-robust and robust controller respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Furthermore,  the time history of the swinging angle α for a non-robust vs robust scenario is illustrated by insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The overall box and whisker  chart includes 150 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Rope Length L1 [cm]  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='7  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5  5  x10−4  Residual Energy Vy  n \x19 \x1a \x1b  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  "  #$ % & \'(  Figure 15: Residual energy at tf of a time-optimal controller at xf = 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1 mm for different rope length L1 varying from 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9  cm to 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1 cm in 15 equally spaced intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The controller is designed for a system with L1 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='5 cm where the non-robust  and robust solution collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The change of swinging angle α for the controller is illustrated by insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The overall box and  whisker chart includes 75 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  19  residual vibrations of the double pendulum system when the length of the pendulum is changed to serve as  a proxy for uncertainties in natural frequencies of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Acknowledgment  The authors acknowledge the support of this work by the US National Science Foundation through  CMMI Award number 2021710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Claude F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Leibovici for his help in  transforming Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (42) and (43) to a polynomial equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
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+page_content='1109/TIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2320231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Fang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Sun, Optimal trajectory planning and tracking control method for overhead cranes, IET Control  Theory & Applications 10 (6) (2016) 692–699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1049/iet-cta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='0809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  [53] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
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+page_content=' Sorensen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Singhose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Dickerson, A controller enabling precise positioning and sway reduction in bridge and  gantry cranes, Control Engineering Practice 15 (7) (2007) 825–837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='conengprac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  [54] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Suksabai,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Waikoonvet,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Chuckpaiwong,  Modelling method investigation of drive and motor for an in-  dustrial  overhead  crane,  IOP  Conference  Series:  Materials  Science  and  Engineering  886  (1)  (2020)  012030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='1088/1757-899X/886/1/012030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  [55] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Knierim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Krieger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Sawodny, Flatness based control of a 3-dof overhead crane with velocity controlled drives,  IFAC Proceedings Volumes 43 (18) (2010) 363–368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3182/20100913-3-US-2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='00083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  22  Appendix  The time-optimal control profile for zone 1 illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' 4 parameterized with two variables T1 and  T2 has a closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For higher order zones, the transcendental constraint equations cannot be  solved in closed form, but can be converted to a polynomial equation as illustrated for zones 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Zone 2  The parameterization of the time-optimal control for Zone 2 leads to the constraints:  2 sin(ωnT1) + sin(ωnT2) = 0  (A-1)  2T2 − 4T1 = xf  Vm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-2)  Let xf/Vm = 2aωn, which gives 2T2 − 4T1 = 2aωn or T2 = 2T1 + aωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-1) becomes 2 sin(ωnT1) +  sin(2ωnT1 + aω2  n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This can be further simplified to:  2 sin (ωnT1) + sin (2ωnT1) cos  �  aω2  n  �  + sin  �  aω2  n  �  cos (2ωnT1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-3)  Let α = sin  �  aω2  n  �  , β = cos  �  aω2  n  �  and t = ωnT1:  2 sin(t) + β sin(2t) + α cos(2t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-4)  Let t = cos−1(z), resulting in the simplified equation:  2 sin(cos−1(z)) + β sin(2cos−1(z)) + α cos(2cos−1(z)) = 0  (A-5)  2  �  1 − z2 + β2z  �  1 − z2 + α  �  2z2 − 1  �  = 0  (A-6)  2  �  1 − z2 (βz + 1) + α  �  2z2 − 1  �  = 0  (A-7)  α  �  2z2 − 1  �  = −2  �  1 − z2 (βz + 1)  (A-8)  α2 �  4z4 − 4z2 + 1  �  = 4  �  1 − z2� �  β2z2 + 2βz + 1  �  (A-9)  �  4α2 + 4β2�  z4 + 8βz3 +  �  −4α2 − 4β2 + 4  �  z2 − 8βz + α2 − 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-10)  Exploiting the knowledge that α2 + β2 = 1, we have:  4z4 + 8βz3 − 8βz + α − 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-11)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-11) the parameter T1 can be calculated by using t = cos−1(z) and T1 = t/ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Assuming  that ωn is given and the user can choose any xf which lies in the bounds of the zone, the quartic equation  provides a solution for the switch time T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' From there, the mid-maneuver time T2 can easily be calculated  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' For illustrative purposes, assume ωn = 2π, xf = 400 mm and Vm = 240 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Since the  discriminant of the quartic equation is negative, we have two real and two complex conjugate roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' We  disregard the complex roots, obtain T1,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='0409 s and T1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='4247 s, and from there T2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='9151 s and  T2,2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='6827 s follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' The time-optimal problem requires the shorter time which is why T2,2 is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  Zone 3  The switch parameterization for Zone 3 can be written as:  3 sin(ωnT1) − sin(ωnT2) = 0  (A-12)  2T2 − 6T1 = xf  Vm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-13)  Let xf/Vm = 2aωn, which gives 2T2 − 6T1 = 2aωn or T2 = 3T1 + aωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-13) becomes −3 sin (ωnT1) +  sin  �  2ωnT1 + aω2  n + ωnT1  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' This can be further simplified to:  −3 sin(ωnT1) + sin (2ωnT1) cos  �  aω2  n + ωnT1  �  + sin  �  aω2  n + ωnT1  �  cos (2ωnT1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-14)  23  Let α = sin  �  aω2  n  �  , β = cos  �  aω2  n  �  and t = ωnT1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-14) can be rewritten as:  −3 sin(t) + sin(2t) (β cos(t) − α sin(t)) + cos(2t) (α cos(t) + β sin(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-15)  Let t = cos−1(z):  −3 sin(cos−1(z)) + sin  �  2 cos−1(z)  � �  β cos(cos−1(z)) − α sin(cos−1(z))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' + cos(2 cos−1(z))  �  α cos(cos−1(z)) + β sin(cos−1(z))  �  = 0  (A-16)  −3  �  1 − z2 + 2z  �  1 − z2  �  βz − α  �  1 − z2  �  +  �  2z2 − 1  � �  αz + β  �  1 − z2  �  = 0  (A-17)  −3  �  1 − z2 + 2βz2�  1 − z2 − 2αz(1 − z2) + (2z2 − 1)αz +  �  2z2 − 1  �  β  �  1 − z2 = 0  (A-18)  �  1 − z2 �  4βz2 − β − 3  �  = −4αz3 + 3αz  (A-19)  �  1 − z2� �  16β2z4 − 8β(β + 3)z2 + β2 + 6β + 9  �  = 16α2z6 − 24α2z4 + 9α2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-20)  With the knowledge that α2 + β2 = 1, we have:  −16z6 + (24 + 24β)z4 + (−30β − 18)z2 + β2 + 6β + 9 = 0  (A-21)  which can be reduced to a cubic equation by introducing z2 = w, resulting in the equation:  −16w3 + (24 + 24β)w2 + (−30β − 18)w + β2 + 6β + 9 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  (A-22)  If we pick ωn = 2π, xf = 600 mm and Vm = 240 mm/s, the only real solution which would satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-12)  is T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='0395 s and therefore T2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='3684 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' Note, t = cos−1(z) supports two real solutions and 4 complex  conjugate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' From there, T1 can be calculated but in this case just one of the real solutions satisfies  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content=' (A-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FAT4oBgHgl3EQf2x4q/content/2301.08716v1.pdf'}
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+Minimal time of magnetization switching in small ferromagnetic
+ellipsoidal samples
+Raphaël Côte∗ 1, Clémentine Courtès † 1, Guillaume Ferrière ‡ 1, and Yannick Privat § 1,2
+1IRMA, Université de Strasbourg, CNRS UMR 7501, Inria, 7 rue René Descartes, 67084
+Strasbourg, France.
+2Institut Universitaire de France (IUF)
+January 11, 2023
+Abstract
+In this paper, we consider a ferromagnetic material of ellipsoidal shape. The associated magnetic
+moment then has two asymptotically stable opposite equilibria, of the form ±m. In order to use
+these materials for memory storage purposes, it is necessary to know how to control the magnetic
+moment. We use as a control variable a spatially uniform external magnetic field and consider the
+question of flipping the magnetic moment, i.e., changing it from the +m configuration to the −m
+one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic
+field used. We therefore include a constraint on the L∞ norm of the controls, assumed to be less
+than a threshold value U. We show that, generically with respect to the dimensions of the ellipsoid,
+there is a minimal value of U for this problem to have a solution. We then characterize it precisely.
+Finally, we investigate some particular configurations associated to geometries enjoying symmetries
+properties and show that in this case the magnetic moment can be controlled in minimal time
+without imposing a threshold condition on U.
+Keywords: ferromagnetic materials, Landau-Lifshitz equation, optimal control, minimal time
+AMS classification: 49J15, 49J30, 35Q60, 78M50.
+1
+Introduction
+1.1
+The Landau-Lifshitz equation for ellipsoidal ferromagnetic samples
+Ferromagnetic materials have come into common use in the last few decades, especially since they are
+found in devices used to store digital information such as magnetic tapes or hard disks, but also in
+magnetic chips called Magnetic Random Access Memory (MRAM). These chips have many advantages
+over their silicon counterparts, in particular that of requiring energy only to change the value bits and
+not to maintain the storage itself. This is probably one of the most challenging applications since it
+opens the door towards new spintronic applications and storage technologies while allowing a very fast
+access to information (see, e.g., [15]).
+∗raphael.cote@unistra.fr
+†clementine.courtes@unistra.fr
+‡guillaume.ferriere@unistra.fr
+§yannick.privat@unistra.fr
+1
+arXiv:2301.03839v1  [math.AP]  10 Jan 2023
+
+The magnetic moment of a ferromagnetic material represented by a domain Ω ⊂ R3 is usually
+modelled as a time-varying vector field
+m : R × Ω → S2,
+where S2 is the unit sphere of R3, the evolution of which is driven by the so-called Landau-Lifshitz
+equation (see [13])
+∂m
+∂t = −m ∧ h(m) − αm ∧ (m ∧ h(m)),
+(1)
+where the effective field h(m) is defined by
+h(m) = 2A∆m + hd(m) + hext
+with α > 0, a constant (in time and space) damping coefficient which is characterized by the material.
+We refer for instance to [12, 6] for additional explanations.
+The constant A > 0 is the exchange
+constant, and can be assumed to be equal to A = 1/2 without loss of generality, with a normalization
+argument. The demagnetizing field hd(m) is the solution of the equations
+�
+div(hd(m) + m) = 0
+curl(hd(m)) = 0
+in D′(R3)
+where m is extended to R3 by 0 outside Ω and D′(R3) denotes the space of distributions on R3. The
+field hext is an external one, for instance it can be an external magnetic field.
+Note that it is possible to complete and specify this physical model by adding other relevant terms,
+for example by taking into account the anisotropic behavior of the crystal that composes the ferromag-
+netic material.
+Finally, it is standard to assume homogeneous Neumann boundary conditions on the magnetization
+on the boundary of Ω.
+In this article, we will consider a ferromagnetic sample of ellipsoidal shape, and the magnetization
+m and external field hext both spatially uniform. Indeed, ellipsoidal domains have been much studied
+in the literature dedicated to ferromagnetism [14, 8, 19]: on the one hand, they cover a large variety
+of geometrical shapes, and on the other hand, they are the only known bodies that can be uniformly
+magnetized in the presence of a spatially uniform inducing field. From the mathematical point of view,
+it is nice to consider such samples because the demagnetizing field hd appearing in the Landau-Lifschitz
+equation can be determined in an explicit way.
+Let us be more precise and clarify the model obtained in this case. In all the following of this article,
+we will denote Ω the ellipsoid of R3 of semiaxes a1 > 0, a2 > 0 and a3 > 0, and a basis (O; e1, e2, e3)
+chosen so that the Cartesian equation of Ω reads
+x2
+a2
+1
++ y2
+a2
+2
++ z2
+a2
+3
+= 1.
+(2)
+An illustration of the ellipsoid Ω is shown in Figure 1. According to [14, 8], for uniform (in space)
+magnetizations m on Ω, the demagnetizing field hd(m) can be explicitly computed and reads
+hd(m) = −Dm,
+with
+D =
+�
+�
+γ1
+0
+0
+0
+γ2
+0
+0
+0
+γ3
+�
+� ,
+where γi (i = 1, 2, 3) denotes a positive constant depending only on the semiaxes a, b and c (we provide
+the precise dependence in Appendix A).
+2
+
+One can easily infer from this result that, provided that the external field hext and the initial
+magnetization m0 are constant in space, so is the magnetic moment m solving the Landau-Lifshitz
+equation (1) completed with homogeneous Neumann boundary conditions.
+As a consequence, the Landau-Lifschitz equation with a time-dependent external magnetic field hext
+reads as the ordinary differential system
+�
+˙m = α (h0(m) − (h0(m) · m)m) − m ∧ h0(m)
+in (0, T)
+m(0) = m0
+(3)
+where the dotted notation ˙m stands for the time derivative of m, h0(m) = −Dm + hext, T > 0,
+m(t) ∈ S2 ⊂ R3, D = diag([γ1, γ2, γ3]) denotes a diagonal matrix with positive coefficients. Up to a
+change of basis, we will also assume without loss of generality that
+0 ≤ γ1 ≤ γ2 ≤ γ3 ≤ 1.
+(4)
+In what follows, we will assume that the ferromagnetic particle is subjected to a spatially uniform
+external magnetic field hext, and we are interested in two asymptotically stable stationary states of the
+resulting system, denoted m. We seek to answer the following question:
+Given a maximum value U of the norm of the field hext at all times, can we determine
+whether there exists such a field flipping the magnetic spin from m to −m in minimal time?
+x
+y
+z
+2a1
+2a2
+2a3
+Figure 1: The ellipsoid shaped ferromagnetic sample
+1.2
+State of the art and structure of the article
+The development of the use of ferromagnetic materials has led to the emergence of new storage pos-
+sibilities, and consequently to a renewed interest of the scientific community around the control of
+EDO/EDP on this topic.
+The use of an external magnetic field to control a ferromagnetic system is a very present issue in
+the literature of the field (see for instance [7, 18, 20, 21]).
+Many works have focused on both the derivation of relevant models, i.e. sufficiently close to the
+physics, but also simple enough to be exploited mathematically, and on the related optimization issues.
+Many studies are devoted to these modeling questions, to the obtaining of exploitable optimality con-
+ditions leading to numerical simulations. Thus, in the same spirit as the present study, the authors of
+3
+
+[5] seek to flip the magnetic spin using electric current injections. Let us mention in the same vein the
+works [10, 11] also addressing similar issues: minimization of the distance to a target state with a fixed
+time horizon, addition of stochastic term in the model, search for a feedback and numerical analysis of
+the considered problems.
+Recent progress has been made in the understanding of the control (exact and approximate) of
+ellipsoidal samples/networks : [9, 4, 1]. Our study has been particularly motivated by [3], in which
+it is notably proved that, when the size of an open bounded by Ω tends to 0, then we find a uniform
+magnification in the domain, which lends itself to the study of ellipsoids.
+Structure of the article.
+In this paper, we are interested in a single ferromagnetic particle of
+ellipsoidal shape in R3. We seek to perform a magnetic moment reversal in minimal time, using an
+external magnetic field as a control of the resulting physical system.
+We model this issue in section 2.1, imposing a maximum L∞ norm on the control translated using
+the parameter U > 0, reflecting the difficulty and cost of using very high magnetic fields.
+In the
+absence of additional symmetries on the geometry of the system, we show the existence of a minimal
+threshold on U for the minimum time problem to have a solution. We refine this result when additional
+symmetries are assumed on the material geometry. The main results of this paper are gathered in the
+section 2.2. The section 3 contains the foundation of the proofs of the main results: indeed, we state
+necessary and sufficient conditions guaranteeing the well-posedness of the time-optimal problem and
+write the necessary conditions of optimality to the first order using the Pontryagin maximum principle.
+The proofs of the main results are contained in the sections 4, 5 and 6. Finally, some numerical
+simulations are listed in the section 7 to illustrate the qualitative behavior of the solutions obtained
+in theoretical theorems. The appendices contain additional information and/or secondary calculations.
+Appendix A contains the calculation of the demagnetizing field in the case of a ferromagnetic ellipsoid
+sample. Appendix B contains the proof that −e1 is indeed the only asymptotically stable state for
+equation (3). Finally, Appendix C contains the calculations of the explicit constants in the case γ1 < γ2.
+Notations.
+In the whole article, |·| denotes the standard euclidean norm on R3 (or Rd), and its inner
+product in Rd is denoted with a dot.
+We are essentially interested in a control problem where the control function is the external field:
+we abide by the usual convention, and denote
+hext = u.
+2
+Existence of a minimal switching time
+Let us recall that, as mentioned in the introduction, we will consider a ferromagnetic sample whose
+shape Ω is the ellipsoid with Cartesian equation (2). The dynamics of the magnetic moment m(·),
+equal to m0 at the initial time, is hence driven by the simplified Landau-Lifshitz equation (3).
+2.1
+Towards an optimal control problem
+The main issue we want to tackle reads
+Given a steady-state m of (3) in S2, can we achieve a reversal by solving an optimal control
+problem, i.e. steering the system from m(0) = m to m(T) = −m while minimizing T?
+In what follows, we will consider particular stationary states: m = ±e1. It is proved in appendix B
+that these equilibria are asymptotically stable when γ1 < γ2. Therefore, they can be used as magnetic
+spin orientation for memory storage purposes. We will denote by u(·) the external (spatially uniform)
+magnetic field imposed on the system. This is the control variable in this problem. The question is
+4
+
+then to ask if it is possible to steer the solution mu of the system (3) associated to the control u(·) and
+to the initial data mu(0) = e1 until mu(T) = −e1, in minimal time.
+Of course, it is necessary to add physical constraints to this problem: if one imposes no restrictions
+on the choice of admissible controls, it is likely that the minimal time problem will have a solution,
+reached by unrealistic controls. For this reason, we will assume in what follows the constraint
+|u(t)| ≤ U
+a.e. in (0, T),
+(CT,U)
+in order to limit the choice of controls to realistic possibilities.
+All in all, the problem we aim at investigating reads as follows.
+Minimal time problem: let U > 0 and assume that m0 = e1. The problem reads
+TU :=
+inf
+(T,u)∈OU
+T,
+(P0)
+where
+OU = {(T, u) | T ∈ R+, u ∈ L∞(0, T) satisfies (CT,U) and mu(T) = −e1},
+with mu, the solution to (3) associated to the control function u(·) and the initial datum e1.
+We will investigate the following issues:
+• Does Problem (P0) have an optimal solution for any value of U > 0 ?
+• How to characterize all the solutions to this problem and understand their geometric dependence
+to the parameters γi, i = 1, 2, 3?
+2.2
+Main results
+First, the minimization problem is indeed well-posed, meaning that the existence of an optimal solution
+is equivalent to the existence of a minimal trajectory.
+Theorem 1. Let U > 0. The following properties are equivalent:
+(i) There exists an optimal pair (TU, u) ∈ OU for Problem (P0).
+(ii) TU is finite.
+(iii) OU is nonempty.
+The behavior of the control system differs greatly depending on the values of the parameters γi and
+more specifically on the values of γ1 and γ2.
+Theorem 2. Assume γ1 < γ2. Then there exists Ucrit > 0 such that
+• for all U ∈ (0, Ucrit], (P0) has no solution.
+• for all U > Ucrit, (P0) has a solution.
+Remark 3. It is notable that the proof of this theorem provides an explicit lower-bound estimate of
+Ucrit. The precise bound is derived in Remark 18
+5
+
+We are now interested in the case where γ1 = γ2, which is not covered by the above result. It is
+interesting to note that in this case, the behavior of the optimized physical system is very different from
+the one described in the Theorem 2. Indeed, this situation of symmetry leads to the fact that there is
+no longer a threshold from which the system is controllable.
+To complete this analysis, we also investigate in the following result the existence of optimal planar
+trajectories. In view of the system symmetry, it is natural to conjecture that the optimal trajectory
+are planar, since all the points in span(e1, e2)∩S2 are stable, and this set is even asymptotically stable.
+Somewhat surprisingly, We show here that this is actually not the case.
+Theorem 4. If γ1 = γ2 ≤ γ3, then the optimal control problem (P0) has a solution whatever the value
+of U > 0, meaning that Ucrit = 0, with the notations of Theorem 2. Furthermore, if γ1 < γ3, the
+optimal trajectory in S2 is not contained in the plane span(e1, e2).
+It is interesting to notice that Theorem 2 can be refined in the particular case where γ1 < γ2 = γ3.
+To this aim, we will deeply exploit the necessary first order optimality conditions provided by the so-
+called Pontryagin Maximum Principle (PMP). We refer to Section 3.3 for a precise statement of such
+conditions.
+Theorem 5. If γ1 ≤ γ2 = γ3, then Ucrit =
+α
+2
+√
+1+α2 (γ2 − γ1). Furthermore, for all U > Ucrit,
+TU =
+π
+√
+1 + α2
+�
+U 2 − U 2
+crit
+.
+Remark 6. In particular, we infer from the result above the following asymptotics:
+TU ∼
+π
+�
+2Ucrit(1 + α2)
+1
+√U − Ucrit
+as U ↘ Ucrit,
+and
+TU ∼
+π
+U
+√
+1 + α2
+as U → +∞.
+Remark 7 (Case of where the shape of the sample is a sphere). In the case where Ω is a sphere, then
+one has γ1 = γ2 = γ3. Then, both conclusions of Theorem 4 and 5 apply, meaning that Ucrit = 0 and
+the optimal time is given by TU = π/(U
+√
+1 + α2). Furthermore, it may be shown that, in that case,
+there exists optimal planar trajectories in each of the hyperplanes span(ei, ej) with i ̸= j. We refer for
+instance to [3, Proof of Prop. 2], whose main argument can be reproduced in our case.
+We end this section by a result on the asymptotic behavior of optimal magnetization trajectories
+as U diverges to +∞. We prove that optimal trajectories tend to be supported on a geodesic on the
+sphere whenever U is large.
+Theorem 8. Let γ1 ≤ γ2 ≤ γ3 and U > Ucrit and m be an optimal trajectory. Let p be its adjoint state.
+Then, if U is large enough, m stays close to the plane V = span(e1, p(TU)) in the following sense: there
+exists U0 > 0 and C > 0 such that for every U > U0 and t ∈ [0, TU],
+∥m(t) − PV m(t)∥ ≤ C
+U ,
+where PV denotes the orthogonal projection onto V .
+6
+
+3
+Minimization and optimality
+3.1
+Proof of Theorem 1: existence of an optimal trajectory
+Let us assume that OU is nonempty. This allows us to consider a minimizing sequence (Tn, un)n∈N ∈ ON
+U,
+and mn ∈ C ([0, Tn] the solution to (3) with field un. By definition, Tn → TU as n → ∞. In what
+follows, we will denote similarly a sequence and any subsequence with a slight abuse of notation, for
+the sake of simplicity.
+Let us introduce the functions �un, �mun defined on [0, 1] by
+�un(s) = un(Tns)
+and
+�mn(s) = mn(Tns).
+Hence, System (3) rewrites
+� ˙�mn = Tn
+�
+α
+�
+�h0( �mn) − (�h0( �mn) · �mn) �mn
+�
+− �mn ∧ �h0( �mn)
+�
+in (0, 1)
+�mn(0) = e1
+(5)
+where �h0( �mn) = −D �mn + �un.
+Similarly, since the sequence (�un)n∈N is bounded in L∞(0, 1), it converges weakly-star in L∞(0, 1)
+up to a subsequence to some element u∗ such that |u∗(·)| ≤ U a.e. in [0, 1] according to the Banach-
+Alaoglu-Bourbaki theorem. Since both ( �mn)n∈N and (�un)n∈N are bounded in L∞([0, 1]), we infer that
+so is ˙�mn according to (3).
+Therefore, the sequence ( �mn)n∈N is bounded in W 1,∞(0, 1) and hence
+converges (up to a subsequence) towards an element �m∗ ∈ W 1,∞(0, 1) in C0([0, 1]) according to the
+Ascoli theorem. In particular, one has necessarily | �m∗(·)| = 1. Now, let us rewrite (5) as the fixed-point
+equation
+∀s ∈ [0, 1],
+�mn(s) = e1 + Tn
+� s
+0
+�
+α
+�
+�h0( �mn) − (�h0( �mn) · �mn) �mn
+�
+− �mn ∧ �h0( �mn)
+�
+dσ
+Observe that the right-hand side is linear with respect to �h0(mn) and that, according to the properties
+above, (�h0( �mn))n∈N converges weakly-star to �h0( �m∗) in L∞(0, 1), where �h0( �m∗) = −D �m∗+�u∗. Letting
+n tend to +∞ in the equation above, we obtain:
+∀s ∈ [0, 1],
+�m∗(s) = e1 + TU
+� s
+0
+�
+α
+�
+�h0( �m∗) − (�h0( �m∗) · �m∗) �m∗�
+− �m∗ ∧ �h0( �m∗)
+�
+dσ.
+Moreover, since ˜mn(1) = −e1 by construction, the convergence in C0([0, 1]) leads to ˜m∗(1) = −e1.
+Taking the previous formula with s = 1, we get
+−2e1 = TU
+� 1
+0
+�
+α
+�
+�h0( �m∗) − (�h0( �m∗) · �m∗) �m∗�
+− �m∗ ∧ �h0( �m∗)
+�
+dσ.
+This proves that TU > 0. Now, let us introduce u∗ as u∗(t) = �u∗(t/TU). By undoing the change of
+variable above, we get that �m∗(t/TU) = mu∗(t) for a.e. t ∈ [0, TU]. Furthermore, mu∗(0) = e1 and
+mu∗(TU) = −e1 since �mn(0) = e1 and �mn(1) = −e1. The converse sense is straightforward and the
+expected conclusion follows.
+Finally, observe that the same reasoning can be reproduced whenever TU is finite.
+3.2
+Sufficient and necessary condition for the existence of an admissible trajectory
+Let (m, u) be a solution to (3) on [0, T], and for t ∈ [0, T], consider the mobile frame B(t) =
+(m(t), ˙e(t), m(t) ∧ ˙e(t)) where ˙e = ˙m/| ˙m|1. According to [3], by observing that
+m ⊥ ˙m,
+˙m ⊥ m ∧ ˙m
+and
+m ⊥ m ∧ (m ∧ Dm),
+1Here, ˙e is merely a notation, and not the time derivative of a previously defined vector e.
+7
+
+one shows easily, by decomposing u(t) into B(t) and writing the equation for u−(u·m)m, the projection
+of u on m⊥, that there exists λ ∈ L∞(0, T) such that
+u =
+1
+1 + α2 (α ˙m + m ∧ ˙m) + Dm − (Dm · m)m + λm.
+(6)
+In fact, λ = u · m.
+Reciprocally, given any function m ∈ W 1,∞([0, T], S2), and any function λ ∈
+L∞([0, T], R), if we define u by (6), then (m, u) is solution to (3). These considerations can be seen as
+a consequence of a flatness property of the main system.
+Again assuming that (m, u) is admissible trajectory, i.e a solution to (3), We infer from (6) that
+u(t) expands as
+u = λm +
+�
+α
+1 + α2 | ˙m| + ˙e · Dm
+�
+˙e +
+�
+Dm · (m ∧ ˙e) +
+1
+1 + α2 | ˙m|
+�
+m ∧ ˙e.
+(7)
+As, a consequence, using that Dm · (m ∧ ˙e) = ˙e · (Dm ∧ m) due the triple product property, we get
+|u|2 = λ2 +
+�
+α
+1 + α2 | ˙m| + ˙e · Dm
+�2
++
+�
+Dm · (m ∧ ˙e) +
+| ˙m|
+1 + α2
+�2
+= λ2 +
+�
+˙e · Dm + α| ˙m|
+1 + α2
+�2
++
+�
+˙e · (Dmu ∧ m) +
+| ˙m|
+1 + α2
+�2
+.
+Clearly, this computation and the previous remarks show that, without loss of generality, we can
+furthermore assume that an optimal trajectory satisfies λ = 0, or equivalently, u · m = 0: we will do
+this in the following.
+Let us introduce, for a given T > 0,
+VT = {m ∈ H1([0, T]; S2) | m(0) = e1 and m(T) = −e1}.
+To investigate the existence of an admissible trajectory, it is then convenient to introduce
+Λ(T) :=
+inf
+λ∈L∞([0,T]) inf
+m∈VT
+sup
+t∈[0,T]
+�
+λ2 +
+�
+˙e · (Dm ∧ m) +
+| ˙m|
+1 + α2
+�2
++
+�
+˙e · Dm + α| ˙m|
+1 + α2
+�2�
+,
+= inf
+m∈VT
+sup
+t∈[0,T]
+��
+˙e · (Dm ∧ m) +
+| ˙m|
+1 + α2
+�2
++
+�
+˙e · Dm + α| ˙m|
+1 + α2
+�2�
+.
+(8)
+We summarize the above discussion in the form of a lemma.
+Lemma 9. The existence of an admissible trajectory for Problem (P0) comes to the existence of
+m ∈ VT such that the function u given by (7) with λ = 0 satisfies ∥u∥L∞([0,T];R3) ≤ U, which is also
+equivalent to Λ(T) ≤ U 2. Also, u satisfies u · m = 0.
+3.3
+Necessary optimality conditions for Problem (P0)
+This problem can be solved by using the Pontryagin maximum principle. The main results of this
+section are gathered in Proposition 11, at the end of this section.
+The Hamiltonian associated to Problem (P0) is
+H : S2 × R3 × {−1, 0} × R3
+→
+R
+(m, p, p0, u)
+�→
+p · (−αm ∧ (m ∧ h0(m)) − m ∧ h0(m)) .
+It can be noted that the dependence of H on the control function u is affine and one has
+H(m, p, p0, u) = p · (−αm ∧ (m ∧ u) − m ∧ u) − p · (−αm ∧ (m ∧ Dm) − m ∧ Dm)
+8
+
+As a first remark, the magnetization stays in S2 = ∂B, where B is the closed unit ball of R3.
+Therefore, our problem is obviously equivalent to the problem with restricted conditions
+TU =
+inf
+(T,u)∈OU
+|m|2−1=0
+T.
+Thus, we use the version of the Pontryagin maximum principle with restricted phase coordinates, as
+stated in [17, Theorem 22]. This theorem is stated for a minimization of an integral with fixed T, but it
+can be easily adapted to the case of a minimal time with classical changes (see for instance the passage
+from Theorem 1 to Theorem 2 in the same reference). With such a statement, we point out that, for
+all m ∈ R3
+∇(|m|2 − 1) = 2m,
+and that
+2m · (−αm ∧ (m ∧ h0(m)) − m ∧ h0(m)) = 0.
+Therefore, in our case, this statement gives the exact same necessary conditions, with an additional
+orthogonality condition for the adjoint state, stated hereafter.
+The first order optimality conditions read as follows: let us denote by (T, u), an optimal pair for
+this problem; there exists an absolutely continuous mapping p : [0, T] → R3 called adjoint state and a
+real number p0 ∈ {0, −1} such that the pair (p, p0) is non-trivial and for almost every t ∈ [0, T], the
+following conditions hold.
+• Adjoint equations. Setting
+F1(m, p)
+:=
+α
+�
+p ∧ (m ∧ Dm) + Dm ∧ (m ∧ p) − D(m ∧ (m ∧ p))
+�
+− Dm ∧ p − D(p ∧ m),
+F2(m, p, u)
+:=
+−α(p ∧ (m ∧ u) + u ∧ (m ∧ p)) + u ∧ p,
+one gets
+˙p = −∂H
+∂m = F1(m, p) + F2(m, p, u).
+(9)
+Remark that, since |m| = 1, one equivalently has
+F1(m, p)
+=
+α(Dp − (Dm · m)p − 2(m · p)Dm − 2(Dm · p)m) − Dm ∧ p − D(p ∧ m),
+F2(m, p, u)
+=
+α(p · m)u + α(u · m)p − 2α(p · u)m + u ∧ p
+• Maximality conditions. For a.e. t ∈ [0, T], u(t) solves the optimization problem
+max
+|v|≤U H(m(t), p(t), p0, v)
+(10)
+and one has at the final time T
+max
+|vT |≤U H(m(T), p(T), p0, vT ) = −p0.
+(11)
+• A useful identity. Since the dynamics only depends on the magnetization m(·) and the control
+u(·), the Hamiltonian functional is constant in time:
+H(m(t), p(t), p0, u(t)) = −p0,
+t ∈ [0, T],
+(12)
+according to (11), by evaluating the expression for t = T.
+9
+
+• An orthogonality condition for the adjoint state. At the final time t = T, the adjoint state p(T)
+is tangent to the boundary |m|2 − 1 = 0 at m(T) = −e1. This condition is thus equivalent to
+p(T) · e1 = 0.
+(13)
+• Orthogonality between u and m. As seen in Lemma 9, u · m = 0 on [0, T].
+Remark 10. Since the initial and final state are fixed, there is no need to impose any transversality
+condition on the adjoint state.
+Let us analyze the conditions (10) and (11).
+The adjoint state p cannot vanish on [0, T].
+Indeed, in the converse case, if there exists t0 ∈
+[0, T] such that p(t0) = 0, it follows from the Cauchy-Lipschitz theorem that p(·) = 0 and by using
+Condition (11), one gets p0 = 0, a contradiction with the non-triviality of the pair (p, p0).
+On condition (10).
+Observe that v �→ H(m(t), p(t), p0, v) is affine with respect to v. According to
+the Karush-Kuhn-Tucker theorem, there exists µ ≥ 0 such that ∇vH(m(t), p(t), p0, u(t)) − µu(t) = 0
+and the slackness condition µ(|u(t)|2 − U 2) = 0 is satisfied.
+If the set I := {|u| < U} is of positive Lebesgue measure, then one has
+α(p(t) − (p(t) · m(t))m(t)) = p(t) ∧ m(t)
+a.e. t ∈ I.
+Taking the scalar product of this identity with p(t) leads to |p(t)|2 = (p(t) · m(t))2 on I. Since p does
+not vanish, it follows from the equality case in the Cauchy-Schwarz inequality that p(t) is proportional
+to m(t). We will show that such a case cannot occur.
+Let us introduce the function ϕ := p − (p · m)m. One can compute that ϕ satisfies the differential
+(linear) relation
+˙ϕ = αDϕ − α(Dm · m)ϕ − Dm ∧ ϕ − D(ϕ ∧ m) + u ∧ ϕ + (m · (Dm ∧ ϕ))m − α(u · ϕ)m
+(14)
+a.e. in [0, T]. This follows from an easy but lengthy computation, and from the fact that the function
+λ given by λ = p · m satisfies
+˙λ = −2α(Dm · m)λ − (Dm ∧ p) · m − α(u · p) + 2α(Dm · p),
+a.e. in [0, T]. We leave the details to the reader.
+Now, let us assume that the set I := {|u(·)| < U} is of positive Lebesgue measure. According to the
+discussion above, there exists a bounded function λ such that p = λm on I, and therefore, ϕ vanishes
+on I. Due to (14) being linear in ϕ, we obtain that ϕ(·) = 0 on [0, T], which means that
+p(T) = (p(T) · m(T))m(T) = (p(T) · e1)e1 = 0,
+from the orthogonality condition (13). But recall that p cannot vanish on [0, T]: we reached a contra-
+diction.
+We conclude that |u| = U a.e. on [0, T]. It follows that
+α(p − (p · m)m) = p ∧ m + µu
+on [0, T],
+and using furthermore that
+|α(p − (p · m)m) − p ∧ m|2 = α2(|p|2 − (p · m)2) + |p ∧ m|2
+10
+
+= (α2 + 1)(|p|2 − (p · m)2),
+one gets an expression of u in terms of p or ϕ:
+u =
+U
+(α2 + 1)1/2
+α(p − (p · m)m) − p ∧ m
+�
+|p|2 − (p · m)2
+=
+U
+(α2 + 1)1/2
+αϕ − ϕ ∧ m
+|ϕ|
+(15)
+In particular, we get
+m ∧ u =
+U
+(α2 + 1)1/2
+αm ∧ ϕ − ϕ
+|ϕ|
+,
+αm ∧ (m ∧ u) =
+U
+(α2 + 1)1/2
+−α2ϕ − αm ∧ ϕ
+|ϕ|
+.
+Substituting those terms in (3) and (14), we get at last
+˙m = m ∧ Dm + m ∧ (m ∧ Dm) + U(α2 + 1)1/2 ϕ
+|ϕ|,
+(16)
+˙ϕ = αDϕ − α(Dm · m)ϕ − Dm ∧ ϕ − D(ϕ ∧ m) + (m · (Dm ∧ ϕ))m − U(α2 + 1)1/2|ϕ|m.
+(17)
+On condition (11).
+By setting p(T) = (0, p2,T , p3,T ) (since p(T) · e1 = 0) and vT = (v1,T , v2,T , v3,T ),
+since m(T) = −e1, this condition also rewrites
+−p0 = max
+|vT |≤U p(T) ·
+�
+�
+0
+αv2,T − v3,T
+αv3,T + v2,T
+�
+� =
+max
+v2
+2,T +v2
+3,T =U2
+�v2,T
+v3,T
+�
+·
+�αp2,T + p3,T
+αp3,T − p2,T
+�
+.
+The Cauchy-Schwarz inequality then implies that −p0 = U
+√
+1 + α2�
+p2
+2,T + p2
+3,T . It follows that p0 =
+−1 (else, the pair (p0, p) would be trivial) and condition (11) finally rewrites:
+U
+�
+1 + α2|ϕ(T)| = −p0 = 1.
+Analysis of the optimality conditions.
+From the previous discussion, u(t) is given by (15) for a.e.
+t ∈ [0, T], leading to
+|ϕ| (α (u − (u · m)m) − m ∧ u) = A
+�
+α2ϕ − αϕ ∧ m − αm ∧ ϕ + m ∧ (ϕ ∧ m)
+�
+= A(α2 + 1)ϕ
+where A =
+U
+(α2+1)1/2 , so that
+H(m(t), p(t), p0, u(t)) = U(α2 + 1)1/2|ϕ| − p · (α (Dm − (Dm · m)m) − m ∧ Dm)
+= U(α2 + 1)1/2|ϕ| − Dm · (α (p − (p · m)m) − p ∧ m)
+= U(α2 + 1)1/2|ϕ| − Dm · (αϕ − ϕ ∧ m)
+= (α2 + 1)1/2|ϕ|
+U
+�
+U 2 − Dm · u
+�
+and we infer that
+|ϕ(t)|
+�
+U 2 − Dm(t) · u(t)
+�
+=
+U
+(α2 + 1)1/2 > 0
+a.e. in [0, T].
+(18)
+11
+
+On condition (12)
+From the previous discussion, we have for any t ∈ [0, T]
+max
+|v|≤U H(m(t), p(t), p0, v) = −p0 = 1,
+(19)
+which leads at t = 0 to
+U
+�
+1 + α2|ϕ(0)| = −p0 = 1.
+(20)
+More generally, with the above expression (15) of u(t) which is the argmax of H, we get
+1 = U
+�
+1 + α2|ϕ| − ϕ ·
+�
+αDm − m ∧ Dm
+�
+.
+(21)
+For the sake of clarity, we sum-up all these informations in the following result.
+Proposition 11 (Necessary first order optimality conditions). Let (T, u) denote an optimal pair for
+Problem (P0). Then, the adjoint state p defined by (9) does not vanish on [0, T] and one has
+u =
+U
+(α2 + 1)1/2
+αϕ − ϕ ∧ m
+|ϕ|
+,
+(22)
+where ϕ is given by ϕ = p − (p · m)m. In particular, one has |u(t)| = U a.e. on [0, T].
+Moreover, ϕ satisfies the differential relation (17) completed by the conditions (18) and
+U
+�
+1 + α2|ϕ(0)| = U
+�
+1 + α2|ϕ(T)| = 1.
+(23)
+Finally, m satisfies (16).
+4
+Proof of Theorem 2
+4.1
+Preliminary results
+We first state preliminary results, in the form of a series of lemmas.
+Lemma 12. For all T < ∞, the map
+L∞(0, T) → W 1,∞(0, T)
+u �→ m solution to (3) with m(0) = e1
+is continuous and locally Lipschitz.
+Proof. Let u1, u2 ∈ L∞(0, T) and m1, m2 the corresponding solution to (3). Define δm := m1 −m2 and
+δu := u1 − u2. Simple, though tedious, calculations provide that δm satisfies
+dδm
+dt
+= α
+�
+−D δm + δu − ((−D δm + δu) · m1)m1
+− ((−Dm2 + u2) · δm)m1 − ((−Dm2 + u2) · m2)δm
+�
+− m1 ∧ (−Dδm + δu) − δm ∧ (−Dm2 + u2).
+in (0, T). Since |m1| = |m2| = 1, we obtain
+����
+dδm
+dt
+���� ≤ ((4α + 2)∥D∥2 + (2α + 1)|u2|) |δm| + (2α + 1)|δu|,
+in (0, T),
+(24)
+12
+
+where ∥·∥2 denotes the operator norm associated to the euclidean norm |·|. Since δm(0) = 0, we have
+for all t ∈ (0, T)
+|δm(t)| ≤
+� t
+0
+����
+dδm
+dt
+����(s) ds ≤ (2α + 1)T∥δu∥L∞ +
+� t
+0
+((4α + 2)∥D∥2 + (2α + 1)∥u2∥L∞) |δm|(s) ds,
+and thus by Gronwall’s lemma,
+|δm(t)| ≤ (2α + 1)T∥δu∥L∞ exp (((4α + 2)∥D∥2 + (2α + 1)∥u2∥L∞)t),
+t ∈ [0, T].
+Using this estimate, and plugging it aslo in (24), we get
+∥δm∥W 1,∞ ≤ C(T, ∥u2∥L∞)∥δu∥L∞,
+and the conclusion follows.
+Lemma 13. If γ1 < γ2, there exists δ > 0 such that for all U > 0, if |m(0) + e1| < δ, then −e1 can be
+reached in finite time with a control u such that |u| ≤ U.
+Proof. Let us introduce m as the solution to (3) with the feedback control term
+u(t) =
+U
+√
+1 + α2
+α(−e1 + (e1 · m(t))m(t)) + e1 ∧ m(t)
+�
+1 − (m(t) · e1)2
+,
+so that the equation on m becomes autonomous, and is well defined as long as m(t) ̸= ±e1. Observing
+that
+�
+m, e1 − (m · e1)m
+�
+1 − (m · e1)2 ,
+m ∧ e1
+�
+1 − (m · e1)2
+�
+is an orthonormal basis, one immediately gets that |u(t)| = U for a.e. t ∈ [0, T].
+Denote m = (m1, m2, m3) the coordinates of m. From (3), the ODEs satisfied by m2 and m3 are
+˙m2 = −α[(γ2 − γ1)m2 − ((γ2 − γ1)m2
+2 + (γ3 − γ1)m3
+3)m2] + (γ1 − γ3)m1m3 + v2,
+˙m3 = −α[(γ3 − γ1)m3 − ((γ2 − γ1)m2
+2 + (γ3 − γ1)m3
+3)m3] − (γ1 − γ2)m1m2 + v3.
+Therefore, by setting �m := (m2, m3), it follows that �m solves the controlled system
+˙˜m = A− ˜m + ξ− + ˜v,
+(25)
+where
+A−
+=
+�−α(γ2 − γ1)
+(γ3 − γ1)
+−(γ2 − γ1)
+−α(γ3 − γ1)
+�
+,
+ξ−
+=
+�α((γ2 − γ1)m2
+2 + (γ3 − γ1)m2
+3)m2 − (γ3 − γ1)(1 + m1)m3
+α((γ2 − γ1)m2
+2 + (γ3 − γ1)m2
+3)m3 + (γ2 − γ1)(1 + m1)m2
+�
+and ˜v = (v2, v3) where v = (v1, v2, v3) = α(u − (u · m)m) − m ∧ u, which means here
+v = U
+�
+1 + α2 (−e1 + m1(t)m(t))
+�
+|m(t)|2 − (m1(t))2
+= U
+�
+1 + α2 (−e1 + m1(t)m(t))
+| ˜m|
+,
+We infer that ˜v = U
+√
+1 + α2m1 ˜m/| ˜m|. Observing that (1 − m1)(1 + m1) = | ˜m|2 yields, as soon as
+m1 ≤ 0,
+|ξ−(t)| ≤ (1 + |α|) δγ+ | ˜m(t)|3
+13
+
+where δγ+ := γ3 − γ1 > 0, and also that
+����˜v + U
+�
+1 + α2 ˜m(t)
+| ˜m|
+���� ≤ U
+�
+1 + α2| ˜m(t)|2.
+With these estimates and by taking the inner product of (25) with ˜m, we get
+1
+2
+d
+dt| ˜m(t)|2 ≤ −U
+�
+1 + α2| ˜m(t)| + (U
+�
+1 + α2 + ∥A−∥)| ˜m(t)|2 + (1 + |α|) δγ+ | ˜m(t)|4,
+and
+d
+dt| ˜m(t)| ≤ −U
+�
+1 + α2 + (U
+�
+1 + α2 + ∥A−∥)| ˜m(t)| + (1 + |α|) δγ+ | ˜m(t)|3.
+Let us introduce δU ∈ (0, 1/2) small enough (depending on U > 0) so that
+(U
+�
+1 + α2 + ∥A−∥)δU + (1 + |α|) δγ+ δ3
+U ≤ U
+√
+1 + α2
+2
+.
+(26)
+Then, if |m(0) + e1| < δU, which gives | ˜m(0)| < δU, one has
+d
+dt| ˜m(t)| ≤ −U
+√
+1 + α2
+2
+< 0,
+as long as | ˜m(t)| < δU. This yields that, for such time intervals, the mapping t �→ | ˜m(t)| is decreasing.
+Therefore, this shows that if δU satisfies (26) and m(0) is such that |m(0)+e1| < δU, then | ˜m(t)| < δU
+for all t ≥ 0 and that ˜m(t) reaches 0 in finite time. In other words, −e1 can be reached in finite time
+with a control u such that |u| ≤ U if m is such that |m + e1| < δU.
+To conclude, it remains to drop the dependency of δ in U. Let us use that −e1 is asymptotically
+stable according to Proposition 24. Therefore, there exists δ > 0 such that, starting from a point
+m(0) chosen so that |m(0) + e1| < δ, we can first let the system evolve without control until we obtain
+|m(TU) + e1| < δU for some finite time TU. From this moment, we know we can reach −e1 in finite
+time, whence the expected conclusion.
+Recall for the sake of readability that the notation TU has been introduced in Section 2.1.
+Lemma 14. Let γ1 < γ2 and U > 0 such that TU < ∞. Then there exists ε > 0 such that TU−ε < ∞.
+Proof. Since TU < ∞ and according to Theorem 1, there exists u∗ ∈ L∞(0, TU) such that m∗(0) = e1
+and m∗(TU) = −e1.
+Now, let us consider m the solution to (3) associated to the control choice
+u = U−ε
+U u∗ for some ε ∈ (0, U) to be defined later. From Lemma 12, we obtain
+∥m − m∗∥W 1,∞(0,TU) ≤ C
+����
+U − ε
+U
+u∗ − u∗
+����
+L∞(0,TU)
+= Cε
+∥u∗∥L∞(0,TU)
+U
+≤ Cε
+for some C > 0.
+Since m∗(TU) = −e1 by definition, we can take ε > 0 small enough so that |m(TU) + e1| < δ, where
+δ > 0 is given by Lemma 13. From this lemma, we know we can reach −e1 in finite time, and since
+|u| ≤ U − ε, this leads to TU−ε < ∞.
+Lemma 15. TU is non-increasing with respect to U > 0. In particular, if TU0 < ∞ for some U0 > 0,
+then TU < ∞ for all U > U0.
+Proof. This property is an immediate consequence of the definition of TU and the fact that the sets OU
+are increasing for the inclusion.
+14
+
+4.2
+Emergence of a threshold
+The following result is the most crucial for concluding. It quantifies the asymptotic stability of e1 for
+the evolution of the magnetization m, with respect to u viewed as a perturbation. It is notable that its
+proof not only highlights the emergence of a threshold but also provides an explicit expression.
+Lemma 16. Let us assume that γ1 < γ2. There exists Ustab > 0 depending only on γ3 − γ1, γ2 − γ1
+and α such that, for any U < Ustab, the following holds. Let (m, u) be a solution of (3) on [0, +∞),
+such that u ∈ L∞([0, ∞)) and ∥u∥L∞ ≤ U. Then for all t ≥ 0, m1(t) ≥ 0.
+In other words, m remains in the hemisphere with pole e1, and in particular, m can not reach −e1.
+One has the same statement if (m, u) are defined on a bounded interval [0, T].
+Proof. Let v = α(u − (u · m)m) − m ∧ u. Then, using that v reads as the sum of two orthogonal terms,
+one has |v|2 ≤ (1 + α2)U 2. Moreover, since |m|2 = 1, there holds
+Dm · m = γ1 + (γ2 − γ1)m2
+2 + (γ3 − γ1)m2
+3.
+As in the proof of Lemma 13, �m solves the controlled system
+˙�m = A �m + ξ + ˜v.
+(27)
+where
+A =
+�−α(γ2 − γ1)
+−(γ3 − γ1)
+γ2 − γ1
+−α(γ3 − γ1)
+�
+,
+ξ =
+�α((γ2 − γ1)m2
+2 + (γ3 − γ1)m2
+3)m2 + (γ3 − γ1)(1 − m1)m3
+α((γ2 − γ1)m2
+2 + (γ3 − γ1)m2
+3)m3 − (γ2 − γ1)(1 − m1)m2
+�
+.
+(28)
+Pay attention to the sign change between A and ξ used here and A− and ξ− introduced in the proof of
+Lemma 13. Let ν ∈ (0, 1] to be fixed later and define
+Tν = inf{t ≥ 0 | | �m(t)| ≥ ν}.
+Our goal is to derive suitable bounds on ˜m, so that for a well chosen ν, Tν = +∞.
+Since m(0) = e1 and m is continuous, we know that Tν > 0.
+Note that one has necessarily
+m1(·) > 0 on (0, Tν). Then, for all t ∈ [0, Tν), using that m is normalized, there holds like previously
+0 ≤ 1 − m1(t) ≤ 1 − m1(t)2 = | �m(t)|2, and therefore
+|ξ(t)| ≤ (1 + |α|) δγ+ | �m(t)|3 ≤ (1 + |α|) δγ+ ν3,
+where δγ+ := γ3 − γ1 ≥ γ2 − γ1 =: δγ− > 0. On the other hand, thanks to the Duhamel formula on
+(27) using the fact that �m(0) = 0, there holds
+�m(t) =
+� t
+0
+exp ((t − s)A)(ξ(s) + ˜v(s)) ds
+for all t ≥ 0. This, together with the previous estimates, drives to
+| �m(t)| ≤
+� t
+0
+∥exp ((t − s)A)∥2
+�
+(1 + |α|)δγ+ν3 +
+�
+1 + α2U
+�
+ds
+≤ (1 + |α|)(δγ+ν3 + U)
+� t
+0
+∥exp ((t − s)A)∥2 ds
+≤ (1 + |α|)(δγ+ν3 + U)
+� t
+0
+∥exp (sA)∥2 ds,
+(29)
+for all t ∈ [0, Tν), where ∥·∥2 still denotes the operator norm associated to the euclidean norm |·|.
+We will now provide an estimate of the norm of the exponential matrix. Recall that the characteristic
+polynomial of A is PA(X) = X2 − Tr(A)X + det(A) with
+det(A) = (1 + α2)δγ− δγ+ > 0,
+Tr(A) = −α(δγ− + δγ+) < 0.
+Its discriminant ∆ reads ∆ = Tr(A)2 − 4 det(A) = α2(δγ+ − δγ−)2 − 4δγ−δγ+.
+To compute the
+eigenvalues of A, we have to distinguish between several cases.
+15
+
+1st case: ∆ > 0.
+Then its eigenvalues are λ± := 1
+2(Tr(A) ±
+√
+∆). Remark that both eigenvalues of
+A are negative (according to the signs of the trace and the determinant above) and different from each
+other, which means that A is diagonalizable. Therefore, we infer2 that
+exp (sA) = esλ+ sA − sλ− I2
+sλ+ − sλ−
++ esλ− sA − sλ+ I2
+sλ− − sλ+
+=
+1
+√
+∆
+�
+esλ+(A − λ− I2) − esλ−(A − λ+ I2)
+�
+=
+1
+√
+∆
+�
+(esλ+ − esλ−)A + (esλ−λ+ − esλ+λ−) I2
+�
+= esλ+
+√
+∆
+�
+(1 − e−s
+√
+∆)A + (e−s
+√
+∆λ+ − λ−) I2
+�
+= sesλ+
+�1 − e−s
+√
+∆
+s
+√
+∆
+A − λ−
+1 − e−s
+√
+∆
+s
+√
+∆
+I2
+�
++ esλ− I2 .
+Thus, using the facts that λ− < λ+ < 0, ∥A∥2 ≥ |λ−| and also that the function f given by f(x) = 1−e−x
+x
+analytically extended to R is uniformly bounded by 1 on [0, ∞), we get
+∥exp (sA)∥2 ≤ esλ+ (s(∥A∥2 + |λ−|) + 1) ≤ esλ+ (2s∥A∥2 + 1) .
+Hence,
+� t
+0
+∥exp (sA)∥2 ds ≤ |λ+|−1(1 − eλ+t) + 2∥A∥2|λ+|−2(1 − (|λ+|t + 1)eλ+t)
+≤ (1 − eλ+t)
+�
+|λ+|−1 + 2∥A∥2|λ+|−2�
+≤ 3(1 − eλ+t)∥A∥2|λ+|−2,
+and according to (29), one has for all t ∈ [0, Tν)
+| �m(t)| ≤ 3∥A∥2|λ+|−2(1 − eλ+t)(1 + |α|)(δγ+ν3 + U)
+To conclude, we will choose U adequately so that the function x �→ 3∥A∥2|λ+|−2(1 + |α|)(δγ+x3 + U)
+admits a fixed point x0 in (0, 1]. This is possible thanks to the next lemma, whose proof is postponed
+to the end of this section for the sake of clarity.
+Lemma 17. Let a, b, c > 0. The function x �→ a−1(bx3 + c) has a fixed point x0 in (0, 1] if, and only
+if c ≤ ax1 − bx3
+1 where x1 = min{1, � a
+3b}.
+Remark that if x1 is as in this lemma, one has ax1 − bx3
+1 ≥
+2
+3ax1 > 0.
+Hence, setting a =
+|λ+|2/(3∥A∥2(1 + |α|)), b = δγ+ and c = U drives us to assume that
+U ≤
+|λ+|2
+3∥A∥2(1 + |α|)x1 − δγ+ x3
+1,
+with
+x1 := min
+�
+1,
+|λ+|
+3
+�
+∥A∥2(1 + |α|)δγ+
+�
+,
+we can take ν = x0 provided by Lemma 17, and the previous estimate leads to
+| �m(t)| ≤ (1 − eλ+t)ν,
+for all t ∈ [0, Tν). A continuity argument then implies that Tν = ∞. In other words, for all t ≥ 0,
+|m1(t)| =
+�
+1 − | ˜m(t)|2 ≥
+�
+1 − (1 − eλ+t)2ν2 > 0. Now, m1 is continuous, so that it keeps a constant
+sign. As m1(0) = 1, m1(t) ≥ 0 for all t ≥ 0, which is the desired conclusion.
+2Here, the Lagrange interpolation formula is used to compute the exponential of A: for every matrix M ∈ Md(C)
+whose spectrum {λi}1≤i≤d consists of distinct eigenvalues, one has
+exp(M) =
+d
+�
+j=1
+eλj �
+i̸=j
+M − λi Id
+λj − λi .
+16
+
+2nd case: ∆ < 0.
+In this case, the eigenvalues are
+λ± := Tr(A) ± i
+√
+−∆
+2
+.
+One more time, the two eigenvalues are distinct, complex conjugate with negative real part. Yet, the
+same decompositions as previously can still be applied and there holds
+exp (sA) = esλ+ sA − sλ− I2
+sλ+ − sλ−
++ esλ− sA − sλ+ I2
+sλ− − sλ+
+=
+1
+i
+√
+−∆
+�
+esλ+(A − λ− I2) − esλ−(A − λ+ I2)
+�
+=
+1
+i
+√
+−∆
+�
+(esλ+ − esλ−)A + (esλ−λ+ − esλ+λ−) I2
+�
+= e
+s
+2 Tr(A)
+i
+√
+−∆
+�
+(e
+is√−∆
+2
+− e− is√−∆
+2
+)A + (λ+e− is√−∆
+2
+− λ−e
+is√−∆
+2
+) I2
+�
+= e
+s
+2 Tr(A)
+√
+−∆
+�
+2 sin
+�s
+√
+−∆
+2
+�
+A +
+�√
+−∆ cos
+�s
+√
+−∆
+2
+�
+− Tr(A) sin
+�s
+√
+−∆
+2
+��
+I2
+�
+= s
+2e
+s
+2 Tr(A)�
+2A − Tr(A) I2
+�
+sinc
+�s
+√
+−∆
+2
+�
++ e
+s
+2 Tr(A) cos
+�s
+√
+−∆
+2
+�
+I2 .
+Thus, we get
+∥exp (sA)∥2 ≤ s
+2e
+s
+2 Tr(A) (2∥A∥2 − Tr(A)) + e
+s
+2 Tr(A) ≤ e
+s
+2 Tr(A) (2s∥A∥2 + 1) ,
+since Tr(A) = λ+ + λ− and |λ±| ≤ ∥A∥2. Hence, following the same way as in the first case, we get
+� t
+0
+∥exp (sA)∥2 ds ≤ (1 − e
+1
+2 Tr(A)t)
+�
+2|Tr(A)|−1 + 8∥A∥2|Tr(A)|−2�
+≤ 12(1 − e
+1
+2 Tr(A)t)∥A∥2|Tr(A)|−2,
+and according to (29), one has for all t ∈ [0, Tν)
+| �m(t)| ≤ 12∥A∥2|Tr(A)|−2(1 − e
+1
+2 Tr(A)t)(1 + |α|)(δγ+ν3 + U)
+Now, by mimicking the reasoning done in the first case, by assuming
+U ≤
+Tr(A)2
+12∥A∥2(1 + |α|)x1 − δγ+ x3
+1,
+with
+x1 := min
+�
+1,
+Tr(A)
+6
+�
+∥A∥2(1 + |α|)δγ+
+�
+,
+and taking ν = x0 given by Lemma 17, the previous estimate leads to
+| �m(t)| ≤ (1 − e
+1
+2 Tr(A)t)ν,
+for all t ∈ [0, Tν). Arguing as in the first case, we infer that Tν = ∞ in this case as well, and then,
+m1(t) > 0 for all t ≥ 0.
+17
+
+3rd case: ∆ = 0.
+In this case, both eigenvalues are equal, one has λ = Tr(A)/2 < 0. Note that, in
+that case, A− 1
+2 Tr(A) I2 is therefore a non-zero nilpotent matrix, and more precisely (A− 1
+2 Tr(A) I2)2 =
+(sA − s
+2 Tr(A) I2)2 = 0. Thus, there holds
+exp (sA) = exp
+�s
+2 Tr(A) I2
+�
+exp (sA − s
+2 Tr(A) I2)
+= e
+s
+2 Tr(A)(I2 +sA − s
+2 Tr(A) I2)
+which yields
+∥exp (sA)∥2 ≤ e
+s
+2 Tr(A)(1 + s(∥A∥2 − 1
+2 Tr(A))) ≤ e
+s
+2 Tr(A)(1 + 2s∥A∥2).
+The computations are then exactly the same ones as in the second case, and the conclusion follows in
+the same fashion.
+Proof of Lemma 17. We are looking for a root x0 ∈ (0, 1] of the polynomial function f given by f(X) =
+bX3 − aX + c, whose derivative 3bX2 − a is negative for X < � a
+3b =: x2 and positive for X > x2.
+The minimum in [0, 1] is therefore reached at x1 and, since f(0) = c > 0, there is a root if and only if
+f(x1) ≤ 0, which corresponds to the assumption in the statement.
+Remark 18. From the proof of Lemma 16, we obtained the following expression for Ustab. Consider
+the matrix A defined there in (28), denote ∆ = Tr(A)2 − 4 det(A) the discriminant of its characteristic
+polynomial, λ± its eigenvalues chosen so that λ+ > λ− whenever ∆ > 0, and δγ+ := γ3 − γ1 > 0.
+Let
+x1(A) :=
+�
+�
+�
+�
+�
+�
+�
+min
+�
+1,
+|λ+|
+3√
+∥A∥2(1+|α|)δγ+
+�
+if ∆ > 0
+min
+�
+1,
+Tr(A)
+6√
+∥A∥2(1+|α|)δγ+
+�
+else.
+Then
+Ustab = Γ(∆) :=
+�
+�
+�
+|λ+|2
+3∥A∥2(1+|α|)x1(A) − δγ+ x1(A)3
+if ∆ > 0
+Tr(A)2
+12∥A∥2(1+|α|)x1(A) − δγ+ x1(A)3
+else.
+Note also that, to complement this result, explicit computations of the quantities involved (like ∥A∥2)
+are provided in Appendix C.
+We now have all the elements to conclude the:
+Proof of Theorem 2. Define
+Ucrit := inf{U | TU < ∞}.
+From Lemma 16, we know that Ucrit > 0. Lemma 15 proves the second point. To investigate the case
+where U = Ucrit, observe that, by definition, (P0) has no solution for all U ∈ (0, Ucrit). Moreover, if
+(P0) had a solution for U = Ucrit, then Lemma 14 would provide a contradiction with respect to the
+definition of Ucrit.
+5
+Cases with symmetry
+In this section, we deal with the two cases when the material satisfies additional symmetry without
+being a sphere (in which case the analysis becomes trivial). They correspond to the cases γ1 = γ2 < γ3
+and γ1 < γ2 = γ3.
+18
+
+5.1
+Proof of Theorem 4 (case γ1 = γ2)
+From Theorem 1, we have to investigate the existence of an admissible trajectory for this problem,
+in other words, the existence of a control u ∈ OU and a time T > 0 such that mu(T) = −e1. This
+property is known to be true as soon as U is large enough according to [3]. But it has to be proved for
+smaller U.
+Let us assume that γ1 = γ2. We will prove that, in that case, infT>0 Λ(T) = 0, (with Λ(T) defined
+by Equation (8)) which will prove that Problem (P0) has a solution whatever the value of U > 0.
+For ε > 0, let us consider a particular trajectory mε of the form mε = (cos(εt), sin(εt), 0). Then, by
+defining
+Fε :=
+�
+˙eε · (Dmε ∧ mε) +
+| ˙mε|
+1 + α2
+�2
++
+�
+˙eε · Dmε + α| ˙mε|
+1 + α2
+�2
+with ˙eε = ˙mε/| ˙mε|, a straightforward computation yields
+Fε =
+ε2
+1 + α2 ≤ ε2.
+We infer that infT>0 Λ(T) ≤ Fε ≤ ε2 whence the conclusion, since ε is arbitrary.
+Let us now prove the last point of this result, assuming that from now on γ1 < γ3. Assume that
+m3(t) = 0 for all t ≥ 0. Then, Dm = γ1m. By contradiction, if such an m is an optimal trajectory,
+Proposition 11 is satisfied, and (16) gives
+˙m = U(α2 + 1)1/2 ϕ
+|ϕ|.
+By taking the third coordinate, we get ϕ3(t) = 0 for all t ≥ 0. Thus, we also get Dϕ = γ1ϕ and (17)
+gives
+˙ϕ = −γ1m ∧ ϕ − D(ϕ ∧ m) − U(α2 + 1)1/2|ϕ|m.
+By taking again the third coordinate, we get
+0 = −γ1(m ∧ ϕ) · e3 − D(ϕ ∧ m) · e3 = (γ3 − γ1)(m ∧ ϕ) · e3 = (γ3 − γ1)(e3 ∧ m) · ϕ.
+Since γ3 > γ1, this proves that (e3 ∧ m) · ϕ = 0. However, at t = 0, this means that
+0 = (e3 ∧ e1)ϕ(0) = ϕ2(0).
+Now ϕ1(0) = ϕ(0) · m(0) = 0, and we obtained ϕ(0) = 0: this is a contradiction with (23).
+5.2
+Proof of Theorem 5 (case γ2 = γ3)
+For this case, we first show that the (PMP) conditions are also sufficient conditions for optimal trajec-
+tories :
+Lemma 19. Let U > Ucrit. Then any trajectory m satisfying the (PMP) ( (9)-(12) with p0 = −1) is
+an optimal trajectory.
+Proof. Let m∗ be an optimal trajectory, and p∗ the associated adjoint state. By definition, they satisfy
+the (PMP) conditions.
+Now, let (m, p) be a trajectory and its adjoint state satisfying the (PMP)
+conditions. Let also ϕ = p − (p · m)m and ϕ∗ = p∗ − (p∗ · m∗)m∗. In particular, we know that ϕ(0)
+satisfies (23) and ϕ(0) ⊥ m(0) = e1, and similarly for ϕ∗ with respect to m∗.
+Thus, there exists
+θ ∈ [0, 2π] such that Rθϕ(0) = ϕ∗(0) where Rθ is the rotation along e1 of angle θ:
+Rθ =
+�
+�
+1
+0
+0
+0
+cos θ
+− sin θ
+0
+sin θ
+cos θ
+�
+� .
+19
+
+On the other hand, since γ2 = γ3, we have DRθ = RθD for all θ, but also Rθf ∧ Rθg = Rθ(f ∧ g)
+for any f, g ∈ R3.
+Last, Rθm∗(0) = Rθe1 = e1.
+Thus, (Rθm, Rθϕ) satisfies the same system of
+ODEs as (m∗, ϕ∗) (i.e. (3)-(9) with u(·) or u∗(·) satisfying (22)) with the same initial data. By the
+Cauchy-Lipschtiz theorem and using the fact that both ϕ and ϕ∗ never vanish thanks to (18), we obtain
+(Rθm, Rθϕ) = (m∗, ϕ∗), i.e. (m, ϕ) = (R−θm∗, R−θϕ∗), and thus the conclusion.
+The following two results exploit in a precise way the (necessary and sufficient) optimality conditions.
+Lemma 20. Let U > 0 and (m, p) satisfy the (PMP) conditions ( (9)-(12) with p0 = −1) and ϕ =
+p − (p · m)m. Then, for every t ≥ 0, ϕ(t) · (e1 ∧ m(t)) = 0.
+Proof. We know that ϕ satisfies (17) and m satisfies (16) with u given by (22). Moreover,
+d
+dt(ϕ · (e1 ∧ m)) = ˙ϕ · (e1 ∧ m) + ϕ · (e1 ∧ ˙m).
+Using the facts that u ⊥ m and m ⊥ (e1 ∧ m), there holds
+˙ϕ · (e1 ∧ m) = αDϕ · (e1 ∧ m) − α(Dm · m)ϕ · (e1 ∧ m) − (Dm ∧ ϕ) · (e1 ∧ m) − D(ϕ ∧ m) · (e1 ∧ m)
++
+U
+√
+1 + α2 (ϕ ∧ ( ϕ
+|ϕ| ∧ m)) · (e1 ∧ m),
+ϕ · (e1 ∧ ˙m) = −αϕ · (e1 ∧ Dm) + α(Dm · m)ϕ · (e1 ∧ m) + ϕ · (e1 ∧ (m ∧ Dm)) + U
+�
+1 + α2ϕ · (e1 ∧ ϕ
+|ϕ|).
+First, we point out that ϕ ⊥ m so that ϕ ∧ (ϕ ∧ m) = −|ϕ|2m, and thus (ϕ ∧ (ϕ ∧ m)) · (e1 ∧ m) = 0.
+Similarly, U
+√
+1 + α2ϕ · (e1 ∧ ϕ
+|ϕ|) = 0. Then, using the triple product formula, we get
+ϕ · (e1 ∧ Dm) = −Dm · (e1 ∧ ϕ)
+(Dm ∧ ϕ) · (e1 ∧ m) = (ϕ ∧ (e1 ∧ m)) · Dm = −(ϕ · e1)(m · Dm),
+(m ∧ Dm) · (e1 ∧ ϕ) = ((e1 ∧ ϕ) ∧ m) · Dm = (m · e1)(ϕ · Dm).
+Moreover, since γ2 = γ3, we know that, for any vector f ∈ R3 such that f · e1 = 0, Df = γ2f = γ3f.
+With the fact that D is symmetric, this leads to
+Dϕ · (e1 ∧ m) = ϕ · D(e1 ∧ m) = γ2ϕ · (e1 ∧ m),
+Dm · (e1 ∧ ϕ) = m · D(e1 ∧ ϕ) = γ2m · (e1 ∧ ϕ) = −γ2ϕ · (e1 ∧ m),
+D(ϕ ∧ m) · (e1 ∧ m) = (ϕ ∧ m) · D(e1 ∧ m) = γ2(ϕ ∧ m) · (e1 ∧ m).
+Last, using again the double product, we have
+ϕ · (e1 ∧ (m ∧ Dm)) = (ϕ · m)(e1 · Dm) − (ϕ · Dm)(e1 · m) = −(ϕ · Dm)(e1 · m).
+But one has then
+(ϕ · e1)(m · Dm) − (ϕ · Dm)(e1 · m) = e1 ·
+�
+(Dm · m)ϕ − (Dm · ϕ)m
+�
+= −e1 · (Dm ∧ (m ∧ ϕ))
+= −Dm · ((m ∧ ϕ) ∧ e1) = −m · D((m ∧ ϕ) ∧ e1)
+= −γ2m · ((m ∧ ϕ) ∧ e1) = γ2(ϕ ∧ m) · (e1 ∧ m).
+This means
+ϕ · (e1 ∧ (m ∧ Dm)) − (Dm ∧ ϕ) · (e1 ∧ m) − D(ϕ ∧ m) · (e1 ∧ m) = 0.
+Hence,
+d
+dt(ϕ · (e1 ∧ m)) = 0.
+The conclusion comes by integration, noticing furthermore that m(0) = e1 and thus e1 ∧ m(0) = 0.
+20
+
+Lemma 21. Let U > 0 and (m, p) satisfy the (PMP) conditions ( (9)-(12) with p0 = −1). Denote
+m = (m1, m2, m3) the coordinates of m, and define t0 := inf{t ≥ 0 | m(t) = ±e1} > 0 (possibly +∞)
+and θ ∈ [0, π] such that m1 = cos θ on [0, t0). Then m1 and θ satisfy on [0, t0)
+˙m1 = α(γ2 − γ1)(1 − m2
+1)m1 − U
+�
+1 + α2
+�
+1 − m2
+1,
+˙θ = −α(γ2 − γ1) sin θ cos θ + U
+�
+1 + α2.
+(30)
+Last, t0 = ∞ if U ≤ Ucrit and t0 = TU if U > Ucrit
+Proof. Let p its adjoint state and ϕ = p−(p·m)m. Then ϕ·(e1 ∧m) = 0 from Lemma 20, which means
+that ϕ is orthogonal to both m and e1 ∧ m for all times in [0, TU]. Moreover, as soon as m(t) ̸= ±e1
+(i.e. as soon as t ∈ (0, TU)), (m(t), e1 ∧m(t), m(t)∧(e1 ∧m(t))) is an orthogonal basis of R3. Therefore,
+ϕ(t) is colinear to m(t) ∧ (e1 ∧ m(t)), i.e. there is λ ∈ C ((0, TU), R) such that
+ϕ = λm ∧ (e1 ∧ m) = λ
+�
+e1 − m1m
+�
+.
+Moreover, from (21), we know that ϕ does not vanish, thus neither does λ, which has a constant sign.
+Then, we also have
+e1 · ϕ(t) = λ(t)(1 − m2
+1)
+and
+|ϕ(t)| = |λ(t)|
+�
+1 − m2
+1,
+which leads to
+e1 · ϕ(t)
+|ϕ(t)| = sign(λ)
+�
+1 − m2
+1,
+with sign(λ) = ±1 constant in time. We also have
+(Dm · m) = γ1m2
+1 + γ2(m2
+2 + m2
+3) = γ2 − (γ2 − γ1)m2
+1,
+e1 · (m ∧ Dm) = Dm · (e1 ∧ m) = m · D(e1 ∧ m) = m · γ2(e1 ∧ m) = 0.
+Therefore, the evolution equation on m1 is
+˙m1 = α(γ2 − γ1)(1 − m2
+1) m1 + sign(λ)U
+�
+1 + α2
+�
+1 − m2
+1.
+(31)
+On the other hand, we know that ˜m = (m2, m3) satisfies at t = 0:
+˙˜m(0) = U
+�
+1 + α2 ˜ϕ(0)
+|ϕ(0)|,
+with ˜ϕ(0) ̸= 0 since ϕ(0) · e1 = 0 and |ϕ(0)| =
+1
+U
+√
+1+α2 > 0. Since ˜m(0) = (0, 0), this means that | ˜m| is
+not vanishing on (0, ε] for some ε > 0 small enough. Since |m|2 = 1, this necessarily means that m2
+1 < 1
+on (0, ε]. Now, we can introduce θ(t) the first angle of the spherical coordinate such that m1 = cos θ,
+and we can assume that θ(0) = 0 and θ(t) > 0 on (0, ε]. The angle θ(t) is then well defined on [0, t0)
+where t0 = min{t > 0 | m(t) = ±e1} and θ(t) ∈ [0, π]. Moreover, since m1 is C 1 (due to (31), for
+example), θ is C 1 on (0, t0) as well. Then, on this interval, we can replace m1 in (31) by its expression
+in terms of θ, which leads to
+− ˙θ sin θ = α(γ2 − γ1) sin2 θ cos θ + sign(λ)U
+�
+1 + α2 sin θ,
+hence
+˙θ = −α(γ2 − γ1) sin θ cos θ − sign(λ)U
+�
+1 + α2.
+21
+
+From this, we also see that θ is C 1 at t = 0 with ˙θ(0) = − sign(λ)U
+√
+1 + α2. As θ ≥ 0 on [0, t0], we
+can easily see that sign(λ) = −1 (otherwise we would have ˙θ(0) < 0). This gives the expected ODEs
+on m1 and θ, but on [0, t0) only.
+To conclude, we shall prove that t0 = ∞ if U ≤ Ucrit or t0 = TU if U > Ucrit, which is equivalent
+to prove that m does not reach e1 again (up to reaching −e1 before), or equivalently that θ does not
+come back to 0 before reaching π. This follows from the fact that θ satisfies an autonomous first-order
+ODE of the form ˙θ = f(θ) with f(0) > 0.
+We are now in position to prove Theorem 5.
+Proof of Theorem 5. Let U > 0 and (m, ϕ) satisfying the (PMP) conditions ((9)-(12) with p0 = −1).
+From Lemma 19 and Theorem 2, we have 2 cases:
+• either U ≤ Ucrit, and then no trajectory reaches −e1 (and so in particular m).
+• either U > Ucrit, and then m reaches −e1.
+Therefore, we shall analyze only the case when U > Ucrit and m is able to reach −e1 and. From Lemma
+21, we can define θ(t) ∈ [0, π] such that m1 = cos θ, and it satisfies (30). Since it is an autonomous ODE
+of the form ˙θ = f(θ) with f(0) > 0, it is easy to prove that θ is able to reach π (which means m1 reaches
+−1 or also m reaches −e1) if and only if f > 0 on [0, π], where f(x) = −α(γ2−γ1) sin x cos x+U
+√
+1 + α2.
+From this,
+f(x) > 0
+∀x ∈ [0, π]
+⇐⇒
+1
+2 sin (2x) < U
+√
+1 + α2
+α(γ2 − γ1)
+∀x ∈ [0, π]
+⇐⇒
+U >
+α
+2
+√
+1 + α2 (γ2 − γ1).
+This gives the desired expression of Ucrit.
+Let us now compute the minimal time in that case. From the ODE (30) satisfied by θ for the
+optimal trajectory, we know that
+TU =
+� π
+0
+dθ
+−α(γ2 − γ1) sin θ cos θ + U
+√
+1 + α2
+=
+� π
+0
+dθ
+− 1
+2α(γ2 − γ1) sin 2θ + U
+√
+1 + α2
+=
+� 2π
+0
+dx
+−α(γ2 − γ1) sin x + 2U
+√
+1 + α2
+=
+1
+2
+√
+1 + α2
+� π
+−π
+dx
+−Ucrit sin x + U .
+With the change of variable y = tan x
+2, so that dx =
+2 dy
+1+y2 , sin x =
+2y
+1+y2 , we get
+TU =
+1
+2
+√
+1 + α2
+� +∞
+−∞
+2 dy
+−2Ucrity + U(1 + y2)
+=
+1
+√
+1 + α2
+� +∞
+−∞
+dy
+U
+�
+y − Ucrit
+U
+�2
++ U2−U2
+crit
+U
+=
+1
+U
+√
+1 + α2
+� +∞
+−∞
+dy
+y2 + U2−U2
+crit
+U2
+22
+
+=
+1
+U
+√
+1 + α2
+�
+U 2 − U 2
+crit
+U 2
+� +∞
+−∞
+dz
+U2−U2
+crit
+U2
+z2 + U2−U2
+crit
+U2
+,
+with the change of variable y =
+�
+U2−U2
+crit
+U2
+z. Therefore,
+TU =
+1
+U
+√
+1 + α2
+�
+U 2
+U 2 − U 2
+crit
+� +∞
+−∞
+dz
+z2 + 1 =
+π
+√
+1 + α2
+�
+U 2 − U 2
+crit
+.
+6
+Proof of Theorem 8: on almost planar trajectories
+Let us first state a result based on tedious computations, whose detail is left to the reader.
+Lemma 22. Let U > Ucrit, m be an optimal trajectory and p its adjoint state.
+Then ζ := p ∧ m = ϕ ∧ m satisfies
+˙ζ = α
+�
+D(m ∧ ζ) ∧ m − (m ∧ ζ) ∧ Dm
+�
++ Dm ∧ ζ − Dζ ∧ m
+Similarly, denote Z := ζ/|ζ|. At every point where ζ does not vanish, one has
+˙Z = PZ⊥
+�
+α
+�
+D(m ∧ Z) ∧ m − (m ∧ Z) ∧ Dm
+�
++ Dm ∧ Z − DZ ∧ m
+�
+,
+(32)
+where PZ⊥ : x �→ x − (Z · x)Z is the projection onto the orthogonal space to Z.
+The proof of the Theorem 8 relies on the following result, establishing the existence of a planar
+trajectory joining e1 to −e1, without any norm condition on the chosen control.
+Lemma 23 (Existence of planar trajectory). For any ε > 0, there exists a trajectory of the form
+m(t) = (m1(t), m2(t), 0) defined on [0, Tε] for some Tε > 0, joining the state e1 to −e1 and such that
+F :=
+�
+˙e · (Dm ∧ m) +
+| ˙m|
+1 + α2
+�2
++
+�
+˙e · Dm + α| ˙m|
+1 + α2
+�2
+≤ 1
+4(γ2 − γ1)2(1 + ε).
+for all t ∈ [0, Tε]. Furthermore, Tε ≲ 1/ε.
+Proof of Lemma 23. Define m1(t) = cos θ(t) and m2(t) = sin θ(t), our goal is to define a suitable
+function θ, such that θ(0) = 0 and θ(Tε) = π.
+Observe that ˙e, Dm, m are coplanar so that ˙e · (Dm ∧ m) = 0. Also
+| ˙m| = | ˙θ|,
+˙e · Dm = (γ2 − γ1) sin(θ) cos(θ) = γ2 − γ1
+2
+sin(2θ).
+Hence
+F =
+1
+(1 + α2)2 | ˙θ|2 +
+�
+γ2 − γ1
+2
+sin(2θ) +
+α| ˙θ|
+1 + α2
+�2
+=
+1
+1 + α2 | ˙θ|2 + α(γ2 − γ1)
+1 + α2
+sin(2θ)| ˙θ| + (γ2 − γ1)2
+4
+sin2(2θ).
+This is a quadratic expression in | ˙θ|. Let us solve F = 1
+4(γ2 − γ1)2(1 + ε): this is a polynomial equation
+of degree 2, whose discriminant reads
+∆ = α2(γ2 − γ1)2
+(1 + α2)2
+sin2(2θ) −
+4
+(1 + α2)2
+(γ2 − γ1)2
+4
+(sin2(2θ) − 1 − ε)
+23
+
+= (γ2 − γ1)2
+(1 + α2)2
+�
+1 + ε + (α2 − 1) sin2(2θ)
+�
+.
+Observe that ∆ > 0 for all θ, so that we can choose
+˙θ = γ2 − γ1
+2
+�
+−α sin(2θ) +
+�
+1 + ε + (α2 − 1) sin2(2θ)
+�
+=: fε(θ)
+As
+fε(θ) ≥ γ2 − γ1
+2
+��
+ε + α2 sin2(2θ) − α sin(2θ)
+�
+≥
+ε
+√
+ε + α2 + α
+> 0,
+we infer that this ODE on θ admits a unique solution θε, strictly increasing such that ˙θε ≳ ε, and so,
+there exists a unique Tε ≲ 1/ε such that θε(Tε) = π. This provides the desired trajectory.
+Denote Uplan = γ2−γ1
+2
+. Lemma 23 shows in particular that Ucrit ≤ Uplan. We are now in position to
+complete the:
+Proof of Theorem 8. Let ζ and Z as in Lemma 22: Z satisfies (32). Observe moreover that the equations
+on ζ and Z remain unchanged if one replaces D into D − λI3 for some λ ∈ R. We can therefore assume
+that the spectral norm of D is ∥D∥ = γ3−γ1
+2
+by taking λ = γ3+γ1
+2
+. Then, since |Z| = |m| = 1 and
+∥PZ⊥∥ = 1, we get
+��� ˙Z
+��� ≤ 2(1 + α)∥D∥ = (1 + α)(γ3 − γ1).
+On the other hand, according to Lemmas 9 and 23, we know that for all U > Uplan, there holds
+TU ≤
+C
+U−Uplan for some constant C > 0. Thus, one has
+|Z(t) − Z(TU)| ≤ (1 + α)(γ3 − γ1)TU ≤ (1 + α)(γ3 − γ1)
+C
+U − Uplan
+.
+(33)
+for all t ∈ [0, TU].We also know that |ζ| = |ϕ|. Thus, by introducing ψ = ϕ/|ϕ|, one gets Z = ψ ∧ m
+and m ∧ Z = ψ since ϕ · m = 0. A straightforward computation yields that the pair (m, ψ) satisfies
+˙ψ = α(Dψ − (Dψ · ψ)ψ) − U
+�
+1 + α2m − Dm ∧ ψ + D(m ∧ ψ)
+− (ψ · D(m ∧ ψ)) ψ − ((m ∧ ψ) · Dm) m,
+˙m = −α (Dm − (Dm · m)m) + m ∧ Dm + U
+�
+1 + α2ψ.
+(34)
+From estimate (33), we infer that, for U large enough,
+∀t ∈ [0, TU],
+|ψ(t) − m ∧ Z(TU)| ≤ C
+U .
+Putting this in Equation (34) for m, we get some constant C > 0 such that for all U large enough and
+t ∈ [0, TU],
+��� ˙m − U
+�
+1 + α2m ∧ Z(TU)
+��� ≤ C,
+which leads to
+| ˙m · Z(TU)| ≤ C.
+Since m(TU) · Z(TU) = 0 by definition and using once again that TU ≤
+C
+U−Uplan , we get for all U large
+enough and t ∈ [0, TU]
+|m · Z(TU)| ≤ C
+U .
+However, p(TU) = ϕ(TU) (since p(TU)·m(TU) = 0 from the orthogonality condition) and m(TU) = −e1,
+and thus the orthogonal space of V is exactly span(Z(TU)), which means that m(t) − PV m(t) =
+(m · Z(TU))Z(TU). The conclusion easily follows.
+24
+
+7
+Conclusion and perspectives
+7.1
+Extension of our results
+It would be natural to extend our study in several directions. On the one hand, we would like to
+complete our study of a single ferromagnetic particle of ellipsoidal shape by studying other criteria,
+and typically a combination of time and cost L2 of control. This problem could read:
+Second version of the optimal control problem: case of L2 constraints. Let λ > 0
+and let us assume that m0 = e1. The problem reads
+Eλ
+U =
+inf
+(T,u)∈OU
+T + λ
+2
+� T
+0
+|u(t)|2 dt,
+(Pλ)
+where mu denotes the solution to (3) associated to the control function u(·),
+or alternatively, if one aims at dropping the effect of the L∞ constraint on the control,
+Modified second version of the optimal control problem: case of L2 constraints.
+Let λ > 0 and let us assume that m0 = e1. The problem reads
+Eλ
+U =
+inf
+(T,u)∈�
+U≥0 OU
+T + λ
+2
+� T
+0
+|u(t)|2 dt,
+(Pλ)
+where mu denotes the solution to (3) associated to the control function u(·).
+Finally, we also plan to study similar issues for more realistic physical systems, for example a
+network of ellipsoidal particles, possibly rectilinear, as in the model introduced in [1].
+7.2
+Numerical illustrations of our results
+We provide hereafter several numerical illustrations of our results. More precisely, we want to determine
+numerically the existence or not of an admissible trajectory connecting e1 to −e1, in accordance with
+what we have found theoretically.
+Let us first notice that a trajectory m can easily be computed
+numerically by solving the ODE (3) with the expression (22) for the control u where the variable ϕ is
+given by (14).
+To initialize both ODE (3) and (14), m(0) = e1 is given, but ϕ(0) is unknown.
+On the one
+hand, we overcome this difficulty by noticing that ϕ(0).e1 = 0, which allows us to have only two
+unknowns: ϕ2(0) and ϕ3(0) to be determined.
+On the other hand, working with the normalized
+variable ψ = ϕ/|ϕ| enables us to reduce the unknowns to only one angle variable ϑ ∈ [0, 2π] such that
+(ψ2(0), ψ3(0)) = (cos(ϑ), sin(ϑ)). ODE (3) and (14) are thus replaced by the system (34).
+Numerically, implement a shooting method to determine ϑ ∈ [0, 2π]: namely, for each ϑ, we solve
+the system (34) on a very large time horizon by a fourth-order Runge-Kutta method and determine if
+the trajectory m reaches −e1 on a certain time T.
+We list below the numerical results, all obtained with α = 0.6. The initial position e1 is represented
+with a red circle on the sphere and the goal −e1 with a green star. Parameters γi, ϑ and control U
+are specified in the caption of each figure. For each one, we have represented the trajectory m on the
+sphere as well as the coordinates of m and of the control u as functions of time.
+First of all, in the non-symmetric case, a threshold on the control appears. If the control is suffi-
+ciently large, there is (at least) an initialization of ψ (i.e at least one angle ϑ) which allows to have an
+admissible trajectory represented in Subfigures 2(a)-2(b). On the contrary, if the control is not large
+enough, no initialization of ψ will give an admissible trajectory. We have represented for instance one
+of them in Subfigure 2(e)-2(f) with a particular ϑ but be aware that they all have the same behavior
+25
+
+(a) Admissible trajectory for ϑ = 0.8976,
+large control U = 10,
+(b) The components of m and u for ϑ = 0.8976, large control U = 10
+(c) Admissible trajectory for ϑ = 2.2440,
+medium control U = 3
+(d) The components of m and u for ϑ = 2.2440, medium control U = 3
+(e) Generic trajectory for ϑ = 0.8976,
+small control U = 0.1
+(f) The components of m and u for ϑ = 0.8976, small control U = 0.1
+Figure 2: Non-symmetric test case with (γ1; γ2; γ3) = (0.2; 0.5; 1), top: with a large control U = 10,
+middle: with a medium control U = 3 and bottom: with a small control U = 0.1
+26
+
+100
+0.75
+0.50
+0.25
+X
+0.00
+0.25
+0.50
+0.75
+-1.00
+100
+0.75
+0.50
+0.25
+1.00
+-0.75
+0.50
+-0.25
+0.00
+=0.50
+Y
+0.25
+0.75
+0.50
+0.75
+-1.00
+1001
+mi
+m3
+0
+ui
+U3
+0
+8
+-2
+-1
+6
+-4
+-2
+4
+-6
+-3
+2
+4
+-8
+m2
+-5
+u2
+-10
+0
+0.0
+0.1
+0.2
+0.0
+0.1
+0.2
+0.0
+0.1
+0.2
+time
+time
+time1.00
+0.75
+0.50
+0.25
+X
+0.00
+-0.25
+-0.50
+-0.75
+-1.00
+1.00
+0.75
+0.50
+-1.00
+0.25
+-0.75
+0.00
+-0.50
+-0.25
+-0.25
+0.00
+-0.50
+0.25
+Z
+-0.75
+0.50
+0.75
+-1.00
+1.001.0
+0.0
+m1
+0.5
+U1
+-0.5
+0.5
+0.0
+-1.0
+0.5
+0.0
+-1.0
+-1.5
+-0.5
+-1.5
+-2.0
+-1.0
+-2.0
+-2.5
+m2
+m3
+-2.5
+-1.5
+U2
+U3
+3.0
+3.0
+0.00
+0.25
+0.50
+0.00
+0.25
+0.50
+0.75
+0.75
+0.00
+0.25
+0.50
+0.75
+time
+time
+time1.00
+0.75
+0.50
+0.25
+X
+0.00
+-0.25
+-0.50
+-0.75
+-1.00
+1.00
+0.75
+0.50
+0.25
+-1.00
+0.00.
+-0.75
+Z
+-0.50
+-0.25
+-0.25
+0.00
+0.50
+Y
+0.25
+-0.75
+0.50
+0.75
+-1.00
+1.001.0
+>
+0.3
+m2
+0.4
+U2
+0.8
+0.2
+0.2
+0.1
+0.6
+m1
+0.0-
+0.0
+Ui
+0.4
+0.1
+-0.2
+0.2
+-0.2
+-0.4
+m3
+0.0
+-0.3
+U3
+0
+20
+40
+20
+40
+0
+20
+40
+time
+time
+timewhatever the initialization of ψ: the trajectory remains in the northern half-sphere without enough
+control.
+Figure 2 is thus a perfect illustration of Theorem 2.
+Note that it also helps to illustrate
+Theorem 8 since the larger U is, the closer the trajectory is to a planar trajectory, as we can see by
+comparing Subfigures 2(c)-2(d) with a medium control and Subfigures 2(a)-2(b) with a larger control.
+(a) Generic trajectory
+(b) The components of m and u
+Figure 3: Non-symmetric test case with (γ1; γ2; γ3) = (0.0; 0.8; 1), ϑ = 2.5646 and a small control
+U = 0.2
+Figure 3 illustrates once again the case of control too weak to reach −e1, for other γi parameters.
+The symmetric case γ1 = γ2 is shown in Figure 4. Even for small controls (U = 0.7 numerically),
+there is (at least) one initialization of ψ leading to an admissible trajectory reaching −e1 in finite
+time. This illustrates well Theorem 4: Ucrit = 0 in this symmetric case. When γ2 < γ3 (Subfigures
+4(a)-4(b)), the admissible trajectories are non planar whereas it is, in the case of a spherical symmetry
+(Subfigures 4(c)-4(d)) without changing anything other than the symmetry of the test case. This is
+again in accordance with the second statement of Theorem 4.
+For the symmetric case γ2 = γ3, we see numerically in Figure 5 that for small values of U, an
+admissible trajectory exists. With the parameters of Figure 5, Theorem 5 gives the following value
+for Ucrit =
+α
+2
+√
+1+α2 (γ2 − γ1) ≃ 0.026, which effectively allows to have admissible trajectories for very
+small values of U. Note also in Subfigure 5(b) that all admissible trajectories reach the target −e1 in
+a time greater than 14. With the values chosen for Figure 5,
+π
+√
+1+α2√
+U2−U2
+crit
+≃ 13.58 corresponds to
+the minimum time determined in Theorem 5. Here again, we notice the non-planar character of the
+trajectory.
+7.3
+Conclusion and perspectives
+The obtained results provide a complete characterization of the question of the magnetic moment
+reversal in minimal time in a simple configuration. Indeed, we have considered here only one ellipsoidal
+particle. In order to approach more realistic configurations, we wish to analyze a model in which several
+ferromagnetic particles of ellipsoidal shape are combined to form a network. We refer for example to
+[2] for a possible model. After having characterized the set of stationary configurations, we will then
+ask ourselves the question of controllability in minimal time, in order to go from one stationary state
+to another.
+27
+
+1.00
+0.75
+0.50
+0.25
+X
+0.00
+-0.25
+0.50
+-0.75
+-1.00
+1.00
+0.75
+0.50
+0.25
+-1.00
+0.00.
+-0.75
+Z
+-0.50
+-0.25
+-0.25
+0.00
+-0.50
+Y
+0.25
+-0.75
+0.50
+0.75
+-1.00
+1.001.0
+m3
+0.4
+0.4
+U3
+0.8
+0.2
+0.2
+0.6
+m1
+0.0
+0.0
+u1
+0.4
+-0.2
+0.2
+0.2
+-0.4
+m2
+0.4
+0.0
+u2
+0
+20
+40
+0
+20
+40
+0
+20
+40
+time
+time
+time(a) Admissible trajectory with γ3 = 1.0
+(b) The components of m and u with γ3 = 1.0
+(c) Admissible trajectory with γ3 = 0.2
+(spherical case)
+(d) The components of m and u with γ3 = 0.2 (spherical case)
+Figure 4: Symmetric test case γ1 = γ2 with (γ1; γ2) = (0.2; 0.2), ϑ = 0.3206 and a small control
+U = 0.7, top: γ3 = 1.0 and bottom: γ3 = 0.2 (spherical case)
+(a) Admissible trajectory
+(b) The components of m and u
+Figure 5: Symmetric γ2 = γ3 test case with (γ1; γ2; γ3) = (0.1; 0.2; 0.2), ϑ = 2.7925 and a small control
+U = 0.2
+28
+
+1.00
+m1
+m2
+0.8
+0.8
+0.75
+U1
+U2
+0.6
+0.50
+0.6
+0.4
+0.25
+0.00
+0.2
+0.4
+-0.25
+0.0
+0.2
+-0.50
+-0.2
+-0.75
+m3
+0.4
+0.0
+U3
+-1.00
+0
+2
+4
+6
+0
+2
+4
+6
+0
+2
+4
+6
+time
+time
+time1.00
+0.75
+0.50
+0.25
+X
+0.00
+-0.25
+-0.50
+-0.75
+-1.00
+00
+1.00-0.75-0.50-0.250.00 0.25 0.50 0.75 1.00
+Z
+Y1.0
+0.7
+1.00
+m1
+m3
+0.8
+0.75
+U1
+0.6
+U3
+0.50
+0.6
+0.5
+0.25
+0.4
+0.4
+0.00
+0.2
+0.3
+-0.25
+0.0
+0.2
+-0.50
+-0.2
+0.1
+0.75
+m2
+-0.4
+U2
+1.00
+0.0
+-0.6
+0
+2
+0
+2
+0
+2
+time
+time
+time1.00
+0.75
+0.50
+0.25
+0.00
+X
+1.00
+0.75
+-0.25
+0.50
+0.25
+-0.50
+V0.00
+-0.75
+-0.25
+-0.50
+-1.00
+-0.75
+-1.00
+1.00 0.75 0.50 0.25 0.00 -0.25-0.50-0.75-1.00
+Z1.00
+m3
+mi
+0.0
+0.05
+0.75
+Ui
+U3
+0.50
+-0.2
+0.00
+0.25
+-0.4
+m2
+0.05
+0.00
+U2
+-0.25
+-0.6
+0.10
+-0.50
+-0.8
+0.15
+-0.75
+-1.00
+-1.0
+0.20 t
+0
+5
+10
+0
+5
+10
+0
+5
+10
+time
+time
+time1.00
+0.75
+0.50
+0.25
+-1.00-
+0.00
+X
+-0.75
+-0.25
+-0.50
+-0.25
+-0.50
+70.00
+-0.75
+0.25
+-1.00
+0.50
+0.75
+1.00
+-1.00-0.75-0.50-0.25 0.00 0.25 0.50 0.75 1.00
+ZAcknowledgements
+The authors were partially supported by the ANR Project MOSICOF. The last author were partially
+supported by the ANR Project TRECOS.
+A
+Computation of the demagnetizing field in a ferromagnetic ellipsoid
+sample
+It is shown in [14, 8] that the demagnetizing tensor D reads D = diag([γ1, γ2, γ3]), where the γi’s are
+given by
+γi = a1a2a3
+2
+� +∞
+0
+dt
+�
+(a1 + t2)(a2
+2 + t2)(a2
+3 + t2)(a2
+i + t2)
+.
+Such an expression can be rewritten in terms of the elliptic integral of the second kind E, defined by
+E(x, p) =
+� x
+0
+(1 − p sin2 θ)1/2 dθ,
+x ∈ R, p ∈ (0, 1).
+If a1 ≥ a2 ≥ a3, then, one has 0 ≤ γ1 ≤ γ2 ≤ γ3 ≤ 1 and these coefficients read
+γ1 = 1 − γ2 − γ3
+γ2 = −
+a3
+a2
+2 − a2
+3
+�
+a3 −
+a1a2
+(a2
+1 − a2
+2)1/2 E
+�a2
+a1
+, a2
+1 − a2
+3
+a2
+1 − a2
+2
+��
+γ3 =
+a2
+a2
+2 − a2
+3
+�
+a2 −
+a1a3
+(a2
+2 − a2
+3)1/2 E
+�a3
+a1
+, a2
+1 − a2
+2
+a2
+1 − a2
+3
+��
+.
+In the case where a1 ≥ a2 = a3 (prolate spheroid), these formula simplify into
+γ1 = −
+a2
+3
+(a2
+1 − a2
+3)3/2
+�
+(a2
+1 − a2
+3)1/2 + a1 argcoth
+�
+a1
+(a2
+1 − a2
+3)1/2
+��
+,
+γ2 = γ3 = 1 − γ1
+2
+.
+In the case where a1 = a2 ≥ a3 (oblate spheroid), these formula simplify into
+γ3 = −
+a2
+1
+(a2
+1 − a2
+3)3/2
+�
+(a2
+1 − a2
+3)1/2 + a3 arctan
+�
+a3
+(a2
+1 − a2
+3)1/2
+�
+− π
+2 a3
+�
+,
+γ1 = γ2 = 1 − γ3
+2
+.
+B
+Stability of steady-states for Eq. (3)
+Let us first notice that, if ¯m is a steady-state of Equation (3) and if no control is applied on this system
+(hext = 0), then by orthogonality of the terms in the right-hand side, it satisfies
+h0( ¯m) = (h0( ¯m) · ¯m) ¯m
+and
+¯m ∧ h0( ¯m) = 0.
+Therefore, ¯m is an eigenfunction of D ¯m and we infer that ¯m = ±ej, j = 1, 2, 3 whenever γ1 < γ2 ≤ γ3.
+Proposition 24 (Asymptotic stability). Let γ1, γ2, γ3 be sorted in ascending order γ1 < γ2 ≤ γ3.
+Then, in the absence of any control hext, the steady-state ±e1 is an asymptotically stable equilibrium
+state for Equation (3). Nevertheless, the steady-states ±e2 and ±e3 are linearly unstable steady-states
+for Equation (3).
+29
+
+Proof. Let h = (h1, h2, h3)T ∈ R3 be a small perturbation such that e1 + h is still an admissible
+magnetization, i.e. on the unit sphere S2 ⊂ R3. We obtain:
+∥e1 + h∥2 = 1 ⇔ (1 + h1)2 + h2
+2 + h2
+3 = 1 ⇔ h1 = −1
+2
+�
+h2
+1 + h2
+2 + h3
+3
+�
+= O(∥h∥2).
+The unknown h1 is therefore of second order and does not occur in a linearized system of the first order.
+By linearizing Equation (3) around the equilibrium state e1, one has, without any control u:
+� ˙h2 = α(γ1 − γ2)h2 + (γ1 − γ3)h3 + O(∥h∥2)
+˙h3 = α(γ1 − γ3)h3 + (γ2 − γ1)h2 + O(∥h∥2)
+(35)
+The Jacobian matrix of the linearized system around e1 is therefore :
+J =
+�α(γ1 − γ2)
+γ1 − γ3
+γ2 − γ1
+α(γ1 − γ3)
+�
+.
+Since γ1 < γ2 ≤ γ3, one has
+det(J) = (α2 + 1)(γ1 − γ2)(γ1 − γ3) > 0
+and
+Tr(J) = α [(γ1 − γ2) + (γ1 − γ3)] < 0.
+We infer that the two eigenvalues of the Jacobian matrix are of negative real parts. The steady state
+e1 is therefore linearly stable and is an hyperbolic point (no eigenvalue with zero real part).
+Hartman Grobman’s theorem [16] allows to conclude about the asymptotic stability of e1 for the
+non-linear Equation (3) without any control u. As for −e1, similar computations give the conclusion.
+Regarding now the stability of ±ek, k = 2, 3, notice that a similar computation drives to the following
+expression of the Jacobian determinant: det J = (α2+1)(γ2−γ1)(γ2−γ3) < 0. The expected conclusion
+follows.
+Remark 25. If γ1 ≤ γ2 ≤ γ3, an eigenvalue of the Jacobian matrix may have a zero real part. In which
+case one can conclude that e1 is linearly (non-asymptotically) stable, but Hartman Grobman’s theorem
+no longer applies to return to the non-linear Equation (3).
+C
+Complement in the case γ1 < γ2: explicit computations of the con-
+stants in the case
+Let us use the notations introduced in Remark 18. We compute
+A∗A =
+�(1 + α2)δγ2
+−
+−2αδγ− δγ+
+−2αδγ− δγ+
+(1 + α2)δγ2
++
+�
+,
+Tr(A∗A) = (1 + α2)(δγ2
+− + δγ2
++) > 0,
+det(A∗A) = (1 − α2)2δγ2
+−δγ2
++ ≥ 0,
+and the discriminant of its characteristic polynomial is
+Tr(A∗A)2 − 4 det(A∗A) = (1 + α2)2(δγ2
+− + δγ2
++)2 − 4(1 − α2)2δγ2
+−δγ2
++
+= (1 + α2)2(δγ2
+− − δγ2
++)2 + 16α2δγ2
+−δγ2
++
+= (1 + α2)2(δγ− − δγ+)2(δγ− + δγ+)2 + 16α2δγ2
+−δγ2
++ > 0.
+and its largest eigenvalue is therefore
+∥A∥2
+2 =
+(1 + α2)(δγ2
+− + δγ2
++) +
+�
+(1 + α2)2(δγ− − δγ+)2(δγ− + δγ+)2 + 16α2δγ2
+−δγ2
++
+2
+.
+30
+
+On the other hand, when ∆ = α2(δγ+ − δγ−)2 − δγ−δγ+ ≥ 0, we have
+λ+ = −α(δγ+ + δγ−) +
+�
+α2(δγ+ − δγ−)2 − δγ−δγ+
+2
+,
+and therefore, with Γ = δγ−1
++ δγ−,
+|λ+|2
+∥A∥2(1 + |α|)δγ+
+=
+1
+√
+2(1 + |α|)
+�
+α(1 + Γ) −
+�
+α2(1 − Γ)2 − Γ
+�2
+�
+(1 + α2)(1 + Γ2) +
+�
+(1 + α2)(Γ − 1)2(1 + Γ)2 + 16α2Γ2
+� 1
+2
+.
+Similarly, when ∆ < 0, we obtain
+|Tr A|2
+∥A∥2(1 + |α|)δγ+
+=
+1
+√
+2(1 + |α|)
+α(1 + Γ)
+�
+(1 + α2)(1 + Γ2) +
+�
+(1 + α2)(Γ − 1)2(1 + Γ)2 + 16α2Γ2
+� 1
+2
+.
+Remark also that ∆ = δγ2
++
+�
+α2(1 − Γ)2 − Γ
+�
+. Therefore, if we define
+˜x0 :=
+�
+�
+�
+|λ+|2
+∥A∥2(1+|α|)δγ+ ,
+if ∆ ≥ 0,
+Tr(A)2
+4∥A∥2(1+|α|)δγ+ ,
+if ∆ < 0,
+and then ˜x1 := min
+�
+1,
+√˜x0
+3
+�
+, we obtain that both of them only depend on Γ and α and thus so is
+µ0 = x0
+3 x1 − x3
+1. Last, the conditions on U becomes U ≤ δγ+ µ0, which is in agreement with the
+invariances on D (invariance by shifting of the γis, invariance by multiplication of the γis with respect
+to a change of time variable and a multiplication of the external field-control).
+References
+[1] Shruti Agarwal, Gilles Carbou, Stéphane Labbé, and Christophe Prieur. Control of a network of magnetic ellipsoidal
+samples. Mathematical Control and Related Fields, 1(2):129–147, 2011.
+[2] Shruti Agarwal, Gilles Carbou, Stéphane Labbé, and Christophe Prieur. Control of a network of magnetic ellipsoidal
+samples. Math. Control Relat. Fields, 1(2):129–147, 2011.
+[3] François Alouges and Karine Beauchard.
+Magnetization switching on small ferromagnetic ellipsoidal samples.
+ESAIM: Control, Optimisation and Calculus of Variations, 15(3):676–711, 2009.
+[4] Francois Alouges, Karine Beauchard, and Mario Sigalotti. Magnetization switching in small ferromagnetic ellipsoidal
+samples. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th
+Chinese Control Conference, pages 2106–2111, 2009.
+[5] Xin An, Ananta K. Majee, Andreas Prohl, and Thanh Tran. Optimal control for a coupled spin-polarized current
+and magnetization system. Adv. Comput. Math., 48(3):Paper No. 28, 40, 2022.
+[6] William Fuller Brown. Micromagnetics. Interscience, 1963.
+[7] Gilles Carbou, Stéphane Labbé, and Emmanuel Trélat. Control of travelling walls in a ferromagnetic nanowire.
+Discrete Contin. Dyn. Syst. Ser. S, 1(1):51–59, 2008.
+[8] Giovanni Di Fratta. The newtonian potential and the demagnetizing factors of the general ellipsoid. Proceedings of
+the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2190):20160197, 2016.
+[9] Shruti Dubey and Sharad Dwivedi. On controllability of a two-dimensional network of ferromagnetic ellipsoidal
+samples. Differential Equations and Dynamical Systems, 27(1):277–297, 2019.
+[10] Thomas Dunst, Markus Klein, Andreas Prohl, and Ailyn Schäfer. Optimal control in evolutionary micromagnetism.
+IMA J. Numer. Anal., 35(3):1342–1380, 2015.
+[11] Thomas Dunst, Ananta K. Majee, Andreas Prohl, and Guy Vallet. On stochastic optimal control in ferromagnetism.
+Arch. Ration. Mech. Anal., 233(3):1383–1440, 2019.
+31
+
+[12] Alex Hubert and Rudolf Schäfer. Magnetic domains: the analysis of magnetic microstructures. Springer Science &
+Business Media, 2008.
+[13] Lev Davidovich Landau, JS Bell, MJ Kearsley, LP Pitaevskii, EM Lifshitz, and JB Sykes.
+Electrodynamics of
+continuous media, volume 8. Elsevier, 2013.
+[14] John A Osborn. Demagnetizing factors of the general ellipsoid. Physical review, 67(11-12):351, 1945.
+[15] Stuart SP Parkin, Masamitsu Hayashi, and Luc Thomas.
+Magnetic domain-wall racetrack memory.
+Science,
+320(5873):190–194, 2008.
+[16] Lawrence Perko. Differential equations and dynamical systems, volume 7. Springer Science & Business Media, 2013.
+[17] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko. The mathematical theory of optimal
+processes. A Pergamon Press Book. The Macmillan Company, New York, 1964. Translated by D. E. Brown.
+[18] Yannick Privat and Emmanuel Trélat. Control and stabilization of steady-states in a finite-length ferromagnetic
+nanowire. ESAIM Control Optim. Calc. Var., 21(2):301–323, 2015.
+[19] Diego Takahashi and Vanderlei C Oliveira Jr.
+Ellipsoids (v1. 0): 3-D magnetic modelling of ellipsoidal bodies.
+Geoscientific Model Development, 10(9):3591–3608, 2017.
+[20] Augusto Visintin. Mathematical models of hysteresis. A survey. In Nonlinear partial differential equations and their
+applications. Collège de France Seminar, Vol. XIII (Paris, 1994/1996), volume 391 of Pitman Res. Notes Math.
+Ser., pages 327–340. Longman, Harlow, 1998.
+[21] Baisheng Yan. On energy-minimization in ferromagnetism controlled by applied fields. Ann. Mat. Pura Appl. (4),
+192(1):115–125, 2013.
+32
+
diff --git a/TdE2T4oBgHgl3EQfXAc0/content/tmp_files/load_file.txt b/TdE2T4oBgHgl3EQfXAc0/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7c862575be875b617180c85af6513856a75bb388
--- /dev/null
+++ b/TdE2T4oBgHgl3EQfXAc0/content/tmp_files/load_file.txt
@@ -0,0 +1,1207 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf,len=1206
+page_content='Minimal time of magnetization switching in small ferromagnetic  ellipsoidal samples  Raphaël Côte∗ 1, Clémentine Courtès † 1, Guillaume Ferrière ‡ 1, and Yannick Privat § 1,2  1IRMA, Université de Strasbourg, CNRS UMR 7501, Inria, 7 rue René Descartes, 67084  Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2Institut Universitaire de France (IUF)  January 11, 2023  Abstract  In this paper, we consider a ferromagnetic material of ellipsoidal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The associated magnetic  moment then has two asymptotically stable opposite equilibria, of the form ±m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In order to use  these materials for memory storage purposes, it is necessary to know how to control the magnetic  moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We use as a control variable a spatially uniform external magnetic field and consider the  question of flipping the magnetic moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=', changing it from the +m configuration to the −m  one, in minimal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Of course, it is necessary to impose restrictions on the external magnetic  field used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We therefore include a constraint on the L∞ norm of the controls, assumed to be less  than a threshold value U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We show that, generically with respect to the dimensions of the ellipsoid,  there is a minimal value of U for this problem to have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We then characterize it precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Finally, we investigate some particular configurations associated to geometries enjoying symmetries  properties and show that in this case the magnetic moment can be controlled in minimal time  without imposing a threshold condition on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Keywords: ferromagnetic materials, Landau-Lifshitz equation, optimal control, minimal time  AMS classification: 49J15, 49J30, 35Q60, 78M50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  The Landau-Lifshitz equation for ellipsoidal ferromagnetic samples  Ferromagnetic materials have come into common use in the last few decades, especially since they are  found in devices used to store digital information such as magnetic tapes or hard disks, but also in  magnetic chips called Magnetic Random Access Memory (MRAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' These chips have many advantages  over their silicon counterparts, in particular that of requiring energy only to change the value bits and  not to maintain the storage itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This is probably one of the most challenging applications since it  opens the door towards new spintronic applications and storage technologies while allowing a very fast  access to information (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=', [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  ∗raphael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='cote@unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='fr  †clementine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='courtes@unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='fr  ‡guillaume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='ferriere@unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='fr  §yannick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='privat@unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='fr  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='03839v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='AP]  10 Jan 2023  The magnetic moment of a ferromagnetic material represented by a domain Ω ⊂ R3 is usually  modelled as a time-varying vector field  m : R × Ω → S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  where S2 is the unit sphere of R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' the evolution of which is driven by the so-called Landau-Lifshitz  equation (see [13])  ∂m  ∂t = −m ∧ h(m) − αm ∧ (m ∧ h(m)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (1)  where the effective field h(m) is defined by  h(m) = 2A∆m + hd(m) + hext  with α > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' a constant (in time and space) damping coefficient which is characterized by the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We refer for instance to [12, 6] for additional explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The constant A > 0 is the exchange  constant, and can be assumed to be equal to A = 1/2 without loss of generality, with a normalization  argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The demagnetizing field hd(m) is the solution of the equations  �  div(hd(m) + m) = 0  curl(hd(m)) = 0  in D′(R3)  where m is extended to R3 by 0 outside Ω and D′(R3) denotes the space of distributions on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The  field hext is an external one, for instance it can be an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Note that it is possible to complete and specify this physical model by adding other relevant terms,  for example by taking into account the anisotropic behavior of the crystal that composes the ferromag-  netic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Finally, it is standard to assume homogeneous Neumann boundary conditions on the magnetization  on the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In this article, we will consider a ferromagnetic sample of ellipsoidal shape, and the magnetization  m and external field hext both spatially uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Indeed, ellipsoidal domains have been much studied  in the literature dedicated to ferromagnetism [14, 8, 19]: on the one hand, they cover a large variety  of geometrical shapes, and on the other hand, they are the only known bodies that can be uniformly  magnetized in the presence of a spatially uniform inducing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From the mathematical point of view,  it is nice to consider such samples because the demagnetizing field hd appearing in the Landau-Lifschitz  equation can be determined in an explicit way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us be more precise and clarify the model obtained in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In all the following of this article,  we will denote Ω the ellipsoid of R3 of semiaxes a1 > 0, a2 > 0 and a3 > 0, and a basis (O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' e1, e2, e3)  chosen so that the Cartesian equation of Ω reads  x2  a2  1  + y2  a2  2  + z2  a2  3  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (2)  An illustration of the ellipsoid Ω is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' According to [14, 8], for uniform (in space)  magnetizations m on Ω, the demagnetizing field hd(m) can be explicitly computed and reads  hd(m) = −Dm,  with  D =  �  �  γ1  0  0  0  γ2  0  0  0  γ3  �  � ,  where γi (i = 1, 2, 3) denotes a positive constant depending only on the semiaxes a, b and c (we provide  the precise dependence in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2  One can easily infer from this result that, provided that the external field hext and the initial  magnetization m0 are constant in space, so is the magnetic moment m solving the Landau-Lifshitz  equation (1) completed with homogeneous Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  As a consequence, the Landau-Lifschitz equation with a time-dependent external magnetic field hext  reads as the ordinary differential system  �  ˙m = α (h0(m) − (h0(m) · m)m) − m ∧ h0(m)  in (0, T)  m(0) = m0  (3)  where the dotted notation ˙m stands for the time derivative of m, h0(m) = −Dm + hext, T > 0,  m(t) ∈ S2 ⊂ R3, D = diag([γ1, γ2, γ3]) denotes a diagonal matrix with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Up to a  change of basis, we will also assume without loss of generality that  0 ≤ γ1 ≤ γ2 ≤ γ3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (4)  In what follows, we will assume that the ferromagnetic particle is subjected to a spatially uniform  external magnetic field hext, and we are interested in two asymptotically stable stationary states of the  resulting system, denoted m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We seek to answer the following question:  Given a maximum value U of the norm of the field hext at all times, can we determine  whether there exists such a field flipping the magnetic spin from m to −m in minimal time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  x  y  z  2a1  2a2  2a3  Figure 1: The ellipsoid shaped ferromagnetic sample  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  State of the art and structure of the article  The development of the use of ferromagnetic materials has led to the emergence of new storage pos-  sibilities, and consequently to a renewed interest of the scientific community around the control of  EDO/EDP on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The use of an external magnetic field to control a ferromagnetic system is a very present issue in  the literature of the field (see for instance [7, 18, 20, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Many works have focused on both the derivation of relevant models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' sufficiently close to the  physics, but also simple enough to be exploited mathematically, and on the related optimization issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Many studies are devoted to these modeling questions, to the obtaining of exploitable optimality con-  ditions leading to numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Thus, in the same spirit as the present study, the authors of  3  [5] seek to flip the magnetic spin using electric current injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let us mention in the same vein the  works [10, 11] also addressing similar issues: minimization of the distance to a target state with a fixed  time horizon, addition of stochastic term in the model, search for a feedback and numerical analysis of  the considered problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Recent progress has been made in the understanding of the control (exact and approximate) of  ellipsoidal samples/networks : [9, 4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Our study has been particularly motivated by [3], in which  it is notably proved that, when the size of an open bounded by Ω tends to 0, then we find a uniform  magnification in the domain, which lends itself to the study of ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Structure of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In this paper, we are interested in a single ferromagnetic particle of  ellipsoidal shape in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We seek to perform a magnetic moment reversal in minimal time, using an  external magnetic field as a control of the resulting physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We model this issue in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1, imposing a maximum L∞ norm on the control translated using  the parameter U > 0, reflecting the difficulty and cost of using very high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In the  absence of additional symmetries on the geometry of the system, we show the existence of a minimal  threshold on U for the minimum time problem to have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We refine this result when additional  symmetries are assumed on the material geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The main results of this paper are gathered in the  section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The section 3 contains the foundation of the proofs of the main results: indeed, we state  necessary and sufficient conditions guaranteeing the well-posedness of the time-optimal problem and  write the necessary conditions of optimality to the first order using the Pontryagin maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The proofs of the main results are contained in the sections 4, 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Finally, some numerical  simulations are listed in the section 7 to illustrate the qualitative behavior of the solutions obtained  in theoretical theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The appendices contain additional information and/or secondary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Appendix A contains the calculation of the demagnetizing field in the case of a ferromagnetic ellipsoid  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Appendix B contains the proof that −e1 is indeed the only asymptotically stable state for  equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Finally, Appendix C contains the calculations of the explicit constants in the case γ1 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In the whole article, |·| denotes the standard euclidean norm on R3 (or Rd), and its inner  product in Rd is denoted with a dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We are essentially interested in a control problem where the control function is the external field:  we abide by the usual convention, and denote  hext = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2  Existence of a minimal switching time  Let us recall that, as mentioned in the introduction, we will consider a ferromagnetic sample whose  shape Ω is the ellipsoid with Cartesian equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The dynamics of the magnetic moment m(·),  equal to m0 at the initial time, is hence driven by the simplified Landau-Lifshitz equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Towards an optimal control problem  The main issue we want to tackle reads  Given a steady-state m of (3) in S2, can we achieve a reversal by solving an optimal control  problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' steering the system from m(0) = m to m(T) = −m while minimizing T?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In what follows, we will consider particular stationary states: m = ±e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It is proved in appendix B  that these equilibria are asymptotically stable when γ1 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore, they can be used as magnetic  spin orientation for memory storage purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We will denote by u(·) the external (spatially uniform)  magnetic field imposed on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This is the control variable in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The question is  4  then to ask if it is possible to steer the solution mu of the system (3) associated to the control u(·) and  to the initial data mu(0) = e1 until mu(T) = −e1, in minimal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Of course, it is necessary to add physical constraints to this problem: if one imposes no restrictions  on the choice of admissible controls, it is likely that the minimal time problem will have a solution,  reached by unrealistic controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' For this reason, we will assume in what follows the constraint  |u(t)| ≤ U  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' in (0, T),  (CT,U)  in order to limit the choice of controls to realistic possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  All in all, the problem we aim at investigating reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Minimal time problem: let U > 0 and assume that m0 = e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The problem reads  TU :=  inf  (T,u)∈OU  T,  (P0)  where  OU = {(T, u) | T ∈ R+, u ∈ L∞(0, T) satisfies (CT,U) and mu(T) = −e1},  with mu, the solution to (3) associated to the control function u(·) and the initial datum e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We will investigate the following issues:  Does Problem (P0) have an optimal solution for any value of U > 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  How to characterize all the solutions to this problem and understand their geometric dependence  to the parameters γi, i = 1, 2, 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Main results  First, the minimization problem is indeed well-posed, meaning that the existence of an optimal solution  is equivalent to the existence of a minimal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The following properties are equivalent:  (i) There exists an optimal pair (TU, u) ∈ OU for Problem (P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (ii) TU is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (iii) OU is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The behavior of the control system differs greatly depending on the values of the parameters γi and  more specifically on the values of γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Assume γ1 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then there exists Ucrit > 0 such that  for all U ∈ (0, Ucrit], (P0) has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  for all U > Ucrit, (P0) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It is notable that the proof of this theorem provides an explicit lower-bound estimate of  Ucrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The precise bound is derived in Remark 18  5  We are now interested in the case where γ1 = γ2, which is not covered by the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It is  interesting to note that in this case, the behavior of the optimized physical system is very different from  the one described in the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Indeed, this situation of symmetry leads to the fact that there is  no longer a threshold from which the system is controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To complete this analysis, we also investigate in the following result the existence of optimal planar  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In view of the system symmetry, it is natural to conjecture that the optimal trajectory  are planar, since all the points in span(e1, e2)∩S2 are stable, and this set is even asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Somewhat surprisingly, We show here that this is actually not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' If γ1 = γ2 ≤ γ3, then the optimal control problem (P0) has a solution whatever the value  of U > 0, meaning that Ucrit = 0, with the notations of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Furthermore, if γ1 < γ3, the  optimal trajectory in S2 is not contained in the plane span(e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  It is interesting to notice that Theorem 2 can be refined in the particular case where γ1 < γ2 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To this aim, we will deeply exploit the necessary first order optimality conditions provided by the so-  called Pontryagin Maximum Principle (PMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We refer to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='3 for a precise statement of such  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' If γ1 ≤ γ2 = γ3, then Ucrit =  α  2  √  1+α2 (γ2 − γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Furthermore, for all U > Ucrit,  TU =  π  √  1 + α2  �  U 2 − U 2  crit  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In particular, we infer from the result above the following asymptotics:  TU ∼  π  �  2Ucrit(1 + α2)  1  √U − Ucrit  as U ↘ Ucrit,  and  TU ∼  π  U  √  1 + α2  as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 7 (Case of where the shape of the sample is a sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In the case where Ω is a sphere, then  one has γ1 = γ2 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, both conclusions of Theorem 4 and 5 apply, meaning that Ucrit = 0 and  the optimal time is given by TU = π/(U  √  1 + α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Furthermore, it may be shown that, in that case,  there exists optimal planar trajectories in each of the hyperplanes span(ei, ej) with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We refer for  instance to [3, Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' 2], whose main argument can be reproduced in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We end this section by a result on the asymptotic behavior of optimal magnetization trajectories  as U diverges to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We prove that optimal trajectories tend to be supported on a geodesic on the  sphere whenever U is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let γ1 ≤ γ2 ≤ γ3 and U > Ucrit and m be an optimal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let p be its adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then, if U is large enough, m stays close to the plane V = span(e1, p(TU)) in the following sense: there  exists U0 > 0 and C > 0 such that for every U > U0 and t ∈ [0, TU],  ∥m(t) − PV m(t)∥ ≤ C  U ,  where PV denotes the orthogonal projection onto V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  6  3  Minimization and optimality  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Proof of Theorem 1: existence of an optimal trajectory  Let us assume that OU is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This allows us to consider a minimizing sequence (Tn, un)n∈N ∈ ON  U,  and mn ∈ C ([0, Tn] the solution to (3) with field un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' By definition, Tn → TU as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In what  follows, we will denote similarly a sequence and any subsequence with a slight abuse of notation, for  the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us introduce the functions �un, �mun defined on [0, 1] by  �un(s) = un(Tns)  and  �mn(s) = mn(Tns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hence, System (3) rewrites  � ˙�mn = Tn  �  α  �  �h0( �mn) − (�h0( �mn) · �mn) �mn  �  − �mn ∧ �h0( �mn)  �  in (0, 1)  �mn(0) = e1  (5)  where �h0( �mn) = −D �mn + �un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Similarly, since the sequence (�un)n∈N is bounded in L∞(0, 1), it converges weakly-star in L∞(0, 1)  up to a subsequence to some element u∗ such that |u∗(·)| ≤ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' in [0, 1] according to the Banach-  Alaoglu-Bourbaki theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since both ( �mn)n∈N and (�un)n∈N are bounded in L∞([0, 1]), we infer that  so is ˙�mn according to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, the sequence ( �mn)n∈N is bounded in W 1,∞(0, 1) and hence  converges (up to a subsequence) towards an element �m∗ ∈ W 1,∞(0, 1) in C0([0, 1]) according to the  Ascoli theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In particular, one has necessarily | �m∗(·)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Now, let us rewrite (5) as the fixed-point  equation  ∀s ∈ [0, 1],  �mn(s) = e1 + Tn  � s  0  �  α  �  �h0( �mn) − (�h0( �mn) · �mn) �mn  �  − �mn ∧ �h0( �mn)  �  dσ  Observe that the right-hand side is linear with respect to �h0(mn) and that, according to the properties  above, (�h0( �mn))n∈N converges weakly-star to �h0( �m∗) in L∞(0, 1), where �h0( �m∗) = −D �m∗+�u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Letting  n tend to +∞ in the equation above, we obtain:  ∀s ∈ [0, 1],  �m∗(s) = e1 + TU  � s  0  �  α  �  �h0( �m∗) − (�h0( �m∗) · �m∗) �m∗�  − �m∗ ∧ �h0( �m∗)  �  dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Moreover, since ˜mn(1) = −e1 by construction, the convergence in C0([0, 1]) leads to ˜m∗(1) = −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Taking the previous formula with s = 1, we get  −2e1 = TU  � 1  0  �  α  �  �h0( �m∗) − (�h0( �m∗) · �m∗) �m∗�  − �m∗ ∧ �h0( �m∗)  �  dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  This proves that TU > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Now, let us introduce u∗ as u∗(t) = �u∗(t/TU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' By undoing the change of  variable above, we get that �m∗(t/TU) = mu∗(t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' t ∈ [0, TU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Furthermore, mu∗(0) = e1 and  mu∗(TU) = −e1 since �mn(0) = e1 and �mn(1) = −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The converse sense is straightforward and the  expected conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Finally, observe that the same reasoning can be reproduced whenever TU is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Sufficient and necessary condition for the existence of an admissible trajectory  Let (m, u) be a solution to (3) on [0, T], and for t ∈ [0, T], consider the mobile frame B(t) =  (m(t), ˙e(t), m(t) ∧ ˙e(t)) where ˙e = ˙m/| ˙m|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' According to [3], by observing that  m ⊥ ˙m,  ˙m ⊥ m ∧ ˙m  and  m ⊥ m ∧ (m ∧ Dm),  1Here, ˙e is merely a notation, and not the time derivative of a previously defined vector e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  7  one shows easily, by decomposing u(t) into B(t) and writing the equation for u−(u·m)m, the projection  of u on m⊥, that there exists λ ∈ L∞(0, T) such that  u =  1  1 + α2 (α ˙m + m ∧ ˙m) + Dm − (Dm · m)m + λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (6)  In fact, λ = u · m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Reciprocally, given any function m ∈ W 1,∞([0, T], S2), and any function λ ∈  L∞([0, T], R), if we define u by (6), then (m, u) is solution to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' These considerations can be seen as  a consequence of a flatness property of the main system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Again assuming that (m, u) is admissible trajectory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e a solution to (3), We infer from (6) that  u(t) expands as  u = λm +  �  α  1 + α2 | ˙m| + ˙e · Dm  �  ˙e +  �  Dm · (m ∧ ˙e) +  1  1 + α2 | ˙m|  �  m ∧ ˙e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (7)  As, a consequence, using that Dm · (m ∧ ˙e) = ˙e · (Dm ∧ m) due the triple product property, we get  |u|2 = λ2 +  �  α  1 + α2 | ˙m| + ˙e · Dm  �2  +  �  Dm · (m ∧ ˙e) +  | ˙m|  1 + α2  �2  = λ2 +  �  ˙e · Dm + α| ˙m|  1 + α2  �2  +  �  ˙e · (Dmu ∧ m) +  | ˙m|  1 + α2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Clearly, this computation and the previous remarks show that, without loss of generality, we can  furthermore assume that an optimal trajectory satisfies λ = 0, or equivalently, u · m = 0: we will do  this in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us introduce, for a given T > 0,  VT = {m ∈ H1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' S2) | m(0) = e1 and m(T) = −e1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To investigate the existence of an admissible trajectory, it is then convenient to introduce  Λ(T) :=  inf  λ∈L∞([0,T]) inf  m∈VT  sup  t∈[0,T]  �  λ2 +  �  ˙e · (Dm ∧ m) +  | ˙m|  1 + α2  �2  +  �  ˙e · Dm + α| ˙m|  1 + α2  �2�  ,  = inf  m∈VT  sup  t∈[0,T]  ��  ˙e · (Dm ∧ m) +  | ˙m|  1 + α2  �2  +  �  ˙e · Dm + α| ˙m|  1 + α2  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (8)  We summarize the above discussion in the form of a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The existence of an admissible trajectory for Problem (P0) comes to the existence of  m ∈ VT such that the function u given by (7) with λ = 0 satisfies ∥u∥L∞([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='R3) ≤ U, which is also  equivalent to Λ(T) ≤ U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Also, u satisfies u · m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='3  Necessary optimality conditions for Problem (P0)  This problem can be solved by using the Pontryagin maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The main results of this  section are gathered in Proposition 11, at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The Hamiltonian associated to Problem (P0) is  H : S2 × R3 × {−1, 0} × R3  →  R  (m, p, p0, u)  �→  p · (−αm ∧ (m ∧ h0(m)) − m ∧ h0(m)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  It can be noted that the dependence of H on the control function u is affine and one has  H(m, p, p0, u) = p · (−αm ∧ (m ∧ u) − m ∧ u) − p · (−αm ∧ (m ∧ Dm) − m ∧ Dm)  8  As a first remark, the magnetization stays in S2 = ∂B, where B is the closed unit ball of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, our problem is obviously equivalent to the problem with restricted conditions  TU =  inf  (T,u)∈OU  |m|2−1=0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Thus, we use the version of the Pontryagin maximum principle with restricted phase coordinates, as  stated in [17, Theorem 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This theorem is stated for a minimization of an integral with fixed T, but it  can be easily adapted to the case of a minimal time with classical changes (see for instance the passage  from Theorem 1 to Theorem 2 in the same reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' With such a statement, we point out that, for  all m ∈ R3  ∇(|m|2 − 1) = 2m,  and that  2m · (−αm ∧ (m ∧ h0(m)) − m ∧ h0(m)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, in our case, this statement gives the exact same necessary conditions, with an additional  orthogonality condition for the adjoint state, stated hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The first order optimality conditions read as follows: let us denote by (T, u), an optimal pair for  this problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' there exists an absolutely continuous mapping p : [0, T] → R3 called adjoint state and a  real number p0 ∈ {0, −1} such that the pair (p, p0) is non-trivial and for almost every t ∈ [0, T], the  following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Setting  F1(m, p)  :=  α  �  p ∧ (m ∧ Dm) + Dm ∧ (m ∧ p) − D(m ∧ (m ∧ p))  �  − Dm ∧ p − D(p ∧ m),  F2(m, p, u)  :=  −α(p ∧ (m ∧ u) + u ∧ (m ∧ p)) + u ∧ p,  one gets  ˙p = −∂H  ∂m = F1(m, p) + F2(m, p, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (9)  Remark that, since |m| = 1, one equivalently has  F1(m, p)  =  α(Dp − (Dm · m)p − 2(m · p)Dm − 2(Dm · p)m) − Dm ∧ p − D(p ∧ m),  F2(m, p, u)  =  α(p · m)u + α(u · m)p − 2α(p · u)m + u ∧ p  Maximality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' For a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' t ∈ [0, T], u(t) solves the optimization problem  max  |v|≤U H(m(t), p(t), p0, v)  (10)  and one has at the final time T  max  |vT |≤U H(m(T), p(T), p0, vT ) = −p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (11)  A useful identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since the dynamics only depends on the magnetization m(·) and the control  u(·), the Hamiltonian functional is constant in time:  H(m(t), p(t), p0, u(t)) = −p0,  t ∈ [0, T],  (12)  according to (11), by evaluating the expression for t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  9  An orthogonality condition for the adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' At the final time t = T, the adjoint state p(T)  is tangent to the boundary |m|2 − 1 = 0 at m(T) = −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This condition is thus equivalent to  p(T) · e1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (13)  Orthogonality between u and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' As seen in Lemma 9, u · m = 0 on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since the initial and final state are fixed, there is no need to impose any transversality  condition on the adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us analyze the conditions (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The adjoint state p cannot vanish on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Indeed, in the converse case, if there exists t0 ∈  [0, T] such that p(t0) = 0, it follows from the Cauchy-Lipschitz theorem that p(·) = 0 and by using  Condition (11), one gets p0 = 0, a contradiction with the non-triviality of the pair (p, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  On condition (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Observe that v �→ H(m(t), p(t), p0, v) is affine with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' According to  the Karush-Kuhn-Tucker theorem, there exists µ ≥ 0 such that ∇vH(m(t), p(t), p0, u(t)) − µu(t) = 0  and the slackness condition µ(|u(t)|2 − U 2) = 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  If the set I := {|u| < U} is of positive Lebesgue measure, then one has  α(p(t) − (p(t) · m(t))m(t)) = p(t) ∧ m(t)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Taking the scalar product of this identity with p(t) leads to |p(t)|2 = (p(t) · m(t))2 on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since p does  not vanish, it follows from the equality case in the Cauchy-Schwarz inequality that p(t) is proportional  to m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We will show that such a case cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us introduce the function ϕ := p − (p · m)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' One can compute that ϕ satisfies the differential  (linear) relation  ˙ϕ = αDϕ − α(Dm · m)ϕ − Dm ∧ ϕ − D(ϕ ∧ m) + u ∧ ϕ + (m · (Dm ∧ ϕ))m − α(u · ϕ)m  (14)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This follows from an easy but lengthy computation, and from the fact that the function  λ given by λ = p · m satisfies  ˙λ = −2α(Dm · m)λ − (Dm ∧ p) · m − α(u · p) + 2α(Dm · p),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Now, let us assume that the set I := {|u(·)| < U} is of positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' According to the  discussion above, there exists a bounded function λ such that p = λm on I, and therefore, ϕ vanishes  on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Due to (14) being linear in ϕ, we obtain that ϕ(·) = 0 on [0, T], which means that  p(T) = (p(T) · m(T))m(T) = (p(T) · e1)e1 = 0,  from the orthogonality condition (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' But recall that p cannot vanish on [0, T]: we reached a contra-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We conclude that |u| = U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It follows that  α(p − (p · m)m) = p ∧ m + µu  on [0, T],  and using furthermore that  |α(p − (p · m)m) − p ∧ m|2 = α2(|p|2 − (p · m)2) + |p ∧ m|2  10  = (α2 + 1)(|p|2 − (p · m)2),  one gets an expression of u in terms of p or ϕ:  u =  U  (α2 + 1)1/2  α(p − (p · m)m) − p ∧ m  �  |p|2 − (p · m)2  =  U  (α2 + 1)1/2  αϕ − ϕ ∧ m  |ϕ|  (15)  In particular, we get  m ∧ u =  U  (α2 + 1)1/2  αm ∧ ϕ − ϕ  |ϕ|  ,  αm ∧ (m ∧ u) =  U  (α2 + 1)1/2  −α2ϕ − αm ∧ ϕ  |ϕ|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Substituting those terms in (3) and (14), we get at last  ˙m = m ∧ Dm + m ∧ (m ∧ Dm) + U(α2 + 1)1/2 ϕ  |ϕ|,  (16)  ˙ϕ = αDϕ − α(Dm · m)ϕ − Dm ∧ ϕ − D(ϕ ∧ m) + (m · (Dm ∧ ϕ))m − U(α2 + 1)1/2|ϕ|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (17)  On condition (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  By setting p(T) = (0, p2,T , p3,T ) (since p(T) · e1 = 0) and vT = (v1,T , v2,T , v3,T ),  since m(T) = −e1, this condition also rewrites  −p0 = max  |vT |≤U p(T) ·  �  �  0  αv2,T − v3,T  αv3,T + v2,T  �  � =  max  v2  2,T +v2  3,T =U2  �v2,T  v3,T  � �αp2,T + p3,T  αp3,T − p2,T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The Cauchy-Schwarz inequality then implies that −p0 = U  √  1 + α2�  p2  2,T + p2  3,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It follows that p0 =  −1 (else, the pair (p0, p) would be trivial) and condition (11) finally rewrites:  U  �  1 + α2|ϕ(T)| = −p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Analysis of the optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  From the previous discussion, u(t) is given by (15) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  t ∈ [0, T], leading to  |ϕ| (α (u − (u · m)m) − m ∧ u) = A  �  α2ϕ − αϕ ∧ m − αm ∧ ϕ + m ∧ (ϕ ∧ m)  �  = A(α2 + 1)ϕ  where A =  U  (α2+1)1/2 , so that  H(m(t), p(t), p0, u(t)) = U(α2 + 1)1/2|ϕ| − p · (α (Dm − (Dm · m)m) − m ∧ Dm)  = U(α2 + 1)1/2|ϕ| − Dm · (α (p − (p · m)m) − p ∧ m)  = U(α2 + 1)1/2|ϕ| − Dm · (αϕ − ϕ ∧ m)  = (α2 + 1)1/2|ϕ|  U  �  U 2 − Dm · u  �  and we infer that  |ϕ(t)|  �  U 2 − Dm(t) · u(t)  �  =  U  (α2 + 1)1/2 > 0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (18)  11  On condition (12)  From the previous discussion, we have for any t ∈ [0, T]  max  |v|≤U H(m(t), p(t), p0, v) = −p0 = 1,  (19)  which leads at t = 0 to  U  �  1 + α2|ϕ(0)| = −p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (20)  More generally, with the above expression (15) of u(t) which is the argmax of H, we get  1 = U  �  1 + α2|ϕ| − ϕ ·  �  αDm − m ∧ Dm  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (21)  For the sake of clarity, we sum-up all these informations in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proposition 11 (Necessary first order optimality conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let (T, u) denote an optimal pair for  Problem (P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, the adjoint state p defined by (9) does not vanish on [0, T] and one has  u =  U  (α2 + 1)1/2  αϕ − ϕ ∧ m  |ϕ|  ,  (22)  where ϕ is given by ϕ = p − (p · m)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In particular, one has |u(t)| = U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Moreover, ϕ satisfies the differential relation (17) completed by the conditions (18) and  U  �  1 + α2|ϕ(0)| = U  �  1 + α2|ϕ(T)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (23)  Finally, m satisfies (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  4  Proof of Theorem 2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Preliminary results  We first state preliminary results, in the form of a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' For all T < ∞, the map  L∞(0, T) → W 1,∞(0, T)  u �→ m solution to (3) with m(0) = e1  is continuous and locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let u1, u2 ∈ L∞(0, T) and m1, m2 the corresponding solution to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Define δm := m1 −m2 and  δu := u1 − u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Simple, though tedious, calculations provide that δm satisfies  dδm  dt  = α  �  −D δm + δu − ((−D δm + δu) · m1)m1  − ((−Dm2 + u2) · δm)m1 − ((−Dm2 + u2) · m2)δm  �  − m1 ∧ (−Dδm + δu) − δm ∧ (−Dm2 + u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  in (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since |m1| = |m2| = 1, we obtain  ����  dδm  dt  ���� ≤ ((4α + 2)∥D∥2 + (2α + 1)|u2|) |δm| + (2α + 1)|δu|,  in (0, T),  (24)  12  where ∥·∥2 denotes the operator norm associated to the euclidean norm |·|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since δm(0) = 0, we have  for all t ∈ (0, T)  |δm(t)| ≤  � t  0  ����  dδm  dt  ����(s) ds ≤ (2α + 1)T∥δu∥L∞ +  � t  0  ((4α + 2)∥D∥2 + (2α + 1)∥u2∥L∞) |δm|(s) ds,  and thus by Gronwall’s lemma,  |δm(t)| ≤ (2α + 1)T∥δu∥L∞ exp (((4α + 2)∥D∥2 + (2α + 1)∥u2∥L∞)t),  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Using this estimate, and plugging it aslo in (24), we get  ∥δm∥W 1,∞ ≤ C(T, ∥u2∥L∞)∥δu∥L∞,  and the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' If γ1 < γ2, there exists δ > 0 such that for all U > 0, if |m(0) + e1| < δ, then −e1 can be  reached in finite time with a control u such that |u| ≤ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let us introduce m as the solution to (3) with the feedback control term  u(t) =  U  √  1 + α2  α(−e1 + (e1 · m(t))m(t)) + e1 ∧ m(t)  �  1 − (m(t) · e1)2  ,  so that the equation on m becomes autonomous, and is well defined as long as m(t) ̸= ±e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Observing  that  �  m, e1 − (m · e1)m  �  1 − (m · e1)2 ,  m ∧ e1  �  1 − (m · e1)2  �  is an orthonormal basis, one immediately gets that |u(t)| = U for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Denote m = (m1, m2, m3) the coordinates of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From (3), the ODEs satisfied by m2 and m3 are  ˙m2 = −α[(γ2 − γ1)m2 − ((γ2 − γ1)m2  2 + (γ3 − γ1)m3  3)m2] + (γ1 − γ3)m1m3 + v2,  ˙m3 = −α[(γ3 − γ1)m3 − ((γ2 − γ1)m2  2 + (γ3 − γ1)m3  3)m3] − (γ1 − γ2)m1m2 + v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' by setting �m := (m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' m3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' it follows that �m solves the controlled system  ˙˜m = A− ˜m + ξ− + ˜v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (25)  where  A−  =  �−α(γ2 − γ1)  (γ3 − γ1)  −(γ2 − γ1)  −α(γ3 − γ1)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  ξ−  =  �α((γ2 − γ1)m2  2 + (γ3 − γ1)m2  3)m2 − (γ3 − γ1)(1 + m1)m3  α((γ2 − γ1)m2  2 + (γ3 − γ1)m2  3)m3 + (γ2 − γ1)(1 + m1)m2  �  and ˜v = (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' v3) where v = (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' v3) = α(u − (u · m)m) − m ∧ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' which means here  v = U  �  1 + α2 (−e1 + m1(t)m(t))  �  |m(t)|2 − (m1(t))2  = U  �  1 + α2 (−e1 + m1(t)m(t))  | ˜m|  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We infer that ˜v = U  √  1 + α2m1 ˜m/| ˜m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Observing that (1 − m1)(1 + m1) = | ˜m|2 yields, as soon as  m1 ≤ 0,  |ξ−(t)| ≤ (1 + |α|) δγ+ | ˜m(t)|3  13  where δγ+ := γ3 − γ1 > 0, and also that  ����˜v + U  �  1 + α2 ˜m(t)  | ˜m|  ���� ≤ U  �  1 + α2| ˜m(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  With these estimates and by taking the inner product of (25) with ˜m, we get  1  2  d  dt| ˜m(t)|2 ≤ −U  �  1 + α2| ˜m(t)| + (U  �  1 + α2 + ∥A−∥)| ˜m(t)|2 + (1 + |α|) δγ+ | ˜m(t)|4,  and  d  dt| ˜m(t)| ≤ −U  �  1 + α2 + (U  �  1 + α2 + ∥A−∥)| ˜m(t)| + (1 + |α|) δγ+ | ˜m(t)|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us introduce δU ∈ (0, 1/2) small enough (depending on U > 0) so that  (U  �  1 + α2 + ∥A−∥)δU + (1 + |α|) δγ+ δ3  U ≤ U  √  1 + α2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (26)  Then, if |m(0) + e1| < δU, which gives | ˜m(0)| < δU, one has  d  dt| ˜m(t)| ≤ −U  √  1 + α2  2  < 0,  as long as | ˜m(t)| < δU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This yields that, for such time intervals, the mapping t �→ | ˜m(t)| is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, this shows that if δU satisfies (26) and m(0) is such that |m(0)+e1| < δU, then | ˜m(t)| < δU  for all t ≥ 0 and that ˜m(t) reaches 0 in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In other words, −e1 can be reached in finite time  with a control u such that |u| ≤ U if m is such that |m + e1| < δU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To conclude, it remains to drop the dependency of δ in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let us use that −e1 is asymptotically  stable according to Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore, there exists δ > 0 such that, starting from a point  m(0) chosen so that |m(0) + e1| < δ, we can first let the system evolve without control until we obtain  |m(TU) + e1| < δU for some finite time TU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From this moment, we know we can reach −e1 in finite  time, whence the expected conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Recall for the sake of readability that the notation TU has been introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let γ1 < γ2 and U > 0 such that TU < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then there exists ε > 0 such that TU−ε < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since TU < ∞ and according to Theorem 1, there exists u∗ ∈ L∞(0, TU) such that m∗(0) = e1  and m∗(TU) = −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Now, let us consider m the solution to (3) associated to the control choice  u = U−ε  U u∗ for some ε ∈ (0, U) to be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From Lemma 12, we obtain  ∥m − m∗∥W 1,∞(0,TU) ≤ C  ����  U − ε  U  u∗ − u∗  ����  L∞(0,TU)  = Cε  ∥u∗∥L∞(0,TU)  U  ≤ Cε  for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Since m∗(TU) = −e1 by definition, we can take ε > 0 small enough so that |m(TU) + e1| < δ, where  δ > 0 is given by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From this lemma, we know we can reach −e1 in finite time, and since  |u| ≤ U − ε, this leads to TU−ε < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' TU is non-increasing with respect to U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In particular, if TU0 < ∞ for some U0 > 0,  then TU < ∞ for all U > U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This property is an immediate consequence of the definition of TU and the fact that the sets OU  are increasing for the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Emergence of a threshold  The following result is the most crucial for concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It quantifies the asymptotic stability of e1 for  the evolution of the magnetization m, with respect to u viewed as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' It is notable that its  proof not only highlights the emergence of a threshold but also provides an explicit expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let us assume that γ1 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' There exists Ustab > 0 depending only on γ3 − γ1, γ2 − γ1  and α such that, for any U < Ustab, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let (m, u) be a solution of (3) on [0, +∞),  such that u ∈ L∞([0, ∞)) and ∥u∥L∞ ≤ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then for all t ≥ 0, m1(t) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In other words, m remains in the hemisphere with pole e1, and in particular, m can not reach −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  One has the same statement if (m, u) are defined on a bounded interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let v = α(u − (u · m)m) − m ∧ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, using that v reads as the sum of two orthogonal terms,  one has |v|2 ≤ (1 + α2)U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Moreover, since |m|2 = 1, there holds  Dm · m = γ1 + (γ2 − γ1)m2  2 + (γ3 − γ1)m2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  As in the proof of Lemma 13, �m solves the controlled system  ˙�m = A �m + ξ + ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (27)  where  A =  �−α(γ2 − γ1)  −(γ3 − γ1)  γ2 − γ1  −α(γ3 − γ1)  �  ,  ξ =  �α((γ2 − γ1)m2  2 + (γ3 − γ1)m2  3)m2 + (γ3 − γ1)(1 − m1)m3  α((γ2 − γ1)m2  2 + (γ3 − γ1)m2  3)m3 − (γ2 − γ1)(1 − m1)m2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (28)  Pay attention to the sign change between A and ξ used here and A− and ξ− introduced in the proof of  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let ν ∈ (0, 1] to be fixed later and define  Tν = inf{t ≥ 0 | | �m(t)| ≥ ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Our goal is to derive suitable bounds on ˜m, so that for a well chosen ν, Tν = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Since m(0) = e1 and m is continuous, we know that Tν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Note that one has necessarily  m1(·) > 0 on (0, Tν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, for all t ∈ [0, Tν), using that m is normalized, there holds like previously  0 ≤ 1 − m1(t) ≤ 1 − m1(t)2 = | �m(t)|2, and therefore  |ξ(t)| ≤ (1 + |α|) δγ+ | �m(t)|3 ≤ (1 + |α|) δγ+ ν3,  where δγ+ := γ3 − γ1 ≥ γ2 − γ1 =: δγ− > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' On the other hand, thanks to the Duhamel formula on  (27) using the fact that �m(0) = 0, there holds  �m(t) =  � t  0  exp ((t − s)A)(ξ(s) + ˜v(s)) ds  for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This, together with the previous estimates, drives to  | �m(t)| ≤  � t  0  ∥exp ((t − s)A)∥2  �  (1 + |α|)δγ+ν3 +  �  1 + α2U  �  ds  ≤ (1 + |α|)(δγ+ν3 + U)  � t  0  ∥exp ((t − s)A)∥2 ds  ≤ (1 + |α|)(δγ+ν3 + U)  � t  0  ∥exp (sA)∥2 ds,  (29)  for all t ∈ [0, Tν), where ∥·∥2 still denotes the operator norm associated to the euclidean norm |·|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We will now provide an estimate of the norm of the exponential matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Recall that the characteristic  polynomial of A is PA(X) = X2 − Tr(A)X + det(A) with  det(A) = (1 + α2)δγ− δγ+ > 0,  Tr(A) = −α(δγ− + δγ+) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Its discriminant ∆ reads ∆ = Tr(A)2 − 4 det(A) = α2(δγ+ − δγ−)2 − 4δγ−δγ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To compute the  eigenvalues of A, we have to distinguish between several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  15  1st case: ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then its eigenvalues are λ± := 1  2(Tr(A) ±  √  ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Remark that both eigenvalues of  A are negative (according to the signs of the trace and the determinant above) and different from each  other, which means that A is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore, we infer2 that  exp (sA) = esλ+ sA − sλ− I2  sλ+ − sλ−  + esλ− sA − sλ+ I2  sλ− − sλ+  =  1  √  ∆  �  esλ+(A − λ− I2) − esλ−(A − λ+ I2)  �  =  1  √  ∆  �  (esλ+ − esλ−)A + (esλ−λ+ − esλ+λ−) I2  �  = esλ+  √  ∆  �  (1 − e−s  √  ∆)A + (e−s  √  ∆λ+ − λ−) I2  �  = sesλ+  �1 − e−s  √  ∆  s  √  ∆  A − λ−  1 − e−s  √  ∆  s  √  ∆  I2  �  + esλ− I2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Thus, using the facts that λ− < λ+ < 0, ∥A∥2 ≥ |λ−| and also that the function f given by f(x) = 1−e−x  x  analytically extended to R is uniformly bounded by 1 on [0, ∞), we get  ∥exp (sA)∥2 ≤ esλ+ (s(∥A∥2 + |λ−|) + 1) ≤ esλ+ (2s∥A∥2 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hence,  � t  0  ∥exp (sA)∥2 ds ≤ |λ+|−1(1 − eλ+t) + 2∥A∥2|λ+|−2(1 − (|λ+|t + 1)eλ+t)  ≤ (1 − eλ+t)  �  |λ+|−1 + 2∥A∥2|λ+|−2�  ≤ 3(1 − eλ+t)∥A∥2|λ+|−2,  and according to (29), one has for all t ∈ [0, Tν)  | �m(t)| ≤ 3∥A∥2|λ+|−2(1 − eλ+t)(1 + |α|)(δγ+ν3 + U)  To conclude, we will choose U adequately so that the function x �→ 3∥A∥2|λ+|−2(1 + |α|)(δγ+x3 + U)  admits a fixed point x0 in (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This is possible thanks to the next lemma, whose proof is postponed  to the end of this section for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let a, b, c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The function x �→ a−1(bx3 + c) has a fixed point x0 in (0, 1] if, and only  if c ≤ ax1 − bx3  1 where x1 = min{1, � a  3b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark that if x1 is as in this lemma, one has ax1 − bx3  1 ≥  2  3ax1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hence, setting a =  |λ+|2/(3∥A∥2(1 + |α|)), b = δγ+ and c = U drives us to assume that  U ≤  |λ+|2  3∥A∥2(1 + |α|)x1 − δγ+ x3  1,  with  x1 := min  �  1,  |λ+|  3  �  ∥A∥2(1 + |α|)δγ+  �  ,  we can take ν = x0 provided by Lemma 17, and the previous estimate leads to  | �m(t)| ≤ (1 − eλ+t)ν,  for all t ∈ [0, Tν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' A continuity argument then implies that Tν = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In other words, for all t ≥ 0,  |m1(t)| =  �  1 − | ˜m(t)|2 ≥  �  1 − (1 − eλ+t)2ν2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Now, m1 is continuous, so that it keeps a constant  sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' As m1(0) = 1, m1(t) ≥ 0 for all t ≥ 0, which is the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  2Here, the Lagrange interpolation formula is used to compute the exponential of A: for every matrix M ∈ Md(C)  whose spectrum {λi}1≤i≤d consists of distinct eigenvalues, one has  exp(M) =  d  �  j=1  eλj �  i̸=j  M − λi Id  λj − λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  16  2nd case: ∆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In this case, the eigenvalues are  λ± := Tr(A) ± i  √  −∆  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  One more time, the two eigenvalues are distinct, complex conjugate with negative real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Yet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='same decompositions as previously can still be applied and there holds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='exp (sA) = esλ+ sA − sλ− I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='sλ+ − sλ−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='+ esλ− sA − sλ+ I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='sλ− − sλ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='esλ+(A − λ− I2) − esλ−(A − λ+ I2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='(esλ+ − esλ−)A + (esλ−λ+ − esλ+λ−) I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='= e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2 Tr(A)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='(e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='is√−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='− e− is√−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=')A + (λ+e− is√−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='− λ−e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='is√−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=') I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='= e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2 Tr(A)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2 sin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='A +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆ cos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='− Tr(A) sin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='= s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2 Tr(A)�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2A − Tr(A) I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='sinc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='+ e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2 Tr(A) cos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='−∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='I2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Thus, we get  ∥exp (sA)∥2 ≤ s  2e  s  2 Tr(A) (2∥A∥2 − Tr(A)) + e  s  2 Tr(A) ≤ e  s  2 Tr(A) (2s∥A∥2 + 1) ,  since Tr(A) = λ+ + λ− and |λ±| ≤ ∥A∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' following the same way as in the first case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' we get  � t  0  ∥exp (sA)∥2 ds ≤ (1 − e  1  2 Tr(A)t)  �  2|Tr(A)|−1 + 8∥A∥2|Tr(A)|−2�  ≤ 12(1 − e  1  2 Tr(A)t)∥A∥2|Tr(A)|−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  and according to (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' one has for all t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Tν)  | �m(t)| ≤ 12∥A∥2|Tr(A)|−2(1 − e  1  2 Tr(A)t)(1 + |α|)(δγ+ν3 + U)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' by mimicking the reasoning done in the first case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' by assuming  U ≤  Tr(A)2  12∥A∥2(1 + |α|)x1 − δγ+ x3  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  with  x1 := min  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Tr(A)  6  �  ∥A∥2(1 + |α|)δγ+  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  and taking ν = x0 given by Lemma 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' the previous estimate leads to  | �m(t)| ≤ (1 − e  1  2 Tr(A)t)ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  for all t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Tν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Arguing as in the first case, we infer that Tν = ∞ in this case as well, and then,  m1(t) > 0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  17  3rd case: ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In this case, both eigenvalues are equal, one has λ = Tr(A)/2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Note that, in  that case, A− 1  2 Tr(A) I2 is therefore a non-zero nilpotent matrix, and more precisely (A− 1  2 Tr(A) I2)2 =  (sA − s  2 Tr(A) I2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Thus, there holds  exp (sA) = exp  �s  2 Tr(A) I2  �  exp (sA − s  2 Tr(A) I2)  = e  s  2 Tr(A)(I2 +sA − s  2 Tr(A) I2)  which yields  ∥exp (sA)∥2 ≤ e  s  2 Tr(A)(1 + s(∥A∥2 − 1  2 Tr(A))) ≤ e  s  2 Tr(A)(1 + 2s∥A∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The computations are then exactly the same ones as in the second case, and the conclusion follows in  the same fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof of Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We are looking for a root x0 ∈ (0, 1] of the polynomial function f given by f(X) =  bX3 − aX + c, whose derivative 3bX2 − a is negative for X < � a  3b =: x2 and positive for X > x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The minimum in [0, 1] is therefore reached at x1 and, since f(0) = c > 0, there is a root if and only if  f(x1) ≤ 0, which corresponds to the assumption in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From the proof of Lemma 16, we obtained the following expression for Ustab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Consider  the matrix A defined there in (28), denote ∆ = Tr(A)2 − 4 det(A) the discriminant of its characteristic  polynomial, λ± its eigenvalues chosen so that λ+ > λ− whenever ∆ > 0, and δγ+ := γ3 − γ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let  x1(A) :=  �  �  �  �  �  �  �  min  �  1,  |λ+|  3√  ∥A∥2(1+|α|)δγ+  �  if ∆ > 0  min  �  1,  Tr(A)  6√  ∥A∥2(1+|α|)δγ+  �  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then  Ustab = Γ(∆) :=  �  �  �  |λ+|2  3∥A∥2(1+|α|)x1(A) − δγ+ x1(A)3  if ∆ > 0  Tr(A)2  12∥A∥2(1+|α|)x1(A) − δγ+ x1(A)3  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Note also that, to complement this result, explicit computations of the quantities involved (like ∥A∥2)  are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We now have all the elements to conclude the:  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Define  Ucrit := inf{U | TU < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  From Lemma 16, we know that Ucrit > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Lemma 15 proves the second point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' To investigate the case  where U = Ucrit, observe that, by definition, (P0) has no solution for all U ∈ (0, Ucrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Moreover, if  (P0) had a solution for U = Ucrit, then Lemma 14 would provide a contradiction with respect to the  definition of Ucrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  5  Cases with symmetry  In this section, we deal with the two cases when the material satisfies additional symmetry without  being a sphere (in which case the analysis becomes trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' They correspond to the cases γ1 = γ2 < γ3  and γ1 < γ2 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Proof of Theorem 4 (case γ1 = γ2)  From Theorem 1, we have to investigate the existence of an admissible trajectory for this problem,  in other words, the existence of a control u ∈ OU and a time T > 0 such that mu(T) = −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This  property is known to be true as soon as U is large enough according to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' But it has to be proved for  smaller U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us assume that γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We will prove that, in that case, infT>0 Λ(T) = 0, (with Λ(T) defined  by Equation (8)) which will prove that Problem (P0) has a solution whatever the value of U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  For ε > 0, let us consider a particular trajectory mε of the form mε = (cos(εt), sin(εt), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, by  defining  Fε :=  �  ˙eε · (Dmε ∧ mε) +  | ˙mε|  1 + α2  �2  +  �  ˙eε · Dmε + α| ˙mε|  1 + α2  �2  with ˙eε = ˙mε/| ˙mε|, a straightforward computation yields  Fε =  ε2  1 + α2 ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We infer that infT>0 Λ(T) ≤ Fε ≤ ε2 whence the conclusion, since ε is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us now prove the last point of this result, assuming that from now on γ1 < γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Assume that  m3(t) = 0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, Dm = γ1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' By contradiction, if such an m is an optimal trajectory,  Proposition 11 is satisfied, and (16) gives  ˙m = U(α2 + 1)1/2 ϕ  |ϕ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  By taking the third coordinate, we get ϕ3(t) = 0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Thus, we also get Dϕ = γ1ϕ and (17)  gives  ˙ϕ = −γ1m ∧ ϕ − D(ϕ ∧ m) − U(α2 + 1)1/2|ϕ|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  By taking again the third coordinate, we get  0 = −γ1(m ∧ ϕ) · e3 − D(ϕ ∧ m) · e3 = (γ3 − γ1)(m ∧ ϕ) · e3 = (γ3 − γ1)(e3 ∧ m) · ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Since γ3 > γ1, this proves that (e3 ∧ m) · ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' However, at t = 0, this means that  0 = (e3 ∧ e1)ϕ(0) = ϕ2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Now ϕ1(0) = ϕ(0) · m(0) = 0, and we obtained ϕ(0) = 0: this is a contradiction with (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Proof of Theorem 5 (case γ2 = γ3)  For this case, we first show that the (PMP) conditions are also sufficient conditions for optimal trajec-  tories :  Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > Ucrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then any trajectory m satisfying the (PMP) ( (9)-(12) with p0 = −1) is  an optimal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let m∗ be an optimal trajectory, and p∗ the associated adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' By definition, they satisfy  the (PMP) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Now, let (m, p) be a trajectory and its adjoint state satisfying the (PMP)  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let also ϕ = p − (p · m)m and ϕ∗ = p∗ − (p∗ · m∗)m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In particular, we know that ϕ(0)  satisfies (23) and ϕ(0) ⊥ m(0) = e1, and similarly for ϕ∗ with respect to m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Thus, there exists  θ ∈ [0, 2π] such that Rθϕ(0) = ϕ∗(0) where Rθ is the rotation along e1 of angle θ:  Rθ =  �  �  1  0  0  0  cos θ  − sin θ  0  sin θ  cos θ  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  19  On the other hand, since γ2 = γ3, we have DRθ = RθD for all θ, but also Rθf ∧ Rθg = Rθ(f ∧ g)  for any f, g ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Last, Rθm∗(0) = Rθe1 = e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Thus, (Rθm, Rθϕ) satisfies the same system of  ODEs as (m∗, ϕ∗) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' (3)-(9) with u(·) or u∗(·) satisfying (22)) with the same initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' By the  Cauchy-Lipschtiz theorem and using the fact that both ϕ and ϕ∗ never vanish thanks to (18), we obtain  (Rθm, Rθϕ) = (m∗, ϕ∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' (m, ϕ) = (R−θm∗, R−θϕ∗), and thus the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The following two results exploit in a precise way the (necessary and sufficient) optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > 0 and (m, p) satisfy the (PMP) conditions ( (9)-(12) with p0 = −1) and ϕ =  p − (p · m)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, for every t ≥ 0, ϕ(t) · (e1 ∧ m(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We know that ϕ satisfies (17) and m satisfies (16) with u given by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Moreover,  d  dt(ϕ · (e1 ∧ m)) = ˙ϕ · (e1 ∧ m) + ϕ · (e1 ∧ ˙m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Using the facts that u ⊥ m and m ⊥ (e1 ∧ m), there holds  ˙ϕ · (e1 ∧ m) = αDϕ · (e1 ∧ m) − α(Dm · m)ϕ · (e1 ∧ m) − (Dm ∧ ϕ) · (e1 ∧ m) − D(ϕ ∧ m) · (e1 ∧ m)  +  U  √  1 + α2 (ϕ ∧ ( ϕ  |ϕ| ∧ m)) · (e1 ∧ m),  ϕ · (e1 ∧ ˙m) = −αϕ · (e1 ∧ Dm) + α(Dm · m)ϕ · (e1 ∧ m) + ϕ · (e1 ∧ (m ∧ Dm)) + U  �  1 + α2ϕ · (e1 ∧ ϕ  |ϕ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  First, we point out that ϕ ⊥ m so that ϕ ∧ (ϕ ∧ m) = −|ϕ|2m, and thus (ϕ ∧ (ϕ ∧ m)) · (e1 ∧ m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Similarly, U  √  1 + α2ϕ · (e1 ∧ ϕ  |ϕ|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, using the triple product formula, we get  ϕ · (e1 ∧ Dm) = −Dm · (e1 ∧ ϕ)  (Dm ∧ ϕ) · (e1 ∧ m) = (ϕ ∧ (e1 ∧ m)) · Dm = −(ϕ · e1)(m · Dm),  (m ∧ Dm) · (e1 ∧ ϕ) = ((e1 ∧ ϕ) ∧ m) · Dm = (m · e1)(ϕ · Dm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Moreover, since γ2 = γ3, we know that, for any vector f ∈ R3 such that f · e1 = 0, Df = γ2f = γ3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  With the fact that D is symmetric, this leads to  Dϕ · (e1 ∧ m) = ϕ · D(e1 ∧ m) = γ2ϕ · (e1 ∧ m),  Dm · (e1 ∧ ϕ) = m · D(e1 ∧ ϕ) = γ2m · (e1 ∧ ϕ) = −γ2ϕ · (e1 ∧ m),  D(ϕ ∧ m) · (e1 ∧ m) = (ϕ ∧ m) · D(e1 ∧ m) = γ2(ϕ ∧ m) · (e1 ∧ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Last, using again the double product, we have  ϕ · (e1 ∧ (m ∧ Dm)) = (ϕ · m)(e1 · Dm) − (ϕ · Dm)(e1 · m) = −(ϕ · Dm)(e1 · m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  But one has then  (ϕ · e1)(m · Dm) − (ϕ · Dm)(e1 · m) = e1 ·  �  (Dm · m)ϕ − (Dm · ϕ)m  �  = −e1 · (Dm ∧ (m ∧ ϕ))  = −Dm · ((m ∧ ϕ) ∧ e1) = −m · D((m ∧ ϕ) ∧ e1)  = −γ2m · ((m ∧ ϕ) ∧ e1) = γ2(ϕ ∧ m) · (e1 ∧ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  This means  ϕ · (e1 ∧ (m ∧ Dm)) − (Dm ∧ ϕ) · (e1 ∧ m) − D(ϕ ∧ m) · (e1 ∧ m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hence,  d  dt(ϕ · (e1 ∧ m)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The conclusion comes by integration, noticing furthermore that m(0) = e1 and thus e1 ∧ m(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  20  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > 0 and (m, p) satisfy the (PMP) conditions ( (9)-(12) with p0 = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Denote  m = (m1, m2, m3) the coordinates of m, and define t0 := inf{t ≥ 0 | m(t) = ±e1} > 0 (possibly +∞)  and θ ∈ [0, π] such that m1 = cos θ on [0, t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then m1 and θ satisfy on [0, t0)  ˙m1 = α(γ2 − γ1)(1 − m2  1)m1 − U  �  1 + α2  �  1 − m2  1,  ˙θ = −α(γ2 − γ1) sin θ cos θ + U  �  1 + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (30)  Last, t0 = ∞ if U ≤ Ucrit and t0 = TU if U > Ucrit  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let p its adjoint state and ϕ = p−(p·m)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then ϕ·(e1 ∧m) = 0 from Lemma 20, which means  that ϕ is orthogonal to both m and e1 ∧ m for all times in [0, TU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Moreover, as soon as m(t) ̸= ±e1  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' as soon as t ∈ (0, TU)), (m(t), e1 ∧m(t), m(t)∧(e1 ∧m(t))) is an orthogonal basis of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore,  ϕ(t) is colinear to m(t) ∧ (e1 ∧ m(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' there is λ ∈ C ((0, TU), R) such that  ϕ = λm ∧ (e1 ∧ m) = λ  �  e1 − m1m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Moreover, from (21), we know that ϕ does not vanish, thus neither does λ, which has a constant sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then, we also have  e1 · ϕ(t) = λ(t)(1 − m2  1)  and  |ϕ(t)| = |λ(t)|  �  1 − m2  1,  which leads to  e1 · ϕ(t)  |ϕ(t)| = sign(λ)  �  1 − m2  1,  with sign(λ) = ±1 constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We also have  (Dm · m) = γ1m2  1 + γ2(m2  2 + m2  3) = γ2 − (γ2 − γ1)m2  1,  e1 · (m ∧ Dm) = Dm · (e1 ∧ m) = m · D(e1 ∧ m) = m · γ2(e1 ∧ m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, the evolution equation on m1 is  ˙m1 = α(γ2 − γ1)(1 − m2  1) m1 + sign(λ)U  �  1 + α2  �  1 − m2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (31)  On the other hand, we know that ˜m = (m2, m3) satisfies at t = 0:  ˙˜m(0) = U  �  1 + α2 ˜ϕ(0)  |ϕ(0)|,  with ˜ϕ(0) ̸= 0 since ϕ(0) · e1 = 0 and |ϕ(0)| =  1  U  √  1+α2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since ˜m(0) = (0, 0), this means that | ˜m| is  not vanishing on (0, ε] for some ε > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since |m|2 = 1, this necessarily means that m2  1 < 1  on (0, ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Now, we can introduce θ(t) the first angle of the spherical coordinate such that m1 = cos θ,  and we can assume that θ(0) = 0 and θ(t) > 0 on (0, ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The angle θ(t) is then well defined on [0, t0)  where t0 = min{t > 0 | m(t) = ±e1} and θ(t) ∈ [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Moreover, since m1 is C 1 (due to (31), for  example), θ is C 1 on (0, t0) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, on this interval, we can replace m1 in (31) by its expression  in terms of θ, which leads to  − ˙θ sin θ = α(γ2 − γ1) sin2 θ cos θ + sign(λ)U  �  1 + α2 sin θ,  hence  ˙θ = −α(γ2 − γ1) sin θ cos θ − sign(λ)U  �  1 + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  21  From this, we also see that θ is C 1 at t = 0 with ˙θ(0) = − sign(λ)U  √  1 + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' As θ ≥ 0 on [0, t0], we  can easily see that sign(λ) = −1 (otherwise we would have ˙θ(0) < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This gives the expected ODEs  on m1 and θ, but on [0, t0) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To conclude, we shall prove that t0 = ∞ if U ≤ Ucrit or t0 = TU if U > Ucrit, which is equivalent  to prove that m does not reach e1 again (up to reaching −e1 before), or equivalently that θ does not  come back to 0 before reaching π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This follows from the fact that θ satisfies an autonomous first-order  ODE of the form ˙θ = f(θ) with f(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We are now in position to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > 0 and (m, ϕ) satisfying the (PMP) conditions ((9)-(12) with p0 = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  From Lemma 19 and Theorem 2, we have 2 cases:  either U ≤ Ucrit, and then no trajectory reaches −e1 (and so in particular m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  either U > Ucrit, and then m reaches −e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, we shall analyze only the case when U > Ucrit and m is able to reach −e1 and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From Lemma  21, we can define θ(t) ∈ [0, π] such that m1 = cos θ, and it satisfies (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Since it is an autonomous ODE  of the form ˙θ = f(θ) with f(0) > 0, it is easy to prove that θ is able to reach π (which means m1 reaches  −1 or also m reaches −e1) if and only if f > 0 on [0, π], where f(x) = −α(γ2−γ1) sin x cos x+U  √  1 + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  From this,  f(x) > 0  ∀x ∈ [0, π]  ⇐⇒  1  2 sin (2x) < U  √  1 + α2  α(γ2 − γ1)  ∀x ∈ [0, π]  ⇐⇒  U >  α  2  √  1 + α2 (γ2 − γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  This gives the desired expression of Ucrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us now compute the minimal time in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' From the ODE (30) satisfied by θ for the  optimal trajectory, we know that  TU =  � π  0  dθ  −α(γ2 − γ1) sin θ cos θ + U  √  1 + α2  =  � π  0  dθ  − 1  2α(γ2 − γ1) sin 2θ + U  √  1 + α2  =  � 2π  0  dx  −α(γ2 − γ1) sin x + 2U  √  1 + α2  =  1  2  √  1 + α2  � π  −π  dx  −Ucrit sin x + U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  With the change of variable y = tan x  2, so that dx =  2 dy  1+y2 , sin x =  2y  1+y2 , we get  TU =  1  2  √  1 + α2  � +∞  −∞  2 dy  −2Ucrity + U(1 + y2)  =  1  √  1 + α2  � +∞  −∞  dy  U  �  y − Ucrit  U  �2  + U2−U2  crit  U  =  1  U  √  1 + α2  � +∞  −∞  dy  y2 + U2−U2  crit  U2  22  =  1  U  √  1 + α2  �  U 2 − U 2  crit  U 2  � +∞  −∞  dz  U2−U2  crit  U2  z2 + U2−U2  crit  U2  ,  with the change of variable y =  �  U2−U2  crit  U2  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore,  TU =  1  U  √  1 + α2  �  U 2  U 2 − U 2  crit  � +∞  −∞  dz  z2 + 1 =  π  √  1 + α2  �  U 2 − U 2  crit  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  6  Proof of Theorem 8: on almost planar trajectories  Let us first state a result based on tedious computations, whose detail is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let U > Ucrit, m be an optimal trajectory and p its adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then ζ := p ∧ m = ϕ ∧ m satisfies  ˙ζ = α  �  D(m ∧ ζ) ∧ m − (m ∧ ζ) ∧ Dm  �  + Dm ∧ ζ − Dζ ∧ m  Similarly, denote Z := ζ/|ζ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' At every point where ζ does not vanish, one has  ˙Z = PZ⊥  �  α  �  D(m ∧ Z) ∧ m − (m ∧ Z) ∧ Dm  �  + Dm ∧ Z − DZ ∧ m  �  ,  (32)  where PZ⊥ : x �→ x − (Z · x)Z is the projection onto the orthogonal space to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The proof of the Theorem 8 relies on the following result, establishing the existence of a planar  trajectory joining e1 to −e1, without any norm condition on the chosen control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Lemma 23 (Existence of planar trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' For any ε > 0, there exists a trajectory of the form  m(t) = (m1(t), m2(t), 0) defined on [0, Tε] for some Tε > 0, joining the state e1 to −e1 and such that  F :=  �  ˙e · (Dm ∧ m) +  | ˙m|  1 + α2  �2  +  �  ˙e · Dm + α| ˙m|  1 + α2  �2  ≤ 1  4(γ2 − γ1)2(1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  for all t ∈ [0, Tε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Furthermore, Tε ≲ 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proof of Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Define m1(t) = cos θ(t) and m2(t) = sin θ(t), our goal is to define a suitable  function θ, such that θ(0) = 0 and θ(Tε) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Observe that ˙e, Dm, m are coplanar so that ˙e · (Dm ∧ m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Also  | ˙m| = | ˙θ|,  ˙e · Dm = (γ2 − γ1) sin(θ) cos(θ) = γ2 − γ1  2  sin(2θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hence  F =  1  (1 + α2)2 | ˙θ|2 +  �  γ2 − γ1  2  sin(2θ) +  α| ˙θ|  1 + α2  �2  =  1  1 + α2 | ˙θ|2 + α(γ2 − γ1)  1 + α2  sin(2θ)| ˙θ| + (γ2 − γ1)2  4  sin2(2θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  This is a quadratic expression in | ˙θ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let us solve F = 1  4(γ2 − γ1)2(1 + ε): this is a polynomial equation  of degree 2, whose discriminant reads  ∆ = α2(γ2 − γ1)2  (1 + α2)2  sin2(2θ) −  4  (1 + α2)2  (γ2 − γ1)2  4  (sin2(2θ) − 1 − ε)  23  = (γ2 − γ1)2  (1 + α2)2  �  1 + ε + (α2 − 1) sin2(2θ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Observe that ∆ > 0 for all θ, so that we can choose  ˙θ = γ2 − γ1  2  �  −α sin(2θ) +  �  1 + ε + (α2 − 1) sin2(2θ)  �  =: fε(θ)  As  fε(θ) ≥ γ2 − γ1  2  ��  ε + α2 sin2(2θ) − α sin(2θ)  �  ≥  ε  √  ε + α2 + α  > 0,  we infer that this ODE on θ admits a unique solution θε, strictly increasing such that ˙θε ≳ ε, and so,  there exists a unique Tε ≲ 1/ε such that θε(Tε) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This provides the desired trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Denote Uplan = γ2−γ1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Lemma 23 shows in particular that Ucrit ≤ Uplan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We are now in position to  complete the:  Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let ζ and Z as in Lemma 22: Z satisfies (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Observe moreover that the equations  on ζ and Z remain unchanged if one replaces D into D − λI3 for some λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We can therefore assume  that the spectral norm of D is ∥D∥ = γ3−γ1  2  by taking λ = γ3+γ1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Then, since |Z| = |m| = 1 and  ∥PZ⊥∥ = 1, we get  ��� ˙Z  ��� ≤ 2(1 + α)∥D∥ = (1 + α)(γ3 − γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  On the other hand, according to Lemmas 9 and 23, we know that for all U > Uplan, there holds  TU ≤  C  U−Uplan for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Thus, one has  |Z(t) − Z(TU)| ≤ (1 + α)(γ3 − γ1)TU ≤ (1 + α)(γ3 − γ1)  C  U − Uplan  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (33)  for all t ∈ [0, TU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='We also know that |ζ| = |ϕ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Thus, by introducing ψ = ϕ/|ϕ|, one gets Z = ψ ∧ m  and m ∧ Z = ψ since ϕ · m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' A straightforward computation yields that the pair (m, ψ) satisfies  ˙ψ = α(Dψ − (Dψ · ψ)ψ) − U  �  1 + α2m − Dm ∧ ψ + D(m ∧ ψ)  − (ψ · D(m ∧ ψ)) ψ − ((m ∧ ψ) · Dm) m,  ˙m = −α (Dm − (Dm · m)m) + m ∧ Dm + U  �  1 + α2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (34)  From estimate (33), we infer that, for U large enough,  ∀t ∈ [0, TU],  |ψ(t) − m ∧ Z(TU)| ≤ C  U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Putting this in Equation (34) for m, we get some constant C > 0 such that for all U large enough and  t ∈ [0, TU],  ��� ˙m − U  �  1 + α2m ∧ Z(TU)  ��� ≤ C,  which leads to  | ˙m · Z(TU)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Since m(TU) · Z(TU) = 0 by definition and using once again that TU ≤  C  U−Uplan , we get for all U large  enough and t ∈ [0, TU]  |m · Z(TU)| ≤ C  U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  However, p(TU) = ϕ(TU) (since p(TU)·m(TU) = 0 from the orthogonality condition) and m(TU) = −e1,  and thus the orthogonal space of V is exactly span(Z(TU)), which means that m(t) − PV m(t) =  (m · Z(TU))Z(TU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The conclusion easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  24  7  Conclusion and perspectives  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Extension of our results  It would be natural to extend our study in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' On the one hand, we would like to  complete our study of a single ferromagnetic particle of ellipsoidal shape by studying other criteria,  and typically a combination of time and cost L2 of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This problem could read:  Second version of the optimal control problem: case of L2 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let λ > 0  and let us assume that m0 = e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The problem reads  Eλ  U =  inf  (T,u)∈OU  T + λ  2  � T  0  |u(t)|2 dt,  (Pλ)  where mu denotes the solution to (3) associated to the control function u(·),  or alternatively, if one aims at dropping the effect of the L∞ constraint on the control,  Modified second version of the optimal control problem: case of L2 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let λ > 0 and let us assume that m0 = e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The problem reads  Eλ  U =  inf  (T,u)∈�  U≥0 OU  T + λ  2  � T  0  |u(t)|2 dt,  (Pλ)  where mu denotes the solution to (3) associated to the control function u(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Finally, we also plan to study similar issues for more realistic physical systems, for example a  network of ellipsoidal particles, possibly rectilinear, as in the model introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Numerical illustrations of our results  We provide hereafter several numerical illustrations of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' More precisely, we want to determine  numerically the existence or not of an admissible trajectory connecting e1 to −e1, in accordance with  what we have found theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Let us first notice that a trajectory m can easily be computed  numerically by solving the ODE (3) with the expression (22) for the control u where the variable ϕ is  given by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  To initialize both ODE (3) and (14), m(0) = e1 is given, but ϕ(0) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  On the one  hand, we overcome this difficulty by noticing that ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e1 = 0, which allows us to have only two  unknowns: ϕ2(0) and ϕ3(0) to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  On the other hand, working with the normalized  variable ψ = ϕ/|ϕ| enables us to reduce the unknowns to only one angle variable ϑ ∈ [0, 2π] such that  (ψ2(0), ψ3(0)) = (cos(ϑ), sin(ϑ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' ODE (3) and (14) are thus replaced by the system (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Numerically, implement a shooting method to determine ϑ ∈ [0, 2π]: namely, for each ϑ, we solve  the system (34) on a very large time horizon by a fourth-order Runge-Kutta method and determine if  the trajectory m reaches −e1 on a certain time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We list below the numerical results, all obtained with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The initial position e1 is represented  with a red circle on the sphere and the goal −e1 with a green star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Parameters γi, ϑ and control U  are specified in the caption of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' For each one, we have represented the trajectory m on the  sphere as well as the coordinates of m and of the control u as functions of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  First of all, in the non-symmetric case, a threshold on the control appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' If the control is suffi-  ciently large, there is (at least) an initialization of ψ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e at least one angle ϑ) which allows to have an  admissible trajectory represented in Subfigures 2(a)-2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' On the contrary, if the control is not large  enough, no initialization of ψ will give an admissible trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We have represented for instance one  of them in Subfigure 2(e)-2(f) with a particular ϑ but be aware that they all have the same behavior  25  (a) Admissible trajectory for ϑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='8976,  large control U = 10,  (b) The components of m and u for ϑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='8976, large control U = 10  (c) Admissible trajectory for ϑ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2440,  medium control U = 3  (d) The components of m and u for ϑ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2440, medium control U = 3  (e) Generic trajectory for ϑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='8976,  small control U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  (f) The components of m and u for ϑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='8976, small control U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='1  Figure 2: Non-symmetric test case with (γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' γ3) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content=' 1), top: with a large control U = 10,  middle: with a medium control U = 3 and bottom: with a small control U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='4  m3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='3  U3  0  20  40  20  40  0  20  40  time  time  timewhatever the initialization of ψ: the trajectory remains in the northern half-sphere without enough  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Figure 2 is thus a perfect illustration of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Note that it also helps to illustrate  Theorem 8 since the larger U is, the closer the trajectory is to a planar trajectory, as we can see by  comparing Subfigures 2(c)-2(d) with a medium control and Subfigures 2(a)-2(b) with a larger control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  (a) Generic trajectory  (b) The components of m and u  Figure 3: Non-symmetric test case with (γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' γ3) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' 1), ϑ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='5646 and a small control  U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='2  Figure 3 illustrates once again the case of control too weak to reach −e1, for other γi parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The symmetric case γ1 = γ2 is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Even for small controls (U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='7 numerically),  there is (at least) one initialization of ψ leading to an admissible trajectory reaching −e1 in finite  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This illustrates well Theorem 4: Ucrit = 0 in this symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' When γ2 < γ3 (Subfigures  4(a)-4(b)), the admissible trajectories are non planar whereas it is, in the case of a spherical symmetry  (Subfigures 4(c)-4(d)) without changing anything other than the symmetry of the test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' This is  again in accordance with the second statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  For the symmetric case γ2 = γ3, we see numerically in Figure 5 that for small values of U, an  admissible trajectory exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' With the parameters of Figure 5, Theorem 5 gives the following value  for Ucrit =  α  2  √  1+α2 (γ2 − γ1) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='026, which effectively allows to have admissible trajectories for very  small values of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Note also in Subfigure 5(b) that all admissible trajectories reach the target −e1 in  a time greater than 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' With the values chosen for Figure 5,  π  √  1+α2√  U2−U2  crit  ≃ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='58 corresponds to  the minimum time determined in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Here again, we notice the non-planar character of the  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='3  Conclusion and perspectives  The obtained results provide a complete characterization of the question of the magnetic moment  reversal in minimal time in a simple configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Indeed, we have considered here only one ellipsoidal  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In order to approach more realistic configurations, we wish to analyze a model in which several  ferromagnetic particles of ellipsoidal shape are combined to form a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We refer for example to  [2] for a possible model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' After having characterized the set of stationary configurations, we will then  ask ourselves the question of controllability in minimal time, in order to go from one stationary state  to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='00  ZAcknowledgements  The authors were partially supported by the ANR Project MOSICOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The last author were partially  supported by the ANR Project TRECOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  A  Computation of the demagnetizing field in a ferromagnetic ellipsoid  sample  It is shown in [14, 8] that the demagnetizing tensor D reads D = diag([γ1, γ2, γ3]), where the γi’s are  given by  γi = a1a2a3  2  � +∞  0  dt  �  (a1 + t2)(a2  2 + t2)(a2  3 + t2)(a2  i + t2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Such an expression can be rewritten in terms of the elliptic integral of the second kind E, defined by  E(x, p) =  � x  0  (1 − p sin2 θ)1/2 dθ,  x ∈ R, p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  If a1 ≥ a2 ≥ a3, then, one has 0 ≤ γ1 ≤ γ2 ≤ γ3 ≤ 1 and these coefficients read  γ1 = 1 − γ2 − γ3  γ2 = −  a3  a2  2 − a2  3  �  a3 −  a1a2  (a2  1 − a2  2)1/2 E  �a2  a1  , a2  1 − a2  3  a2  1 − a2  2  ��  γ3 =  a2  a2  2 − a2  3  �  a2 −  a1a3  (a2  2 − a2  3)1/2 E  �a3  a1  , a2  1 − a2  2  a2  1 − a2  3  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In the case where a1 ≥ a2 = a3 (prolate spheroid), these formula simplify into  γ1 = −  a2  3  (a2  1 − a2  3)3/2  �  (a2  1 − a2  3)1/2 + a1 argcoth  �  a1  (a2  1 − a2  3)1/2  ��  ,  γ2 = γ3 = 1 − γ1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  In the case where a1 = a2 ≥ a3 (oblate spheroid), these formula simplify into  γ3 = −  a2  1  (a2  1 − a2  3)3/2  �  (a2  1 − a2  3)1/2 + a3 arctan  �  a3  (a2  1 − a2  3)1/2  �  − π  2 a3  �  ,  γ1 = γ2 = 1 − γ3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  B  Stability of steady-states for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' (3)  Let us first notice that, if ¯m is a steady-state of Equation (3) and if no control is applied on this system  (hext = 0), then by orthogonality of the terms in the right-hand side, it satisfies  h0( ¯m) = (h0( ¯m) · ¯m) ¯m  and  ¯m ∧ h0( ¯m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Therefore, ¯m is an eigenfunction of D ¯m and we infer that ¯m = ±ej, j = 1, 2, 3 whenever γ1 < γ2 ≤ γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Proposition 24 (Asymptotic stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let γ1, γ2, γ3 be sorted in ascending order γ1 < γ2 ≤ γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Then, in the absence of any control hext, the steady-state ±e1 is an asymptotically stable equilibrium  state for Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Nevertheless, the steady-states ±e2 and ±e3 are linearly unstable steady-states  for Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  29  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Let h = (h1, h2, h3)T ∈ R3 be a small perturbation such that e1 + h is still an admissible  magnetization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' on the unit sphere S2 ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We obtain:  ∥e1 + h∥2 = 1 ⇔ (1 + h1)2 + h2  2 + h2  3 = 1 ⇔ h1 = −1  2  �  h2  1 + h2  2 + h3  3  �  = O(∥h∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  The unknown h1 is therefore of second order and does not occur in a linearized system of the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  By linearizing Equation (3) around the equilibrium state e1, one has, without any control u:  � ˙h2 = α(γ1 − γ2)h2 + (γ1 − γ3)h3 + O(∥h∥2)  ˙h3 = α(γ1 − γ3)h3 + (γ2 − γ1)h2 + O(∥h∥2)  (35)  The Jacobian matrix of the linearized system around e1 is therefore :  J =  �α(γ1 − γ2)  γ1 − γ3  γ2 − γ1  α(γ1 − γ3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Since γ1 < γ2 ≤ γ3, one has  det(J) = (α2 + 1)(γ1 − γ2)(γ1 − γ3) > 0  and  Tr(J) = α [(γ1 − γ2) + (γ1 − γ3)] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  We infer that the two eigenvalues of the Jacobian matrix are of negative real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The steady state  e1 is therefore linearly stable and is an hyperbolic point (no eigenvalue with zero real part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Hartman Grobman’s theorem [16] allows to conclude about the asymptotic stability of e1 for the  non-linear Equation (3) without any control u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' As for −e1, similar computations give the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Regarding now the stability of ±ek, k = 2, 3, notice that a similar computation drives to the following  expression of the Jacobian determinant: det J = (α2+1)(γ2−γ1)(γ2−γ3) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' The expected conclusion  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' If γ1 ≤ γ2 ≤ γ3, an eigenvalue of the Jacobian matrix may have a zero real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In which  case one can conclude that e1 is linearly (non-asymptotically) stable, but Hartman Grobman’s theorem  no longer applies to return to the non-linear Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  C  Complement in the case γ1 < γ2: explicit computations of the con-  stants in the case  Let us use the notations introduced in Remark 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' We compute  A∗A =  �(1 + α2)δγ2  −  −2αδγ− δγ+  −2αδγ− δγ+  (1 + α2)δγ2  +  �  ,  Tr(A∗A) = (1 + α2)(δγ2  − + δγ2  +) > 0,  det(A∗A) = (1 − α2)2δγ2  −δγ2  + ≥ 0,  and the discriminant of its characteristic polynomial is  Tr(A∗A)2 − 4 det(A∗A) = (1 + α2)2(δγ2  − + δγ2  +)2 − 4(1 − α2)2δγ2  −δγ2  +  = (1 + α2)2(δγ2  − − δγ2  +)2 + 16α2δγ2  −δγ2  +  = (1 + α2)2(δγ− − δγ+)2(δγ− + δγ+)2 + 16α2δγ2  −δγ2  + > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  and its largest eigenvalue is therefore  ∥A∥2  2 =  (1 + α2)(δγ2  − + δγ2  +) +  �  (1 + α2)2(δγ− − δγ+)2(δγ− + δγ+)2 + 16α2δγ2  −δγ2  +  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  30  On the other hand, when ∆ = α2(δγ+ − δγ−)2 − δγ−δγ+ ≥ 0, we have  λ+ = −α(δγ+ + δγ−) +  �  α2(δγ+ − δγ−)2 − δγ−δγ+  2  ,  and therefore, with Γ = δγ−1  + δγ−,  |λ+|2  ∥A∥2(1 + |α|)δγ+  =  1  √  2(1 + |α|)  �  α(1 + Γ) −  �  α2(1 − Γ)2 − Γ  �2  �  (1 + α2)(1 + Γ2) +  �  (1 + α2)(Γ − 1)2(1 + Γ)2 + 16α2Γ2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Similarly, when ∆ < 0, we obtain  |Tr A|2  ∥A∥2(1 + |α|)δγ+  =  1  √  2(1 + |α|)  α(1 + Γ)  �  (1 + α2)(1 + Γ2) +  �  (1 + α2)(Γ − 1)2(1 + Γ)2 + 16α2Γ2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Remark also that ∆ = δγ2  +  �  α2(1 − Γ)2 − Γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Therefore, if we define  ˜x0 :=  �  �  �  |λ+|2  ∥A∥2(1+|α|)δγ+ ,  if ∆ ≥ 0,  Tr(A)2  4∥A∥2(1+|α|)δγ+ ,  if ∆ < 0,  and then ˜x1 := min  �  1,  √˜x0  3  �  , we obtain that both of them only depend on Γ and α and thus so is  µ0 = x0  3 x1 − x3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Last, the conditions on U becomes U ≤ δγ+ µ0, which is in agreement with the  invariances on D (invariance by shifting of the γis, invariance by multiplication of the γis with respect  to a change of time variable and a multiplication of the external field-control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content=' ESAIM Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+page_content='  Ellipsoids (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' 0): 3-D magnetic modelling of ellipsoidal bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Geoscientific Model Development, 10(9):3591–3608, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  [20] Augusto Visintin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Mathematical models of hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' In Nonlinear partial differential equations and their  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Collège de France Seminar, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' XIII (Paris, 1994/1996), volume 391 of Pitman Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Notes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=', pages 327–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Longman, Harlow, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  [21] Baisheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' On energy-minimization in ferromagnetism controlled by applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' Pura Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content=' (4),  192(1):115–125, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
+page_content='  32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE2T4oBgHgl3EQfXAc0/content/2301.03839v1.pdf'}
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+ASYMPTOTIC EXPANSIONS RELATING TO THE DISTRIBUTION OF
+THE LENGTH OF LONGEST INCREASING SUBSEQUENCES
+FOLKMAR BORNEMANN
+Abstract. We study the distribution of the length of longest increasing subsequences in
+random permutations of n integers as n grows large and establish an asymptotic expansion
+in powers of n−1/3. Whilst the limit distribution was already shown by Baik, Deift and
+Johansson to be the GUE Tracy–Widom distribution F, we find explicit analytic expressions
+of the first few finite-size correction terms as linear combinations of higher order derivatives
+of F with rational polynomial coefficients. Our proof replaces Johansson’s de-Poissonization,
+which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of
+Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is
+subject to a tameness hypothesis concerning complex zeros of the analytically continued
+Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft
+edge transition law of LUE is studied, which is then transformed into an expansion of the
+Poissonized length distribution for large intensities. Finally, expansions of Stirling-type
+approximations and of the expected value and variance of the length distribution are given.
+1. Introduction
+The length Ln(σ) of longest increasing subsequences1 of permutations σ on {1, 2, . . . , n}
+becomes a discrete random variable when the permutations are drawn randomly with uniform
+distribution. This way the problem of enumerating all permutations σ that satisfy Ln(σ) ⩽ l
+gets encoded in the discrete probability distribution P(Ln ⩽ l). The present paper studies an
+asymptotic expansion of this distribution when n grows large. As there are relations to KPZ
+growth models (directly so for the PNG model with droplet initial condition, see [28, 54, 55]
+and [31, Chap. 10]), we expect our findings to have a bearing there, too.
+Prior work. We start by recalling some fundamental results and notions. More details and
+references can be found in the outstanding surveys and monographs [3, 10, 57, 61].
+Ulam’s problem. The study of the behavior as n grows large dates back to Ulam [66] in 1961,
+who mentioned that Monte-Carlo computations of Neighbor would indicate E(Ln) ≈ 1.7√n.
+Ulam continued by asking: “Another question of interest would be to find the distribution of
+the length of the maximum monotone subsequence around this average.”
+Refined numerical experiments by Baer and Brock [5] in 1968 suggested that E(Ln) ∼ 2√n
+might be the precise leading order. In a 1970 lecture, Hammersley [36] presented a proof,
+based on subadditive ergodic theory, that the limit c = limn→∞ E(Ln)/√n exists. Finally, in
+1977, Vershik and Kerov [68] as well as Logan and Shepp [45] succeeded in proving c = 2.
+Poissonization. A major tool used by Hammersley was a random process that is, basically,
+equivalent to the following Poissonization of the random variable Ln: by drawing from the
+different permutation groups independently and by taking Nr ∈ {0, 1, 2, . . .} to be a further
+independent random variable with a Poisson distribution of intensity r > 0, the combined
+random variable LNr is distributed according to
+P(LNr ⩽ l) = e−r
+∞
+�
+n=0
+P(Ln ⩽ l)rn
+n! =: P(r; l).
+2010 Mathematics Subject Classification. 05A16, 60B20, 30D15, 30E15, 33C10.
+Key words and phrases. random permutations, random matrices, asymptotics, analytic de-Poissonization.
+1Defined as the maximum of all k for which there are 1 ⩽ i1 < i2 < · · · < ik ⩽ n with σi1 < σi2 < · · · < σik.
+1
+arXiv:2301.02022v1  [math.PR]  5 Jan 2023
+
+2
+FOLKMAR BORNEMANN
+The entire function2 P(z; l) is the Poisson generating function of the sequence P(Ln ⩽ l)
+(n = 0, 1, . . .) and f(z; l) := ezP(z; l) is the corresponding exponential generating function.
+As it turns out, it is much simpler to analyze the Poissonized distribution of LNr as the
+intensity r grows large than the original distribution of Ln as n grows large.
+There is, however, also a way back from LNr to Ln: namely, the expected value of the
+Poisson distribution being E(Nr) = r, combined with some level of concentration, suggests
+P(Ln ⩽ l) ≈ P(n; l)
+when n → ∞ while l is kept near the mode of the distribution. Being a Tauberian result, such
+a de-Poissonization is subject to additional conditions, which we will discuss in a moment.
+Starting in the early 1990s the Poisson generating function P(z; l) (or the exponential one
+to the same end) has been represented in terms of one of the following interrelated forms:
+• a Toeplitz determinant in terms of modified Bessel functions [35],
+• Fredholm determinants of various (discrete) integral operators [8, 9, 18, 19, 42],
+• a unitary group integral [56].
+A particular case of those representations plays a central role in our study: namely3
+(1)
+P(r; l) = Ehard
+2
+(4r; l),
+where4 Ehard
+2
+(s; ν) denotes the probability that, in the hard-edge scaling limit, the scaled
+smallest eigenvalue of the Laguerre unitary ensemble (LUE) with real parameter ν > 0 is
+bounded from below by s ⩾ 0. This probability is known to be given in terms of a Fredholm
+determinant (see [30]):
+(2)
+Ehard
+2
+(s; ν) = det(I − KBessel
+ν
+)
+��
+L2(0,s),
+where Kν denotes the Bessel kernel in x, y ⩾ 0 (for the integral formula see [65, Eq. (2.2)]):
+(3) KBessel
+ν
+(x, y) := Jν(√x)√yJ′
+ν(√y) − Jν(√y)√xJ′
+ν(√x)
+2(x − y)
+= 1
+4
+� 1
+0
+Jν(√σx)Jν(√σy) dσ.
+Obviously, the singularities at the diagonal x = y are removable.
+The work of Tracy and Widom [65] establishes that the Fredholm determinant (2) can
+be expressed in terms of Painlevé III. Recently, based on Okamoto’s Hamiltonian σ-PIII′
+framework, Forrester and Mays [33] used that connection to compile a table of the exact
+rational values of P(Ln ⩽ l) for up to n = 700;5 whereas in our work [16], based on an
+equivalent representation in terms of a Chazy I equation, we have compiled such a table6 for
+up to n = 1000.
+In their seminal 1999 work [7], by relating the representation of P(z; l) in terms of the
+Toeplitz determinant to the machinery of Riemann–Hilbert problems and studying the
+underlying double-scaling limit by the Deift–Zhou method of steepest descent, Baik, Deift
+and Johansson answered Ulam’s question and proved that, for t being any fixed real number,
+(4)
+lim
+r→∞ P
+�LNr − 2√r
+r1/6
+⩽ t
+�
+= F(t),
+where F is the GUE Tracy–Widom distribution: that is, the distribution which expresses,
+among many other limit laws, the probability that in the soft-edge scaling limit of the Gaussian
+unitary ensemble (GUE) the scaled largest eigenvalue is bounded from above by t. As for the
+2Throughout the paper we will use n as an integer n ⩾ 0, r as a corresponding real variable r > 0 (intensity)
+and z as its continuation into the complex plane.
+3A derivation from the group integral is found in [18, §2] and from the Toeplitz determinant in [32, Eq. (3.33)].
+4Throughout the paper, we will use l as an integer l ⩾ 0 and ν as a corresponding real variable ν > 0, which
+is used whenever an expression of l generalizes to non-integer arguments.
+5Previously, by combinatorial means, Baer and Brock [5] had compiled a table for up to n = 36, supplemented
+later by Odlyzko and Rains [47, 48] with the cases n = 60, 90, 120. The cases n = 30, 60, 90 got printed in [46].
+6Available for download at https://box-m3.ma.tum.de/f/7c4f8cb22f5d425f8cff/.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+3
+Poissonized length distribution itself, the Tracy–Widom distribution can be represented in
+terms of a Fredholm determinant (see [30]): namely
+(5)
+F(t) = det(I − K0)|L2(t,∞)
+where K0 denotes the Airy kernel in x, y ∈ R (for the integral formula see [64, Eq. (4.5)]):
+(6)
+K0(x, y) := Ai(x) Ai′(y) − Ai′(x) Ai(y)
+x − y
+=
+� ∞
+0
+Ai(x + σ) Ai(y + σ) dσ.
+Obviously, also in this case the singularities at the diagonal x = y are removable. Since the
+limit distribution in (4) is continuous, by a standard Tauberian follow-up [67, Lemma 2.1] of
+the Portmanteau theorem the limit law holds uniformly in t.
+In 2003, Borodin and Forrester gave an alternative proof of (4) which is based on studying
+the hard-to-soft edge transition of LUE for ν → ∞ in form of the limit law [18, Thm. 1]
+(7)
+lim
+ν→∞ Ehard
+2
+��
+ν − t(ν/2)1/3�2; ν
+�
+= F(t)
+(see also [31, §10.8.4]), which will be the starting point of our study. Still, there are other
+proofs of (4) based on representations in terms of Fredholm determinants of further (discrete)
+integral operators; for expositions and references see the monographs [10, 57].
+De-Poissonization. In the literature, the de-Poissonization of the limit law (4) has so far
+been based exclusively on variants of the following lemma (cf. [10, Cor. 2.5], originally stated
+as [41, Lemma 2.5]), which uses monotonicity as the underlying Tauberian condition.
+Lemma 1.1 (Johansson’s de-Poissonization lemma [41]). Suppose the sequence an of proba-
+bilities 0 ⩽ an ⩽ 1 satisfies the monotonicity condition an+1 ⩽ an for all n = 0, 1, 2, . . . and
+denote its Poisson generating function by
+(8)
+P(z) = e−z
+∞
+�
+n=0
+an
+zn
+n! .
+Then, for s ⩾ 1 and n ⩾ 2 :7 P
+�
+n + 2√sn log n
+�
+− n−s ⩽ an ⩽ P
+�
+n − 2√sn log n
+�
++ n−s.
+After establishing the Tauberian condition of monotonicity and applying a variant of
+Lemma 1.1 to (4), Baik, Deift and Johansson [7, Thm. 1.1] got
+(9)
+lim
+n→∞ P
+�Ln − 2√n
+n1/6
+⩽ t
+�
+= F(t),
+which holds uniformly in t for the same reasons as given above. (The simple calculations
+based on Lemma 1.1 are given in [10, p. 239]; note that the uniformity of the limit law (4) is
+used there without explicitly saying so.) Adding tail estimates to the picture, those authors
+were also able to lift the limit law to the moments and got, expanding on Ulam’s problem,
+that the expected value satisfies [7, Thm. 1.2]
+(10)
+E(Ln) = 2√n + M1n1/6 + o(n1/6),
+M1 =
+� ∞
+−∞
+tF ′(t) dt ≈ −1.7711.
+Expansions. To our knowledge, only for the Poissonized limit law (4) a finite-size correction
+term has been rigorously established prior to the present paper:8 namely, as a by-product
+along the way of their study of the limiting distribution of maximal crossings and nestings
+of Poissonized random matchings, Baik and Jenkins [11, Thm. 1.3] obtained (using the
+7Note the trade-off between sharper error terms ∓n−s and less sharp perturbations of n by ±2√sn log n.
+8Expansions of probability distributions are sometimes called Edgeworth expansions in reference to the
+classical one for the central limit theorem. In random matrix theory quite a variety of such expansions, or at
+least some precise estimates of convergence rates, have been studied: e.g., for the soft-edge scaling limits of
+the Gauss and Laguerre ensembles [21, 22, 27, 44] and of the Jacobi ensembles [43], for the hard-edge scaling
+limit of the Laguerre ensembles [15, 26, 34, 53], for the bulk scaling limit of the circular ensembles [17], and
+for various joint probability distributions [1, 11, 12, 28, 59, 70].
+
+4
+FOLKMAR BORNEMANN
+machinery of Riemann–Hilbert problems and Painlevé representations of the Tracy–Widom
+distribution), as r → ∞ with t being any fixed real number,
+(11a)
+P
+�LNr − 2√r
+r1/6
+⩽ t
+�
+= F
+�
+t(r)�
+− 1
+10
+�
+F ′′(t) + t2
+6 F ′(t)
+�
+r−1/3 + O(r−1/2),
+where (with ⌊·⌋ denoting the Gauss bracket)
+(11b)
+t(r) := ⌊2√r + tr1/6⌋ − 2√r
+r1/6
+.
+However, even if there is enough uniformity in this result and the option to Taylor expand the
+Poisson generating function P(r) at n with a uniform bound while l is kept near the mode
+of the distributions (see Sect. 5.1 for details on this option), the sandwiching in Johansson’s
+de-Poissonization Lemma 1.1 does not allow us to obtain a result better than (cf. [11, §9])
+(12)
+P
+�Ln − 2√n
+n1/6
+⩽ t
+�
+= F
+�
+t(n)�
++ O
+�
+n−1/6 log n
+�
+.
+In their recent study of finite-size effects, Forrester and Mays [33, Prop. 1.1] gave a different
+proof of (11) based on the Bessel kernel determinant (2). Moreover, suggested by exact data
+for n = 700 and a Monte-Carlo simulation for n = 20 000 they were led to conjecture [33,
+Conj. 4.2]
+(13)
+P
+�Ln − 2√n
+n1/6
+⩽ t
+�
+= F
+�
+t(n)�
++ F D
+1 (t)n−1/3 + · · ·
+with the approximate graphical form of F D
+1 (t) displayed in [33, Fig. 7].
+The presence of the Gauss bracket in t(r) and t(n), while keeping t at other places of the
+expansions (11) and (13), causes undesirable effects in the error terms (see Remark 4.1 below
+for a detailed discussion). Therefore, in our work [16] on a Stirling-type formula approximating
+the distribution P(Ln ⩽ l), we suggested to use the integer l in the continuous expansion
+terms instead of introducing the continuous variable t into the discrete distribution in the
+first place, with the latter variant turning the discrete distribution into a piecewise constant
+function of t. By introducing the scaling
+tν(r) := ν − 2√r
+r1/6
+(r > 0)
+we were led (based on numerical experiments using the Stirling-type approximation for n
+getting as large as 1010), to conjecture the expansion
+P(Ln ⩽ l) = F(t) + F D
+1 (t) n−1/3 + F D
+2 (t) n−2/3 + O(n−1)
+���
+t=tl(n),
+displaying the graphical form of the functions F D
+1 (t), F D
+2 (t) in the left panels of [16, Figs. 4/6].
+Moreover, as a note added in proof (see [16, Eq. (11)]), we announced that inserting the Baik–
+Jenkins expansion (11) into the Stirling-type formula and using its (numerically observed)
+apparent order O(n−2/3) of approximation would yield the functional form of F D
+1 to be
+(14)
+F D
+1 (t) = − 1
+10
+�
+6F ′′(t) + t2
+6 F ′(t)
+�
+.
+The quest for a proof, and for a similar expression for F D
+2 , motivated our present work.
+The new findings of the paper. In the analysis of algorithms in theoretical computer
+science, or the enumeration of combinatorial structures to the same end, the original enumer-
+ation problem is often represented in form of recurrences or functional/differential equations.
+For instance, this situation arises in a large class of algorithms involving a splitting process,
+trees, or hashing. Embedding such processes into a Poisson process9 often leads to more
+tractable equations, so that sharp tools for a subsequent de-Poissonization were developed in
+9As a heuristic principle in probability and combinatorics, Poissonization was popularized by Aldous’ book [2].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+5
+the 1990s; for references and details see [39, 40, 63] and Appendix A.1. In particular, if the
+Poisson generating function P(z) of a sequence of real an > 0, as defined in (8), is an entire
+function, an application of the saddle point method to the Cauchy integral
+an = n!
+2πi
+�
+P(z)ez dz
+zn+1
+yields, under suitable growth conditions on P(z) as z → ∞ in the complex plane, the Jasz10
+expansion
+an ∼ P(n) +
+∞
+�
+j=2
+bj(n)P (j)(n),
+where the polynomial coefficients bj(n) are the diagonal Poisson–Charlier polynomials (that
+is, with intensity r = n). In Appendix A.1 we give a heuristic derivation of that expansion
+and recall, in the detailed form of Thm. A.1, a specific analytic de-Poissonization result from
+the comprehensive memoir [39] of Jacquet and Szpankowski—a result which applies to a
+family of Poisson generating functions at once, providing uniform error bounds.
+Now, the difficult part of applying Thm. A.1 is checking the Tauberian growth conditions in
+the complex plane, which are required to hold uniformly for the family of Poisson generating
+functions (recall that, in the case of the longest increasing subsequence problem, P(z) = P(z; l)
+depends on the integer l near the mode of the length distribution). After observing a
+striking similarity of those growth conditions with the notion of H-admissibility for the
+corresponding exponential generating function (as introduced by Hayman in his memoir [37]
+on the generalization of Stirling’s formula), a closer look at the proof of Hayman’s [37,
+Thm. XI] revealed the following result (see Thm. A.3 for a precise quantitative statement):
+The family of all entire functions of genus zero which have, for some ϵ > 0,
+no zeros in the sector |arg z| ⩽ π/2 + ϵ satisfies a universal bound that implies
+Tauberian growth conditions suitable for analytic de-Poissonization.
+On the other hand, in our work [16, Thm. 2.2] on Stirling-type formulae for the problem
+of longest increasing subsequences, when proving the H-admissibility of the exponential
+generating functions f(z; l) = ezP(z; l) (for each l), we had established, based on the repre-
+sentation [56] of P(z; l) as a group integral:
+For any integer l ⩾ 0 and any δ > 0, the exponential generating function
+f(z; l) is an entire function of genus zero having at most finitely many zeros
+in the sector |arg z| ⩽ π − δ, none of them being real.
+Under the reasonable assumption (supported by numerical experiments) that those finitely
+many complex zeros do not come too close to the real axis and do not grow too fast as
+n → ∞ while l stays near the mode of the length distribution, the uniformity of the Tauberian
+growth conditions can be preserved (see Corollary A.1 for the technical details). We call this
+assumption the tameness hypothesis11 concerning the zeros of the family of P(z; l).
+Subject to the tameness hypothesis, the main result of the present paper, Thm. 5.1, gives
+the asymptotic expansion
+P(Ln ⩽ l) = F(t) +
+m
+�
+j=1
+F D
+j (t) n−j/3 + O
+�
+n−(m+1)/3� ����
+t=tl(n)
+,
+which is uniformly valid when n, l → ∞ while tl(n) stays bounded.12 Here, the F D
+j
+are
+smooth functions and, for technical reasons that are explained in Section 3, the integer m ⩾ 0
+cannot be chosen freely but has to be limited to the range m ⩽ m∗ = 23. We conjecture that
+this restriction is artificial and can be removed.
+10Dubbed so in [29, §VIII.18] to compliment the seminal work of Jacquet and Szpankowski [39].
+11Proving it seems to be rather difficult, though—at least we were lacking the methodology to do so.
+12This is meant, in fact, when we say that l stays near the mode of the length distribution.
+
+6
+FOLKMAR BORNEMANN
+Finally, now without any detour via the Stirling-type formula, Thm. 5.1 confirms that the
+expansion term F D
+1 is given by (14) and yields the striking formula (see (110) for F D
+3 (t))
+F D
+2 (t) =
+�
+− 139
+350 + 2t3
+1575
+�
+F ′(t) +
+�
+− 43t
+350 +
+t4
+7200
+�
+F ′′(t) + t2
+100F ′′′(t) + 9
+50F (4)(t).
+Organization of the paper. In Sect. 2 we start with a careful discussion of expansions of
+perturbed Airy kernel determinants. We stress the importance of such kernel expansions to
+be differentiable (i.e., one can differentiate into the error term) to easily lift the error bounds
+to trace norms. The subtle, but fundamental difficulty of such a lift seems to have been
+missed, more often than not, in the existing literature on convergence rates and expansions
+of limit laws in random matrix theory (notable exceptions are, e.g., [27, 43, 44]).
+In the rather lengthy Sect. 3 we study the asymptotic expansion of the Borodin–Forrester
+hard-to-edge transition law (7). It is based on a uniform version of Olver’s asymptotic
+expansion of Bessel functions of large order in the transition region, which we discuss in
+Appendix A.3. In Sect. 3 we lay the foundational work for the concrete functional form of all
+subsequent finite-size correction terms. We reduce the complexity of computing these terms
+by using a coordinate transform on the level of kernels to simplify the kernel expansion—a
+coordinate transform which gets subsequently reversed on the level of the distributions.13 As
+yet another application of that technique we simplify in Sect. 3.4 the finite-size correction
+term of Choup [21] to the soft-edge limit law of GUE and LUE.
+In Sect. 4 we expand the Poissonized length distribution, thereby generalizing the result (12)
+of Baik and Jenkins. In Sect. 4 we also discuss the potentially detrimental effect of using
+Gauss brackets, as in (12), alongside with the continuous variable t in the expansion terms.
+In Sect. 5 we state and prove the main result of the paper: the expansion of the Baik–Deift–
+Johansson limit law (9) of the length distribution. Here we use the Jasz expansion of analytic
+de-Poissonization (as detailed in Appendix A.1). The universal bounds for entire functions of
+genus zero, which are used to prove the Tauberian growth conditions in the complex plane, and
+their relation to the theory of H-admissibility are prepared for in Appendix A.2. Additionally,
+in Sect. 5 we discuss the modifications that apply to the discrete density P(Ln = l) (that is,
+to the PDF of the length distribution).
+In Sect. 6 we apply our findings to the asymptotic expansion of the Stirling-type formula
+which we introduced in [16] as an accurate tool for the numerical approximation of the length
+distribution. Subject to the tameness hypothesis, we prove the observation [16, Eq. (8b)]
+about a leading O(n−2/3) error of that formula.
+Finally, in Sect. 7 we study the asymptotic expansion of the expected value and variance.
+Based on a reasonable hypothesis about some uniformity in the tail bounds, we add several
+more concrete expansion terms to the Baik–Deift–Johansson solution (10) of Ulam’s problem.
+2. Expansions of perturbed Airy kernel determinants
+In Sect. 3 we will establish, with m being some non-negative integer and t0 some real
+number, kernel expansions of the form
+(15) Kh(x, y) = K0(x, y)+
+m
+�
+j=1
+hjKj(x, y)+hm+1Rm(x, y; h), Rm+1,h(x, y) = O
+�
+e−(x+y)�
+,
+which are
+• uniform for x, y ⩾ t0 as h → 0+;
+• repeatedly differentiable w.r.t. x, y as uniform expansions under the same conditions.
+Here, Kh is a family of smooth kernels, K0 denotes the Airy kernel (6) and the Kj(x, y) are
+finite sums of rank one kernels u(x)v(y), with factors u(ξ), v(ξ) of the functional form
+(16)
+p(ξ) Ai(ξ)
+or
+p(ξ) Ai′(ξ),
+13In [15] we applied a similar transformation “trick” to the expansion of the hard-edge limit law of LUE.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+7
+where p(ξ) is a polynomial in ξ. Since the existing literature tends to neglect the issue of
+estimating trace norms in terms of kernel bounds, this section aims at establishing a relatively
+easy framework for lifting such an expansion to one of the Fredholm determinant.
+2.1. Bounds on the kernels and induced trace class operators. Bounds on the kernels
+and on the trace norms of the induced integral operators can be deduced from the estimates14
+|p(ξ)| · max
+�
+| Ai(ξ)|, | Ai′(ξ)|
+�
+⩽ ape−ξ
+(ξ ∈ R),
+where the constant ap does only depend on the polynomial p (note that we can take ap = 1
+when p(ξ) ≡ 1). This way we get from (6)
+|K0(x, y)| ⩽
+� ∞
+0
+| Ai(x + σ) Ai(y + σ)| dσ ⩽ e−(x+y)
+� ∞
+0
+e−2σ dσ = 1
+2e−(x+y)
+(x, y ∈ R)
+and, for 1 ⩽ j ⩽ m, constants cj such that
+|Kj(x, y)| ⩽ cje−(x+y)
+(x, y ∈ R).
+We denote the induced integral operators on L2(t, ∞) in bold face (suppressing the dependence
+on t in the notation) and the corresponding spaces of trace class and Hilbert–Schmidt operators
+by J p(t, ∞) with p = 1 and p = 2. Now, the Airy operator K0 (being by (6) the square of
+the Hilbert–Schmidt operator At with kernel Ai(x + y − t)) and the Kj (j ⩾ 1) (being finite
+rank operators) are trace class. Their trace norms are bounded by
+∥K0∥J 1(t,∞) ⩽ ∥At∥2
+J 2(t,∞) =
+� ∞
+t
+� ∞
+t
+| Ai(x + y − t)|2 dx dy
+=
+� ∞
+0
+� ∞
+0
+| Ai(x + y + t)|2 dx dy ⩽ e−2t
+� ∞
+0
+� ∞
+0
+e−2(x+y) dx dy = 1
+4e−2t
+(t ∈ R),
+and likewise, because of
+∥u ⊗ v∥J 1(t,∞) ⩽ ∥u∥L2(t,∞)∥v∥L2(t,∞) ⩽ apaq
+2
+e−2t
+(t ∈ R)
+for factors u(ξ), v(ξ) of the form (16) with polynomials p and q, there are constants c∗
+j s.t.
+∥Kj∥J 1(t,∞) ⩽ c∗
+je−2t
+(1 ⩽ j ⩽ m, t ∈ R).
+On the other hand, as there is, in general, no direct relation between kernel bounds and
+bounds of the trace norm of induced integral operators (see [60, p. 25]), lifting the error term
+in the kernel expansion (15) to trace norm is less straightforward.15
+By taking the explicitly assumed differentiability of the kernel expansion into account,
+there is a constant c∗ such that
+Sm+1,h(x, y) := ∂yRm+1,h(x, y),
+|Sm+1,h(x, y)| ⩽ c∗e−(x+y)
+holds true for all x, y ⩾ t0 as 0 < h ⩽ h0 (h0 chosen sufficiently small). If we denote an
+indicator function by χ, integration gives
+Rm+1,h(x, y) = −
+� ∞
+t
+Sm+1,h(x, σ)eσ/2 · e−σ/2χ[t,σ](y) dσ,
+14This bound, chosen for convenience but not for optimality, follows from the superexponential decay of the
+Airy function and its derivative as ξ → +∞ and the bounds O
+�(−ξ)−1/4� and O
+�(−ξ)1/4� as ξ → −∞, cf.
+the expansions (103) and [52, Eq. (9.7.9/10)].
+15This subtle technical point has frequently been missed in the literature when lifting kernel expansion to
+trace class operators. (E.g., the argument given in [21, p. 12] for lifting a kernel expansion to the Edgeworth
+expansion of the largest eigenvalue distribution of GUE and LUE lacks in that respect. It can be made
+rigorous when supplemented by the estimates given here; see Thm. 2.1. Another rigorous approach can be
+found in the work of Johnstone [43, 44].)
+
+8
+FOLKMAR BORNEMANN
+which shows that Rm+1,h is the product of two Hilbert–Schmidt operators and thus trace
+class with a norm bounded by
+∥Rm+1,h∥2
+J 1(t,∞) ⩽
+�� ∞
+t
+� ∞
+t
+Sm+1,h(x, y)2ey dx dy
+�
+·
+�� ∞
+t
+� x
+t
+e−x dx dy
+�
+⩽ c2
+∗
+�� ∞
+t
+� ∞
+t
+e−2x−y dx dy
+�
+·
+�� ∞
+t
+� x
+t
+e−x dx dy
+�
+= c2
+∗
+2 e−4t.
+We have thus lifted the kernel expansion (15) to an operator expansion in J 1(t, ∞), namely
+(17)
+Kh = K0 +
+m
+�
+j=1
+hjKj + hm+1Rm+1,h,
+∥Rm+1,h∥J 1(t,∞) = O
+�
+e−2t�
+,
+uniformly valid for t ⩾ t0 as h → 0+.
+2.2. Fredholm determinants. If a continuous kernel K(x, y) has a weighted uniform bound
+sup
+x,y⩾t0
+ex+y |K(x, y)| ⩽ M < ∞,
+then the Fredholm determinant
+det(I − K)|L2(t,s) :=
+∞
+�
+m=0
+(−1)m
+m!
+� s
+t
+· · ·
+� s
+t
+m
+det
+j,k=1 K(xj, xk) dx1 · · · dxm
+is well defined for t < s ⩽ ∞; cf. [4, §3.4] (writing the integrals in terms of the weighted
+measure e−x dx). If the induced integral operator K on L2(t, ∞) is trace class, the Fredholm
+determinant can be expressed in terms of the operator determinant. In fact, by the orthogonal
+decomposition
+L2(t, ∞) = L2(t, s) ⊕ L2(s, ∞)
+and denoting by Ps the orthogonal projection onto the first component (i.e., multiplication
+of L2-functions by the indicator function of the interval (t, s)) we have (cf. [60, Chap. 3])
+(18)
+det(I − K)|L2(t,s) = det(I − KPs).
+For lifting the operator expansion (17) to one for the (Fredholm) determinants it suffices to
+restrict ourselves to the case s = ∞. This is because a block decomposition of Kh shows
+∥Kh − KhPs∥J 1(t,∞) = ∥ ˜Kh∥J 1(s,∞)
+where ˜Kh denotes the induced integral operator on the subspace L2(s, ∞) ⊂ L2(t, ∞). Now,
+the local Lipschitz continuity of the operator determinant [60, Thm. 3.4] implies
+|det(I − KhPs) − det(I − Kh)| ⩽ ∥ ˜Kh∥J 1(s,∞) exp
+�
+2∥Kh∥J 1(t,∞) + 1
+�
+= O
+�
+e−2s�
+where, by Sect. 2.1, the bound holds uniformly for t0 ⩽ t < s ⩽ ∞ as h → 0.
+2.3. Expansions of the operator determinants. Plemelj’s formula gives, for trace class
+perturbations E bounded by ∥E∥J 1(t,∞) < 1, the convergent series expansion (cf., e.g., [60,
+Eq. (5.12)])
+(19)
+det(I − E) = exp
+�
+−
+∞
+�
+n=1
+n−1 tr(En)
+�
+= 1 − tr E + 1
+2
+�
+(tr E)2 − tr(E2)
+�
++ O(∥E∥3
+1).
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+9
+Thus, since I − K0 is invertible with a uniformly bounded inverse as t ⩾ t0,16 we have
+det(I − Kh) = det(I − K0) det(I − Eh),
+Eh :=
+m
+�
+j=1
+hj(I − K0)−1Kj + hm+1(I − K0)−1Rm+1,h,
+with the trace norm of Eh being bounded as follows (using the results of Sect. 2.1 and
+observing that the trace class forms an ideal within the bounded operators):
+∥Eh∥J 1(t,∞) ⩽ ∥(I − K0)−1∥
+h
+1 − h · O
+�
+e−2t�
+= h · O(e−2t),
+uniformly for t ⩾ t0 as h → 0+. By (19) this implies, uniformly under the same conditions,
+det
+�
+I − Eh
+�
+= 1 +
+m
+�
+j=1
+dj(t)hj + hm+1 · O(e−2t).
+The functions dj(t) depend smoothly on t and satisfy dj = O(e−2t); the first two of them are
+d1(t) = − tr
+�
+(I − K0)−1K1
+�
+,
+d2(t) = 1
+2
+�
+tr
+�
+(I − K0)−1K1
+��2 − 1
+2 tr
+�
+((I − K0)−1K1)2�
+− tr
+�
+(I − K0)−1K2
+�
+.
+Since the Kj (j ⩾ 1) are finite rank operators, these trace expressions can be recast in terms
+of resolvent kernels and integral traces.
+Taking the bound 0 ⩽ F(t) ⩽ 1 of the Tracy–Widom distribution (being a probability
+distribution) into account, the results of Sect. 2 can be summarized in form of the following:
+Theorem 2.1. Let Kh(x, y) be a continuous kernel, K0 the Airy kernel (6) and let the
+Kj(x, y) be finite sums of rank one kernels with factors of the form (16). If, for some fixed
+non-negative integer m and some real number t0, there is a kernel expansions of the form
+Kh(x, y) = K0(x, y) +
+m
+�
+j=1
+hjKj(x, y) + hm+1 · O
+�
+e−(x+y)�
+,
+which holds uniformly for x, y ⩾ t0 as h → 0+ and which can be repeatedly differentiated
+w.r.t. x and y as uniform expansions, then the Fredholm determinant of Kh on (t, s) satisfies
+(20)
+det(I − Kh)|L2(t,s) = F(t) +
+m
+�
+j=1
+Gj(t)hj + hm+1 · O(e−2t) + O(e−2s)
+uniformly for t0 ⩽ t < s ⩽ ∞ as h → 0. Here, F denotes the Tracy–Widom distribution (5)
+and the Gj(t) are smooth functions depending on the kernels K1, . . . , Kj, satisfying the (right)
+tail bounds Gj = O(e−2t). The first two are
+G1(t) = −F(t) · tr
+�
+(I − K0)−1K1
+���
+L2(t,∞) ,
+G2(t) = F(t) ·
+�1
+2
+�
+tr
+�
+(I − K0)−1K1
+���
+L2(t,∞)
+�2
+− 1
+2 tr
+�
+((I − K0)−1K1)2���
+L2(t,∞) − tr
+�
+(I − K0)−1K2
+���
+L2(t,∞)
+�
+,
+where (I − K0)−1 is understood as a resolvent kernel and the traces as integrals. The determi-
+nantal expansion (20) can repeatedly be differentiated w.r.t. t and s, preserving uniformity.
+16The Airy kernel K0 induces a symmetric positive definite integral operator K0 on L2(t, ∞). Its norm as a
+bounded operator is thus is given by the spectral radius, which stays below 1 uniformly as t ⩾ t0, cf. [64]:
+∥K0∥ = ρ(K0) ⩽ c(t0) < 1.
+By functional calculus we thus get the uniform bound ∥(I − K0)−1∥ =
+1
+1−ρ(K0) ⩽
+1
+1−c(t0).
+
+10
+FOLKMAR BORNEMANN
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+-1
+-0.75
+-0.5
+-0.25
+0
+0.25
+0.5
+Figure 1. Plots of F1(t) (left panel) and F2(t) (middle panel) as in (24). The right panel
+shows F3(t) as in (108) (black solid line) with the approximations (25) for ν = 100 (red
+dotted line) and ν = 800 (green dashed line): the close agreement validates the functional
+forms (24) and (108). Details about the numerical method can be found in [13, 14, 16, 17].
+3. Expansion of the Hard-to-Soft Edge Transition
+In this section we prove an expansion for the hard-to-soft transition limit (7). To avoid
+notational clutter, we use the quantity
+(21)
+hν := 2−1/3ν−2/3
+and study expansions in powers of hν as hν → 0+. The transform s = φν(t) used in the
+transition limit can briefly be written as
+(22)
+φν(t) = ν2(1 − hνt)2.
+For technical reasons related to the proof of Lemma 3.2 (which uses the divisibility of a
+certain sequence of polynomials that has only been checked by inspection up to m∗), we have
+to impose a bound
+m∗ = 23
+on the number m of expansion terms in all of the main results of the present paper. We
+conjecture that Lemma 3.2 is true without such a restriction, though.
+Theorem 3.1. There holds the hard-to-soft edge transition expansion
+(23)
+Ehard
+2
+(φν(t); ν) = F(t) +
+m
+�
+j=1
+Fj(t)hj
+ν + hm+1
+ν
+· O(e−3t/2),
+which is uniformly valid when t0 ⩽ t < h−1
+ν
+as hν → 0+, m being any fixed integer in the
+range 0 ⩽ m ⩽ m∗ and t0 any fixed real number. Preserving uniformity, the expansion can be
+repeatedly differentiated w.r.t. the variable t. Here the Fj are certain smooth functions; the
+first two are
+F1(t) = 3t2
+10 F ′(t) − 1
+5F ′′(t),
+(24a)
+F2(t) =
+� 2
+175 + 32t3
+175
+�
+F ′(t) +
+�
+− 16t
+175 + 9t4
+200
+�
+F ′′(t) − 3t2
+50 F ′′′(t) + 1
+50F (4)(t),
+(24b)
+while a similar expression for F3 is displayed in (108).
+It is rewarding to validate intriguing formulae such as (24b/c) and (108) by numerical
+methods: Fig. 1 plots the functions F1(t), F2(t), F3(t) next to the approximation
+(25)
+F3(t) ≈ h−3
+ν
+�
+Ehard
+2
+(φν(t); ν) − F(t) − F1(t)hν − F2(t)h2
+ν
+�
+for ν = 100 and ν = 800: the close matching with F D
+3 (t) as displayed by the latter is a
+very strong testament of correctness of (24) and (108) (in fact, some slips in preliminary
+calculations have been caught looking at plots which exhibited mismatches).
+The proof of Thm. 3.1 is split into several steps and will be concluded in Sects. 3.2 and 3.3.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+11
+3.1. Kernel expansions. We start with an auxiliary result.
+Lemma 3.1. Define for h > 0 and x, y < h−1 the function
+(26)
+Φ(x, y; h) :=
+�
+(1 − hx)(1 − hy)
+1 − h(x + y)/2
+.
+This function Φ satisfies the bound
+(27)
+0 < Φ(x, y; h) ⩽ 1
+and has the convergent power series expansion
+(28a)
+Φ(x, y; h) = 1 − (x − y)2
+∞
+�
+k=2
+rk(x, y)hk
+where the rk(x, y) are certain homogeneous symmetric rational17 polynomials of degree k − 2,
+the first few of them being
+(28b)
+r2(x, y) = 1
+8,
+r3(x, y) = 1
+8(x + y),
+r4(x, y) =
+1
+128
+�
+13(x2 + y2) + 22xy
+�
+.
+The series converges uniformly for x, y < (1 − δ)h−1, δ being any fixed real positive number.
+Proof. The bound (27) is the inequality of arithmetic and geometric means for the two
+positive real quantities 1 − hx and 1 − hy. By using
+lim
+y→x
+1
+(x − y)2
+�
+Φ(x, y; h) − 1
+�
+= −1
+8
+�
+h
+1 − hx
+�2
+,
+the analyticity of Φ(x, y; h) w.r.t. h, and the scaling law
+Φ(λ−1x, λ−1y; λh) = Φ(x, y; h)
+(λ > 0)
+we deduce the claims about the form und uniformity of the power series expansion (28).
+□
+Because of the representation (2) of Ehard
+2
+(s; ν) in terms of a Fredholm determinant of the
+Bessel kernel (3), we have to expand the induced transformation of that kernel Kν.
+Lemma 3.2. The change of variables s = φν(t), mapping t < h−1
+ν
+monotonically decreasing
+to s > 0, induces the symmetrically transformed Bessel kernel
+ˆKBessel
+ν
+(x, y) :=
+�
+φ′ν(x)φ′ν(y) KBessel
+ν
+(φν(x), φν(y)).
+(29a)
+There holds the kernel expansion
+ˆKBessel
+ν
+(x, y) = K0(x, y) +
+m
+�
+j=1
+Kj(x, y)hj
+ν + hm+1
+ν
+· O
+�
+e−(x+y)�
+,
+(29b)
+which is uniformly valid when t0 ⩽ x, y < h−1
+ν
+as hν → 0+, m being any fixed integer in the
+range 0 ⩽ m ⩽ m∗ and t0 any fixed real number. Here the Kj, j = 1, . . . , m∗, are certain
+finite rank kernels of the form
+Kj(x, y) =
+�
+κ,λ∈{0,1}
+pj,κλ(x, y) Ai(κ)(x) Ai(λ)(y)
+where pj,κλ(x, y) are rational polynomials; the first two kernels are
+(30a)
+K1(x, y) = 1
+10
+�
+− 3
+�
+x2 + xy + y2�
+Ai(x) Ai(y)
++ 2
+�
+Ai(x) Ai′(y) + Ai′(x) Ai(y)
+�
++ 3(x + y) Ai′(x) Ai′(y)
+�
+17Throughout the paper the term “rational polynomial” is used for polynomials with rational coefficients.
+
+12
+FOLKMAR BORNEMANN
+and
+(30b)
+K2(x, y) =
+1
+1400
+��
+− 235
+�
+x3 + y3�
+− 319xy(x + y) + 56
+�
+Ai(x) Ai(y)
++
+�
+63(x4 + x3y − x2y2 − xy3 − y4) − 55x + 239y
+�
+Ai(x) Ai′(y)
++
+�
+63(−x4 − x3y − x2y2 + xy3 + y4) + 239x − 55y
+�
+Ai′(x) Ai(y)
++
+�
+340(x2 + y2) + 256xy
+�
+Ai′(x) Ai′(y)
+�
+.
+Preserving uniformity, the kernel expansion (29) can repeatedly be differentiated w.r.t. x, y.
+Proof. By using the function Φ(x, y; h) as defined in (26) and writing
+φν(t) = ων(t)2,
+ων(t) = ν(1 − hνt),
+aν(x, y) = (1 − hνy) ·
+1
+√2hν
+Jν
+�
+ων(x)
+�
+· d
+dy
+1
+√2hν
+Jν
+�
+ων(y)
+�
+we can factor the transformed Bessel kernel in the simple form
+ˆKBessel
+ν
+(x, y) = Φ(x, y; hν) · aν(x, y) − aν(y, x)
+x − y
+,
+noting, by symmetry, the removability of the singularities at x = y of the second factor.
+First, if x or y is between 3
+4 · h−1
+ν
+and h−1
+ν , using the bound 0 < Φ ⩽ 1 (see (27)) one can
+argue as in the proof of Lemma A.1: since at least one of the Bessel factors is of the form
+J(κ)
+ν
+(z) with 0 ⩽ z ⩽ 1/4, which plainly falls into the superexponentially decaying region
+as ν → ∞, and since at least one of the Airy factors of each term of the expansion is also
+superexponentially decaying as ν → ∞, the transformed Bessel kernel and the expansion
+terms in (29) get completely absorbed into the error term (bounding the other factors as in
+Sect. 2.1)
+hm+1
+ν
+· O
+�
+e−(x+y)�
+.
+Here, the removable singularities at x = y are dealt with by using the differentiability of the
+corresponding bounds (or by extending to the complex domain and using Cauchy’s integral
+formula as in the proof of [18, Prop. 8]).
+Therefore, we may suppose from now on that t0 ⩽ x, y ⩽ 3
+4 · h−1
+ν . By Lemma 3.1, in this
+range of x and y, the power series expansion
+(31)
+Φ(x, y; hν) = 1 − (x − y)2
+∞
+�
+k=2
+rk(x, y)hk
+ν
+converges uniformly. Here, the rk(x, y) are certain homogeneous symmetric rational polyno-
+mials of degree k − 2; the first of them being r2(x, y) = 1/8.
+Next, we rewrite the uniform version of the large order expansion of Bessel functions in
+the transition region, as given in Lemma A.1, in the form
+(32)
+1
+√2hν
+Jν
+�
+ων(t)
+�
+=
+�
+1 + hνpm(t; hν)
+�
+Ai(t) + hνqm(t; hν) Ai′(t) + hm+1
+ν
+O(e−t),
+where the estimate of the remainder is uniform for t0 ⩽ t < h−1
+ν
+as hν → 0+ and
+pm(t; h) = 21/3
+m−1
+�
+k=0
+Ak+1(−t/21/3) (21/3h)k,
+qm(t; h) = 22/3
+m−1
+�
+k=0
+Bk+1(−t/21/3) (21/3h)k,
+with the polynomials Ak(τ) and Bk(τ) from (101). It follows from Remark A.3 that pm(t; h)
+and qm(t; h) are rational polynomials in t and h, starting with
+p2(t; h) = 2
+10t +
+h
+1400(63t5 + 120t2),
+q2(t; h) = 3
+10t2 +
+h
+1400(340t3 + 40).
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+13
+Also given in Lemma A.1, under the same conditions, the expansion (32) can be repeatedly
+differentiated w.r.t t while preserving uniformity. From this we obtain, using the Airy
+differential equation Ai′′(ξ) = ξ Ai(ξ), that uniformly (given the range x and y)18
+aν(x, y) =
+�
+κ,λ∈{0,1}
+pm
+κλ(x, y; hν) Ai(κ)(x) Ai(λ)(y) + hm+1
+ν
+O(e−(x+y)),
+where
+pm
+00(x, y; h) = h(1 − hy)(1 + hpm(x; h))
+�
+yqm(y; h) + ∂ypm(y; h)
+�
+,
+pm
+01(x, y; h) = (1 − hy)(1 + hpm(x; h))
+�
+1 + h(pm(y; h) + ∂yqm(y; h))
+�
+,
+pm
+10(x, y; h) = h2(1 − hy)qm(x; h)
+�
+yqm(y; h) + ∂ypm(y; h)
+�
+,
+pm
+11(x, y; h) = h(1 − hy)qm(x; h)
+�
+1 + h(pm(y; h) + ∂yqm(y; h))
+�
+are rational polynomials in x, y and h. In particular, those factorizations show
+pm
+00(x, y; h) = O(h),
+pm
+11(x, y; h) = O(h),
+pm
+01(x, y; h) = 1 + O(h),
+pm
+10(x, y; h) = O(h2).
+If we denote by ˆpm
+κλ(x, y; h) the polynomials obtained from pm
+κλ(x, y; h) after dropping all
+powers of h that have an exponent larger than m (thus contributing terms to the expansion
+that get absorbed in the error term), we obtain
+aν(x, y) = Ai(x) Ai′(y) +
+�
+κ,λ∈{0,1}
+ˆpm
+κλ(x, y; hν) Ai(κ)(x) Ai(λ)(y) + hm+1
+ν
+O(e−(x+y)),
+with a polynomial expansion
+ˆpm
+κλ(x, y; h) =
+m
+�
+j=1
+ˆpj,κλ(x, y)hj
+whose coefficient polynomials ˆpj,κλ(x, y), being the unique expansion coefficients as hν → 0+,
+are now independent of m. Hence, the anti-symmetrization of aν(x, y) satisfies the uniform
+expansion (given the range of x and y)
+(33)
+aν(x, y) − aν(y, x) =
+�
+Ai(x) Ai′(y) − Ai′(x) Ai(y)
+�
++
+�
+κ,λ∈{0,1}
+ˆqm
+κλ(x, y; hν) Ai(κ)(x) Ai(λ)(y) + hm+1
+ν
+O(e−(x+y))
+with the polynomial expansion
+ˆqm
+κλ(x, y; h) =
+m
+�
+j=1
+ˆqj,κλ(x, y)hj,
+where
+ˆqj,00(x, y) = ˆpj,00(x, y) − ˆpj,00(y, x),
+ˆqj,11(x, y) = ˆpj,11(x, y) − ˆpj,11(y, x)
+ˆqj,01(x, y) = ˆpj,01(x, y) − ˆpj,10(y, x),
+ˆqj,10(x, y) = −qj,01(y, x).
+Since the rational polynomials ˆqj,00(x, y) and ˆqj,11(x, y) are anti-symmetric in x, y, they
+factor in the form
+(x − y) × (symmetric rational polynomial in x and y);
+18Because of the superexponential decay (103) of the Airy function Ai(t) and its derivative Ai′(t) as t → +∞,
+cross terms with the remainder are uniformly estimated in the form
+polynomial(x) · Ai(κ)(x) · O
+�
+e−y�
+= O
+�
+e−(x+y)�
+.
+
+14
+FOLKMAR BORNEMANN
+the first few cases are
+ˆq1,00(x, y) = − 3
+10(x − y)(x2 + xy + y2),
+ˆq1,11(x, y) = 3
+10(x − y)(x + y),
+ˆq2,00(x, y) =
+1
+1400(x − y)
+�
+− 235
+�
+x3 + y3�
+− 319xy(x + y) + 56
+�
+,
+ˆq2,11(x, y) =
+1
+1400(x − y)
+�
+340
+�
+x2 + y2�
++ 256xy
+�
+.
+Even though there is no straightforward structural reason for the rational polynomials ˆqj,01
+(and thus ˆqj,10) to be divisible by x − y as well, an inspection19 of the first cases reveals this
+to be true for at least j = 1, . . . , m∗; the first two of them being
+ˆq1,01(x, y) = 2
+10(x − y),
+ˆq2,01(x, y) =
+1
+1400(x − y)
+�
+63
+�
+x4 + x3y − x2y2 − xy3 − y4�
++ 120x + 64y
+�
+.
+Now, by restricting ourselves to the explicitly checked cases m ⩽ m∗, we denote by qm
+κλ(x, y; h)
+the polynomials obtained from ˆqm
+κλ(x, y; h) after division by the factor x − y. Since (33) is an
+expansion of an anti-symmetric function with anti-symmetric remainder which can repeatedly
+be differentiated, division by x−y yields removable singularities at x = y and does not change
+the character of the expansion (see also the argument given in the proof of [18, Prop. 8]):
+aν(x, y) − aν(y, x)
+x − y
+= K0(x, y) +
+�
+κ,λ∈{0,1}
+qm
+κλ(x, y; hν) Ai(κ)(x) Ai(λ)(y) + hm+1
+ν
+O
+�
+e−(x+y)�
+.
+The lemma now follows by multiplying this expansion with (31), noting that the terms
+−rk(x, y)(x − y)2K0(x, y) = −rk(x, y)(x − y)
+�
+Ai(x) Ai′(y) − Ai′(x) Ai(y)
+�
+also take the form asserted for the terms in the kernels Kj (j ⩾ 1).
+Finally, since all the expansions can repeatedly be differentiated under the same conditions
+while preserving their uniformity, the same holds for the resulting expansion of the kernel.
+□
+Remark 3.1. The case m = 0 of Lemma 3.2, i.e.,
+ˆKBessel
+ν
+(x, y) = K0(x, y) + hν · O
+�
+e−(x+y)�
+is the β = 2 case of [18, Eq. (4.8)] in the work of Borodin and Forrester. There, in [18, Prop. 8]
+it is stated that this expansion would be uniformly valid for x, y ⩾ t0. However, stated in
+such a generality, it is not correct (see Fn. 38) and, in fact, similar to our proof given above,
+their proof is restricted to the range t0 ⩽ x, y < h−1
+ν , which completely suffices to address
+the hard-to-soft edge transition. (See Remark A.4 for yet another issue with [18, Prop. 8].)
+To reduce the complexity of calculating the functional form of the first two finite-size
+correction terms in the hard-to-soft edge transition (23), we consider a second kernel transform.
+Lemma 3.3. For h > 0, the Airy kernel K0 and the first two expansion kernels K1, K2
+from Lemma 3.2 we consider
+Kh(x, y) = K0(x, y) + K1(x, y)h + K2(x, y)h2
+19See Remark A.3 for the computation of the polynomials Ak, Bk and thus qj,01(x, y)—a Mathematica
+notebook comes with the sources of the arXiv version of the present paper. Since the divisibility of ˆqj,01(x, y)
+by x − y depends very much on the specific rational coefficients of the polynomials Ak, Bk, this is probably
+not just a contingent fact. We conjecture it to be true for all j; a proof, however, would have to uncover some
+hidden symmetry.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+15
+and the transformation,20 where ζ(z) is defined as in Sect. A.3,
+(34)
+s = ψ−1
+h (t) := 2−1/3h−1ζ(1 − ht).
+Then t = ψh(s) maps s ∈ R monotonically increasing to −∞ < t < h−1, with t ⩽ µh−1,
+µ = 0.94884 · · · , when s ⩽ 2h−1, and induces the symmetrically transformed kernel
+˜Kh(x, y) :=
+�
+ψ′
+h(x)ψ′
+h(y) Kh(ψh(x), ψh(y))
+(35a)
+which expands as
+˜Kh(x, y) = K0(x, y) + ˜K1(x, y)h + ˜K2(x, y)h2 + h3 · O
+�
+e−(x+y)�
+,
+(35b)
+uniformly in s0 ⩽ x, y ⩽ 2h−1 as h → 0+, s0 being a fixed real number. Here, ˜K1 and ˜K2 are
+˜K1 = 1
+5(Ai ⊗ Ai′ + Ai′ ⊗ Ai)
+(35c)
+˜K2 =
+1
+350
+�
+55(Ai ⊗ Ai′′′ + Ai′′′ ⊗ Ai) − 51(Ai′ ⊗ Ai′′ + Ai′′ ⊗ Ai′) − 96 Ai ⊗ Ai
+�
+.
+(35d)
+Preserving uniformity, the kernel expansion can repeatedly be differentiated w.r.t. x, y.
+Proof. Reversing the power series (104) gives
+t = ψh(s) = s − 3s2
+10 h − s3
+350h2 + · · · ,
+which is uniformly convergent for s0 ⩽ s ⩽ 2h−1 since 1 − ht stays bounded away from zero
+there. A routine calculation with truncated power series gives formula (35c) for ˜K1 and
+˜K2(x, y) =
+1
+350
+�
+14 Ai(x) Ai(y) + (−51x + 55y) Ai(x) Ai′(y) + (55x − 51y) Ai′(x) Ai(y)
+�
+.
+The Airy differential equation ξ Ai(ξ) = Ai′′(ξ) implies the replacement rule
+(36)
+ξj Ai(k)(ξ) = ξj−1 Ai(k+2)(ξ) − kξj−1 Ai(k−1)(ξ)
+(j ⩾ 1, k ⩾ 0)
+which, if repeatedly applied to a kernel of the given structure, allows us to absorb any powers
+of x and y into higher order derivatives of Ai. This process yields the asserted form of ˜K2,
+which will be the preferred form in course of the calculations in Sect. 3.3.
+Since we stay within the range of uniformity of the power series expansions and calculations
+with truncated powers series are amenable to repeated differentiation, the result now follows
+from the bounds given in Sect. 2.1.
+□
+3.2. Proof of the general form of the expansion. By Lemma 3.2 and Thm. 2.1 we get
+(the Fredholm determinants are seen to be equal by transforming the integrals)
+Ehard
+2
+(φν(t); ν) = det(I − KBessel
+ν
+)|L2(0,φν(t)) = det(I − ˆKBessel
+ν
+)|L2(t,h−1
+ν )
+= F(t) +
+m
+�
+j=1
+Fj(t)hj
+ν + hm+1
+ν
+O(e−2t) + O(e−2h−1
+ν ),
+uniformly for t0 ⩽ t < h−1
+ν
+as hν → 0+; preserving uniformity, this expansion can be
+repeatedly differentiated w.r.t. the variable t. By Thm. 2.1, the Fj(t) are certain smooth
+functions that can be expressed in terms of traces of integral operators of the form given in
+Thm. 2.1. Observing
+e−2h−1
+ν
+< e−h−1
+ν /2e−3t/2 = hm+1
+ν
+O(e−3t/2)
+(hν → 0+)
+we can combine the two error terms as hm+1
+ν
+O(e−3t/2). This finishes the proof of (23).
+20Note that for z = 1 − hνt we thus get νz = ων(t) and ν2/3ζ = s in Olver’s expansion (102). As it turns out,
+by using this transformation, the kernel expansion simplifies in the same fashion also for m ⩾ 2, cf. Fn. 39.
+
+16
+FOLKMAR BORNEMANN
+3.3. Functional form of F1(t) and F2(t). Instead of calculating F1, F2 directly from the
+formulae in Thm. 2.1 applied to the kernels K1, K2 in (30a/b) we will reduce them to the
+corresponding functions ˜F1, ˜F2 induced by the much simpler kernels ˜K1, ˜K2 in (35c/d).
+3.3.1. Functional form of ˜F1(t) and ˜F2(t). First, by writing
+ujk(t) = tr
+�
+(I − K0)−1 Ai(j) ⊗ Ai(k) ���
+L2(t,∞)
+and observing (the symmetry of the resolvent kernel implies the symmetry ujk(t) = ukj(t))
+tr
+�
+(I − K0)−1 ˜K1
+���
+L2(t,∞) = 2
+5u10(t),
+tr
+�
+((I − K0)−1 ˜K1)2���
+L2(t,∞) = 2
+25
+�
+u00(t)u11(t) + u10(t)2�
+,
+tr
+�
+(I − K0)−1 ˜K2
+���
+L2(t,∞) =
+1
+175
+�
+− 48u00(t) − 51u21(t) + 55u30(t)
+�
+,
+the formulae of Thm. 2.1 applied to ˜K1 and ˜K2 give
+˜F1(t) = −2
+5F(t)u10(t),
+(37a)
+˜F2(t) = F(t)
+� 48
+175u00(t) − 1
+25
+�
+u00(t)u11(t) − u10(t)2�
++ 51
+175u21(t) − 11
+35u30(t)
+�
+.
+(37b)
+We recall from [16, Remark 3.1] that the simple recursion
+u′
+jk(t) = uj+1,k(t) + uj,k+1(t) − uj0(t)uk0(t)
+yields similar formulae for the first few derivatives of the distribution F(t):
+(38)
+F ′(t) = F(t) · u00(t),
+F ′′(t) = 2F(t) · u10(t),
+F ′′′(t) = 2F(t) ·
+�
+u11(t) + u20(t)
+�
+,
+F (4)(t) = 2F(t) ·
+�
+u00(t)u11(t) − u10(t)2 + 3u21(t) + u30(t)
+�
+.
+By a linear elimination of the terms u00(t), u10(t), u00(t)u11(t) − u10(t)2 we obtain, as an
+intermediate step,
+(39a)
+˜F1(t) = −1
+5F ′′(t),
+˜F2(t) = 48
+175F ′(t) − 1
+50F (4)(t) + 24
+175F(t)
+�
+3u21(t) − 2u30(t)
+�
+.
+Second, to simplify even further, we refer to the full power of the general Tracy–Widom
+theory (i.e., representing F in terms of Painlevé II): by advancing its set of formulae, Shinault
+and Tracy [59, p. 68] showed, in course of an explicit inspection of each single case in the
+range 0 ⩽ j + k ⩽ 8, that the functions F(t) · ujk(t) are linear combinations of the form
+(39b)
+p1(t)F ′(t) + p2(t)F ′′(t) + · · · + pj+k+1(t)F (j+k+1)(t)
+with rational polynomials pκ(t) (depending on j, k), and conjectured this structure to be
+true for all j, k. In particular, their table has the entries
+F(t) · u21(t) = −1
+4F ′(t) + 1
+8F (4)(t),
+F(t) · u30(t) = 7
+12F ′(t) + t
+3F ′′(t) + 1
+24F (4)(t).
+This way we get, rather unexpectedly, the simple and short form21
+(39c)
+˜F2(t) =
+2
+175F ′(t) − 16t
+175F ′′(t) + 1
+50F (4)(t);
+a similar expression for an accordingly constructed ˜F3 is displayed in (107).
+21Note that a direct application of the table in [59, p. 68] to (37b) produces a far less appealing result, namely
+˜F2(t) =
+19
+1050F ′(t) + tF ′(t)2
+75F(t) − 11t
+105F ′′(t) + F ′′(t)2
+100F(t) − F ′(t)F ′′′(t)
+75F(t)
++
+7
+300F (4)(t),
+which is no longer linear in F.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+17
+3.3.2. Lifting to the functional form of F1(t) and F2(t). The relation between F1(t), F2(t)
+and their counterparts with a tilde is established by Lemma 3.3. By using the notation
+introduced there, with t being any fixed real number, the expansion parameter h sufficiently
+small and s = ψ−1
+h (t), Thm. 2.1 yields (the Fredholm determinants are seen to be equal by
+transforming the integrals)
+(40)
+det(I − Kh)|L2(t, µh−1) = F(t) + F1(t)h + F2(t)h2 + O(h3)
+= det(I − ˜Kh)|L2(s, 2h−1) = F(s) + ˜F1(s)h + ˜F2(s)h2 + O(h3),
+where we have absorbed the exponentially small contributions O(e−2µh−1) and O(e−4h−1)
+into the O(h3) error term. Using the power series (104), that is,
+s = 2−1/3h−1ζ(1 − ht) = t + 3t2
+10 h + 32t3
+175 h2 + · · · ,
+we get by Taylor expansion
+F(s) = F(t) + 3t2
+10 F ′(t)h +
+�32t3
+175 F ′(t) + 9t4
+200F ′′(t)
+�
+h2 + O(h3),
+˜F1(s) = ˜F1(t) + 3t2
+10
+˜F ′
+1(t)h + O(h2),
+˜F2(s) = ˜F2(t) + O(h).
+By plugging this into (40) and comparing coefficients we obtain
+F1(t) = ˜F1(t) + 3t2
+10 F ′(t),
+F2(t) = ˜F2(t) + 3t2
+10
+˜F ′
+1(t) + 32t3
+175 F ′(t) + 9t4
+200F ′′(t).
+Combined with (39), this finishes the proof of (24).
+3.4. Simplifying the form of Choup’s Edgeworth expansions. When, instead of the
+detour via ˜K1, Thm. 2.1 is directly applied to the kernel K1 in (30a), we get
+F1(t) = −1
+5F ′′(t) + 3
+10F(t) tr
+�
+(I − K0)−1L
+���
+L2(t,∞),
+where
+(41a)
+L(x, y) = (x2 + xy + y2) Ai(x) Ai(y) − (x + y) Ai′(x) Ai′(y).
+Now, a comparison with (24a) proves the useful formula22
+(41b)
+F(t) tr
+�
+(I − K0)−1L
+���
+L2(t,∞) = t2F ′(t).
+As an application to the existing literature, this formula helps us to simplify the results
+obtained by Choup for the soft-edge limit expansions of GUE and LUE: that is, when
+studying the distribution of the largest eigenvalue distribution function in GUEn and LUEn,ν
+(dimension n, parameter ν) as n → ∞. In fact, since the kernel L appears in the first finite-size
+correction term of a corresponding kernel expansion [21, Thm. 1.2/1.3], lifting that expansion
+to the Fredholm determinant by Thm. 2.1 allows us to recast [21, Thm. 1.4] in a simplified
+22Note that our derivation of this formula does only depend on Fredholm determinants and does not use
+any representation in terms of Painlevé II. Based on Painlevé representations, it has been derived, implicitly
+though, in the recent work of Forrester and Mays [33]: see Eqns. (1.16), (1.19), (2.17) and (2.29) there. A
+further alternative derivation follows from observing that, by repeated application of (36),
+tr
+�
+(I − K0)−1L
+���
+L2(t,∞) = −2u10(t) + u22(t) − 2u31(t) + 2u40(t)
+and using the table for the functions F(t) · ujk(t) (0 ⩽ j + k ⩽ 8) compiled in [59, p. 68] (which is based on
+an extension of formulae of the Tracy–Widom theory that represents F in terms of Painlevé II).
+
+18
+FOLKMAR BORNEMANN
+form: namely, denoting the maximum eigenvalues by λG
+n and λL
+n,ν, we obtain, locally uniform
+in t as n → ∞,
+P
+�
+λG
+n ⩽
+√
+2n + t · 2−1/2n−1/6�
+= F(t) + n−2/3
+40
+�
+2t2F ′(t) − 3F ′′(t)
+�
++ O(n−1),
+(42)
+P
+�
+λL
+n,ν ⩽ 4n + 2ν + t · 2(2n)1/3�
+= F(t) − 21/3n−2/3
+10
+�
+t2F ′(t) + F ′′(t)
+�
++ O(n−1);
+(43)
+a result, which answers a question suggested by Baik and Jenkins [11, p. 4367].
+4. Expansion of the Poissonized Length Distribution
+The Poissonization of the length distribution requires the hard-to-soft edge transition of
+Thm. 3.1 to be applied to the probability distribution Ehard
+2
+(4r; ν) (for integer ν = l, but we
+consider the case of general ν > 0 first). For large intensities r the mode of this distribution
+is located in the range of those parameters ν for which the scaled variable
+(44a)
+tν(r) := ν − 2√r
+r1/6
+(r > 0).
+stays bounded. It is convenient to note that tν(r) satisfies the differential equation
+(44b)
+t′
+ν(r) = −r−2/3 − r−1
+6 tν(r)
+(r > 0).
+In these terms we get the following theorem.
+Theorem 4.1. There holds the expansion
+(45)
+Ehard
+2
+(4r; ν) = F(t) +
+m
+�
+j=1
+F P
+j (t) r−j/3 + r−(m+1)/3 · O
+�
+e−t�����
+t=tν(r)
+,
+which is uniformly valid when r, ν → ∞ subject to
+t0 ⩽ tν(r) ⩽ r1/3,
+with m being any fixed integer in the range 0 ⩽ m ⩽ m∗ and t0 any fixed real number.
+Preserving uniformity, the expansion can be repeatedly differentiated w.r.t. the variable r.
+Here the F P
+j
+are certain smooth functions; the first two are23
+F P
+1 (t) = − t2
+60F ′(t) − 1
+10F ′′(t),
+(47a)
+F P
+2 (t) =
+� 1
+350 + 2t3
+1575
+�
+F ′(t) +
+� 11t
+1050 +
+t4
+7200
+�
+F ′′(t) + t2
+600F ′′′(t) +
+1
+200F (4)(t),
+(47b)
+while a similar expression for F P
+3 (t) is displayed in (109).
+Proof. For r, ν > 0 (i.e., equivalently t > −2r1/3 and s < h−1
+ν ) the transformations
+4r = φν(s),
+t = tν(r),
+are inverted by the expressions
+(48)
+s =
+t
+�
+1 + t
+2r−1/3�1/3 ,
+hν =
+r−1/3
+2
+�
+1 + t
+2r−1/3�2/3 .
+For t0 ⩽ t ⩽ r1/3 we get
+s0 := ( 2
+3)1/3t0 ⩽ ( 2
+3)1/3t ⩽ s < h−1
+ν
+23To validate formulae (47a/b) and (109), Fig. 2 plots F P
+3 (t) next to the approximation
+(46)
+F P
+3
+�
+tν(r)
+�
+≈ r−1�
+Ehard
+2
+(4r; ν) − F(t) − F P
+1 (t)r−1/3 − F P
+2 (t)r−2/3� ����
+t=tν(r)
+.
+for r = 250 and r = 2000, varying ν in such a way that t = tν(r) covers the interval [−6, 2].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+19
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+-0.07
+-0.06
+-0.05
+-0.04
+-0.03
+-0.02
+-0.01
+0
+0.01
+0.02
+0.03
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+-0.02
+-0.015
+-0.01
+-0.005
+0
+0.005
+0.01
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+10-3
+Figure 2. Plots of F P
+1 (t) (left panel) and F P
+2 (t) (middle panel) as in (47). The right panel
+shows F P
+3 (t) as in (109) (black solid line) with the approximations (46) for r = 250 (red
+dotted line) and r = 2000 (green dashed line); the parameter ν has been varied such that
+tν(r) covers the range of t displayed here. Note that the functions F P
+j (t) (j = 1, 2, 3) are
+by about a factor of 10 to 100 smaller in scale than their counterparts in Fig. 1.
+and observe that in this range of t the expressions in (48) expand as uniformly convergent
+power series in powers of r−1/3, starting with
+s = t − t2
+6 r−1/3 + t3
+18r−2/3 + · · · ,
+hν = 1
+2r−1/3 − t
+6r−2/3 + · · · .
+If we plug these uniformly convergent power series into the uniform expansion of Thm. (3.1),
+Ehard
+2
+(4r; ν) = Ehard
+2
+(φν(s); ν) = F(s) +
+m
+�
+j=1
+F1(s)hj
+ν + hm+1
+ν
+O(e−3s/2),
+we obtain the asserted form of the expansion (45) (as well as the claim about the repeated
+differentiability), simplifying the exponential error term by observing that (3/2)2/3 > 1. In
+particular, the first two correction terms in (45) are thus
+F P
+1 (t) = 1
+2F1(t) − t2
+6 F ′(t),
+F P
+2 (t) = 1
+4F2(t) − t
+6F1(t) − t2
+12F ′
+1(t) + t3
+18F ′(t) + t4
+72F ′′(t).
+Together with the expressions given in (24) this yields the functional form asserted in (47).
+□
+By (1), specializing Thm. 4.1 to the case of integer parameter ν = l yields the expansion
+(49)
+P
+�
+LNr ⩽ l
+�
+= F(t) +
+m
+�
+j=1
+F P
+j (t) r−j/3 + r−(m+1)/3 · O
+�
+e−t�����
+t=tl(r)
+which is uniformly valid under the conditions stated there.
+Remark 4.1. In the literature, scalings are often applied to the probability distribution rather
+than to the expansion terms. Since LNr is a an integer-valued random variable, one has to
+exercise some care with the scaled distribution function being piecewise constant. Namely,
+for t ∈ R being any fixed number, one has
+P
+�LNr − 2√r
+r1/6
+⩽ t
+�
+= P
+�
+LNr ⩽ l
+�
+,
+l =
+�
+2√r + tr1/6�
+,
+where ⌊·⌋ denotes the Gauss bracket. Thus, by defining
+t(r) =
+�
+2√r + tr1/6�
+− 2√r
+r1/6
+
+20
+FOLKMAR BORNEMANN
+and noting that t(r) stays bounded when r → ∞ while t is fixed, (49) takes the form
+(50)
+P
+�LNr − 2√r
+r1/6
+⩽ t
+�
+= F
+�
+t(r)�
++
+m
+�
+j=1
+F P
+j
+�
+t(r)�
+r−j/3 + O
+�
+r−(m+1)/3�
+(r → ∞).
+If one chooses to re-introduce the continuous variable t in (parts of) the expansion terms,
+one has to take into account that
+(51)
+t(r) = t + O(r−1/6)
+(r → ∞)
+where the exponent −1/6 in the error term is sharp. For example, this gives (as previously
+obtained by Baik and Jenkins [11, Thm. 1.3] using the technology of Riemann–Hilbert
+problems to prove the expansion and Painlevé representations to put F P
+1 into the simple
+functional form (47a))
+(52)
+P
+�LNr − 2√r
+r1/6
+⩽ t
+�
+= F
+�
+t(r)�
++ F P
+1 (t) r−1/3 + O(r−1/2)
+(r → ∞),
+where the O(r−1/2) error term is governed by the Gauss bracket in (51) and cannot be improved
+upon—completely dominating the order O(r−2/3) correction term in (50). Therefore, claiming
+an O(r−2/3) error term to hold in (52) as stated in [33, Prop. 1.1] neglects the effect of the
+Gauss bracket.24
+5. De-Poissonization and the Expansion of the Length Distribution
+5.1. Expansion of the CDF. In this section we prove (subject to a tameness hypothesis
+on the zeros of the generating functions in a sector of the complex plane) an expansion
+of the CDF P(Ln ⩽ l) of the length distribution near its mode. The general form of such
+an expansion was conjectured in the recent papers [16, 33] where approximations of the
+graphical form of the first few terms were provided (see [16, Figs. 4/6] and [33, Fig.7]). Here,
+for the first time, we give the functional form of these terms. The underlying tool is analytic
+de-Poissonization, a technique that was developed in the 1990s in theoretical computer science
+and analytic combinatorics.
+To prepare for the application of analytic de-Poissonization in the form of the Jacquet–
+Szpankowski Thm. A.1, we consider any fixed compact interval [t0, t1] and a sequence of
+integers ln → ∞ such that
+(53)
+t0 ⩽ t∗
+n := tln(n) ⩽ t1
+(n = 1, 2, 3, . . .).
+When n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 with n0 large enough (depending only on t0, t1)
+we thus get the uniform bounds25
+2√r + (t0 − 1)r1/6 ⩽ ln ⩽ 2√r + (t1 + 1)r1/6.
+24Furthermore, the right panel of [33, Fig. 3] is not showing an approximation of the O(r−2/3) term in (50),
+let alone in (52), but instead an approximation of the O(ν−4/3) term in the auxiliary expansion
+Ehard
+2
+��
+ν − t( ν
+2 )1/3 + t2
+6 ( ν
+2 )−1/3�2
+; ν
+�
+= F(t) + ˆF1(t) ( ν
+2 )−2/3 + ˆF2(t) ( ν
+2 )−4/3 + O(ν−2)
+(ν → ∞),
+cf. [33, Eqs. (2.3/2.33)]. Now, Thm. 3.1 and the formulae in (47) yield the simple relations
+ˆF1(t) = F P
+1 (t),
+ˆF2(t) = F P
+2 (t) + t
+3F P
+1 (t),
+which are consistent with [33, Fig. 3]; the additional term tF P
+1 (t)/3 explains the different shape of ˆF2(t), as
+displayed in the right panel there, when compared to F P
+2 (t), as shown in the middle panel of Fig. 2 here.
+25Observe that
+2√r + tr1/6 = 2√n + tn1/6 + O(n1/10)
+uniformly for t0 ⩽ t ⩽ t1 and n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+21
+We write the induced Poisson generating function, and exponential generating function, of
+the length distribution as
+(54)
+Pk(z) := P(z; lk) = e−z
+∞
+�
+n=0
+P(Ln ⩽ lk)zn
+n! ,
+fk(z) := ezPk(z).
+By (1) we have Pn(r) = Ehard
+2
+(4r; ln) for real r > 0, so that Thm. 4.1 (see also (49)) gives
+the expansion
+(55)
+Pn(r) = F(t) +
+m
+�
+j=1
+F P
+j (t) r−j/3 + O(r−(m+1)/3)
+����
+t=tln(r)
+,
+uniformly valid when n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞, m being any fixed integer in the
+range 0 ⩽ m ⩽ m∗. Here, the implied constant in the error term depends only on t0, t1,
+but not on the specific sequence ln. Preserving uniformity, the expansion can be repeatedly
+differentiated w.r.t. the variable r. In particular, using the differential equation (44b) we get
+that P (j)
+n (n) expands in powers of n−1/3, starting with a leading order term of the form
+(56a)
+P (j)
+n (n) = (−1)jF (j)(t∗
+n)n−2j/3 + O(n−(2j+1)/3)
+(n → ∞);
+the first specific cases being (see (72) for P ′
+n(n))
+Pn(n) = F(t∗
+n) + F P
+1 (t)n−1/3 + F P
+2 (t)n−2/3 + O(n−1)
+����
+t=t∗n
+,
+(56b)
+P ′′
+n(n) = F ′′(t∗
+n)n−4/3 +
+�
+F P
+1
+′′(t) + 5
+6F ′(t) + t
+3F ′′(t)
+�
+n−5/3 + O(n−2)
+����
+t=t∗n
+,
+(56c)
+P (4)
+n (n) = F (4)(t)n−8/3 + O(n−3)
+����
+t=t∗n
+;
+(56d)
+the implied constants in the error terms depend only on t0, t1.
+We recall from the results of [16, Sect. 2] (note the slight differences in notation), and the
+proofs given there, that the exponential generating functions fn(z) are entire functions of genus
+zero having, for each 0 < ϵ < π/2, only finitely many zeros26 in the sector |arg z| ⩽ π/2 + ϵ.
+If we denote the real auxiliary functions (cf. Def. A.1) of fn(r) = erPn(r) by an(r) and bn(r),
+the expansion (55), and its derivatives based on (44b), give (cf. also (73))
+(57)
+an(r) = r + O(r1/3),
+bn(r) = r + O(r2/3),
+uniformly valid when n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞; the implied constants in the error
+terms depend only on t0, t1.
+Now, numerical experiments strongly hint at the property that the zeros of the exponential
+generating functions fn(z) in the sectors |arg z| ⩽ π/2+ϵ satisfy a uniform tameness condition
+as in Def. A.2 (see also Remark A.2): the zeros are neither coming too close to the positive
+real axis nor are they getting too large. Proving this property seems to be rather difficult,
+though—at least we were lacking the methodology to do so. Given this state of affairs, the
+results on the expansions of the length distribution will be subject to the following:
+Tameness hypothesis. For any real t0 < t1 and any sequence of integers ln → ∞ satisfy-
+ing (53) the zeros of the induced family fn(z) of exponential generating functions (54) are
+uniformly tame (see Def. A.2), with parameters and implied constants only depending on t0
+and t1.
+26Because of fn(r) > 0 for r > 0, the real zeros of fn are negative and the complex ones are coming in
+conjugate pairs.
+
+22
+FOLKMAR BORNEMANN
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.3
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+Figure 3. Plots of F D
+1 (t) (left panel) and F D
+2 (t) (middle panel) as in (60); both agree with
+the numerical prediction of their graphical form given in the left panels of [16, Figs. 4/6].
+The right panel shows F D
+3 (t) as in (110) (black solid line) with the approximations (59) for
+n = 250 (red +), n = 500 (green ◦) and n = 1000 (blue •); the integer l has been varied
+such that tl(n) spreads over the range of t displayed here. Evaluation of (59) uses the table
+of exact values of P(Ln ⩽ l) up to n = 1000 that was compiled in [16].
+Theorem 5.1. Let t0 < t1 be any real numbers and assume the tameness hypothesis. Then
+there holds the expansion
+(58)
+P(Ln ⩽ l) = F(t) +
+m
+�
+j=1
+F D
+j (t) n−j/3 + O(n−(m+1)/3)
+����
+t=tl(n)
+,
+which is uniformly valid when n, l → ∞ subject to t0 ⩽ tl(n) ⩽ t1 with m being any fixed
+integer in the range 0 ⩽ m ⩽ m∗. Here the F D
+j
+are certain smooth functions; the first two
+are27
+F D
+1 (t) = − t2
+60F ′(t) − 3
+5F ′′(t),
+(60a)
+F D
+2 (t) =
+�
+− 139
+350 + 2t3
+1575
+�
+F ′(t) +
+�
+− 43t
+350 +
+t4
+7200
+�
+F ′′(t) + t2
+100F ′′′(t) + 9
+50F (4)(t),
+(60b)
+while a similar expression for F D
+3 (t) is displayed in (110).
+Proof. Following up the preparations preceding the formulation of the theorem, the tameness
+hypothesis allows us to apply Corollary A.1, bounding fn(z) = ezPn(z) by
+(61)
+��fn(reiθ)
+�� ⩽
+�
+�
+�
+2fn(r)e− 1
+2 θ2r,
+0 ⩽ |θ| ⩽ r−2/5,
+2fn(r)e− 1
+2 r1/5,
+r−2/5 ⩽ |θ| ⩽ π,
+for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 when n0 is sufficiently large (depending on t0, t1).
+Using the trivial bounds (for r > 0 and |θ| ⩽ π)
+0 ⩽ fn(r) ⩽ er,
+0 ⩽ Pn(r) ⩽ 1,
+1 − 1
+2θ2 ⩽ cos θ,
+the first case in (61) can be recast in form of the bound
+��Pn(reiθ)
+�� ⩽ 2Pn(r)er
+�
+1−cos θ− 1
+2 θ2�
+⩽ 2,
+which proves condition (I) of Thm. A.1 with B = 2, D = 1, β = 0 and δ = 2/5; whereas the
+second case implies
+|fn(neiθ)| ⩽ 2fn(n)e− 1
+2 n1/5 ⩽ 2 exp
+�
+n − 1
+2n1/5�
+,
+27To validate the expansion (58) and the formulae (60a/b), (110), Fig. 3 plots F D
+3 (t) next to the approximation
+(59)
+F D
+3
+�
+tl(n)
+�
+≈ n−1�
+P(Ln ⩽ l) − F(t) − F D
+1 (t)n−1/3 − F D
+2 (t)n−2/3� ����
+t=tl(n)
+.
+for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl(n) spreads over [−6, 3].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+23
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+Figure 4. Plots of F ∗
+1 (t) (left panel) and F ∗
+2 (t) (middle panel) as in (65); both agree with
+the numerical prediction of their graphical form given in the right panels of [16, Figs. 4/6].
+The right panel shows F ∗
+3 as in (111) (black solid line) with the approximations (64) for
+n = 250 (red +), n = 500 (green ◦) and n = 1000 (blue •); the integer l has been varied
+such that tl−1/2(n) spreads over the range of t displayed here. Evaluation of (64) uses the
+table of exact values of P(Ln = l) up to n = 1000 that was compiled in [16].
+which proves condition (O) of Thm. A.1 with A = 1/2, C = 0, α = 1/5 and γ = 0. Hence,
+there holds the Jasz expansion (92), namely
+P(Ln ⩽ ln) = Pn(n) +
+M
+�
+j=2
+bj(n)P (j)
+n (n) + O(n−(M+1)/5)
+for any M = 0, 1, 2, . . . as n ⩾ n1; here n1 and the implied constant depend on t0, t1.
+By noting that the diagonal Poisson–Charlier polynomials bj have degree ⩽ ⌊j/2⌋ and by
+choosing M large enough, the expansions (56) of P (j)
+n (n) in terms of powers of n−1/3 yield
+that there are smooth functions F D
+j
+such that
+P(Ln ⩽ ln) = F(t) +
+m
+�
+j=1
+F D
+j (t) n−j/3 + O(n−(m+1)/3)
+����
+t=t∗n
+as n → ∞; m being any integer in the range 0 ⩽ m ⩽ m∗. Given the uniformity of the bound
+for fixed t0 and t1, we can replace ln by l and t∗
+n by tl(n) as long as we respect t0 ⩽ tl(n) ⩽ t1.
+This finishes the proof of (58).
+The first two functions F D
+1 and F D
+2 (t) can be determined using the particular case (93) of
+the Jasz expansion from Example A.1 (which applies here because of (56a)), namely
+P(Ln ⩽ ln) = Pn(n) − n
+2 P ′′
+n(n) + n2
+8 P (4)
+n (n) + O(n−1).
+Inserting the formulae displayed in (56) we thus obtain
+F D
+1 (t) = F P
+1 (t) − 1
+2F ′′(t)
+(62a)
+F D
+2 (t) = F P
+2 (t) − 1
+2F P
+1
+′′(t) − 5
+12F ′(t) − t
+6F ′′(t) + 1
+8F (4)(t).
+(62b)
+Together with the expressions given in (47) this yields the functional form asserted in (60).
+□
+5.2. Expansion of the PDF. Subject to its assumptions, Thm. 5.1 implies for the PDF of
+the length distribution that
+P(Ln = l) = P(Ln ⩽ l) − P(Ln ⩽ l − 1)
+=
+�
+F(tl(n)) − F(tl−1(n))
+�
++
+m
+�
+j=1
+�
+F D
+j (tl(n)) − F D
+j (tl−1(n))
+�
+n−j/3 + O(n−(m+1)/3).
+
+24
+FOLKMAR BORNEMANN
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+ℙ
+Mo
+50
+52
+54
+56
+58
+60
+62
+64
+66
+68
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+ℙ
+Mh
+Figure 5. The exact discrete length distribution P(Ln = l) (blue bars centered at the
+integers l) vs. the asymptotic expansion (63) for m = 0 (the Baik–Deift–Johansson limit,
+dotted line) and for m = 2 (the limit with the first two finite-size correction terms added,
+solid line). Left: n = 100; right: n = 1000. The expansions are displayed as functions of the
+continuous variable ν, evaluating the right-hand-side of (63) in t = tν−1/2(n). The exact
+values are from the table compiled in [16]. Note that a graphically accurate continuous
+approximation of the discrete distribution must intersect the bars right in the middle of
+their top sides: this is, indeed, the case for m = 2 (except at the left tail for n = 100). In
+contrast, the uncorrected limit law (m = 0) is noticeable inaccurate for this range of n.
+Applying the central differencing formula (which is, basically, just a Taylor expansion for
+smooth G centered at the midpoint)
+G(t + h) − G(t) = hG′(t + h/2) + h3
+24G′′′(t + h/2) +
+h5
+1920G(5)(t + h/2) + · · · ,
+with increment h = n−1/6, we immediately get the following corollary of Thm. 5.1.
+Corollary 5.1. Let t0 < t1 be any real numbers and assume the tameness hypothesis. Then
+there holds the expansion
+(63)
+n1/6 P(Ln = l) = F ′(t) +
+m
+�
+j=1
+F ∗
+j (t)n−j/3 + O(n−(m+1)/3)
+����
+t=tl−1/2(n)
+,
+which is uniformly valid when n, l → ∞ subject to the constraint t0 ⩽ tl−1/2(n) ⩽ t1 with m
+being any fixed integer in the range 0 ⩽ m ⩽ m∗. Here the F ∗
+j are certain smooth functions;
+the first two are28
+F ∗
+1 (t) = − t
+30F ′(t) − t2
+60F ′′(t) − 67
+120F ′′′(t),
+(65a)
+F ∗
+2 (t) = 2t2
+525F ′(t) +
+�
+− 629
+1200 + 23t3
+12600
+�
+F ′′(t) +
+�
+− 899t
+8400 +
+t4
+7200
+�
+F ′′′(t)
+(65b)
++ 67t2
+7200F (4)(t) + 1493
+9600F (5)(t),
+while a similar expression for F ∗
+3 (t) is displayed in (111).
+Remark 5.1. The case m = 0 of Corollary 5.1 gives
+n1/6 P(Ln = l) = F ′(tl−1/2(n)) + O(n−1/3),
+28To validate the expansion (63) and the formulae (65a/b), (111), Fig. 4 plots F ∗
+3 (t) next to the approximation
+(64)
+F ∗
+3
+�
+tl−1/2(n)
+�
+≈ n7/6�
+P(Ln = l) − F ′(t)n−1/6 − F ∗
+1 (t)n−1/2 − F ∗
+2 (t)n−5/6� ����
+t=tl−1/2(n)
+.
+for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl−1/2(n) spreads over [−6, 3].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+25
+where the exponent in the error term cannot be improved. By noting
+tl−1/2(n) = tl(n) − 1
+2n−1/6
+we understand that, for fixed large n, visualizing the discrete length distribution near its
+mode by plotting the points
+�
+tl(n), n1/6 P(Ln = l)
+�
+next to the graph (t, F ′(t)) introduces a perceivable bias: namely, all points are shifted by an
+amount of n−1/6/2 to the right of the graph. Exactly such a bias can be observed in the first
+ever published plot of the PDF vs. the density of the Tracy–Widom distribution by Odlyzko
+and Rains in [48, Fig. 1]: the Monte-Carlo data for n = 106 display a consistent shift by 0.05.
+6. Expansions of Stirling-Type Formulae
+In our work [16] we advocated the use of a Stirling-type formula to approximate the length
+distribution for larger n (because of being much more efficient and accurate than Monte-Carlo
+simulations). To recall some of our findings there, let us denote the exponential generating
+function and its Poisson counterpart simply by
+f(z) =
+∞
+�
+n=0
+P(Ln ⩽ l)zn
+n! ,
+P(z) = e−zf(z),
+suppressing the dependence on the integer parameter l from the notation for the sake of
+brevity. It was shown in [16, Thm. 2.2] that the entire function f is H-admissible so that
+there is the normal approximation (see Def. A.1 and Thm. A.2)
+(66)
+P(Ln ⩽ l) =
+n!f(r)
+rn�
+2πb(r)
+�
+exp
+�
+−(n − a(r))2
+2b(r)
+�
++ o(1)
+�
+(r → ∞)
+uniformly in n = 0, 1, 2, . . . while l is any fixed integer. Here, a(r) and b(r) are the real
+auxiliary functions
+a(r) = rf′(r)
+f(r) ,
+b(r) = ra′(r).
+We consider the two cases r = rn, a(rn) = n and r = n. After dividing (66) by the classical
+Stirling factor (which does not change anything of substance; see Remark 6.1)
+(67)
+τn :=
+n!
+√
+2πn
+� e
+n
+�n
+∼ 1 + n−1
+12 + n−2
+288 + · · · ,
+we get after some re-arranging of terms the Stirling-type formula Sn,l (r = rn) and the
+simplified Stirling-type formula ˜Sn,l (r = n):
+Sn,l :=
+P(rn)
+�
+b(rn)/n
+exp
+�
+n Λ
+�rn − n
+n
+��
+,
+Λ(h) = h − log(1 + h),
+(68a)
+˜Sn,l :=
+P(n)
+�
+b(n)/n
+exp
+�
+−(n − a(n))2
+2b(n)
+�
+.
+(68b)
+As shown in [16], both approximations are amenable for a straightforward numerical evaluation
+using the tools developed in [13, 14, 17]. For fixed l, the normal approximation (66) implies
+P(Ln ⩽ l) = Sn,l · (1 + o(1))
+(n → ∞);
+but numerical experiments reported in [16, Fig. 3, Eq. (8b)] suggest that there holds
+P(Ln ⩽ l) = Sn,l + O(n−2/3)
+
+26
+FOLKMAR BORNEMANN
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.12
+-0.1
+-0.08
+-0.06
+-0.04
+-0.02
+0
+0.02
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.02
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+Figure 6. Left panel: plots of ˜F S
+2 (t) (solid line) and ˜F S
+2 (t) (dash-dotted line) as in (70). The
+middle and right panel show the approximations of F S
+3 (t) and ˜F S
+3 (t) in (77) for n = 250
+(red +), n = 500 (green ◦) and n = 1000 (blue •); the integer l has been varied such that
+tl(n) spreads over the range of the variable t displayed here; the dotted line displays a
+polynomial fit to the data points of degree 30 to help visualizing their joint graphical form.
+Evaluation of (77) uses the table of exact values of P(Ln ⩽ l) up to n = 1000 that was
+compiled in [16].
+uniformly when n, l → ∞ while tl(n) stays bounded. Subject to the tameness hypothesis of
+Thm. 5.1 we prove this observation as well as its counterpart for the simplified Stirling-type
+formula, thereby unveiling the functional form of the error term O(n−2/3):29
+Theorem 6.1. Let t0 < t1 be any real numbers and assume the tameness hypothesis. Then,
+for the Stirling-type formula Sn,l and its simplification ˜Sn,l, there hold the expansions (note
+that both are starting at j = 2)
+P(Ln ⩽ l) = Sn,l +
+m
+�
+j=2
+F S
+j
+�
+tl(n)
+�
+n−j/3 + O(n−(m+1)/3),
+(69a)
+P(Ln ⩽ l) = ˜Sn,l +
+m
+�
+j=2
+˜F S
+j
+�
+tl(n)
+�
+n−j/3 + O(n−(m+1)/3),
+(69b)
+which are uniformly valid when n, l → ∞ subject to t0 ⩽ tl(n) ⩽ t1 with m being any fixed
+integer in the range 0 ⩽ m ⩽ m∗. Here the F S
+j and ˜F S
+j are certain smooth functions; the first
+of them being30
+F S
+2 (t) = −3
+4
+F ′(t)4
+F(t)3 + 3
+2
+F ′(t)2F ′′(t)
+F(t)2
+− 3
+8
+F ′′(t)2
+F(t) − 1
+2
+F ′(t)F ′′′(t)
+F(t)
++ 1
+8F (4)
+2 (t),
+(70a)
+˜F S
+2 (t) = −1
+2F ′(t) + 1
+4
+F ′(t)4
+F(t)3 − 3
+8
+F ′′(t)2
+F(t) + 1
+8F (4)
+2 (t).
+(70b)
+The solution rn of the equation a(rn) = n, required to evaluate Sn,l, satisfies the expansion31
+rn = n + F ′�
+tl(n)
+�
+F
+�
+tl(n)
+� n1/3 + O(1),
+which is uniformly valid under the same conditions.
+29Note that the expansions (74) for ˜Sn,l and (76) for Sn,l given in the proof do not require the tameness
+hypothesis. It is only required to facilitate the comparison with the result of Thm. 5.1, which then yields (69).
+30The functional form of the terms F S
+2 , ˜F S
+2 differs significantly from the one of corresponding terms in the
+previous theorems. Though they still share the form
+F(t) ·
+�
+rational polynomial in u00(t), u10(t), u11(t), u20(t), u21(t), u30(t)
+�
+,
+using the algorithmic ideas underlying the tabulation of F(t) · ujk(t) (0 ⩽ j + k ⩽ 8) in [59, p. 68] one can
+show that F S
+2 (t) and ˜F S
+2 do not simplify to the form (39b) of a linear combination of derivatives of F with
+(rational) polynomial coefficients.
+31This provides excellent initial guesses for solving a(rn) = n by iteration; cf. [16, Sect. 3.4].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+27
+Proof. We restrict ourselves to the case m = 2, focussing on the concrete functional form of
+the expansion terms; nevertheless the general form of the expansions (69) should become
+clear along the way.
+Preparatory steps. Because of P(r) = Ehard
+2
+(4r; l) (using the notation preceding Thm. 6.1),
+Thm. 4.1 gives that
+(71)
+P(r) = F(t) + F P
+1 (t) · r−1/3 + F P
+2 (t) · r−2/3 + O(r−1)
+���
+t=tl(r) ,
+which is uniformly valid when r, l → ∞ subject to the constraint t0 ⩽ tl(r) ⩽ t1 (the same
+constraint applies to the expansions to follow). Preserving uniformity, the expansion can be
+repeatedly differentiated w.r.t. the variable r, which yields by using the differential equation
+(44b) satisfied by tl(r) (cf. also (56a))
+(72)
+P ′(r) = −F ′(t)r−2/3 −
+�
+F P
+1
+′(t) + t
+6F ′(t)
+�
+r−1 + O(r−4/3)
+����
+t=tl(r)
+.
+Recalling f(r) = erP(r), we thus get
+(73a)
+a(r) = r + rP ′(r)
+P(r) = r + a1(t)r1/3 + a2(t) + O(r−1/3)
+���
+t=tl(r)
+with the coefficient functions
+(73b)
+a1(t) = −F ′(t)
+F(t) ,
+a2(t) = −
+1
+F(t)
+�
+F P
+1
+′(t) + t
+6F ′(t)
+�
++ F P
+1 (t)F ′(t)
+F(t)2
+;
+a further differentiation yields
+(73c)
+b(r) = ra′(r) = r − a′
+1(t)r2/3 +
+�1
+3a1(t) − t
+6a′
+1(t) − a′
+2(t)
+�
+r1/3 + O(1)
+����
+t=tl(r)
+.
+The simplified Stirling-type formula. Here we have r = n and we write t∗ := tl(n) to be
+brief. By inserting the expansions (71) and (73) into the expression (68b), we obtain after a
+routine calculation with truncated power series and collecting terms as in (62) that
+(74)
+˜Sn,l = F(t∗) + F D
+1 (t∗)n−1/3 +
+�
+F D
+2 (t∗) − ˜F S
+2 (t∗)
+�
+n−2/3 + O(n−1),
+where the remaining ˜F S
+2 (t) is given by (70b); a subtraction from (58) yields (69b).
+The Stirling-type formula. Here we have r = rn and we have to distinguish between t∗ and
+tl(r) = t∗ · (n/r)1/6 + 2
+√n − √r
+r1/6
+.
+By inserting the expansion
+(75a)
+rn = n + r1(t∗)n1/3 + r2(t∗) + O(n−1/3)
+into tn(rn) and a(rn) we obtain
+tl(rn) = t∗ − r1(t∗)n−1/3 −
+�
+r2(t∗) + t∗
+6 r1(t∗)
+�
+n−2/3 + O(n−1)
+(75b)
+a(rn) = n + (a1(t∗) + r1(t∗))n1/3 +
+�
+a2(t∗) + r2(t∗) − r1(t∗)a′
+1(t∗)
+�
++ O(n−1/3).
+(75c)
+Thus the solution of a(rn) = n, which by Thm. A.2 is unique, leads to the relations
+(75d)
+r1(t) = −a1(t),
+r2(t) = −a2(t) − a1(t)a′
+1(t).
+By inserting, first, the expansions (75) into the expansions (71) and (73) for the particular
+choice r = rn and, next, the thus obtained results into the expression (68a), we obtain after
+a routine calculation with truncated power series and collecting terms as in (62) that
+(76)
+Sn,l = F(t∗) + F D
+1 (t∗)n−1/3 +
+�
+F D
+2 (t∗) − F S
+2 (t∗)
+�
+n−2/3 + O(n−1),
+where the remaining F S
+2 (t) is given by (70a); a subtraction from (58) yields (69a).
+□
+
+28
+FOLKMAR BORNEMANN
+To validate the expansions (69) and the formulae (70a/b), Fig. 6 plots the approximations
+F S
+3
+�
+tl(n)
+�
+≈ n−1�
+P(Ln ⩽ l) − Sn,l − F S
+2
+�
+tl(n)
+�
+n−2/3�
+,
+(77a)
+˜F S
+3
+�
+tl(n)
+�
+≈ n−1�
+P(Ln ⩽ l) − ˜Sn,l − ˜F S
+2
+�
+tl(n)
+�
+n−2/3�
+(77b)
+for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl(n) spreads
+over [−6, 3]. The plot suggests the following observations:
+• Apparently there holds ˜F S
+2 (t) < F S
+2 (t) < 0 for t ∈ [−6, 3], which if generally true
+would imply
+P(Ln ⩽ l) < Sn,l < ˜Sn,l
+for n being sufficiently large and l near the mode of the distribution. This one-sided
+approximation of the length distribution by the Stirling formula Sn,l from above is
+also clearly visible in [16, Tables 1/2].
+• Comparing F S
+2 (t) in Fig. 6 to F D
+2 (t) in Fig. 3 shows that the maximum error
+max
+l=1,...,n
+���P(Ln ⩽ l) −
+�
+F(t) + F D
+1 (t)n−1/3���
+t=tl(n)
+��� ≈ n−2/3∥F D
+2 ∥∞ ≈ 0.25n−2/3
+of approximating the length distribution by the first finite-size correction in Thm. 5.1
+is about an order of magnitude larger than the maximum error of the Stirling-type
+formula,
+max
+l=1,...,n |P(Ln ⩽ l) − Sn,l| ≈ n−2/3∥F S
+2 ∥∞ ≈ 0.031n−2/3.
+This property of the Stirling-type formula was already observed in [16, Fig. 3] and
+was used there to approximate the graphical form of F D
+2 (t) (see [16, Fig. 6]).
+Remark 6.1. If one includes the classical Stirling factor (67) into the Stirling-type formula
+by replacing (68a) with the unmodified normal approximation (66), that is, with
+S∗
+n,l := τn
+P(rn)
+�
+b(rn)/n
+exp
+�
+n Λ
+�rn − n
+n
+��
+= τnSn,l,
+Thm. 6.1 would remain valid: in fact, multiplication of (69a) by the expansion (67) of τn in
+powers of n−1 gives, by taking (58) into account,
+P(Ln ⩽ l) = S∗
+n,l +
+m
+�
+j=2
+F S∗
+j
+�
+tl(n)
+�
+n−j/3 + O(n−(m+1)/3),
+where the first two coefficient functions are
+F S∗
+2 (t) = F S
+2 (t),
+F S∗
+3 (t) = F S
+3 (t) − 1
+12F(t),
+. . . .
+Because of limt→+∞ F(t) = 1, we would loose the decay of F S
+3 (t) for large t, leaving us
+with a non-zero residual value coming from the classical Stirling factor. For this reason, we
+recommend dropping the factor τn, thereby resolving an ambiguity expressed in [16, Fn. 28].
+7. Expansions of Expected Value and Variance
+Lifting the expansion (63) of the PDF of the length distribution to one of the expected
+value and variance requires a control of the tails (of the distribution itself and of the expansion
+terms) which, at least right now, we can only conjecture to hold true.
+To get to a reasonable conjecture, we recall the tail estimates for the discrete distribution
+(see [7, Eqn. (9.6/9.12)]),
+P(Ln = l) ⩽ Ce−c|tl(n)|3
+(tl(n) ⩽ t0 < 0),
+P(Ln = l) ⩽ Ce−c|tl(n)|3/5
+(0 < t1 ⩽ tl(n)),
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+29
+when n is large enough with c > 0 being some absolute constant and C > 0 a constant that
+depends on t0, t1.
+On the other hand, from Thm. 5.1 and its proof we see that the F ∗
+j (t) take the form
+(78)
+F(t) ·
+�
+rational polynomial in terms of the form tr((I − K0)−1K)|L2(t,∞)
+�
+,
+where the kernels K are finite sums of rank one kernels with factors of the form (16). The
+results of Sect. 2.3 thus show that the F ∗
+j (t) are exponentially decaying when t → ∞.
+Now, looking at the left tail, the (heuristic) estimate of the largest eigenvalue of the Airy
+operator K0 on L2(t, ∞) as given in the work of Tracy and Widom [64, Eq. (1.23)] shows a
+superexponential growth bound of the operator norm
+∥(I − K0)−1∥ ⩽ C|t|−3/4ec|t|3/2
+(t ⩽ t0).
+This, together with the superexponential decay (see [6, Cor. 1.3] and [24, Thm. 1] for the
+specific constants)
+0 ⩽ F(t) ⩽ C|t|−1/8e−c|t|3
+(t ⩽ t0)
+of the Tracy–Widom distribution itself, and with an at most polynomial growth of the trace
+norms ∥K|J 1(t,∞) as t → −∞, shows that the bounds for the discrete distribution find a
+counterpart for the expansion terms F ∗
+j (t):
+(79)
+|F ∗
+j (t)| ⩽ Ce−c|t|3
+(t ⩽ t0 < 0),
+|F ∗
+j (t)| ⩽ Ce−c|t|
+(0 < t1 ⩽ t).
+Thus, assuming an additional amount of uniformity that would allow us to absorb the
+exponentially small tails in the error term of Corollary 5.1, we conjecture the following:
+Uniform Tails Hypothesis. The expansion (63) can be sharpened to include the tails in
+the form
+(80)
+n1/6 P(Ln = l) = F ′(t) +
+m
+�
+j=1
+F ∗
+j (t)n−j/3 + n−(m+1)/3 · O
+�
+e−c|t|3/5� ����
+t=tl−1/2(n)
+,
+uniformly valid in l = 1, . . . , n as n → ∞.
+We now follow the ideas sketched in our work [16, §4.3] on the Stirling-type formula. By
+shift and rescale, the expected value of Ln can be written in the form
+E(Ln) =
+n
+�
+l=1
+l · P(Ln = l) = 2√n + 1
+2 +
+n
+�
+l=1
+tl−1/2(n) · n1/6 P(Ln = l).
+Inserting the expansion (80) of the uniform tail hypothesis gives, since its error term is
+uniformly summable,
+(81)
+E(Ln) = 2√n + 1
+2 +
+m
+�
+j=0
+µ(n)
+j
+n1/6−j/3 + O(n−(m+1)/3),
+with coefficients (still depending on n, though), writing F ∗
+0 := F ′,
+µ(n)
+j
+:= n−1/6
+n
+�
+l=1
+tl−1/2(n)F ∗
+j
+�
+tl−1/2(n)
+�
+.
+By the tail estimates (79) we have, writing a := −n−1/6(2√n + 1
+2) and h := n−1/6,
+µ(n)
+j
+= h
+∞
+�
+l=−∞
+(a + lh)F ∗
+j (a + lh) + O(e−cn1/3).
+Now, based on a precise description of its pole field in [38], it is known that the Hastings–
+McLeod solution of Painlevé II and a fortiori, by the Tracy–Widom theory [64], also F and
+its derivatives can be continued analytically to the strip |ℑz| < 2.9. Therefore, we assume:
+
+30
+FOLKMAR BORNEMANN
+Uniform Strip Hypothesis. The F ∗
+j (j = 1, . . . , m) extend analytically to a strip |ℑz| ⩽ s
+of the complex z-plane, uniformly converging to 0 as z → ∞ in that strip.
+Under that hypothesis, a classical result about the rectangular rule in quadrature theory
+(see, e.g., [23, Eq. (3.4.14)]) gives
+h
+∞
+�
+l=−∞
+(a + lh)F ∗
+j (a + lh) =
+� ∞
+−∞
+F ∗
+j (t) dt + O(e−πs/h).
+Thus the µ(n)
+j
+and their limit quantities
+µj =
+� ∞
+−∞
+tF ∗
+j (t) dt
+differ only by an exponential small error of at most O(e−cn1/6), which can be absorbed in
+the error term of (81); an illustration of such a rapid convergence is given in [16, Table 3] for
+the case j = 0.
+The functional form of F ∗
+0 = F ′, F ∗
+1 and F ∗
+2 , namely being a linear combination of higher
+order derivatives of F with polynomial coefficients (see (65)), allows us to express µ0, µ1, µ2
+in terms of the moments
+Mj :=
+� ∞
+−∞
+tjF ′(t) dt
+of the Tracy–Widom distribution F. In fact, repeated integration by parts yields the simpli-
+fying rule (where k ⩾ 1)
+� ∞
+−∞
+tj F (k)(t) dt =
+�
+�
+�
+�
+�
+(−1)k−1j!
+(j − k + 1)!Mj−k+1
+k ⩽ j + 1,
+0
+otherwise.
+Repeated application of that rule proves, in summary, the following contribution to Ulam’s
+problem about the expected value when n grows large.
+Theorem 7.1. Let m be any fixed integer in the range 0 ⩽ m ⩽ m∗. Then, under the uniform
+tails hypothesis and the uniform strip hypothesis there holds, as n → ∞,
+(82)
+E(Ln) = 2√n + 1
+2 +
+m
+�
+j=0
+µjn1/6−j/3 + O(n−1/6−m/3),
+where the constants µj are given by
+µ0 =
+� ∞
+−∞
+tF ′(t) dt,
+µj =
+� ∞
+−∞
+tF ∗
+j (t) dt
+(j = 1, 2, . . .).
+The first few cases can be expressed in terms of the moments of the Tracy–Widom distribution:
+(83)
+µ0 = M1,
+µ1 = 1
+60M2,
+µ2 = 89
+350 −
+1
+1400M3,
+while a similar expression for µ3 is displayed in (112); highly accurate numerical values are
+listed in Table 1.
+Likewise, by shift and rescale, the variance of Ln can be written in the form
+Var(Ln) =
+n
+�
+l=1
+l2 · P(Ln = l) − E(Ln)2
+= n1/3
+n
+�
+l=1
+tl−1/2(n)2 · P(Ln = l) −
+�
+E(Ln) − 2√n − 1
+2
+�2
+.
+By inserting the expansions (80), (82) and arguing as for Thm. 7.1 we get the following:
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+31
+Table 1. Highly accurate values of µ0, . . . , µ3 and ν0, . . . , ν3 as computed from (83), (85)
+and from (112), (113) based on values for Mj obtained as in [16, Table 3] (cf. Prähofer’s
+values for M1, . . . , M4, published in [59, p. 70]). For the values of µ4, µ5 and ν4, ν5 see the
+supplementary material mentioned in Fn. 39.
+j
+Mj
+µj
+νj
+0
+1.00000 00000 00000 00000 · · ·
+−1.77108 68074 11601 62598 · · ·
+0.81319 47928 32957 84477 · · ·
+1
+−1.77108 68074 11601 62598 · · ·
+0.06583 23878 70339 62521 · · ·
+−1.20720 50777 85797 46901 · · ·
+2
+3.94994 32722 20377 51300 · · ·
+0.26122 27462 52162 60525 · · ·
+0.56715 66368 69744 43503 · · ·
+3
+−9.71184 47530 27647 35361 · · ·
+−0.11938 39067 94582 09131 · · ·
+0.01669 21858 10456 60764 · · ·
+4
+26.02543 54268 39994 56536 · · ·
+−0.00483 35524 95005 83878 · · ·
+−0.12447 09934 16776 05579 · · ·
+5
+−74.20410 74434 81824 47477 · · ·
+0.01222 78407 77590 95405 · · ·
+−0.00293 40551 03931 43008 · · ·
+Corollary 7.1. Let m be any fixed integer in the range 0 ⩽ m ⩽ m∗. Then, under the
+uniform tails hypothesis and the uniform strip hypothesis there holds, as n → ∞,
+(84)
+Var(Ln) =
+m
+�
+j=0
+νjn1/3−j/3 + O(n−m/3),
+with certain constants νj. The first few cases can be expressed in terms of the moments of
+the Tracy–Widom distribution
+(85)
+ν0 = −M2
+1 + M2,
+ν1 = −67
+60 + 1
+30
+�
+− M1M2 + M3
+�
+,
+ν2 = − 57
+175M1 +
+1
+700M1M3 −
+1
+3600M2
+2 −
+29
+25200M4,
+while a similar expression for ν3 is displayed in (113); highly accurate numerical values are
+listed in Table 1.
+The expansions of expected value and variance can be cross-validated by looking at the
+numerical values for the coefficients µ1, µ2, µ3 and ν1, ν2, ν3 that we predicted in [16, §4.3]:
+those values were computed by fitting, in high precision arithmetic, expansions (back then
+only conjectured) of the form (82) with m = 9 and (84) with m = 8 to the exact tabulated
+data for n = 500, . . . , 1000. A decision about which digits were to be considered correct was
+made by comparing the result against a similar computation for n = 600, . . . , 1000. As it
+turns out, the predictions of [16, §4.3] agree to all the decimal places shown there (that is, to
+7, 7, 6 and 9, 6, 4 places) with the theory-based, highly accurate values given in Table 1.
+A. Appendix: Variations on the Saddle Point Method
+A.1. Analytic de-Poissionization and the Jasz expansion. In their comprehensive
+1998 memoir [39], Jacquet and Szpankowski gave a detailed study of what they termed
+analytic de-Poissonization (in form of a useful repackaging of the saddle point method),
+proving a selection of asymptotic expansions and applying them to various asymptotic
+problems in analytic algorithmics and combinatorics (with generating functions given in
+terms of functional equations amenable for checking the Tauberian growth conditions in the
+complex plane). Expositions with a selection of further applications can be found in [40,
+Sect. 7.2] and [63, Chap. 10].
+Formal derivation of the Jasz expansion. Following the ideas of [39, Remark 3] let us start
+with a purely formal derivation to motivate the algebraic form of the expansion. Suppose
+that the Poisson generating function
+P(z) = e−z
+∞
+�
+n=0
+an
+zn
+n!
+
+32
+FOLKMAR BORNEMANN
+of a sequence an is an entire function and consider some r > 0. If we write the power series
+expansion of P(z), centered at z = r, in the operator form
+P(z) = e(z−r)DP(r),
+where D denotes differentiation w.r.t. the variable r, we get by Cauchy’s formula (with a
+contour encircling z = 0 counter-clockwise with index one)
+(86)
+an = n!
+2πi
+�
+P(z)ez dz
+zn+1 = e−rD
+� n!
+2πi
+�
+ez(D+1) dz
+zn+1
+�
+P(r) = e−rD(D + 1)nP(r).
+By the Cauchy product of power series
+(87)
+e−rx(x + 1)n =
+∞
+�
+j=0
+cj(n; r)xj,
+cj(n; r) :=
+j
+�
+k=0
+�n
+k
+�(−r)j−k
+(j − k)! ,
+we get from (86) the formal expansion
+(88)
+an ∼
+∞
+�
+j=0
+cj(n; r)P (j)(r).
+Note that the coefficients cj(n; r) are polynomials of degree j in n and r. From (87) one
+easily verifies that they satisfy the three-term recurrence
+(j + 1)cj+1(n; r) + (j + r − n)cj(n; r) + rcj−1(n; r) = 0
+(j = 0, 1, 2, . . .)
+with initial data c0(n; r) = 1 and c1(n; r) = n − r.
+Remark A.1. From (87) and [62, §2.81] one immediately gets that the cj(n; r) are, up to
+normalization, the Poisson–Charlier polynomials:
+∞
+�
+n=0
+cj(n; r)ck(n; r)e−rrn
+n!
+= δjk
+rj
+j!
+(j, k = 0, 1, 2, . . .),
+so that they are orthogonal w.r.t. the Poisson distribution of intensity r > 0. In particular,
+[62, Eq. (2.81.6)] gives (with L(ν)
+k (x) the Laguerre polynomials) the representation
+cj(n; r) = L(n−j)
+j
+(r).
+Things simplify for the particular choice r = n which is suggested by the expected value of
+the Poisson distribution (cf. Lemma 1.1). The corresponding polynomials bj(n) := cj(n; n),
+which we call the diagonal Poisson–Charlier polynomials, satisfy the three-term recurrence
+(89)
+b0(n) = 1,
+b1(n) = 0,
+(j + 1)bj+1(n) + jbj(n) + nbj−1(n) = 0
+(j = 0, 1, 2, . . .).
+From this we infer inductively that
+bj(0) = 0,
+deg bj ⩽ ⌊j/2⌋
+(j = 1, 2, . . .).
+Now the formal expansion (86) becomes what is dubbed the Jasz expansion in [29, §VIII.18]:
+(90)
+an ∼ P(n) +
+∞
+�
+j=2
+bj(n)P (j)(n)
+= P(n) − n
+2 P ′′(n) + n
+3 P ′′′(n) +
+�n2
+8 − n
+4
+�
+P (4)(n)
++
+�
+−n2
+6 + n
+5
+�
+P (5)(n) +
+�
+−n3
+48 + 13n2
+72
+− n
+6
+�
+P (6)(n) + · · · .
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+33
+Diagonal analytic de-Poissonization. Jacquet and Szpankowski were able to prove that the
+expansion (90) can be made rigorous if the Poisson generating function satisfies a Tauberian
+condition in form of a growth condition at the essential singularity at z = ∞ in the complex
+plane. In fact, this can be cast to accomodate the needs of double scaling limits in a uniform
+fashion: for a two-parameter family of coefficients an,k one expands the diagonal term an,n
+by, first, applying the Jasz expansion w.r.t. to n for k fixed and, then, selecting k = n only
+afterwards (a process that is called diagonal de-Poissonization in [39]).
+The following theorem is a particular case of [39, Thm. 4] (with Ψ = 1 and the modifications
+discussed preceding [39, Eq. (27)]). It repackages the saddle point method (cf. [20, Chap. 5]
+and [69, Chap. VI]) for the asymptotic evaluation of the Cauchy integral
+(91)
+an = n!
+2πi
+�
+P(z)ez dz
+zn+1
+in a far more directly applicable fashion. Concerning the asserted uniform bounds of the
+implied constants, see the beginning of [39, §5.2].
+Theorem A.1 (Jacquet–Szpankowski 1998). Let a family of entire Poisson generating
+functions of the form
+Pk(z) = e−z
+∞
+�
+n=0
+an,k
+zn
+n!
+(k = 0, 1, 2, . . .)
+satisfy the following two conditions32 for n ⩾ n0 where A, B, C, D, α, β, γ, δ are some constants
+with A, α > 0 and 0 ⩽ δ < 1/2:
+(I) If |r − n| ⩽ Dn1−δ and |θ| ⩽ Dr−δ then |Pn(reiθ)| ⩽ Bnβ.
+(O) If |θ| > Dn−δ then |Pn(neiθ) exp(neiθ)| ⩽ Cnγ exp(n − Anα).
+Then, for any m = 0, 1, 2, . . . there holds, when n ⩾ n1 with n1 large enough,
+(92)
+an,n = Pn(n) +
+m
+�
+j=2
+bj(n)P (j)
+n (n) + O
+�
+nβ−(m+1)(1−2δ)�
+,
+where the bj(n) are the diagonal Poisson–Charlier polynomials (89) which have degree ⩽ ⌊j/2⌋
+and satisfy bj(0) = 0 (j ⩾ 1). The implied constant in (92) and the constant n1 depend only
+on n0 and the constants entering the conditions (I) and (O).
+Example A.1. In the proof of Thm. 5.1 we use Thm. A.1 in the particular case β = 0, δ = 2/5
+for a family of Poisson generating functions with (cf. (56a))
+P (j)
+n (n) = O(n−2j/3).
+For m = 4 the expansion (92) is then given by (compare with (90))
+an,n = Pn(n) − n
+2 P ′′
+n(n) + n
+3 P ′′′
+n (n) +
+�n2
+8 − n
+4
+�
+P (4)
+n (n) + O(n−1).
+Since the terms nP ′′′
+n (n)/3 = O(n−1) and −nP (4)
+n (n)/4 = O(n−5/3) get absorbed in the error
+term O(n−1), and can therefore be dropped, the Jasz expansion simplifies in that case to
+(93)
+an,n = Pn(n) − n
+2 P ′′
+n(n) + n2
+8 P (4)
+n (n) + O(n−1).
+32Here, (I) means “inside” and (O) “outside” with respect to the “polynomial cone” {z = reiθ : |θ| ⩽ Dr−δ}.
+
+34
+FOLKMAR BORNEMANN
+A.2. H-admissibility and Hayman’s Theorem XI. In his 1956 memoir [37] on a gener-
+alization of Stirling’s formula, Hayman gave a related but different repackaging of the saddle
+point method for the asymptotic evaluation of the Cauchy integral (91) by introducing the
+notion of H-admissible functions. We collect estimates given in course of the proofs of some
+of Hayman’s theorems that will help us to establish the conditions (I) and (O) required for
+applying analytic de-Poissonization in form of Thm. A.1.
+Definition A.1 (Hayman [37, p. 68]). An entire function f(z) is said to be H-admissible if
+the following four conditions are satisfied:
+– [positivity] for sufficiently large r > 0, there holds f(r) > 0; inducing there the real
+functions (which we call the auxiliary functions associated with f)
+a(r) = rf′(r)
+f(r) ,
+b(r) = ra′(r);
+by Hadamard’s convexity theorem a(r) is monotonely increasing and b(r) is positive.
+– [capture] b(r) → ∞ as r → ∞;
+– [locality] for some function 0 < θ0(r) < π there holds33
+f(reiθ) = f(r)eiθa(r)−θ2b(r)/2 (1 + o(1))
+(r → ∞, |θ| ⩽ θ0(r));
+– [decay] for the angles in the complement there holds
+f(reiθ) = o(f(r))
+�
+b(r)
+(r → ∞, θ0(r) ⩽ |θ| ⩽ π).
+Instead of providing an asymptotic expansion (with an additive error term) as in Thm. A.1,
+H-admissibility gives just a versatile leading order term of an in form of a normal approxi-
+mation. However, the error term is multiplicative then.
+Theorem A.2 (Hayman [37, Thm. I, Cor. II]). Let f be an entire H-admissible function
+with Maclaurin series
+f(z) =
+∞
+�
+n=0
+anzn
+(z ∈ C).
+Then:
+I. [normal approximation] There holds, uniformly in n ∈ N0 = {0, 1, 2, . . .}, that
+(94)
+anrn
+f(r) =
+1
+�
+2πb(r)
+�
+exp
+�
+−(n − a(r))2
+2b(r)
+�
++ o(1)
+�
+(r → ∞).
+II. [Stirling-type formula] For n sufficiently large, it follows from the positivity and
+capture conditions of H-admissibility that a(rn) = n has a unique solution rn such
+that rn → ∞ as n → ∞ and therefore, by the normal approximation (94), there holds
+(95)
+an =
+f(rn)
+rnn
+�
+2πb(rn)
+(1 + o(1))
+(n → ∞).
+For the probabilistic content of the normal approximation (94) see, e.g., [25] and [16,
+Remark 2.1].
+We observe the similarity of the locality and decay conditions to the conditions (I) and
+(O) in the Jacquet–Szpankowski Thm. A.1. In fact, in establishing the H-admissibility of
+certain families of functions, Hayman proved estimates that allow us to infer the validity of
+conditions (I) and (O). A striking example is given by the following theorem, which gives
+uniform bounds for a class of functions that is of particular interest to our study.
+33As is customary in asymyptotic analysis in the complex plane, we understand such asymptotics (and similar
+expansions with o- or O-terms) to hold uniformly in the stated angular segments for all r ⩾ r0 with some
+sufficiently large r0 > 0.
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+35
+Theorem A.3 (Hayman [37, Thm. XI]). Let f be an entire function of genus zero, having
+for some ϵ > 0 no zeros in the sector |arg z| ⩽ π/2 + ϵ. If f satisfies the positivity condition
+of Def. A.1, then there is the universal bound
+(96)
+��f(reiθ)
+�� ⩽
+�
+�
+�
+2f(r)e− 1
+2 θ2b(r),
+0 ⩽ |θ| ⩽ b(r)−2/5,
+2f(r)e− 1
+2 b(r)1/5,
+b(r)−2/5 ⩽ |θ| ⩽ π,
+which is valid when b(r) is large enough to ensure 8b(r)−1/5 csc2(ϵ/2) csc(ϵ) ⩽ log 2. Hence,
+if f also satisfies the capture condition of Def. A.1, then it is H-admissible.
+Proof. Since the bound (96) is hidden in the two-page long proof of [37, Thm. XI] (only the
+H-admissibility is stated explicitly there), we collect the details here. First, [37, Eq. (15.6)]
+states that, if |θ| ⩽ 1/4, then
+log f(reiθ) = log f(r) + iθa(r) − 1
+2θ2b(r) + ϵ(r, θ)
+where the error term is bounded by
+|ϵ(r, θ)| ⩽ c(ϵ) · b(r) |θ|3,
+c(ϵ) := 8 csc2(ϵ/2) csc(ϵ).
+Now, for b(r) large enough to ensure
+b(r)−1/5 ⩽ c(ϵ)−1 log 2 ⩽ min
+�√
+2ϵ, 1/2
+�
+we thus get with 0 < θ0(r) := b(r)−2/5 ⩽ min
+�
+2ϵ, 1/4
+�
+and 0 ⩽ |θ| ⩽ θ0(r) that
+log
+��f(reiθ)
+�� = ℜ log f(reiθ) = log f(r) − 1
+2θ2b(r) + ℜϵ(r, θ),
+��ℜϵ(r, θ)
+�� ⩽ log 2.
+Exponentiation gives, for 0 ⩽ |θ| ⩽ θ0(r),
+��f(reiθ)
+�� ⩽ 2f(r)e− 1
+2 θ2b(r).
+Next, if we combine this estimate with [37, Lemma 8] we get, since θ0(r) ⩽ 2ϵ, that
+��f(reiθ)
+�� ⩽
+��f(reiθ0(r))
+�� ⩽ 2f(r)e− 1
+2 θ0(r)2b(r) = 2f(r)e− 1
+2 b(r)1/5
+(θ0(r) ⩽ |θ| ⩽ π)
+which finishes the proof of the universal bound (96).
+□
+If, instead of having no zeros in the sector |arg z| ⩽ π/2 + ϵ at all, the entire function
+f has a finite number of them, Thm. A.3 remains valid but the lower bound on b(r) will
+now depend on these finitely many zeros. To restore uniformity we consider families of such
+functions whose zeros satisfy the following tameness condition.
+Definition A.2. Let fn be a family of entire functions such that, for some fixed ϵ > 0, each
+of them has finitely many zeros (listed according to their multiplicities)
+zn,1, . . . , zn,mn
+in the sector |arg z| ⩽ π/2 + ϵ, none of them being a positive real number. We call these zeros
+uniformly tame (w.r.t. the positive real axis and w.r.t. infinity) if there are some constants
+1/5 < µ ⩽ 1/3 and ν > 0 such that the family of polynomials
+(97)
+pn(z) = (z − zn,1) · · · (z − zn,mn)
+satisfies
+(98)
+�
+r d
+dr
+�2
+log pn(r) = −
+mn
+�
+j=1
+rzn,j
+(r − zn,j)2 = O(r1−µ),
+|pn(reiθ)| = pn(r)(1 + O(r−ν)),
+uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.
+
+36
+FOLKMAR BORNEMANN
+Remark A.2. Note that a single function f would satisfy condition (98) with error terms of
+the form O(r−1) in both places. Therefore the tameness condition allows us to accommodate
+a significant growth of the implied constants in these O(r−1) terms as n → ∞: in the first
+case because of zeros of fn getting close to the positive real axis and in the second case
+because of them getting large.
+Corollary A.1. Let fn be a family of entire functions of genus zero with positive Maclaurin
+coefficients such that, for some fixed ϵ > 0, each of them has a most finitely many zeros in
+the sector |arg z| ⩽ π/2 + ϵ. If these zeros are uniformly tame in the sense of Def. A.2 and if
+the auxiliary functions belonging to fn satisfy
+(99)
+bn(r) = r + O(r2/3)
+(r → ∞),
+uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞, then there holds the bound
+(100)
+��fn(reiθ)
+�� ⩽
+�
+�
+�
+2fn(r)e− 1
+2 θ2r,
+0 ⩽ |θ| ⩽ r−2/5,
+2fn(r)e− 1
+2 r1/5,
+r−2/5 ⩽ |θ| ⩽ π,
+for all n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0, n0 being sufficiently large. Here n0 depends only
+on the parameters of the tameness condition and the implied constants in (98) and (99).
+Proof. Factoring out the finitely many zeros of fn in the sector |arg z| ⩽ π/2 + ϵ by using
+the polynomials (97), we have
+fn(z) = f∗
+n(z) · pn(z)
+where f∗
+n is an entire function of genus zero that has no zeros in that sector. Since fn(r) > 0
+for r > 0 and the leading coefficient of the polynomial pn(z) is one, f∗
+n satisfies the positivity
+condition of Def. A.1. Denoting the auxiliary functions of f∗
+n by a∗
+n and b∗
+n, the tameness
+condition (98) yields
+bn(r) = b∗
+n(r) +
+�
+r d
+dr
+�2
+log pn(r) = b∗
+n(r) + O(r1−µ),
+|pn(reiθ)| = pn(r)(1 + O(r−ν)),
+uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.
+By (99) this gives b∗
+n(r) = r(1 + O(r−µ)), so that by Thm. A.3 (its proof shows that we
+can take a factor 3/2 instead of 2 if log 2 is replaced by log(3/2) in the lower bound on b(r))
+��f∗
+n(reiθ)
+�� ⩽
+�
+�
+�
+3
+2f∗
+n(r)e− 1
+2 θ2b∗
+n(r),
+0 ⩽ |θ| ⩽ b∗
+n(r)−2/5,
+3
+2f∗
+n(r)e− 1
+2 b∗
+n(r)1/5,
+b∗
+n(r)−2/5 ⩽ |θ| ⩽ π,
+for n large enough to ensure 8b∗
+n(r)−1/5 csc2(ϵ/2) csc(ϵ) ⩽ log(3/2). We write this briefly as
+��f∗
+n(reiθ)
+�� ⩽ 3
+2f∗
+n(r) exp
+�
+− 1
+2 min
+�
+θ2b∗
+n(r), b∗
+n(r)1/5��
+for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 where n0 is large enough (just depending on the
+parameters and the implied constants in the tameness condition). If we multiply this bound
+by
+|pn(reiθ)| = pn(r)(1 + O(r−ν))
+and use b∗
+n(r) = r(1 + O(r−µ)) to infer
+min
+�
+θ2b∗
+n(r), b∗
+n(r)1/5�
+= min(θ2r, r1/5) · (1 + O(r−µ)) = min(θ2r, r1/5) + O(r1/5−µ),
+we obtain the asserted estimate in the compact form
+��fn(reiθ)
+�� ⩽ 2fn(r) exp
+�
+− 1
+2 min
+�
+θ2r, r1/5��
+for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0, where n0 is large enough.
+□
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+37
+A.3. Bessel functions of large order in the transition region. In the 1950s F. Olver
+started a systematic and exhaustive study of asymptotic expansions of the Bessel functions
+Jν(z) for large order ν and argument z. For the transition region34 z = ν + τν1/3 he obtained
+from applying the saddle point method to integral representations of Sommerfeld’s type the
+asymptotic expansion [49, Eq. (3.1)] (cf. also [52, §10.19(iii)])
+(101a)
+Jν(ν + τν1/3) ∼ 21/3
+ν1/3 Ai(−21/3τ)
+∞
+�
+k=0
+Ak(τ)
+ν2k/3 + 22/3
+ν1/3 Ai′(−21/3τ)
+∞
+�
+k=1
+Bk(τ)
+ν2k/3
+valid when |arg ν| ⩽ π/2 − δ < π with τ being any fixed complex number. Here, Ak(τ) and
+Bk(τ) are certain rational polynomials of increasing degree; the first few are [49, Eq. (2.42)]35
+A0(τ) = 1,
+A1(τ) = −1
+5τ,
+A2(τ) = − 9
+100τ 5 + 3
+35τ 2,
+(101b)
+B0(τ) = 0,
+B1(τ) = 3
+10τ 2,
+B2(τ) = −17
+70τ 3 + 1
+70.
+(101c)
+Remark A.3. The sequence [49, Eqs. (2.10), (2.14), (2.18), (2.38), (2.40)] of formulae in Olver’s
+1952 paper gives an actual method36 to calculate Ak(τ) and Bk(τ) (combining reversion and
+nesting of power series with recursive formulae). The degrees of Ak are the positive integers
+congruent to 0, 1 mod 5 (starting with deg A1 = 1) and the degrees of Bk are the positive
+integers congruent to 2, 3 mod 5 (starting with deg B1 = 2). In both families of polynomials
+the coefficients of τ m are zero when m is not congruent mod 3 to the degree.
+As stated in [49, p. 422], the expansion (101) can be repeatedly differentiated with respect
+to τ, valid under the same conditions. For a modern account of differentiability w.r.t. τ
+and ν, adding uniformity for τ from any compact real set, see the recent work of Sher [58,
+Prop. 2.8] which is based on the (microlocal) theory of so-called polyhomogeneous conormal
+joint asymptotic expansions.
+The purposes of Sect. 3 require to identify a larger region of real τ where the expansion (101)
+is uniform as ν → ∞ through positive real values. To this end we use the uniform asymptotic
+expansions of Bessel functions for large order ν, pioneered by Olver [50] in 1954 by analyzing
+turning points of the Bessel differential equation (cf. [51, Chap. 11] and [69, Chap. VIII]):
+(102a)
+Jν(νz) ∼
+�
+4ζ
+1 − z2
+�1/4 �
+Ai(ν2/3ζ)
+ν1/3
+∞
+�
+k=0
+A∗
+k(ζ)
+ν2k
++ Ai′(ν2/3ζ)
+ν5/3
+∞
+�
+k=0
+B∗
+k(ζ)
+ν2k
+�
+,
+uniformly for z ∈ (0, ∞) as ν → ∞. Here, the parameters and coefficients are, for 0 < z < 1,
+(102b)
+2
+3ζ3/2 = log
+�
+1 +
+√
+1 − z2
+z
+�
+−
+�
+1 − z2,
+and
+A∗
+k(ζ) =
+2k
+�
+j=0
+�3
+2
+�j
+vjζ−3j/2U2k−j
+�
+(1 − z2)−1/2�
+,
+(102c)
+B∗
+k(ζ) = −ζ−1/2
+2k+1
+�
+j=0
+�3
+2
+�j
+ujζ−3j/2U2k−j+1
+�
+(1 − z2)−1/2�
+,
+(102d)
+34Where Jν(ν + τν1/3) changes at about τ ≈ 0 from being superexponentially small (to the left) to being
+oscillatory (to the right).
+35Note that we keep the indexing of the polynomials Bk as in [49], which differs from [52, §10.19(iii)].
+36By a Mathematica implementation we extended (and reproduced) Olver’s original table [49, Eq. (2.42)] of
+A0, . . . , An and B0, . . . , Bn from n = 4 to n = 23 in about 180 hours computing time. The polynomials A23
+and B23 of degree 56 and 57 exhibit rational coefficients that are ratios of integers with up to 65 digits.
+
+38
+FOLKMAR BORNEMANN
+where the Uk(x) are recursively defined rational polynomials of degree 3k (cf. [50, Eq. (2.19)])
+and uk, vk (u0 = v0 = 1) are the rational coefficients of the asymptotic expansions of the
+Airy function and its derivative in a sector containing the positive real axis:
+(103)
+Ai(z) ∼
+e−ξ
+2√πz1/4
+∞
+�
+k=0
+(−1)k uk
+ξk ,
+Ai′(z) ∼ −z1/4e−ξ
+2√π
+∞
+�
+k=0
+(−1)k vk
+ξk ,
+ξ = 2
+3z3/2,
+as z → ∞ within |arg z| ⩽ π − δ. Note that ζ = ζ(z) can be continued analytically to the
+z-plane cut along the negative real axis;37 A∗
+k(ζ) and B∗
+k(ζ) can be continued accordingly.
+As stated in [50, p. 342], valid under the same conditions while preserving uniformity, the
+expansion can be repeatedly differentiated with respect to z.
+In particular, with 0 < δ < 1 fixed, the power series expansion
+(104)
+2−1/3ζ = (1 − z) + 3
+10(1 − z)2 + 32
+175(1 − z)3 + · · ·
+converges uniformly for |1 − z| ⩽ 1 − δ (because of the logarithmic singularity at z = 0 the
+radius of convergence of this series is exactly 1, so that this range of uniformity cannot be
+extended). If we put
+νz = ν + τν1/3, i.e.,
+z = 1 + τν−2/3,
+plugging the uniformly convergent series ζ(z) into the uniform large ν expansion (102)
+recovers the form of the transition region expansion (101) and proves that it holds uniformly
+for
+|τ| ⩽ (1 − δ)ν2/3
+as ν → ∞ through positive real values.
+At the expense of considerably larger error terms, this result can be extended as follows:
+Lemma A.1. For any non-negative integer m and any real τ0 there holds, as ν → ∞ through
+positive real values,
+(105)
+Jν(ν + τν1/3) = 21/3 Ai(−21/3τ)
+m
+�
+k=0
+Ak(τ)
+ν(2k+1)/3 + 22/3 Ai′(−21/3τ)
+m
+�
+k=1
+Bk(τ)
+ν(2k+1)/3
++ ν−1−2m/3 · O
+�
+exp(21/3τ)
+�
+,
+uniformly for −ν2/3 < τ ⩽ τ0. Here, Ak(τ) and Bk(τ) are the rational polynomials in (101).
+Preserving uniformity, the expansion (105) can be repeatedly differentiated w.r.t. τ.
+Proof. Let us write
+Jν(ν + τν1/3) = Em(ν; τ) + Rm(ν; τ),
+where Em denotes the sum of the expansion terms in (105) and Rm is the remainder. We
+split the range of τ into the two parts
+(I): − 3
+4ν2/3 ⩽ τ ⩽ τ0,
+(II): − ν2/3 < τ ⩽ −3
+4ν2/3.
+In part (I), as argued above for δ = 1/4, the expansion (101) is uniformly valid, that is,
+Rm(ν; τ) = ν−1−2m/3 · O
+�
+Am+1(τ) Ai
+�
+− 21/3τ
+��
++ ν−1−2m/3O
+�
+Bm+1(τ) Ai′ �
+− 21/3τ
+��
+uniformly for these τ. Now, the superexponential decay of the Airy function Ai(x) and its
+derivative as x → ∞ through positive values, as displayed in the expansions (103), imply the
+asserted uniform bound
+Rm(ν; τ) = ν−1−2m/3 · O
+�
+exp(21/3τ)
+�
+in part (I) of the range of τ.
+37In particular, for positive real z, the thus defined ζ(z) is a strictly monotonically decreasing real function
+with limz→0+ ζ(z) = +∞, ζ(1) = 0 and limz→+∞ ζ(z) = −∞; cf. [52, Eq. (10.20.3)].
+
+ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES
+39
+On the other hand, in part (II) of the range of τ, we infer from (103) that for 0 < ϵ < 1/2
+Em(ν; τ) = O
+�
+exp(−(3ν2/3/4)1+ϵ�
+= ν−1−2m/3 · O
+�
+exp(21/3τ)
+�
+.
+We now show that also
+(106)
+Jν
+�
+ν + τν1/3�
+= ν−1−2m/3 · O
+�
+exp(21/3τ)
+�
+uniformly in part (II) of the range of τ, so that all terms in (105) are absorbed in the asserted
+error term. Here we observe
+0 < z = 1 + τν2/3 ⩽ 1
+4,
+1.095 · · · ⩽ 2
+3ζ3/2 < ∞,
+so that the leading order terms in (102) and (103) yield the bound
+Jν(νz) ∼ ν−1/2
+�
+4
+1 − z2
+�1/4
+(ν2/3ζ)1/4 Ai(ν2/3ζ) = ν−1/2 · O
+�
+exp(− 2
+3ζ3/2ν)
+�
+,
+uniformly for the τ in (II). Because of 2
+3ζ3/2 ⩾ 1.095 and −ν ⩽ −21/3ν2/3 < 21/3τ for ν ⩾ 2,
+this bound can be relaxed, as required, to
+Jν
+�
+ν + τν1/3�
+= Jν(νz) = ν−1−2m/3 · O
+�
+exp(21/3τ)
+�
+.
+Finally, the claim about the derivatives follows from the repeated differentiability of the
+uniform expansion (102) and the differential equation of the Airy function, Ai′′(x) = x Ai(x)
+(so that the general form of the expansions underlying the proof does not change).
+□
+Remark A.4. The cases m = 0 and m = 1 of Lemma A.1 have previously been stated as
+[18, Eq. (4.11)] and [33, Eq. (2.10)]. However, the proofs given there are incomplete: in [18,
+p. 2978] the power series (104) is used up to the boundary of its circle of convergence, so
+that uniformity becomes an issue; whereas in [33, p. 7] it is claimed that Olver’s transition
+expansion (101) would be uniform w.r.t. τ ∈ (−∞, τ0], which is not the case.38
+B. Supplement: Expansion Terms for m = 3
+Put to the extreme, with the help of a CAS such as Mathematica, the methods of the
+present paper can be used to calculate the concrete functional form of the expansion terms for
+up to m = 7 and larger. We refrain, however, from giving any of the computational details39
+and just tabulate the results for m = 3 here.
+Auxiliary Transformation. The transformation of Lemma 3.3 supplements (39) by
+(107)
+˜F3(t) = 64t
+7875F ′(t) − 24t2
+875 F ′′(t) − 122
+7875F ′′′(t) + 16t
+875F (4)(t) −
+1
+750F (6)(t).
+Hard-to-soft edge transition. As a supplement to (24) we have
+(108)
+F3(t) =
+� 64t
+7875 + 1037t4
+7875
+�
+F ′(t) +
+�
+− 9t2
+175 + 48t5
+875
+�
+F ′′(t)
++
+�
+− 122
+7875 − 8t3
+125 + 9t6
+2000
+�
+F ′′′(t) +
+� 16t
+875 − 9t4
+1000
+�
+F (4)(t) + 3t2
+500F (5)(t) −
+1
+750F (6)(t);
+a plot is shown in the right panel of Fig. 1.
+38Besides that the principal branch of Jν(z) (ν ̸∈ Z) is not defined at negative real z, there is a counter-example
+for ν = n being a positive integer: choosing τ = 0 in (101) gives to leading order
+(−1)nJn(−n) = Jn(n) ∼ 21/3n−1/3 Ai(0)
+(n → ∞),
+which differs significantly from applying (101) formally to τ = −2n2/3 (for which n + τn1/3 = −n).
+39A Mathematica notebook with the results for up to m = 7 comes with the sources of the arXiv version of
+the present paper. As for m = 1, 2, we observe also for m = 3, . . . , 7 that the expressions for ˜Fm (and a fortiori
+for Fm, F P
+m, F D
+m , F ∗
+m) take the form of a linear combination of higher order derivatives of the Tracy–Widom
+distribution F with certain rational polynomials as coefficients: we conjecture that this is generally the case.
+
+40
+FOLKMAR BORNEMANN
+Poissonized length distribution. As a supplement to (47) we have
+(109)
+F P
+3 (t) = −
+�
+t
+1125 +
+41t4
+283500
+�
+F ′(t) −
+� 11t2
+6300 +
+t5
+47250
+�
+F ′′(t)
+−
+�
+61
+31500+ 19t3
+63000+
+t6
+1296000
+�
+F ′′′(t)−
+� 11t
+10500+
+t4
+72000
+�
+F (4)(t)−
+t2
+12000F (5)(t)−
+1
+6000F (6)(t);
+a plot is shown in the right panel of Fig. 2.
+CDF of length distribution. As a supplement to (60) we have
+(110)
+F D
+3 (t) = −
+� 562t
+7875 +
+41t4
+283500
+�
+F ′(t) +
+� t2
+300 −
+t5
+47250
+�
+F ′′(t)
++
+� 5137
+15750 + 9t3
+7000 −
+t6
+1296000
+�
+F ′′′(t)+
+� 129t
+1750 −
+t4
+12000
+�
+F (4)(t)− 3t2
+1000F (5)(t)− 9
+250F (6)(t);
+a plot is shown in the right panel of Fig. 3.
+PDF of length distribution. As a supplement to (65) we have
+(111)
+F ∗
+3 (t) = −
+� 373
+5250 + 41t3
+70875
+�
+F ′(t) −
+� 1781t
+28000 +
+71t4
+283500
+�
+F ′′(t)
++
+� 63t2
+8000 −
+13t5
+504000
+�
+F ′′′(t) +
+� 41473
+112000 + 13t3
+12096 −
+t6
+1296000
+�
+F (4)(t)
++
+� 131057t
+2016000 −
+67t4
+864000
+�
+F (5)(t) − 1493t2
+576000F (6)(t) − 232319
+8064000F (7)(t);
+a plot is shown in the right panel of Fig. 4.
+Expected Value. As a supplement to (83) we have
+(112)
+µ3 = 538
+7875M1 +
+281
+4536000M4;
+a highly accurate numerical value is displayed in Table 1.
+Variance. As a supplement to (85) we have
+(113)
+ν3 = −1076
+7875M2
+1 −
+281
+2268000M1M4 + 893
+7875M2 +
+1
+42000M2M3 +
+227
+2268000M5;
+a highly accurate numerical value is displayed in Table 1.
+Acknowledgements. I would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge
+(UK), for support and hospitality during the 2022 program “Applicable resurgent asymptotics: towards a
+universal theory (ARA2)” where work on the present paper was undertaken. This work was supported by
+EPSRC grant no EP/R014604/1.
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+Email address: bornemann@tum.de
+
diff --git a/VNA0T4oBgHgl3EQfEv9v/content/tmp_files/load_file.txt b/VNA0T4oBgHgl3EQfEv9v/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3587b378df1a2da9127d25eee23d2e124c8a5f65
--- /dev/null
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@@ -0,0 +1,1931 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf,len=1930
+page_content='ASYMPTOTIC EXPANSIONS RELATING TO THE DISTRIBUTION OF  THE LENGTH OF LONGEST INCREASING SUBSEQUENCES  FOLKMAR BORNEMANN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We study the distribution of the length of longest increasing subsequences in  random permutations of n integers as n grows large and establish an asymptotic expansion  in powers of n−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Whilst the limit distribution was already shown by Baik, Deift and  Johansson to be the GUE Tracy–Widom distribution F, we find explicit analytic expressions  of the first few finite-size correction terms as linear combinations of higher order derivatives  of F with rational polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Our proof replaces Johansson’s de-Poissonization,  which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of  Jacquet and Szpankowski, which is based on growth conditions in the complex plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' it is  subject to a tameness hypothesis concerning complex zeros of the analytically continued  Poissonized length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In a preparatory step an expansion of the hard-to-soft  edge transition law of LUE is studied, which is then transformed into an expansion of the  Poissonized length distribution for large intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Finally, expansions of Stirling-type  approximations and of the expected value and variance of the length distribution are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Introduction  The length Ln(σ) of longest increasing subsequences1 of permutations σ on {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , n}  becomes a discrete random variable when the permutations are drawn randomly with uniform  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This way the problem of enumerating all permutations σ that satisfy Ln(σ) ⩽ l  gets encoded in the discrete probability distribution P(Ln ⩽ l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The present paper studies an  asymptotic expansion of this distribution when n grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As there are relations to KPZ  growth models (directly so for the PNG model with droplet initial condition, see [28, 54, 55]  and [31, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 10]), we expect our findings to have a bearing there, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We start by recalling some fundamental results and notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' More details and  references can be found in the outstanding surveys and monographs [3, 10, 57, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Ulam’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The study of the behavior as n grows large dates back to Ulam [66] in 1961,  who mentioned that Monte-Carlo computations of Neighbor would indicate E(Ln) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='7√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Ulam continued by asking: “Another question of interest would be to find the distribution of  the length of the maximum monotone subsequence around this average.”  Refined numerical experiments by Baer and Brock [5] in 1968 suggested that E(Ln) ∼ 2√n  might be the precise leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In a 1970 lecture, Hammersley [36] presented a proof,  based on subadditive ergodic theory, that the limit c = limn→∞ E(Ln)/√n exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Finally, in  1977, Vershik and Kerov [68] as well as Logan and Shepp [45] succeeded in proving c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Poissonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A major tool used by Hammersley was a random process that is, basically,  equivalent to the following Poissonization of the random variable Ln: by drawing from the  different permutation groups independently and by taking Nr ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='} to be a further  independent random variable with a Poisson distribution of intensity r > 0, the combined  random variable LNr is distributed according to  P(LNr ⩽ l) = e−r  ∞  �  n=0  P(Ln ⩽ l)rn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' =: P(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 05A16, 60B20, 30D15, 30E15, 33C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' random permutations, random matrices, asymptotics, analytic de-Poissonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  1Defined as the maximum of all k for which there are 1 ⩽ i1 < i2 < · · · < ik ⩽ n with σi1 < σi2 < · · · < σik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02022v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='PR]  5 Jan 2023  2  FOLKMAR BORNEMANN  The entire function2 P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) is the Poisson generating function of the sequence P(Ln ⩽ l)  (n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=') and f(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) := ezP(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) is the corresponding exponential generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  As it turns out, it is much simpler to analyze the Poissonized distribution of LNr as the  intensity r grows large than the original distribution of Ln as n grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  There is, however, also a way back from LNr to Ln: namely, the expected value of the  Poisson distribution being E(Nr) = r, combined with some level of concentration, suggests  P(Ln ⩽ l) ≈ P(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l)  when n → ∞ while l is kept near the mode of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Being a Tauberian result, such  a de-Poissonization is subject to additional conditions, which we will discuss in a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Starting in the early 1990s the Poisson generating function P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) (or the exponential one  to the same end) has been represented in terms of one of the following interrelated forms:  a Toeplitz determinant in terms of modified Bessel functions [35],  Fredholm determinants of various (discrete) integral operators [8, 9, 18, 19, 42],  a unitary group integral [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  A particular case of those representations plays a central role in our study: namely3  (1)  P(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) = Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l),  where4 Ehard  2  (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) denotes the probability that, in the hard-edge scaling limit, the scaled  smallest eigenvalue of the Laguerre unitary ensemble (LUE) with real parameter ν > 0 is  bounded from below by s ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This probability is known to be given in terms of a Fredholm  determinant (see [30]):  (2)  Ehard  2  (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = det(I − KBessel  ν  )  ��  L2(0,s),  where Kν denotes the Bessel kernel in x, y ⩾ 0 (for the integral formula see [65, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2)]):  (3) KBessel  ν  (x, y) := Jν(√x)√yJ′  ν(√y) − Jν(√y)√xJ′  ν(√x)  2(x − y)  = 1  4  � 1  0  Jν(√σx)Jν(√σy) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Obviously, the singularities at the diagonal x = y are removable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The work of Tracy and Widom [65] establishes that the Fredholm determinant (2) can  be expressed in terms of Painlevé III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Recently, based on Okamoto’s Hamiltonian σ-PIII′  framework, Forrester and Mays [33] used that connection to compile a table of the exact  rational values of P(Ln ⩽ l) for up to n = 700;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5 whereas in our work [16], based on an  equivalent representation in terms of a Chazy I equation, we have compiled such a table6 for  up to n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In their seminal 1999 work [7], by relating the representation of P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) in terms of the  Toeplitz determinant to the machinery of Riemann–Hilbert problems and studying the  underlying double-scaling limit by the Deift–Zhou method of steepest descent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Baik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Deift  and Johansson answered Ulam’s question and proved that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' for t being any fixed real number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (4)  lim  r→∞ P  �LNr − 2√r  r1/6  ⩽ t  �  = F(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  where F is the GUE Tracy–Widom distribution: that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the distribution which expresses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  among many other limit laws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the probability that in the soft-edge scaling limit of the Gaussian  unitary ensemble (GUE) the scaled largest eigenvalue is bounded from above by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As for the  2Throughout the paper we will use n as an integer n ⩾ 0, r as a corresponding real variable r > 0 (intensity)  and z as its continuation into the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  3A derivation from the group integral is found in [18, §2] and from the Toeplitz determinant in [32, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='33)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  4Throughout the paper, we will use l as an integer l ⩾ 0 and ν as a corresponding real variable ν > 0, which  is used whenever an expression of l generalizes to non-integer arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  5Previously, by combinatorial means, Baer and Brock [5] had compiled a table for up to n = 36, supplemented  later by Odlyzko and Rains [47, 48] with the cases n = 60, 90, 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The cases n = 30, 60, 90 got printed in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  6Available for download at https://box-m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='de/f/7c4f8cb22f5d425f8cff/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  3  Poissonized length distribution itself, the Tracy–Widom distribution can be represented in  terms of a Fredholm determinant (see [30]): namely  (5)  F(t) = det(I − K0)|L2(t,∞)  where K0 denotes the Airy kernel in x, y ∈ R (for the integral formula see [64, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5)]):  (6)  K0(x, y) := Ai(x) Ai′(y) − Ai′(x) Ai(y)  x − y  =  � ∞  0  Ai(x + σ) Ai(y + σ) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Obviously, also in this case the singularities at the diagonal x = y are removable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since the  limit distribution in (4) is continuous, by a standard Tauberian follow-up [67, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1] of  the Portmanteau theorem the limit law holds uniformly in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In 2003, Borodin and Forrester gave an alternative proof of (4) which is based on studying  the hard-to-soft edge transition of LUE for ν → ∞ in form of the limit law [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1]  (7)  lim  ν→∞ Ehard  2  ��  ν − t(ν/2)1/3�2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν  �  = F(t)  (see also [31, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4]), which will be the starting point of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Still, there are other  proofs of (4) based on representations in terms of Fredholm determinants of further (discrete)  integral operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' for expositions and references see the monographs [10, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  De-Poissonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In the literature, the de-Poissonization of the limit law (4) has so far  been based exclusively on variants of the following lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [10, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5], originally stated  as [41, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5]), which uses monotonicity as the underlying Tauberian condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 (Johansson’s de-Poissonization lemma [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Suppose the sequence an of proba-  bilities 0 ⩽ an ⩽ 1 satisfies the monotonicity condition an+1 ⩽ an for all n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' and  denote its Poisson generating function by  (8)  P(z) = e−z  ∞  �  n=0  an  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Then, for s ⩾ 1 and n ⩾ 2 :7 P  �  n + 2√sn log n  �  − n−s ⩽ an ⩽ P  �  n − 2√sn log n  �  + n−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  After establishing the Tauberian condition of monotonicity and applying a variant of  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 to (4), Baik, Deift and Johansson [7, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1] got  (9)  lim  n→∞ P  �Ln − 2√n  n1/6  ⩽ t  �  = F(t),  which holds uniformly in t for the same reasons as given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (The simple calculations  based on Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 are given in [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 239];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' note that the uniformity of the limit law (4) is  used there without explicitly saying so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=') Adding tail estimates to the picture, those authors  were also able to lift the limit law to the moments and got, expanding on Ulam’s problem,  that the expected value satisfies [7, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2]  (10)  E(Ln) = 2√n + M1n1/6 + o(n1/6),  M1 =  � ∞  −∞  tF ′(t) dt ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='7711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' To our knowledge, only for the Poissonized limit law (4) a finite-size correction  term has been rigorously established prior to the present paper:8 namely, as a by-product  along the way of their study of the limiting distribution of maximal crossings and nestings  of Poissonized random matchings, Baik and Jenkins [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3] obtained (using the  7Note the trade-off between sharper error terms ∓n−s and less sharp perturbations of n by ±2√sn log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  8Expansions of probability distributions are sometimes called Edgeworth expansions in reference to the  classical one for the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In random matrix theory quite a variety of such expansions, or at  least some precise estimates of convergence rates, have been studied: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', for the soft-edge scaling limits of  the Gauss and Laguerre ensembles [21, 22, 27, 44] and of the Jacobi ensembles [43], for the hard-edge scaling  limit of the Laguerre ensembles [15, 26, 34, 53], for the bulk scaling limit of the circular ensembles [17], and  for various joint probability distributions [1, 11, 12, 28, 59, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  4  FOLKMAR BORNEMANN  machinery of Riemann–Hilbert problems and Painlevé representations of the Tracy–Widom  distribution), as r → ∞ with t being any fixed real number,  (11a)  P  �LNr − 2√r  r1/6  ⩽ t  �  = F  �  t(r)�  − 1  10  �  F ′′(t) + t2  6 F ′(t)  �  r−1/3 + O(r−1/2),  where (with ⌊·⌋ denoting the Gauss bracket)  (11b)  t(r) := ⌊2√r + tr1/6⌋ − 2√r  r1/6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  However, even if there is enough uniformity in this result and the option to Taylor expand the  Poisson generating function P(r) at n with a uniform bound while l is kept near the mode  of the distributions (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 for details on this option), the sandwiching in Johansson’s  de-Poissonization Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 does not allow us to obtain a result better than (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [11, §9])  (12)  P  �Ln − 2√n  n1/6  ⩽ t  �  = F  �  t(n)�  + O  �  n−1/6 log n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In their recent study of finite-size effects, Forrester and Mays [33, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1] gave a different  proof of (11) based on the Bessel kernel determinant (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Moreover, suggested by exact data  for n = 700 and a Monte-Carlo simulation for n = 20 000 they were led to conjecture [33,  Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2]  (13)  P  �Ln − 2√n  n1/6  ⩽ t  �  = F  �  t(n)�  + F D  1 (t)n−1/3 + · · ·  with the approximate graphical form of F D  1 (t) displayed in [33, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The presence of the Gauss bracket in t(r) and t(n), while keeping t at other places of the  expansions (11) and (13), causes undesirable effects in the error terms (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 below  for a detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Therefore, in our work [16] on a Stirling-type formula approximating  the distribution P(Ln ⩽ l), we suggested to use the integer l in the continuous expansion  terms instead of introducing the continuous variable t into the discrete distribution in the  first place, with the latter variant turning the discrete distribution into a piecewise constant  function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By introducing the scaling  tν(r) := ν − 2√r  r1/6  (r > 0)  we were led (based on numerical experiments using the Stirling-type approximation for n  getting as large as 1010), to conjecture the expansion  P(Ln ⩽ l) = F(t) + F D  1 (t) n−1/3 + F D  2 (t) n−2/3 + O(n−1)  ���  t=tl(n),  displaying the graphical form of the functions F D  1 (t), F D  2 (t) in the left panels of [16, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Moreover, as a note added in proof (see [16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (11)]), we announced that inserting the Baik–  Jenkins expansion (11) into the Stirling-type formula and using its (numerically observed)  apparent order O(n−2/3) of approximation would yield the functional form of F D  1 to be  (14)  F D  1 (t) = − 1  10  �  6F ′′(t) + t2  6 F ′(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The quest for a proof, and for a similar expression for F D  2 , motivated our present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The new findings of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In the analysis of algorithms in theoretical computer  science, or the enumeration of combinatorial structures to the same end, the original enumer-  ation problem is often represented in form of recurrences or functional/differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For instance, this situation arises in a large class of algorithms involving a splitting process,  trees, or hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Embedding such processes into a Poisson process9 often leads to more  tractable equations, so that sharp tools for a subsequent de-Poissonization were developed in  9As a heuristic principle in probability and combinatorics, Poissonization was popularized by Aldous’ book [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  5  the 1990s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' for references and details see [39, 40, 63] and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In particular, if the  Poisson generating function P(z) of a sequence of real an > 0, as defined in (8), is an entire  function, an application of the saddle point method to the Cauchy integral  an = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2πi  �  P(z)ez dz  zn+1  yields, under suitable growth conditions on P(z) as z → ∞ in the complex plane, the Jasz10  expansion  an ∼ P(n) +  ∞  �  j=2  bj(n)P (j)(n),  where the polynomial coefficients bj(n) are the diagonal Poisson–Charlier polynomials (that  is, with intensity r = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 we give a heuristic derivation of that expansion  and recall, in the detailed form of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, a specific analytic de-Poissonization result from  the comprehensive memoir [39] of Jacquet and Szpankowski—a result which applies to a  family of Poisson generating functions at once, providing uniform error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, the difficult part of applying Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 is checking the Tauberian growth conditions in  the complex plane, which are required to hold uniformly for the family of Poisson generating  functions (recall that, in the case of the longest increasing subsequence problem, P(z) = P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l)  depends on the integer l near the mode of the length distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' After observing a  striking similarity of those growth conditions with the notion of H-admissibility for the  corresponding exponential generating function (as introduced by Hayman in his memoir [37]  on the generalization of Stirling’s formula), a closer look at the proof of Hayman’s [37,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' XI] revealed the following result (see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 for a precise quantitative statement):  The family of all entire functions of genus zero which have, for some ϵ > 0,  no zeros in the sector |arg z| ⩽ π/2 + ϵ satisfies a universal bound that implies  Tauberian growth conditions suitable for analytic de-Poissonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  On the other hand, in our work [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2] on Stirling-type formulae for the problem  of longest increasing subsequences, when proving the H-admissibility of the exponential  generating functions f(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) = ezP(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) (for each l), we had established, based on the repre-  sentation [56] of P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) as a group integral:  For any integer l ⩾ 0 and any δ > 0, the exponential generating function  f(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) is an entire function of genus zero having at most finitely many zeros  in the sector |arg z| ⩽ π − δ, none of them being real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Under the reasonable assumption (supported by numerical experiments) that those finitely  many complex zeros do not come too close to the real axis and do not grow too fast as  n → ∞ while l stays near the mode of the length distribution, the uniformity of the Tauberian  growth conditions can be preserved (see Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 for the technical details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We call this  assumption the tameness hypothesis11 concerning the zeros of the family of P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Subject to the tameness hypothesis, the main result of the present paper, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, gives  the asymptotic expansion  P(Ln ⩽ l) = F(t) +  m  �  j=1  F D  j (t) n−j/3 + O  �  n−(m+1)/3� ����  t=tl(n)  ,  which is uniformly valid when n, l → ∞ while tl(n) stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12 Here, the F D  j  are  smooth functions and, for technical reasons that are explained in Section 3, the integer m ⩾ 0  cannot be chosen freely but has to be limited to the range m ⩽ m∗ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We conjecture that  this restriction is artificial and can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  10Dubbed so in [29, §VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='18] to compliment the seminal work of Jacquet and Szpankowski [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  11Proving it seems to be rather difficult, though—at least we were lacking the methodology to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  12This is meant, in fact, when we say that l stays near the mode of the length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  6  FOLKMAR BORNEMANN  Finally, now without any detour via the Stirling-type formula, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 confirms that the  expansion term F D  1 is given by (14) and yields the striking formula (see (110) for F D  3 (t))  F D  2 (t) =  �  − 139  350 + 2t3  1575  �  F ′(t) +  �  − 43t  350 +  t4  7200  �  F ′′(t) + t2  100F ′′′(t) + 9  50F (4)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2 we start with a careful discussion of expansions of  perturbed Airy kernel determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We stress the importance of such kernel expansions to  be differentiable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', one can differentiate into the error term) to easily lift the error bounds  to trace norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The subtle, but fundamental difficulty of such a lift seems to have been  missed, more often than not, in the existing literature on convergence rates and expansions  of limit laws in random matrix theory (notable exceptions are, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', [27, 43, 44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In the rather lengthy Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 we study the asymptotic expansion of the Borodin–Forrester  hard-to-edge transition law (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It is based on a uniform version of Olver’s asymptotic  expansion of Bessel functions of large order in the transition region, which we discuss in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 we lay the foundational work for the concrete functional form of all  subsequent finite-size correction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We reduce the complexity of computing these terms  by using a coordinate transform on the level of kernels to simplify the kernel expansion—a  coordinate transform which gets subsequently reversed on the level of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='13 As  yet another application of that technique we simplify in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4 the finite-size correction  term of Choup [21] to the soft-edge limit law of GUE and LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4 we expand the Poissonized length distribution, thereby generalizing the result (12)  of Baik and Jenkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4 we also discuss the potentially detrimental effect of using  Gauss brackets, as in (12), alongside with the continuous variable t in the expansion terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5 we state and prove the main result of the paper: the expansion of the Baik–Deift–  Johansson limit law (9) of the length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here we use the Jasz expansion of analytic  de-Poissonization (as detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The universal bounds for entire functions of  genus zero, which are used to prove the Tauberian growth conditions in the complex plane, and  their relation to the theory of H-admissibility are prepared for in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Additionally,  in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5 we discuss the modifications that apply to the discrete density P(Ln = l) (that is,  to the PDF of the length distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6 we apply our findings to the asymptotic expansion of the Stirling-type formula  which we introduced in [16] as an accurate tool for the numerical approximation of the length  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Subject to the tameness hypothesis, we prove the observation [16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (8b)]  about a leading O(n−2/3) error of that formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 7 we study the asymptotic expansion of the expected value and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Based on a reasonable hypothesis about some uniformity in the tail bounds, we add several  more concrete expansion terms to the Baik–Deift–Johansson solution (10) of Ulam’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansions of perturbed Airy kernel determinants  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 we will establish, with m being some non-negative integer and t0 some real  number, kernel expansions of the form  (15) Kh(x, y) = K0(x, y)+  m  �  j=1  hjKj(x, y)+hm+1Rm(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h), Rm+1,h(x, y) = O  �  e−(x+y)�  ,  which are  uniform for x, y ⩾ t0 as h → 0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  repeatedly differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' x, y as uniform expansions under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Here, Kh is a family of smooth kernels, K0 denotes the Airy kernel (6) and the Kj(x, y) are  finite sums of rank one kernels u(x)v(y), with factors u(ξ), v(ξ) of the functional form  (16)  p(ξ) Ai(ξ)  or  p(ξ) Ai′(ξ),  13In [15] we applied a similar transformation “trick” to the expansion of the hard-edge limit law of LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  7  where p(ξ) is a polynomial in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since the existing literature tends to neglect the issue of  estimating trace norms in terms of kernel bounds, this section aims at establishing a relatively  easy framework for lifting such an expansion to one of the Fredholm determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Bounds on the kernels and induced trace class operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Bounds on the kernels  and on the trace norms of the induced integral operators can be deduced from the estimates14  |p(ξ)| · max  �  | Ai(ξ)|, | Ai′(ξ)|  �  ⩽ ape−ξ  (ξ ∈ R),  where the constant ap does only depend on the polynomial p (note that we can take ap = 1  when p(ξ) ≡ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This way we get from (6)  |K0(x, y)| ⩽  � ∞  0  | Ai(x + σ) Ai(y + σ)| dσ ⩽ e−(x+y)  � ∞  0  e−2σ dσ = 1  2e−(x+y)  (x, y ∈ R)  and, for 1 ⩽ j ⩽ m, constants cj such that  |Kj(x, y)| ⩽ cje−(x+y)  (x, y ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We denote the induced integral operators on L2(t, ∞) in bold face (suppressing the dependence  on t in the notation) and the corresponding spaces of trace class and Hilbert–Schmidt operators  by J p(t, ∞) with p = 1 and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Now, the Airy operator K0 (being by (6) the square of  the Hilbert–Schmidt operator At with kernel Ai(x + y − t)) and the Kj (j ⩾ 1) (being finite  rank operators) are trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Their trace norms are bounded by  ∥K0∥J 1(t,∞) ⩽ ∥At∥2  J 2(t,∞) =  � ∞  t  � ∞  t  | Ai(x + y − t)|2 dx dy  =  � ∞  0  � ∞  0  | Ai(x + y + t)|2 dx dy ⩽ e−2t  � ∞  0  � ∞  0  e−2(x+y) dx dy = 1  4e−2t  (t ∈ R),  and likewise, because of  ∥u ⊗ v∥J 1(t,∞) ⩽ ∥u∥L2(t,∞)∥v∥L2(t,∞) ⩽ apaq  2  e−2t  (t ∈ R)  for factors u(ξ), v(ξ) of the form (16) with polynomials p and q, there are constants c∗  j s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ∥Kj∥J 1(t,∞) ⩽ c∗  je−2t  (1 ⩽ j ⩽ m, t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  On the other hand, as there is, in general, no direct relation between kernel bounds and  bounds of the trace norm of induced integral operators (see [60, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 25]), lifting the error term  in the kernel expansion (15) to trace norm is less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  By taking the explicitly assumed differentiability of the kernel expansion into account,  there is a constant c∗ such that  Sm+1,h(x, y) := ∂yRm+1,h(x, y),  |Sm+1,h(x, y)| ⩽ c∗e−(x+y)  holds true for all x, y ⩾ t0 as 0 < h ⩽ h0 (h0 chosen sufficiently small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If we denote an  indicator function by χ, integration gives  Rm+1,h(x, y) = −  � ∞  t  Sm+1,h(x, σ)eσ/2 · e−σ/2χ[t,σ](y) dσ,  14This bound, chosen for convenience but not for optimality, follows from the superexponential decay of the  Airy function and its derivative as ξ → +∞ and the bounds O  �(−ξ)−1/4� and O  �(−ξ)1/4� as ξ → −∞, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  the expansions (103) and [52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='9/10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  15This subtle technical point has frequently been missed in the literature when lifting kernel expansion to  trace class operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', the argument given in [21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 12] for lifting a kernel expansion to the Edgeworth  expansion of the largest eigenvalue distribution of GUE and LUE lacks in that respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It can be made  rigorous when supplemented by the estimates given here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Another rigorous approach can be  found in the work of Johnstone [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=')  8  FOLKMAR BORNEMANN  which shows that Rm+1,h is the product of two Hilbert–Schmidt operators and thus trace  class with a norm bounded by  ∥Rm+1,h∥2  J 1(t,∞) ⩽  �� ∞  t  � ∞  t  Sm+1,h(x, y)2ey dx dy  � �� ∞  t  � x  t  e−x dx dy  �  ⩽ c2  ∗  �� ∞  t  � ∞  t  e−2x−y dx dy  � �� ∞  t  � x  t  e−x dx dy  �  = c2  ∗  2 e−4t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We have thus lifted the kernel expansion (15) to an operator expansion in J 1(t, ∞), namely  (17)  Kh = K0 +  m  �  j=1  hjKj + hm+1Rm+1,h,  ∥Rm+1,h∥J 1(t,∞) = O  �  e−2t�  ,  uniformly valid for t ⩾ t0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Fredholm determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If a continuous kernel K(x, y) has a weighted uniform bound  sup  x,y⩾t0  ex+y |K(x, y)| ⩽ M < ∞,  then the Fredholm determinant  det(I − K)|L2(t,s) :=  ∞  �  m=0  (−1)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  � s  t  · ·  � s  t  m  det  j,k=1 K(xj, xk) dx1 · · · dxm  is well defined for t < s ⩽ ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [4, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4] (writing the integrals in terms of the weighted  measure e−x dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If the induced integral operator K on L2(t, ∞) is trace class, the Fredholm  determinant can be expressed in terms of the operator determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In fact, by the orthogonal  decomposition  L2(t, ∞) = L2(t, s) ⊕ L2(s, ∞)  and denoting by Ps the orthogonal projection onto the first component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', multiplication  of L2-functions by the indicator function of the interval (t, s)) we have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [60, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3])  (18)  det(I − K)|L2(t,s) = det(I − KPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For lifting the operator expansion (17) to one for the (Fredholm) determinants it suffices to  restrict ourselves to the case s = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This is because a block decomposition of Kh shows  ∥Kh − KhPs∥J 1(t,∞) = ∥ ˜Kh∥J 1(s,∞)  where ˜Kh denotes the induced integral operator on the subspace L2(s, ∞) ⊂ L2(t, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Now,  the local Lipschitz continuity of the operator determinant [60, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4] implies  |det(I − KhPs) − det(I − Kh)| ⩽ ∥ ˜Kh∥J 1(s,∞) exp  �  2∥Kh∥J 1(t,∞) + 1  �  = O  �  e−2s�  where, by Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, the bound holds uniformly for t0 ⩽ t < s ⩽ ∞ as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansions of the operator determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Plemelj’s formula gives, for trace class  perturbations E bounded by ∥E∥J 1(t,∞) < 1, the convergent series expansion (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', [60,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12)])  (19)  det(I − E) = exp  �  −  ∞  �  n=1  n−1 tr(En)  �  = 1 − tr E + 1  2  �  (tr E)2 − tr(E2)  �  + O(∥E∥3  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  9  Thus, since I − K0 is invertible with a uniformly bounded inverse as t ⩾ t0,16 we have  det(I − Kh) = det(I − K0) det(I − Eh),  Eh :=  m  �  j=1  hj(I − K0)−1Kj + hm+1(I − K0)−1Rm+1,h,  with the trace norm of Eh being bounded as follows (using the results of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 and  observing that the trace class forms an ideal within the bounded operators):  ∥Eh∥J 1(t,∞) ⩽ ∥(I − K0)−1∥  h  1 − h · O  �  e−2t�  = h · O(e−2t),  uniformly for t ⩾ t0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By (19) this implies, uniformly under the same conditions,  det  �  I − Eh  �  = 1 +  m  �  j=1  dj(t)hj + hm+1 · O(e−2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The functions dj(t) depend smoothly on t and satisfy dj = O(e−2t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first two of them are  d1(t) = − tr  �  (I − K0)−1K1  �  ,  d2(t) = 1  2  �  tr  �  (I − K0)−1K1  ��2 − 1  2 tr  �  ((I − K0)−1K1)2�  − tr  �  (I − K0)−1K2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Since the Kj (j ⩾ 1) are finite rank operators, these trace expressions can be recast in terms  of resolvent kernels and integral traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Taking the bound 0 ⩽ F(t) ⩽ 1 of the Tracy–Widom distribution (being a probability  distribution) into account, the results of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2 can be summarized in form of the following:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let Kh(x, y) be a continuous kernel, K0 the Airy kernel (6) and let the  Kj(x, y) be finite sums of rank one kernels with factors of the form (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If, for some fixed  non-negative integer m and some real number t0, there is a kernel expansions of the form  Kh(x, y) = K0(x, y) +  m  �  j=1  hjKj(x, y) + hm+1 · O  �  e−(x+y)�  ,  which holds uniformly for x, y ⩾ t0 as h → 0+ and which can be repeatedly differentiated  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' x and y as uniform expansions, then the Fredholm determinant of Kh on (t, s) satisfies  (20)  det(I − Kh)|L2(t,s) = F(t) +  m  �  j=1  Gj(t)hj + hm+1 · O(e−2t) + O(e−2s)  uniformly for t0 ⩽ t < s ⩽ ∞ as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, F denotes the Tracy–Widom distribution (5)  and the Gj(t) are smooth functions depending on the kernels K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , Kj, satisfying the (right)  tail bounds Gj = O(e−2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The first two are  G1(t) = −F(t) · tr  �  (I − K0)−1K1  ���  L2(t,∞) ,  G2(t) = F(t) ·  �1  2  �  tr  �  (I − K0)−1K1  ���  L2(t,∞)  �2  − 1  2 tr  �  ((I − K0)−1K1)2���  L2(t,∞) − tr  �  (I − K0)−1K2  ���  L2(t,∞)  �  ,  where (I − K0)−1 is understood as a resolvent kernel and the traces as integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The determi-  nantal expansion (20) can repeatedly be differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' t and s, preserving uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  16The Airy kernel K0 induces a symmetric positive definite integral operator K0 on L2(t, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Its norm as a  bounded operator is thus is given by the spectral radius, which stays below 1 uniformly as t ⩾ t0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [64]:  ∥K0∥ = ρ(K0) ⩽ c(t0) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By functional calculus we thus get the uniform bound ∥(I − K0)−1∥ =  1  1−ρ(K0) ⩽  1  1−c(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  10  FOLKMAR BORNEMANN  6  5  4  3  2  1  0  1  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6  6  5  4  3  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5  6  5  4  3  2  1  0  1  2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Plots of F1(t) (left panel) and F2(t) (middle panel) as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The right panel  shows F3(t) as in (108) (black solid line) with the approximations (25) for ν = 100 (red  dotted line) and ν = 800 (green dashed line): the close agreement validates the functional  forms (24) and (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Details about the numerical method can be found in [13, 14, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansion of the Hard-to-Soft Edge Transition  In this section we prove an expansion for the hard-to-soft transition limit (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' To avoid  notational clutter, we use the quantity  (21)  hν := 2−1/3ν−2/3  and study expansions in powers of hν as hν → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The transform s = φν(t) used in the  transition limit can briefly be written as  (22)  φν(t) = ν2(1 − hνt)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For technical reasons related to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 (which uses the divisibility of a  certain sequence of polynomials that has only been checked by inspection up to m∗), we have  to impose a bound  m∗ = 23  on the number m of expansion terms in all of the main results of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We  conjecture that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 is true without such a restriction, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' There holds the hard-to-soft edge transition expansion  (23)  Ehard  2  (φν(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = F(t) +  m  �  j=1  Fj(t)hj  ν + hm+1  ν  O(e−3t/2),  which is uniformly valid when t0 ⩽ t < h−1  ν  as hν → 0+, m being any fixed integer in the  range 0 ⩽ m ⩽ m∗ and t0 any fixed real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Preserving uniformity, the expansion can be  repeatedly differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here the Fj are certain smooth functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the  first two are  F1(t) = 3t2  10 F ′(t) − 1  5F ′′(t),  (24a)  F2(t) =  � 2  175 + 32t3  175  �  F ′(t) +  �  − 16t  175 + 9t4  200  �  F ′′(t) − 3t2  50 F ′′′(t) + 1  50F (4)(t),  (24b)  while a similar expression for F3 is displayed in (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  It is rewarding to validate intriguing formulae such as (24b/c) and (108) by numerical  methods: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1 plots the functions F1(t), F2(t), F3(t) next to the approximation  (25)  F3(t) ≈ h−3  ν  �  Ehard  2  (φν(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) − F(t) − F1(t)hν − F2(t)h2  ν  �  for ν = 100 and ν = 800: the close matching with F D  3 (t) as displayed by the latter is a  very strong testament of correctness of (24) and (108) (in fact, some slips in preliminary  calculations have been caught looking at plots which exhibited mismatches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 is split into several steps and will be concluded in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Kernel expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We start with an auxiliary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Define for h > 0 and x, y < h−1 the function  (26)  Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) :=  �  (1 − hx)(1 − hy)  1 − h(x + y)/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  This function Φ satisfies the bound  (27)  0 < Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) ⩽ 1  and has the convergent power series expansion  (28a)  Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 1 − (x − y)2  ∞  �  k=2  rk(x, y)hk  where the rk(x, y) are certain homogeneous symmetric rational17 polynomials of degree k − 2,  the first few of them being  (28b)  r2(x, y) = 1  8,  r3(x, y) = 1  8(x + y),  r4(x, y) =  1  128  �  13(x2 + y2) + 22xy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The series converges uniformly for x, y < (1 − δ)h−1, δ being any fixed real positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The bound (27) is the inequality of arithmetic and geometric means for the two  positive real quantities 1 − hx and 1 − hy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By using  lim  y→x  1  (x − y)2  �  Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) − 1  �  = −1  8  �  h  1 − hx  �2  ,  the analyticity of Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h, and the scaling law  Φ(λ−1x, λ−1y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' λh) = Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  (λ > 0)  we deduce the claims about the form und uniformity of the power series expansion (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  Because of the representation (2) of Ehard  2  (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) in terms of a Fredholm determinant of the  Bessel kernel (3), we have to expand the induced transformation of that kernel Kν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The change of variables s = φν(t), mapping t < h−1  ν  monotonically decreasing  to s > 0, induces the symmetrically transformed Bessel kernel  ˆKBessel  ν  (x, y) :=  �  φ′ν(x)φ′ν(y) KBessel  ν  (φν(x), φν(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (29a)  There holds the kernel expansion  ˆKBessel  ν  (x, y) = K0(x, y) +  m  �  j=1  Kj(x, y)hj  ν + hm+1  ν  O  �  e−(x+y)�  ,  (29b)  which is uniformly valid when t0 ⩽ x, y < h−1  ν  as hν → 0+, m being any fixed integer in the  range 0 ⩽ m ⩽ m∗ and t0 any fixed real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here the Kj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , m∗, are certain  finite rank kernels of the form  Kj(x, y) =  �  κ,λ∈{0,1}  pj,κλ(x, y) Ai(κ)(x) Ai(λ)(y)  where pj,κλ(x, y) are rational polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first two kernels are  (30a)  K1(x, y) = 1  10  �  − 3  �  x2 + xy + y2�  Ai(x) Ai(y)  + 2  �  Ai(x) Ai′(y) + Ai′(x) Ai(y)  �  + 3(x + y) Ai′(x) Ai′(y)  �  17Throughout the paper the term “rational polynomial” is used for polynomials with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  12  FOLKMAR BORNEMANN  and  (30b)  K2(x, y) =  1  1400  ��  − 235  �  x3 + y3�  − 319xy(x + y) + 56  �  Ai(x) Ai(y)  +  �  63(x4 + x3y − x2y2 − xy3 − y4) − 55x + 239y  �  Ai(x) Ai′(y)  +  �  63(−x4 − x3y − x2y2 + xy3 + y4) + 239x − 55y  �  Ai′(x) Ai(y)  +  �  340(x2 + y2) + 256xy  �  Ai′(x) Ai′(y)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Preserving uniformity, the kernel expansion (29) can repeatedly be differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By using the function Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) as defined in (26) and writing  φν(t) = ων(t)2,  ων(t) = ν(1 − hνt),  aν(x, y) = (1 − hνy) ·  1  √2hν  Jν  �  ων(x)  �  d  dy  1  √2hν  Jν  �  ων(y)  �  we can factor the transformed Bessel kernel in the simple form  ˆKBessel  ν  (x, y) = Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) · aν(x, y) − aν(y, x)  x − y  ,  noting, by symmetry, the removability of the singularities at x = y of the second factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  First, if x or y is between 3  4 · h−1  ν  and h−1  ν , using the bound 0 < Φ ⩽ 1 (see (27)) one can  argue as in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1: since at least one of the Bessel factors is of the form  J(κ)  ν  (z) with 0 ⩽ z ⩽ 1/4, which plainly falls into the superexponentially decaying region  as ν → ∞, and since at least one of the Airy factors of each term of the expansion is also  superexponentially decaying as ν → ∞, the transformed Bessel kernel and the expansion  terms in (29) get completely absorbed into the error term (bounding the other factors as in  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1)  hm+1  ν  O  �  e−(x+y)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Here, the removable singularities at x = y are dealt with by using the differentiability of the  corresponding bounds (or by extending to the complex domain and using Cauchy’s integral  formula as in the proof of [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Therefore, we may suppose from now on that t0 ⩽ x, y ⩽ 3  4 · h−1  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, in this  range of x and y, the power series expansion  (31)  Φ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) = 1 − (x − y)2  ∞  �  k=2  rk(x, y)hk  ν  converges uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, the rk(x, y) are certain homogeneous symmetric rational polyno-  mials of degree k − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first of them being r2(x, y) = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Next, we rewrite the uniform version of the large order expansion of Bessel functions in  the transition region, as given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, in the form  (32)  1  √2hν  Jν  �  ων(t)  �  =  �  1 + hνpm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν)  �  Ai(t) + hνqm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) Ai′(t) + hm+1  ν  O(e−t),  where the estimate of the remainder is uniform for t0 ⩽ t < h−1  ν  as hν → 0+ and  pm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 21/3  m−1  �  k=0  Ak+1(−t/21/3) (21/3h)k,  qm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 22/3  m−1  �  k=0  Bk+1(−t/21/3) (21/3h)k,  with the polynomials Ak(τ) and Bk(τ) from (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It follows from Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 that pm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  and qm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) are rational polynomials in t and h, starting with  p2(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 2  10t +  h  1400(63t5 + 120t2),  q2(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 3  10t2 +  h  1400(340t3 + 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  13  Also given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, under the same conditions, the expansion (32) can be repeatedly  differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t t while preserving uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' From this we obtain, using the Airy  differential equation Ai′′(ξ) = ξ Ai(ξ), that uniformly (given the range x and y)18  aν(x, y) =  �  κ,λ∈{0,1}  pm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) Ai(κ)(x) Ai(λ)(y) + hm+1  ν  O(e−(x+y)),  where  pm  00(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = h(1 − hy)(1 + hpm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h))  �  yqm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) + ∂ypm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  �  ,  pm  01(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = (1 − hy)(1 + hpm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h))  �  1 + h(pm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) + ∂yqm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h))  �  ,  pm  10(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = h2(1 − hy)qm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  �  yqm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) + ∂ypm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  �  ,  pm  11(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = h(1 − hy)qm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  �  1 + h(pm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) + ∂yqm(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h))  �  are rational polynomials in x, y and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In particular, those factorizations show  pm  00(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = O(h),  pm  11(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = O(h),  pm  01(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = 1 + O(h),  pm  10(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) = O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  If we denote by ˆpm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) the polynomials obtained from pm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) after dropping all  powers of h that have an exponent larger than m (thus contributing terms to the expansion  that get absorbed in the error term), we obtain  aν(x, y) = Ai(x) Ai′(y) +  �  κ,λ∈{0,1}  ˆpm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) Ai(κ)(x) Ai(λ)(y) + hm+1  ν  O(e−(x+y)),  with a polynomial expansion  ˆpm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) =  m  �  j=1  ˆpj,κλ(x, y)hj  whose coefficient polynomials ˆpj,κλ(x, y), being the unique expansion coefficients as hν → 0+,  are now independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Hence, the anti-symmetrization of aν(x, y) satisfies the uniform  expansion (given the range of x and y)  (33)  aν(x, y) − aν(y, x) =  �  Ai(x) Ai′(y) − Ai′(x) Ai(y)  �  +  �  κ,λ∈{0,1}  ˆqm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) Ai(κ)(x) Ai(λ)(y) + hm+1  ν  O(e−(x+y))  with the polynomial expansion  ˆqm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) =  m  �  j=1  ˆqj,κλ(x, y)hj,  where  ˆqj,00(x, y) = ˆpj,00(x, y) − ˆpj,00(y, x),  ˆqj,11(x, y) = ˆpj,11(x, y) − ˆpj,11(y, x)  ˆqj,01(x, y) = ˆpj,01(x, y) − ˆpj,10(y, x),  ˆqj,10(x, y) = −qj,01(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Since the rational polynomials ˆqj,00(x, y) and ˆqj,11(x, y) are anti-symmetric in x, y, they  factor in the form  (x − y) × (symmetric rational polynomial in x and y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  18Because of the superexponential decay (103) of the Airy function Ai(t) and its derivative Ai′(t) as t → +∞,  cross terms with the remainder are uniformly estimated in the form  polynomial(x) · Ai(κ)(x) · O  �  e−y�  = O  �  e−(x+y)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  14  FOLKMAR BORNEMANN  the first few cases are  ˆq1,00(x, y) = − 3  10(x − y)(x2 + xy + y2),  ˆq1,11(x, y) = 3  10(x − y)(x + y),  ˆq2,00(x, y) =  1  1400(x − y)  �  − 235  �  x3 + y3�  − 319xy(x + y) + 56  �  ,  ˆq2,11(x, y) =  1  1400(x − y)  �  340  �  x2 + y2�  + 256xy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Even though there is no straightforward structural reason for the rational polynomials ˆqj,01  (and thus ˆqj,10) to be divisible by x − y as well, an inspection19 of the first cases reveals this  to be true for at least j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , m∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first two of them being  ˆq1,01(x, y) = 2  10(x − y),  ˆq2,01(x, y) =  1  1400(x − y)  �  63  �  x4 + x3y − x2y2 − xy3 − y4�  + 120x + 64y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, by restricting ourselves to the explicitly checked cases m ⩽ m∗, we denote by qm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h)  the polynomials obtained from ˆqm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' h) after division by the factor x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since (33) is an  expansion of an anti-symmetric function with anti-symmetric remainder which can repeatedly  be differentiated, division by x−y yields removable singularities at x = y and does not change  the character of the expansion (see also the argument given in the proof of [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 8]):  aν(x, y) − aν(y, x)  x − y  = K0(x, y) +  �  κ,λ∈{0,1}  qm  κλ(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' hν) Ai(κ)(x) Ai(λ)(y) + hm+1  ν  O  �  e−(x+y)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The lemma now follows by multiplying this expansion with (31), noting that the terms  −rk(x, y)(x − y)2K0(x, y) = −rk(x, y)(x − y)  �  Ai(x) Ai′(y) − Ai′(x) Ai(y)  �  also take the form asserted for the terms in the kernels Kj (j ⩾ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Finally, since all the expansions can repeatedly be differentiated under the same conditions  while preserving their uniformity, the same holds for the resulting expansion of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The case m = 0 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=',  ˆKBessel  ν  (x, y) = K0(x, y) + hν · O  �  e−(x+y)�  is the β = 2 case of [18, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='8)] in the work of Borodin and Forrester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' There, in [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 8]  it is stated that this expansion would be uniformly valid for x, y ⩾ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' However, stated in  such a generality, it is not correct (see Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 38) and, in fact, similar to our proof given above,  their proof is restricted to the range t0 ⩽ x, y < h−1  ν , which completely suffices to address  the hard-to-soft edge transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (See Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4 for yet another issue with [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=')  To reduce the complexity of calculating the functional form of the first two finite-size  correction terms in the hard-to-soft edge transition (23), we consider a second kernel transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For h > 0, the Airy kernel K0 and the first two expansion kernels K1, K2  from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 we consider  Kh(x, y) = K0(x, y) + K1(x, y)h + K2(x, y)h2  19See Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 for the computation of the polynomials Ak, Bk and thus qj,01(x, y)—a Mathematica  notebook comes with the sources of the arXiv version of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since the divisibility of ˆqj,01(x, y)  by x − y depends very much on the specific rational coefficients of the polynomials Ak, Bk, this is probably  not just a contingent fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We conjecture it to be true for all j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' a proof, however, would have to uncover some  hidden symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  15  and the transformation,20 where ζ(z) is defined as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3,  (34)  s = ψ−1  h (t) := 2−1/3h−1ζ(1 − ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Then t = ψh(s) maps s ∈ R monotonically increasing to −∞ < t < h−1, with t ⩽ µh−1,  µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='94884 · · · , when s ⩽ 2h−1, and induces the symmetrically transformed kernel  ˜Kh(x, y) :=  �  ψ′  h(x)ψ′  h(y) Kh(ψh(x), ψh(y))  (35a)  which expands as  ˜Kh(x, y) = K0(x, y) + ˜K1(x, y)h + ˜K2(x, y)h2 + h3 · O  �  e−(x+y)�  ,  (35b)  uniformly in s0 ⩽ x, y ⩽ 2h−1 as h → 0+, s0 being a fixed real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, ˜K1 and ˜K2 are  ˜K1 = 1  5(Ai ⊗ Ai′ + Ai′ ⊗ Ai)  (35c)  ˜K2 =  1  350  �  55(Ai ⊗ Ai′′′ + Ai′′′ ⊗ Ai) − 51(Ai′ ⊗ Ai′′ + Ai′′ ⊗ Ai′) − 96 Ai ⊗ Ai  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (35d)  Preserving uniformity, the kernel expansion can repeatedly be differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Reversing the power series (104) gives  t = ψh(s) = s − 3s2  10 h − s3  350h2 + · · · ,  which is uniformly convergent for s0 ⩽ s ⩽ 2h−1 since 1 − ht stays bounded away from zero  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A routine calculation with truncated power series gives formula (35c) for ˜K1 and  ˜K2(x, y) =  1  350  �  14 Ai(x) Ai(y) + (−51x + 55y) Ai(x) Ai′(y) + (55x − 51y) Ai′(x) Ai(y)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The Airy differential equation ξ Ai(ξ) = Ai′′(ξ) implies the replacement rule  (36)  ξj Ai(k)(ξ) = ξj−1 Ai(k+2)(ξ) − kξj−1 Ai(k−1)(ξ)  (j ⩾ 1, k ⩾ 0)  which, if repeatedly applied to a kernel of the given structure, allows us to absorb any powers  of x and y into higher order derivatives of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This process yields the asserted form of ˜K2,  which will be the preferred form in course of the calculations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Since we stay within the range of uniformity of the power series expansions and calculations  with truncated powers series are amenable to repeated differentiation, the result now follows  from the bounds given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Proof of the general form of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 we get  (the Fredholm determinants are seen to be equal by transforming the integrals)  Ehard  2  (φν(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = det(I − KBessel  ν  )|L2(0,φν(t)) = det(I − ˆKBessel  ν  )|L2(t,h−1  ν )  = F(t) +  m  �  j=1  Fj(t)hj  ν + hm+1  ν  O(e−2t) + O(e−2h−1  ν ),  uniformly for t0 ⩽ t < h−1  ν  as hν → 0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' preserving uniformity, this expansion can be  repeatedly differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, the Fj(t) are certain smooth  functions that can be expressed in terms of traces of integral operators of the form given in  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Observing  e−2h−1  ν  < e−h−1  ν /2e−3t/2 = hm+1  ν  O(e−3t/2)  (hν → 0+)  we can combine the two error terms as hm+1  ν  O(e−3t/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This finishes the proof of (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  20Note that for z = 1 − hνt we thus get νz = ων(t) and ν2/3ζ = s in Olver’s expansion (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As it turns out,  by using this transformation, the kernel expansion simplifies in the same fashion also for m ⩾ 2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  16  FOLKMAR BORNEMANN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Functional form of F1(t) and F2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Instead of calculating F1, F2 directly from the  formulae in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 applied to the kernels K1, K2 in (30a/b) we will reduce them to the  corresponding functions ˜F1, ˜F2 induced by the much simpler kernels ˜K1, ˜K2 in (35c/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Functional form of ˜F1(t) and ˜F2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' First, by writing  ujk(t) = tr  �  (I − K0)−1 Ai(j) ⊗ Ai(k) ���  L2(t,∞)  and observing (the symmetry of the resolvent kernel implies the symmetry ujk(t) = ukj(t))  tr  �  (I − K0)−1 ˜K1  ���  L2(t,∞) = 2  5u10(t),  tr  �  ((I − K0)−1 ˜K1)2���  L2(t,∞) = 2  25  �  u00(t)u11(t) + u10(t)2�  ,  tr  �  (I − K0)−1 ˜K2  ���  L2(t,∞) =  1  175  �  − 48u00(t) − 51u21(t) + 55u30(t)  �  ,  the formulae of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 applied to ˜K1 and ˜K2 give  ˜F1(t) = −2  5F(t)u10(t),  (37a)  ˜F2(t) = F(t)  � 48  175u00(t) − 1  25  �  u00(t)u11(t) − u10(t)2�  + 51  175u21(t) − 11  35u30(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (37b)  We recall from [16, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1] that the simple recursion  u′  jk(t) = uj+1,k(t) + uj,k+1(t) − uj0(t)uk0(t)  yields similar formulae for the first few derivatives of the distribution F(t):  (38)  F ′(t) = F(t) · u00(t),  F ′′(t) = 2F(t) · u10(t),  F ′′′(t) = 2F(t) ·  �  u11(t) + u20(t)  �  ,  F (4)(t) = 2F(t) ·  �  u00(t)u11(t) − u10(t)2 + 3u21(t) + u30(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By a linear elimination of the terms u00(t), u10(t), u00(t)u11(t) − u10(t)2 we obtain, as an  intermediate step,  (39a)  ˜F1(t) = −1  5F ′′(t),  ˜F2(t) = 48  175F ′(t) − 1  50F (4)(t) + 24  175F(t)  �  3u21(t) − 2u30(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Second, to simplify even further, we refer to the full power of the general Tracy–Widom  theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', representing F in terms of Painlevé II): by advancing its set of formulae, Shinault  and Tracy [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 68] showed, in course of an explicit inspection of each single case in the  range 0 ⩽ j + k ⩽ 8, that the functions F(t) · ujk(t) are linear combinations of the form  (39b)  p1(t)F ′(t) + p2(t)F ′′(t) + · · · + pj+k+1(t)F (j+k+1)(t)  with rational polynomials pκ(t) (depending on j, k), and conjectured this structure to be  true for all j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In particular, their table has the entries  F(t) · u21(t) = −1  4F ′(t) + 1  8F (4)(t),  F(t) · u30(t) = 7  12F ′(t) + t  3F ′′(t) + 1  24F (4)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  This way we get, rather unexpectedly, the simple and short form21  (39c)  ˜F2(t) =  2  175F ′(t) − 16t  175F ′′(t) + 1  50F (4)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a similar expression for an accordingly constructed ˜F3 is displayed in (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  21Note that a direct application of the table in [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 68] to (37b) produces a far less appealing result, namely  ˜F2(t) =  19  1050F ′(t) + tF ′(t)2  75F(t) − 11t  105F ′′(t) + F ′′(t)2  100F(t) − F ′(t)F ′′′(t)  75F(t)  +  7  300F (4)(t),  which is no longer linear in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Lifting to the functional form of F1(t) and F2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The relation between F1(t), F2(t)  and their counterparts with a tilde is established by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By using the notation  introduced there, with t being any fixed real number, the expansion parameter h sufficiently  small and s = ψ−1  h (t), Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 yields (the Fredholm determinants are seen to be equal by  transforming the integrals)  (40)  det(I − Kh)|L2(t, µh−1) = F(t) + F1(t)h + F2(t)h2 + O(h3)  = det(I − ˜Kh)|L2(s, 2h−1) = F(s) + ˜F1(s)h + ˜F2(s)h2 + O(h3),  where we have absorbed the exponentially small contributions O(e−2µh−1) and O(e−4h−1)  into the O(h3) error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Using the power series (104), that is,  s = 2−1/3h−1ζ(1 − ht) = t + 3t2  10 h + 32t3  175 h2 + · · · ,  we get by Taylor expansion  F(s) = F(t) + 3t2  10 F ′(t)h +  �32t3  175 F ′(t) + 9t4  200F ′′(t)  �  h2 + O(h3),  ˜F1(s) = ˜F1(t) + 3t2  10  ˜F ′  1(t)h + O(h2),  ˜F2(s) = ˜F2(t) + O(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By plugging this into (40) and comparing coefficients we obtain  F1(t) = ˜F1(t) + 3t2  10 F ′(t),  F2(t) = ˜F2(t) + 3t2  10  ˜F ′  1(t) + 32t3  175 F ′(t) + 9t4  200F ′′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Combined with (39), this finishes the proof of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Simplifying the form of Choup’s Edgeworth expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' When, instead of the  detour via ˜K1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 is directly applied to the kernel K1 in (30a), we get  F1(t) = −1  5F ′′(t) + 3  10F(t) tr  �  (I − K0)−1L  ���  L2(t,∞),  where  (41a)  L(x, y) = (x2 + xy + y2) Ai(x) Ai(y) − (x + y) Ai′(x) Ai′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, a comparison with (24a) proves the useful formula22  (41b)  F(t) tr  �  (I − K0)−1L  ���  L2(t,∞) = t2F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  As an application to the existing literature, this formula helps us to simplify the results  obtained by Choup for the soft-edge limit expansions of GUE and LUE: that is, when  studying the distribution of the largest eigenvalue distribution function in GUEn and LUEn,ν  (dimension n, parameter ν) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In fact, since the kernel L appears in the first finite-size  correction term of a corresponding kernel expansion [21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3], lifting that expansion  to the Fredholm determinant by Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 allows us to recast [21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4] in a simplified  22Note that our derivation of this formula does only depend on Fredholm determinants and does not use  any representation in terms of Painlevé II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Based on Painlevé representations, it has been derived, implicitly  though, in the recent work of Forrester and Mays [33]: see Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='16), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='19), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='29) there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A  further alternative derivation follows from observing that, by repeated application of (36),  tr  �  (I − K0)−1L  ���  L2(t,∞) = −2u10(t) + u22(t) − 2u31(t) + 2u40(t)  and using the table for the functions F(t) · ujk(t) (0 ⩽ j + k ⩽ 8) compiled in [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 68] (which is based on  an extension of formulae of the Tracy–Widom theory that represents F in terms of Painlevé II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  18  FOLKMAR BORNEMANN  form: namely, denoting the maximum eigenvalues by λG  n and λL  n,ν, we obtain, locally uniform  in t as n → ∞,  P  �  λG  n ⩽  √  2n + t · 2−1/2n−1/6�  = F(t) + n−2/3  40  �  2t2F ′(t) − 3F ′′(t)  �  + O(n−1),  (42)  P  �  λL  n,ν ⩽ 4n + 2ν + t · 2(2n)1/3�  = F(t) − 21/3n−2/3  10  �  t2F ′(t) + F ′′(t)  �  + O(n−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (43)  a result, which answers a question suggested by Baik and Jenkins [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4367].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansion of the Poissonized Length Distribution  The Poissonization of the length distribution requires the hard-to-soft edge transition of  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 to be applied to the probability distribution Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) (for integer ν = l, but we  consider the case of general ν > 0 first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For large intensities r the mode of this distribution  is located in the range of those parameters ν for which the scaled variable  (44a)  tν(r) := ν − 2√r  r1/6  (r > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It is convenient to note that tν(r) satisfies the differential equation  (44b)  t′  ν(r) = −r−2/3 − r−1  6 tν(r)  (r > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In these terms we get the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' There holds the expansion  (45)  Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = F(t) +  m  �  j=1  F P  j (t) r−j/3 + r−(m+1)/3 · O  �  e−t�����  t=tν(r)  ,  which is uniformly valid when r, ν → ∞ subject to  t0 ⩽ tν(r) ⩽ r1/3,  with m being any fixed integer in the range 0 ⩽ m ⩽ m∗ and t0 any fixed real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Preserving uniformity, the expansion can be repeatedly differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Here the F P  j  are certain smooth functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first two are23  F P  1 (t) = − t2  60F ′(t) − 1  10F ′′(t),  (47a)  F P  2 (t) =  � 1  350 + 2t3  1575  �  F ′(t) +  � 11t  1050 +  t4  7200  �  F ′′(t) + t2  600F ′′′(t) +  1  200F (4)(t),  (47b)  while a similar expression for F P  3 (t) is displayed in (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For r, ν > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', equivalently t > −2r1/3 and s < h−1  ν ) the transformations  4r = φν(s),  t = tν(r),  are inverted by the expressions  (48)  s =  t  �  1 + t  2r−1/3�1/3 ,  hν =  r−1/3  2  �  1 + t  2r−1/3�2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For t0 ⩽ t ⩽ r1/3 we get  s0 := ( 2  3)1/3t0 ⩽ ( 2  3)1/3t ⩽ s < h−1  ν  23To validate formulae (47a/b) and (109), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2 plots F P  3 (t) next to the approximation  (46)  F P  3  �  tν(r)  �  ≈ r−1�  Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) − F(t) − F P  1 (t)r−1/3 − F P  2 (t)r−2/3� ����  t=tν(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  for r = 250 and r = 2000, varying ν in such a way that t = tν(r) covers the interval [−6, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  19  6  5  4  3  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='03  6  5  4  3  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='005  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01  6  5  4  3  2  1  0  1  2  10  8  6  4  2  0  2  4  10-3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Plots of F P  1 (t) (left panel) and F P  2 (t) (middle panel) as in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The right panel  shows F P  3 (t) as in (109) (black solid line) with the approximations (46) for r = 250 (red  dotted line) and r = 2000 (green dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the parameter ν has been varied such that  tν(r) covers the range of t displayed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Note that the functions F P  j (t) (j = 1, 2, 3) are  by about a factor of 10 to 100 smaller in scale than their counterparts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  and observe that in this range of t the expressions in (48) expand as uniformly convergent  power series in powers of r−1/3, starting with  s = t − t2  6 r−1/3 + t3  18r−2/3 + · · · ,  hν = 1  2r−1/3 − t  6r−2/3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  If we plug these uniformly convergent power series into the uniform expansion of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1),  Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = Ehard  2  (φν(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν) = F(s) +  m  �  j=1  F1(s)hj  ν + hm+1  ν  O(e−3s/2),  we obtain the asserted form of the expansion (45) (as well as the claim about the repeated  differentiability), simplifying the exponential error term by observing that (3/2)2/3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In  particular, the first two correction terms in (45) are thus  F P  1 (t) = 1  2F1(t) − t2  6 F ′(t),  F P  2 (t) = 1  4F2(t) − t  6F1(t) − t2  12F ′  1(t) + t3  18F ′(t) + t4  72F ′′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Together with the expressions given in (24) this yields the functional form asserted in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  By (1), specializing Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 to the case of integer parameter ν = l yields the expansion  (49)  P  �  LNr ⩽ l  �  = F(t) +  m  �  j=1  F P  j (t) r−j/3 + r−(m+1)/3 · O  �  e−t�����  t=tl(r)  which is uniformly valid under the conditions stated there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In the literature, scalings are often applied to the probability distribution rather  than to the expansion terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since LNr is a an integer-valued random variable, one has to  exercise some care with the scaled distribution function being piecewise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Namely,  for t ∈ R being any fixed number, one has  P  �LNr − 2√r  r1/6  ⩽ t  �  = P  �  LNr ⩽ l  �  ,  l =  �  2√r + tr1/6�  ,  where ⌊·⌋ denotes the Gauss bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Thus, by defining  t(r) =  �  2√r + tr1/6�  − 2√r  r1/6  20  FOLKMAR BORNEMANN  and noting that t(r) stays bounded when r → ∞ while t is fixed, (49) takes the form  (50)  P  �LNr − 2√r  r1/6  ⩽ t  �  = F  �  t(r)�  +  m  �  j=1  F P  j  �  t(r)�  r−j/3 + O  �  r−(m+1)/3�  (r → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  If one chooses to re-introduce the continuous variable t in (parts of) the expansion terms,  one has to take into account that  (51)  t(r) = t + O(r−1/6)  (r → ∞)  where the exponent −1/6 in the error term is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For example, this gives (as previously  obtained by Baik and Jenkins [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3] using the technology of Riemann–Hilbert  problems to prove the expansion and Painlevé representations to put F P  1 into the simple  functional form (47a))  (52)  P  �LNr − 2√r  r1/6  ⩽ t  �  = F  �  t(r)�  + F P  1 (t) r−1/3 + O(r−1/2)  (r → ∞),  where the O(r−1/2) error term is governed by the Gauss bracket in (51) and cannot be improved  upon—completely dominating the order O(r−2/3) correction term in (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Therefore, claiming  an O(r−2/3) error term to hold in (52) as stated in [33, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1] neglects the effect of the  Gauss bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' De-Poissonization and the Expansion of the Length Distribution  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansion of the CDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In this section we prove (subject to a tameness hypothesis  on the zeros of the generating functions in a sector of the complex plane) an expansion  of the CDF P(Ln ⩽ l) of the length distribution near its mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The general form of such  an expansion was conjectured in the recent papers [16, 33] where approximations of the  graphical form of the first few terms were provided (see [16, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4/6] and [33, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here,  for the first time, we give the functional form of these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The underlying tool is analytic  de-Poissonization, a technique that was developed in the 1990s in theoretical computer science  and analytic combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  To prepare for the application of analytic de-Poissonization in the form of the Jacquet–  Szpankowski Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, we consider any fixed compact interval [t0, t1] and a sequence of  integers ln → ∞ such that  (53)  t0 ⩽ t∗  n := tln(n) ⩽ t1  (n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  When n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 with n0 large enough (depending only on t0, t1)  we thus get the uniform bounds25  2√r + (t0 − 1)r1/6 ⩽ ln ⩽ 2√r + (t1 + 1)r1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  24Furthermore, the right panel of [33, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3] is not showing an approximation of the O(r−2/3) term in (50),  let alone in (52), but instead an approximation of the O(ν−4/3) term in the auxiliary expansion  Ehard  2  ��  ν − t( ν  2 )1/3 + t2  6 ( ν  2 )−1/3�2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ν  �  = F(t) + ˆF1(t) ( ν  2 )−2/3 + ˆF2(t) ( ν  2 )−4/3 + O(ν−2)  (ν → ∞),  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [33, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='33)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Now, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 and the formulae in (47) yield the simple relations  ˆF1(t) = F P  1 (t),  ˆF2(t) = F P  2 (t) + t  3F P  1 (t),  which are consistent with [33, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the additional term tF P  1 (t)/3 explains the different shape of ˆF2(t), as  displayed in the right panel there, when compared to F P  2 (t), as shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  25Observe that  2√r + tr1/6 = 2√n + tn1/6 + O(n1/10)  uniformly for t0 ⩽ t ⩽ t1 and n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  21  We write the induced Poisson generating function, and exponential generating function, of  the length distribution as  (54)  Pk(z) := P(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' lk) = e−z  ∞  �  n=0  P(Ln ⩽ lk)zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ,  fk(z) := ezPk(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By (1) we have Pn(r) = Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ln) for real r > 0, so that Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 (see also (49)) gives  the expansion  (55)  Pn(r) = F(t) +  m  �  j=1  F P  j (t) r−j/3 + O(r−(m+1)/3)  ����  t=tln(r)  ,  uniformly valid when n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞, m being any fixed integer in the  range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, the implied constant in the error term depends only on t0, t1,  but not on the specific sequence ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Preserving uniformity, the expansion can be repeatedly  differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In particular, using the differential equation (44b) we get  that P (j)  n (n) expands in powers of n−1/3, starting with a leading order term of the form  (56a)  P (j)  n (n) = (−1)jF (j)(t∗  n)n−2j/3 + O(n−(2j+1)/3)  (n → ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  the first specific cases being (see (72) for P ′  n(n))  Pn(n) = F(t∗  n) + F P  1 (t)n−1/3 + F P  2 (t)n−2/3 + O(n−1)  ����  t=t∗n  ,  (56b)  P ′′  n(n) = F ′′(t∗  n)n−4/3 +  �  F P  1  ′′(t) + 5  6F ′(t) + t  3F ′′(t)  �  n−5/3 + O(n−2)  ����  t=t∗n  ,  (56c)  P (4)  n (n) = F (4)(t)n−8/3 + O(n−3)  ����  t=t∗n  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (56d)  the implied constants in the error terms depend only on t0, t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We recall from the results of [16, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2] (note the slight differences in notation), and the  proofs given there, that the exponential generating functions fn(z) are entire functions of genus  zero having, for each 0 < ϵ < π/2, only finitely many zeros26 in the sector |arg z| ⩽ π/2 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  If we denote the real auxiliary functions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1) of fn(r) = erPn(r) by an(r) and bn(r),  the expansion (55), and its derivatives based on (44b), give (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' also (73))  (57)  an(r) = r + O(r1/3),  bn(r) = r + O(r2/3),  uniformly valid when n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the implied constants in the error  terms depend only on t0, t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, numerical experiments strongly hint at the property that the zeros of the exponential  generating functions fn(z) in the sectors |arg z| ⩽ π/2+ϵ satisfy a uniform tameness condition  as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 (see also Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2): the zeros are neither coming too close to the positive  real axis nor are they getting too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Proving this property seems to be rather difficult,  though—at least we were lacking the methodology to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Given this state of affairs, the  results on the expansions of the length distribution will be subject to the following:  Tameness hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For any real t0 < t1 and any sequence of integers ln → ∞ satisfy-  ing (53) the zeros of the induced family fn(z) of exponential generating functions (54) are  uniformly tame (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2), with parameters and implied constants only depending on t0  and t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  26Because of fn(r) > 0 for r > 0, the real zeros of fn are negative and the complex ones are coming in  conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  22  FOLKMAR BORNEMANN  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Plots of F D  1 (t) (left panel) and F D  2 (t) (middle panel) as in (60);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' both agree with  the numerical prediction of their graphical form given in the left panels of [16, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The right panel shows F D  3 (t) as in (110) (black solid line) with the approximations (59) for  n = 250 (red +), n = 500 (green ◦) and n = 1000 (blue •);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the integer l has been varied  such that tl(n) spreads over the range of t displayed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Evaluation of (59) uses the table  of exact values of P(Ln ⩽ l) up to n = 1000 that was compiled in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let t0 < t1 be any real numbers and assume the tameness hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Then  there holds the expansion  (58)  P(Ln ⩽ l) = F(t) +  m  �  j=1  F D  j (t) n−j/3 + O(n−(m+1)/3)  ����  t=tl(n)  ,  which is uniformly valid when n, l → ∞ subject to t0 ⩽ tl(n) ⩽ t1 with m being any fixed  integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here the F D  j  are certain smooth functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first two  are27  F D  1 (t) = − t2  60F ′(t) − 3  5F ′′(t),  (60a)  F D  2 (t) =  �  − 139  350 + 2t3  1575  �  F ′(t) +  �  − 43t  350 +  t4  7200  �  F ′′(t) + t2  100F ′′′(t) + 9  50F (4)(t),  (60b)  while a similar expression for F D  3 (t) is displayed in (110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Following up the preparations preceding the formulation of the theorem, the tameness  hypothesis allows us to apply Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, bounding fn(z) = ezPn(z) by  (61)  ��fn(reiθ)  �� ⩽  �  �  �  2fn(r)e− 1  2 θ2r,  0 ⩽ |θ| ⩽ r−2/5,  2fn(r)e− 1  2 r1/5,  r−2/5 ⩽ |θ| ⩽ π,  for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 when n0 is sufficiently large (depending on t0, t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Using the trivial bounds (for r > 0 and |θ| ⩽ π)  0 ⩽ fn(r) ⩽ er,  0 ⩽ Pn(r) ⩽ 1,  1 − 1  2θ2 ⩽ cos θ,  the first case in (61) can be recast in form of the bound  ��Pn(reiθ)  �� ⩽ 2Pn(r)er  �  1−cos θ− 1  2 θ2�  ⩽ 2,  which proves condition (I) of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 with B = 2, D = 1, β = 0 and δ = 2/5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' whereas the  second case implies  |fn(neiθ)| ⩽ 2fn(n)e− 1  2 n1/5 ⩽ 2 exp  �  n − 1  2n1/5�  ,  27To validate the expansion (58) and the formulae (60a/b), (110), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 plots F D  3 (t) next to the approximation  (59)  F D  3  �  tl(n)  �  ≈ n−1�  P(Ln ⩽ l) − F(t) − F D  1 (t)n−1/3 − F D  2 (t)n−2/3� ����  t=tl(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl(n) spreads over [−6, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  23  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Plots of F ∗  1 (t) (left panel) and F ∗  2 (t) (middle panel) as in (65);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' both agree with  the numerical prediction of their graphical form given in the right panels of [16, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The right panel shows F ∗  3 as in (111) (black solid line) with the approximations (64) for  n = 250 (red +), n = 500 (green ◦) and n = 1000 (blue •);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the integer l has been varied  such that tl−1/2(n) spreads over the range of t displayed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Evaluation of (64) uses the  table of exact values of P(Ln = l) up to n = 1000 that was compiled in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  which proves condition (O) of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 with A = 1/2, C = 0, α = 1/5 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Hence,  there holds the Jasz expansion (92), namely  P(Ln ⩽ ln) = Pn(n) +  M  �  j=2  bj(n)P (j)  n (n) + O(n−(M+1)/5)  for any M = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' as n ⩾ n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' here n1 and the implied constant depend on t0, t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By noting that the diagonal Poisson–Charlier polynomials bj have degree ⩽ ⌊j/2⌋ and by  choosing M large enough, the expansions (56) of P (j)  n (n) in terms of powers of n−1/3 yield  that there are smooth functions F D  j  such that  P(Ln ⩽ ln) = F(t) +  m  �  j=1  F D  j (t) n−j/3 + O(n−(m+1)/3)  ����  t=t∗n  as n → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' m being any integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Given the uniformity of the bound  for fixed t0 and t1, we can replace ln by l and t∗  n by tl(n) as long as we respect t0 ⩽ tl(n) ⩽ t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  This finishes the proof of (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The first two functions F D  1 and F D  2 (t) can be determined using the particular case (93) of  the Jasz expansion from Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 (which applies here because of (56a)), namely  P(Ln ⩽ ln) = Pn(n) − n  2 P ′′  n(n) + n2  8 P (4)  n (n) + O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Inserting the formulae displayed in (56) we thus obtain  F D  1 (t) = F P  1 (t) − 1  2F ′′(t)  (62a)  F D  2 (t) = F P  2 (t) − 1  2F P  1  ′′(t) − 5  12F ′(t) − t  6F ′′(t) + 1  8F (4)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (62b)  Together with the expressions given in (47) this yields the functional form asserted in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansion of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Subject to its assumptions, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 implies for the PDF of  the length distribution that  P(Ln = l) = P(Ln ⩽ l) − P(Ln ⩽ l − 1)  =  �  F(tl(n)) − F(tl−1(n))  �  +  m  �  j=1  �  F D  j (tl(n)) − F D  j (tl−1(n))  �  n−j/3 + O(n−(m+1)/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  24  FOLKMAR BORNEMANN  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25  ℙ  Mo  50  52  54  56  58  60  62  64  66  68  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='16  ℙ  Mh  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The exact discrete length distribution P(Ln = l) (blue bars centered at the  integers l) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the asymptotic expansion (63) for m = 0 (the Baik–Deift–Johansson limit,  dotted line) and for m = 2 (the limit with the first two finite-size correction terms added,  solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Left: n = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' right: n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The expansions are displayed as functions of the  continuous variable ν, evaluating the right-hand-side of (63) in t = tν−1/2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The exact  values are from the table compiled in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Note that a graphically accurate continuous  approximation of the discrete distribution must intersect the bars right in the middle of  their top sides: this is, indeed, the case for m = 2 (except at the left tail for n = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In  contrast, the uncorrected limit law (m = 0) is noticeable inaccurate for this range of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Applying the central differencing formula (which is, basically, just a Taylor expansion for  smooth G centered at the midpoint)  G(t + h) − G(t) = hG′(t + h/2) + h3  24G′′′(t + h/2) +  h5  1920G(5)(t + h/2) + · · · ,  with increment h = n−1/6, we immediately get the following corollary of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let t0 < t1 be any real numbers and assume the tameness hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Then  there holds the expansion  (63)  n1/6 P(Ln = l) = F ′(t) +  m  �  j=1  F ∗  j (t)n−j/3 + O(n−(m+1)/3)  ����  t=tl−1/2(n)  ,  which is uniformly valid when n, l → ∞ subject to the constraint t0 ⩽ tl−1/2(n) ⩽ t1 with m  being any fixed integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here the F ∗  j are certain smooth functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  the first two are28  F ∗  1 (t) = − t  30F ′(t) − t2  60F ′′(t) − 67  120F ′′′(t),  (65a)  F ∗  2 (t) = 2t2  525F ′(t) +  �  − 629  1200 + 23t3  12600  �  F ′′(t) +  �  − 899t  8400 +  t4  7200  �  F ′′′(t)  (65b)  + 67t2  7200F (4)(t) + 1493  9600F (5)(t),  while a similar expression for F ∗  3 (t) is displayed in (111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The case m = 0 of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 gives  n1/6 P(Ln = l) = F ′(tl−1/2(n)) + O(n−1/3),  28To validate the expansion (63) and the formulae (65a/b), (111), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4 plots F ∗  3 (t) next to the approximation  (64)  F ∗  3  �  tl−1/2(n)  �  ≈ n7/6�  P(Ln = l) − F ′(t)n−1/6 − F ∗  1 (t)n−1/2 − F ∗  2 (t)n−5/6� ����  t=tl−1/2(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl−1/2(n) spreads over [−6, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  25  where the exponent in the error term cannot be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By noting  tl−1/2(n) = tl(n) − 1  2n−1/6  we understand that, for fixed large n, visualizing the discrete length distribution near its  mode by plotting the points  �  tl(n), n1/6 P(Ln = l)  �  next to the graph (t, F ′(t)) introduces a perceivable bias: namely, all points are shifted by an  amount of n−1/6/2 to the right of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Exactly such a bias can be observed in the first  ever published plot of the PDF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the density of the Tracy–Widom distribution by Odlyzko  and Rains in [48, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1]: the Monte-Carlo data for n = 106 display a consistent shift by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansions of Stirling-Type Formulae  In our work [16] we advocated the use of a Stirling-type formula to approximate the length  distribution for larger n (because of being much more efficient and accurate than Monte-Carlo  simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' To recall some of our findings there, let us denote the exponential generating  function and its Poisson counterpart simply by  f(z) =  ∞  �  n=0  P(Ln ⩽ l)zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ,  P(z) = e−zf(z),  suppressing the dependence on the integer parameter l from the notation for the sake of  brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It was shown in [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2] that the entire function f is H-admissible so that  there is the normal approximation (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2)  (66)  P(Ln ⩽ l) =  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='f(r)  rn�  2πb(r)  �  exp  �  −(n − a(r))2  2b(r)  �  + o(1)  �  (r → ∞)  uniformly in n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' while l is any fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, a(r) and b(r) are the real  auxiliary functions  a(r) = rf′(r)  f(r) ,  b(r) = ra′(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We consider the two cases r = rn, a(rn) = n and r = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' After dividing (66) by the classical  Stirling factor (which does not change anything of substance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1)  (67)  τn :=  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  √  2πn  � e  n  �n  ∼ 1 + n−1  12 + n−2  288 + · · · ,  we get after some re-arranging of terms the Stirling-type formula Sn,l (r = rn) and the  simplified Stirling-type formula ˜Sn,l (r = n):  Sn,l :=  P(rn)  �  b(rn)/n  exp  �  n Λ  �rn − n  n  ��  ,  Λ(h) = h − log(1 + h),  (68a)  ˜Sn,l :=  P(n)  �  b(n)/n  exp  �  −(n − a(n))2  2b(n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (68b)  As shown in [16], both approximations are amenable for a straightforward numerical evaluation  using the tools developed in [13, 14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For fixed l, the normal approximation (66) implies  P(Ln ⩽ l) = Sn,l · (1 + o(1))  (n → ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  but numerical experiments reported in [16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (8b)] suggest that there holds  P(Ln ⩽ l) = Sn,l + O(n−2/3)  26  FOLKMAR BORNEMANN  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  6  5  4  3  2  1  0  1  2  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='05  6  5  4  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='16  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Left panel: plots of ˜F S  2 (t) (solid line) and ˜F S  2 (t) (dash-dotted line) as in (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The  middle and right panel show the approximations of F S  3 (t) and ˜F S  3 (t) in (77) for n = 250  (red +), n = 500 (green ◦) and n = 1000 (blue •);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the integer l has been varied such that  tl(n) spreads over the range of the variable t displayed here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the dotted line displays a  polynomial fit to the data points of degree 30 to help visualizing their joint graphical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Evaluation of (77) uses the table of exact values of P(Ln ⩽ l) up to n = 1000 that was  compiled in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  uniformly when n, l → ∞ while tl(n) stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Subject to the tameness hypothesis of  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 we prove this observation as well as its counterpart for the simplified Stirling-type  formula, thereby unveiling the functional form of the error term O(n−2/3):29  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let t0 < t1 be any real numbers and assume the tameness hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Then,  for the Stirling-type formula Sn,l and its simplification ˜Sn,l, there hold the expansions (note  that both are starting at j = 2)  P(Ln ⩽ l) = Sn,l +  m  �  j=2  F S  j  �  tl(n)  �  n−j/3 + O(n−(m+1)/3),  (69a)  P(Ln ⩽ l) = ˜Sn,l +  m  �  j=2  ˜F S  j  �  tl(n)  �  n−j/3 + O(n−(m+1)/3),  (69b)  which are uniformly valid when n, l → ∞ subject to t0 ⩽ tl(n) ⩽ t1 with m being any fixed  integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here the F S  j and ˜F S  j are certain smooth functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first  of them being30  F S  2 (t) = −3  4  F ′(t)4  F(t)3 + 3  2  F ′(t)2F ′′(t)  F(t)2  − 3  8  F ′′(t)2  F(t) − 1  2  F ′(t)F ′′′(t)  F(t)  + 1  8F (4)  2 (t),  (70a)  ˜F S  2 (t) = −1  2F ′(t) + 1  4  F ′(t)4  F(t)3 − 3  8  F ′′(t)2  F(t) + 1  8F (4)  2 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (70b)  The solution rn of the equation a(rn) = n, required to evaluate Sn,l, satisfies the expansion31  rn = n + F ′�  tl(n)  �  F  �  tl(n)  � n1/3 + O(1),  which is uniformly valid under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  29Note that the expansions (74) for ˜Sn,l and (76) for Sn,l given in the proof do not require the tameness  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It is only required to facilitate the comparison with the result of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, which then yields (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  30The functional form of the terms F S  2 , ˜F S  2 differs significantly from the one of corresponding terms in the  previous theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Though they still share the form  F(t) ·  �  rational polynomial in u00(t), u10(t), u11(t), u20(t), u21(t), u30(t)  �  ,  using the algorithmic ideas underlying the tabulation of F(t) · ujk(t) (0 ⩽ j + k ⩽ 8) in [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 68] one can  show that F S  2 (t) and ˜F S  2 do not simplify to the form (39b) of a linear combination of derivatives of F with  (rational) polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  31This provides excellent initial guesses for solving a(rn) = n by iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [16, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  27  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We restrict ourselves to the case m = 2, focussing on the concrete functional form of  the expansion terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' nevertheless the general form of the expansions (69) should become  clear along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Preparatory steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Because of P(r) = Ehard  2  (4r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' l) (using the notation preceding Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1),  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 gives that  (71)  P(r) = F(t) + F P  1 (t) · r−1/3 + F P  2 (t) · r−2/3 + O(r−1)  ���  t=tl(r) ,  which is uniformly valid when r, l → ∞ subject to the constraint t0 ⩽ tl(r) ⩽ t1 (the same  constraint applies to the expansions to follow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Preserving uniformity, the expansion can be  repeatedly differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable r, which yields by using the differential equation  (44b) satisfied by tl(r) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' also (56a))  (72)  P ′(r) = −F ′(t)r−2/3 −  �  F P  1  ′(t) + t  6F ′(t)  �  r−1 + O(r−4/3)  ����  t=tl(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Recalling f(r) = erP(r), we thus get  (73a)  a(r) = r + rP ′(r)  P(r) = r + a1(t)r1/3 + a2(t) + O(r−1/3)  ���  t=tl(r)  with the coefficient functions  (73b)  a1(t) = −F ′(t)  F(t) ,  a2(t) = −  1  F(t)  �  F P  1  ′(t) + t  6F ′(t)  �  + F P  1 (t)F ′(t)  F(t)2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a further differentiation yields  (73c)  b(r) = ra′(r) = r − a′  1(t)r2/3 +  �1  3a1(t) − t  6a′  1(t) − a′  2(t)  �  r1/3 + O(1)  ����  t=tl(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The simplified Stirling-type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here we have r = n and we write t∗ := tl(n) to be  brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By inserting the expansions (71) and (73) into the expression (68b), we obtain after a  routine calculation with truncated power series and collecting terms as in (62) that  (74)  ˜Sn,l = F(t∗) + F D  1 (t∗)n−1/3 +  �  F D  2 (t∗) − ˜F S  2 (t∗)  �  n−2/3 + O(n−1),  where the remaining ˜F S  2 (t) is given by (70b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' a subtraction from (58) yields (69b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The Stirling-type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here we have r = rn and we have to distinguish between t∗ and  tl(r) = t∗ · (n/r)1/6 + 2  √n − √r  r1/6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By inserting the expansion  (75a)  rn = n + r1(t∗)n1/3 + r2(t∗) + O(n−1/3)  into tn(rn) and a(rn) we obtain  tl(rn) = t∗ − r1(t∗)n−1/3 −  �  r2(t∗) + t∗  6 r1(t∗)  �  n−2/3 + O(n−1)  (75b)  a(rn) = n + (a1(t∗) + r1(t∗))n1/3 +  �  a2(t∗) + r2(t∗) − r1(t∗)a′  1(t∗)  �  + O(n−1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (75c)  Thus the solution of a(rn) = n, which by Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 is unique, leads to the relations  (75d)  r1(t) = −a1(t),  r2(t) = −a2(t) − a1(t)a′  1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By inserting, first, the expansions (75) into the expansions (71) and (73) for the particular  choice r = rn and, next, the thus obtained results into the expression (68a), we obtain after  a routine calculation with truncated power series and collecting terms as in (62) that  (76)  Sn,l = F(t∗) + F D  1 (t∗)n−1/3 +  �  F D  2 (t∗) − F S  2 (t∗)  �  n−2/3 + O(n−1),  where the remaining F S  2 (t) is given by (70a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' a subtraction from (58) yields (69a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  28  FOLKMAR BORNEMANN  To validate the expansions (69) and the formulae (70a/b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6 plots the approximations  F S  3  �  tl(n)  �  ≈ n−1�  P(Ln ⩽ l) − Sn,l − F S  2  �  tl(n)  �  n−2/3�  ,  (77a)  ˜F S  3  �  tl(n)  �  ≈ n−1�  P(Ln ⩽ l) − ˜Sn,l − ˜F S  2  �  tl(n)  �  n−2/3�  (77b)  for n = 250, n = 500 and n = 1000, varying the integer l in such a way that t = tl(n) spreads  over [−6, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The plot suggests the following observations:  Apparently there holds ˜F S  2 (t) < F S  2 (t) < 0 for t ∈ [−6, 3], which if generally true  would imply  P(Ln ⩽ l) < Sn,l < ˜Sn,l  for n being sufficiently large and l near the mode of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This one-sided  approximation of the length distribution by the Stirling formula Sn,l from above is  also clearly visible in [16, Tables 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Comparing F S  2 (t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6 to F D  2 (t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 shows that the maximum error  max  l=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=',n  ���P(Ln ⩽ l) −  �  F(t) + F D  1 (t)n−1/3���  t=tl(n)  ��� ≈ n−2/3∥F D  2 ∥∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='25n−2/3  of approximating the length distribution by the first finite-size correction in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1  is about an order of magnitude larger than the maximum error of the Stirling-type  formula,  max  l=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=',n |P(Ln ⩽ l) − Sn,l| ≈ n−2/3∥F S  2 ∥∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='031n−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  This property of the Stirling-type formula was already observed in [16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3] and  was used there to approximate the graphical form of F D  2 (t) (see [16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If one includes the classical Stirling factor (67) into the Stirling-type formula  by replacing (68a) with the unmodified normal approximation (66), that is, with  S∗  n,l := τn  P(rn)  �  b(rn)/n  exp  �  n Λ  �rn − n  n  ��  = τnSn,l,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 would remain valid: in fact, multiplication of (69a) by the expansion (67) of τn in  powers of n−1 gives, by taking (58) into account,  P(Ln ⩽ l) = S∗  n,l +  m  �  j=2  F S∗  j  �  tl(n)  �  n−j/3 + O(n−(m+1)/3),  where the first two coefficient functions are  F S∗  2 (t) = F S  2 (t),  F S∗  3 (t) = F S  3 (t) − 1  12F(t),  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Because of limt→+∞ F(t) = 1, we would loose the decay of F S  3 (t) for large t, leaving us  with a non-zero residual value coming from the classical Stirling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For this reason, we  recommend dropping the factor τn, thereby resolving an ambiguity expressed in [16, Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expansions of Expected Value and Variance  Lifting the expansion (63) of the PDF of the length distribution to one of the expected  value and variance requires a control of the tails (of the distribution itself and of the expansion  terms) which, at least right now, we can only conjecture to hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  To get to a reasonable conjecture, we recall the tail estimates for the discrete distribution  (see [7, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12)]),  P(Ln = l) ⩽ Ce−c|tl(n)|3  (tl(n) ⩽ t0 < 0),  P(Ln = l) ⩽ Ce−c|tl(n)|3/5  (0 < t1 ⩽ tl(n)),  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  29  when n is large enough with c > 0 being some absolute constant and C > 0 a constant that  depends on t0, t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  On the other hand, from Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 and its proof we see that the F ∗  j (t) take the form  (78)  F(t) ·  �  rational polynomial in terms of the form tr((I − K0)−1K)|L2(t,∞)  �  ,  where the kernels K are finite sums of rank one kernels with factors of the form (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The  results of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 thus show that the F ∗  j (t) are exponentially decaying when t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, looking at the left tail, the (heuristic) estimate of the largest eigenvalue of the Airy  operator K0 on L2(t, ∞) as given in the work of Tracy and Widom [64, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='23)] shows a  superexponential growth bound of the operator norm  ∥(I − K0)−1∥ ⩽ C|t|−3/4ec|t|3/2  (t ⩽ t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  This, together with the superexponential decay (see [6, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3] and [24, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1] for the  specific constants)  0 ⩽ F(t) ⩽ C|t|−1/8e−c|t|3  (t ⩽ t0)  of the Tracy–Widom distribution itself, and with an at most polynomial growth of the trace  norms ∥K|J 1(t,∞) as t → −∞, shows that the bounds for the discrete distribution find a  counterpart for the expansion terms F ∗  j (t):  (79)  |F ∗  j (t)| ⩽ Ce−c|t|3  (t ⩽ t0 < 0),  |F ∗  j (t)| ⩽ Ce−c|t|  (0 < t1 ⩽ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Thus, assuming an additional amount of uniformity that would allow us to absorb the  exponentially small tails in the error term of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, we conjecture the following:  Uniform Tails Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The expansion (63) can be sharpened to include the tails in  the form  (80)  n1/6 P(Ln = l) = F ′(t) +  m  �  j=1  F ∗  j (t)n−j/3 + n−(m+1)/3 · O  �  e−c|t|3/5� ����  t=tl−1/2(n)  ,  uniformly valid in l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , n as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We now follow the ideas sketched in our work [16, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3] on the Stirling-type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' By  shift and rescale, the expected value of Ln can be written in the form  E(Ln) =  n  �  l=1  l · P(Ln = l) = 2√n + 1  2 +  n  �  l=1  tl−1/2(n) · n1/6 P(Ln = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Inserting the expansion (80) of the uniform tail hypothesis gives, since its error term is  uniformly summable,  (81)  E(Ln) = 2√n + 1  2 +  m  �  j=0  µ(n)  j  n1/6−j/3 + O(n−(m+1)/3),  with coefficients (still depending on n, though), writing F ∗  0 := F ′,  µ(n)  j  := n−1/6  n  �  l=1  tl−1/2(n)F ∗  j  �  tl−1/2(n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By the tail estimates (79) we have, writing a := −n−1/6(2√n + 1  2) and h := n−1/6,  µ(n)  j  = h  ∞  �  l=−∞  (a + lh)F ∗  j (a + lh) + O(e−cn1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, based on a precise description of its pole field in [38], it is known that the Hastings–  McLeod solution of Painlevé II and a fortiori, by the Tracy–Widom theory [64], also F and  its derivatives can be continued analytically to the strip |ℑz| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Therefore, we assume:  30  FOLKMAR BORNEMANN  Uniform Strip Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The F ∗  j (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , m) extend analytically to a strip |ℑz| ⩽ s  of the complex z-plane, uniformly converging to 0 as z → ∞ in that strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Under that hypothesis, a classical result about the rectangular rule in quadrature theory  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', [23, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='14)]) gives  h  ∞  �  l=−∞  (a + lh)F ∗  j (a + lh) =  � ∞  −∞  F ∗  j (t) dt + O(e−πs/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Thus the µ(n)  j  and their limit quantities  µj =  � ∞  −∞  tF ∗  j (t) dt  differ only by an exponential small error of at most O(e−cn1/6), which can be absorbed in  the error term of (81);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' an illustration of such a rapid convergence is given in [16, Table 3] for  the case j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The functional form of F ∗  0 = F ′, F ∗  1 and F ∗  2 , namely being a linear combination of higher  order derivatives of F with polynomial coefficients (see (65)), allows us to express µ0, µ1, µ2  in terms of the moments  Mj :=  � ∞  −∞  tjF ′(t) dt  of the Tracy–Widom distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In fact, repeated integration by parts yields the simpli-  fying rule (where k ⩾ 1)  � ∞  −∞  tj F (k)(t) dt =  �  �  �  �  �  (−1)k−1j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (j − k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='Mj−k+1  k ⩽ j + 1,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Repeated application of that rule proves, in summary, the following contribution to Ulam’s  problem about the expected value when n grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let m be any fixed integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Then, under the uniform  tails hypothesis and the uniform strip hypothesis there holds, as n → ∞,  (82)  E(Ln) = 2√n + 1  2 +  m  �  j=0  µjn1/6−j/3 + O(n−1/6−m/3),  where the constants µj are given by  µ0 =  � ∞  −∞  tF ′(t) dt,  µj =  � ∞  −∞  tF ∗  j (t) dt  (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The first few cases can be expressed in terms of the moments of the Tracy–Widom distribution:  (83)  µ0 = M1,  µ1 = 1  60M2,  µ2 = 89  350 −  1  1400M3,  while a similar expression for µ3 is displayed in (112);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' highly accurate numerical values are  listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Likewise, by shift and rescale, the variance of Ln can be written in the form  Var(Ln) =  n  �  l=1  l2 · P(Ln = l) − E(Ln)2  = n1/3  n  �  l=1  tl−1/2(n)2 · P(Ln = l) −  �  E(Ln) − 2√n − 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By inserting the expansions (80), (82) and arguing as for Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 we get the following:  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  31  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Highly accurate values of µ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , µ3 and ν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , ν3 as computed from (83), (85)  and from (112), (113) based on values for Mj obtained as in [16, Table 3] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Prähofer’s  values for M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , M4, published in [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For the values of µ4, µ5 and ν4, ν5 see the  supplementary material mentioned in Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  j  Mj  µj  νj  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='00000 00000 00000 00000 · · ·  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='77108 68074 11601 62598 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='81319 47928 32957 84477 · · ·  1  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='77108 68074 11601 62598 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='06583 23878 70339 62521 · · ·  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='20720 50777 85797 46901 · · ·  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='94994 32722 20377 51300 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='26122 27462 52162 60525 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='56715 66368 69744 43503 · · ·  3  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='71184 47530 27647 35361 · · ·  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='11938 39067 94582 09131 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01669 21858 10456 60764 · · ·  4  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='02543 54268 39994 56536 · · ·  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='00483 35524 95005 83878 · · ·  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='12447 09934 16776 05579 · · ·  5  −74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='20410 74434 81824 47477 · · ·  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='01222 78407 77590 95405 · · ·  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='00293 40551 03931 43008 · · ·  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let m be any fixed integer in the range 0 ⩽ m ⩽ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Then, under the  uniform tails hypothesis and the uniform strip hypothesis there holds, as n → ∞,  (84)  Var(Ln) =  m  �  j=0  νjn1/3−j/3 + O(n−m/3),  with certain constants νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The first few cases can be expressed in terms of the moments of  the Tracy–Widom distribution  (85)  ν0 = −M2  1 + M2,  ν1 = −67  60 + 1  30  �  − M1M2 + M3  �  ,  ν2 = − 57  175M1 +  1  700M1M3 −  1  3600M2  2 −  29  25200M4,  while a similar expression for ν3 is displayed in (113);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' highly accurate numerical values are  listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The expansions of expected value and variance can be cross-validated by looking at the  numerical values for the coefficients µ1, µ2, µ3 and ν1, ν2, ν3 that we predicted in [16, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3]:  those values were computed by fitting, in high precision arithmetic, expansions (back then  only conjectured) of the form (82) with m = 9 and (84) with m = 8 to the exact tabulated  data for n = 500, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A decision about which digits were to be considered correct was  made by comparing the result against a similar computation for n = 600, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As it  turns out, the predictions of [16, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3] agree to all the decimal places shown there (that is, to  7, 7, 6 and 9, 6, 4 places) with the theory-based, highly accurate values given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Appendix: Variations on the Saddle Point Method  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Analytic de-Poissionization and the Jasz expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In their comprehensive  1998 memoir [39], Jacquet and Szpankowski gave a detailed study of what they termed  analytic de-Poissonization (in form of a useful repackaging of the saddle point method),  proving a selection of asymptotic expansions and applying them to various asymptotic  problems in analytic algorithmics and combinatorics (with generating functions given in  terms of functional equations amenable for checking the Tauberian growth conditions in the  complex plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Expositions with a selection of further applications can be found in [40,  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2] and [63, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Formal derivation of the Jasz expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Following the ideas of [39, Remark 3] let us start  with a purely formal derivation to motivate the algebraic form of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Suppose  that the Poisson generating function  P(z) = e−z  ∞  �  n=0  an  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  32  FOLKMAR BORNEMANN  of a sequence an is an entire function and consider some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If we write the power series  expansion of P(z), centered at z = r, in the operator form  P(z) = e(z−r)DP(r),  where D denotes differentiation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the variable r, we get by Cauchy’s formula (with a  contour encircling z = 0 counter-clockwise with index one)  (86)  an = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2πi  �  P(z)ez dz  zn+1 = e−rD  � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2πi  �  ez(D+1) dz  zn+1  �  P(r) = e−rD(D + 1)nP(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By the Cauchy product of power series  (87)  e−rx(x + 1)n =  ∞  �  j=0  cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r)xj,  cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) :=  j  �  k=0  �n  k  �(−r)j−k  (j − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ,  we get from (86) the formal expansion  (88)  an ∼  ∞  �  j=0  cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r)P (j)(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Note that the coefficients cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) are polynomials of degree j in n and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' From (87) one  easily verifies that they satisfy the three-term recurrence  (j + 1)cj+1(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) + (j + r − n)cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) + rcj−1(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) = 0  (j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=')  with initial data c0(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) = 1 and c1(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) = n − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' From (87) and [62, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='81] one immediately gets that the cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) are, up to  normalization, the Poisson–Charlier polynomials:  ∞  �  n=0  cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r)ck(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r)e−rrn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  = δjk  rj  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (j, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ),  so that they are orthogonal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the Poisson distribution of intensity r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In particular,  [62, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6)] gives (with L(ν)  k (x) the Laguerre polynomials) the representation  cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' r) = L(n−j)  j  (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Things simplify for the particular choice r = n which is suggested by the expected value of  the Poisson distribution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The corresponding polynomials bj(n) := cj(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' n),  which we call the diagonal Poisson–Charlier polynomials, satisfy the three-term recurrence  (89)  b0(n) = 1,  b1(n) = 0,  (j + 1)bj+1(n) + jbj(n) + nbj−1(n) = 0  (j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  From this we infer inductively that  bj(0) = 0,  deg bj ⩽ ⌊j/2⌋  (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now the formal expansion (86) becomes what is dubbed the Jasz expansion in [29, §VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='18]:  (90)  an ∼ P(n) +  ∞  �  j=2  bj(n)P (j)(n)  = P(n) − n  2 P ′′(n) + n  3 P ′′′(n) +  �n2  8 − n  4  �  P (4)(n)  +  �  −n2  6 + n  5  �  P (5)(n) +  �  −n3  48 + 13n2  72  − n  6  �  P (6)(n) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  33  Diagonal analytic de-Poissonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Jacquet and Szpankowski were able to prove that the  expansion (90) can be made rigorous if the Poisson generating function satisfies a Tauberian  condition in form of a growth condition at the essential singularity at z = ∞ in the complex  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In fact, this can be cast to accomodate the needs of double scaling limits in a uniform  fashion: for a two-parameter family of coefficients an,k one expands the diagonal term an,n  by, first, applying the Jasz expansion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' to n for k fixed and, then, selecting k = n only  afterwards (a process that is called diagonal de-Poissonization in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The following theorem is a particular case of [39, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4] (with Ψ = 1 and the modifications  discussed preceding [39, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (27)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' It repackages the saddle point method (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [20, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5]  and [69, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' VI]) for the asymptotic evaluation of the Cauchy integral  (91)  an = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  2πi  �  P(z)ez dz  zn+1  in a far more directly applicable fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Concerning the asserted uniform bounds of the  implied constants, see the beginning of [39, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 (Jacquet–Szpankowski 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let a family of entire Poisson generating  functions of the form  Pk(z) = e−z  ∞  �  n=0  an,k  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=')  satisfy the following two conditions32 for n ⩾ n0 where A, B, C, D, α, β, γ, δ are some constants  with A, α > 0 and 0 ⩽ δ < 1/2:  (I) If |r − n| ⩽ Dn1−δ and |θ| ⩽ Dr−δ then |Pn(reiθ)| ⩽ Bnβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (O) If |θ| > Dn−δ then |Pn(neiθ) exp(neiθ)| ⩽ Cnγ exp(n − Anα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Then, for any m = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' there holds, when n ⩾ n1 with n1 large enough,  (92)  an,n = Pn(n) +  m  �  j=2  bj(n)P (j)  n (n) + O  �  nβ−(m+1)(1−2δ)�  ,  where the bj(n) are the diagonal Poisson–Charlier polynomials (89) which have degree ⩽ ⌊j/2⌋  and satisfy bj(0) = 0 (j ⩾ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The implied constant in (92) and the constant n1 depend only  on n0 and the constants entering the conditions (I) and (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 we use Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 in the particular case β = 0, δ = 2/5  for a family of Poisson generating functions with (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (56a))  P (j)  n (n) = O(n−2j/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For m = 4 the expansion (92) is then given by (compare with (90))  an,n = Pn(n) − n  2 P ′′  n(n) + n  3 P ′′′  n (n) +  �n2  8 − n  4  �  P (4)  n (n) + O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Since the terms nP ′′′  n (n)/3 = O(n−1) and −nP (4)  n (n)/4 = O(n−5/3) get absorbed in the error  term O(n−1), and can therefore be dropped, the Jasz expansion simplifies in that case to  (93)  an,n = Pn(n) − n  2 P ′′  n(n) + n2  8 P (4)  n (n) + O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  32Here, (I) means “inside” and (O) “outside” with respect to the “polynomial cone” {z = reiθ : |θ| ⩽ Dr−δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  34  FOLKMAR BORNEMANN  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' H-admissibility and Hayman’s Theorem XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In his 1956 memoir [37] on a gener-  alization of Stirling’s formula, Hayman gave a related but different repackaging of the saddle  point method for the asymptotic evaluation of the Cauchy integral (91) by introducing the  notion of H-admissible functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We collect estimates given in course of the proofs of some  of Hayman’s theorems that will help us to establish the conditions (I) and (O) required for  applying analytic de-Poissonization in form of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 (Hayman [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 68]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' An entire function f(z) is said to be H-admissible if  the following four conditions are satisfied:  – [positivity] for sufficiently large r > 0, there holds f(r) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' inducing there the real  functions (which we call the auxiliary functions associated with f)  a(r) = rf′(r)  f(r) ,  b(r) = ra′(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  by Hadamard’s convexity theorem a(r) is monotonely increasing and b(r) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  – [capture] b(r) → ∞ as r → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  – [locality] for some function 0 < θ0(r) < π there holds33  f(reiθ) = f(r)eiθa(r)−θ2b(r)/2 (1 + o(1))  (r → ∞, |θ| ⩽ θ0(r));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  – [decay] for the angles in the complement there holds  f(reiθ) = o(f(r))  �  b(r)  (r → ∞, θ0(r) ⩽ |θ| ⩽ π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Instead of providing an asymptotic expansion (with an additive error term) as in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1,  H-admissibility gives just a versatile leading order term of an in form of a normal approxi-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' However, the error term is multiplicative then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 (Hayman [37, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' I, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' II]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let f be an entire H-admissible function  with Maclaurin series  f(z) =  ∞  �  n=0  anzn  (z ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Then:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [normal approximation] There holds, uniformly in n ∈ N0 = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' }, that  (94)  anrn  f(r) =  1  �  2πb(r)  �  exp  �  −(n − a(r))2  2b(r)  �  + o(1)  �  (r → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [Stirling-type formula] For n sufficiently large, it follows from the positivity and  capture conditions of H-admissibility that a(rn) = n has a unique solution rn such  that rn → ∞ as n → ∞ and therefore, by the normal approximation (94), there holds  (95)  an =  f(rn)  rnn  �  2πb(rn)  (1 + o(1))  (n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  For the probabilistic content of the normal approximation (94) see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', [25] and [16,  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We observe the similarity of the locality and decay conditions to the conditions (I) and  (O) in the Jacquet–Szpankowski Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In fact, in establishing the H-admissibility of  certain families of functions, Hayman proved estimates that allow us to infer the validity of  conditions (I) and (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A striking example is given by the following theorem, which gives  uniform bounds for a class of functions that is of particular interest to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  33As is customary in asymyptotic analysis in the complex plane, we understand such asymptotics (and similar  expansions with o- or O-terms) to hold uniformly in the stated angular segments for all r ⩾ r0 with some  sufficiently large r0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  35  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 (Hayman [37, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' XI]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let f be an entire function of genus zero, having  for some ϵ > 0 no zeros in the sector |arg z| ⩽ π/2 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If f satisfies the positivity condition  of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, then there is the universal bound  (96)  ��f(reiθ)  �� ⩽  �  �  �  2f(r)e− 1  2 θ2b(r),  0 ⩽ |θ| ⩽ b(r)−2/5,  2f(r)e− 1  2 b(r)1/5,  b(r)−2/5 ⩽ |θ| ⩽ π,  which is valid when b(r) is large enough to ensure 8b(r)−1/5 csc2(ϵ/2) csc(ϵ) ⩽ log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Hence,  if f also satisfies the capture condition of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1, then it is H-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since the bound (96) is hidden in the two-page long proof of [37, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' XI] (only the  H-admissibility is stated explicitly there), we collect the details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' First, [37, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='6)]  states that, if |θ| ⩽ 1/4, then  log f(reiθ) = log f(r) + iθa(r) − 1  2θ2b(r) + ϵ(r, θ)  where the error term is bounded by  |ϵ(r, θ)| ⩽ c(ϵ) · b(r) |θ|3,  c(ϵ) := 8 csc2(ϵ/2) csc(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Now, for b(r) large enough to ensure  b(r)−1/5 ⩽ c(ϵ)−1 log 2 ⩽ min  �√  2ϵ, 1/2  �  we thus get with 0 < θ0(r) := b(r)−2/5 ⩽ min  �  2ϵ, 1/4  �  and 0 ⩽ |θ| ⩽ θ0(r) that  log  ��f(reiθ)  �� = ℜ log f(reiθ) = log f(r) − 1  2θ2b(r) + ℜϵ(r, θ),  ��ℜϵ(r, θ)  �� ⩽ log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Exponentiation gives, for 0 ⩽ |θ| ⩽ θ0(r),  ��f(reiθ)  �� ⩽ 2f(r)e− 1  2 θ2b(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Next, if we combine this estimate with [37, Lemma 8] we get, since θ0(r) ⩽ 2ϵ, that  ��f(reiθ)  �� ⩽  ��f(reiθ0(r))  �� ⩽ 2f(r)e− 1  2 θ0(r)2b(r) = 2f(r)e− 1  2 b(r)1/5  (θ0(r) ⩽ |θ| ⩽ π)  which finishes the proof of the universal bound (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  If, instead of having no zeros in the sector |arg z| ⩽ π/2 + ϵ at all, the entire function  f has a finite number of them, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 remains valid but the lower bound on b(r) will  now depend on these finitely many zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' To restore uniformity we consider families of such  functions whose zeros satisfy the following tameness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let fn be a family of entire functions such that, for some fixed ϵ > 0, each  of them has finitely many zeros (listed according to their multiplicities)  zn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , zn,mn  in the sector |arg z| ⩽ π/2 + ϵ, none of them being a positive real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We call these zeros  uniformly tame (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the positive real axis and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' infinity) if there are some constants  1/5 < µ ⩽ 1/3 and ν > 0 such that the family of polynomials  (97)  pn(z) = (z − zn,1) · · · (z − zn,mn)  satisfies  (98)  �  r d  dr  �2  log pn(r) = −  mn  �  j=1  rzn,j  (r − zn,j)2 = O(r1−µ),  |pn(reiθ)| = pn(r)(1 + O(r−ν)),  uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  36  FOLKMAR BORNEMANN  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Note that a single function f would satisfy condition (98) with error terms of  the form O(r−1) in both places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Therefore the tameness condition allows us to accommodate  a significant growth of the implied constants in these O(r−1) terms as n → ∞: in the first  case because of zeros of fn getting close to the positive real axis and in the second case  because of them getting large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let fn be a family of entire functions of genus zero with positive Maclaurin  coefficients such that, for some fixed ϵ > 0, each of them has a most finitely many zeros in  the sector |arg z| ⩽ π/2 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If these zeros are uniformly tame in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='2 and if  the auxiliary functions belonging to fn satisfy  (99)  bn(r) = r + O(r2/3)  (r → ∞),  uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞, then there holds the bound  (100)  ��fn(reiθ)  �� ⩽  �  �  �  2fn(r)e− 1  2 θ2r,  0 ⩽ |θ| ⩽ r−2/5,  2fn(r)e− 1  2 r1/5,  r−2/5 ⩽ |θ| ⩽ π,  for all n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0, n0 being sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here n0 depends only  on the parameters of the tameness condition and the implied constants in (98) and (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Factoring out the finitely many zeros of fn in the sector |arg z| ⩽ π/2 + ϵ by using  the polynomials (97), we have  fn(z) = f∗  n(z) · pn(z)  where f∗  n is an entire function of genus zero that has no zeros in that sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Since fn(r) > 0  for r > 0 and the leading coefficient of the polynomial pn(z) is one, f∗  n satisfies the positivity  condition of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Denoting the auxiliary functions of f∗  n by a∗  n and b∗  n, the tameness  condition (98) yields  bn(r) = b∗  n(r) +  �  r d  dr  �2  log pn(r) = b∗  n(r) + O(r1−µ),  |pn(reiθ)| = pn(r)(1 + O(r−ν)),  uniformly in n − n3/5 ⩽ r ⩽ n + n3/5 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  By (99) this gives b∗  n(r) = r(1 + O(r−µ)), so that by Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 (its proof shows that we  can take a factor 3/2 instead of 2 if log 2 is replaced by log(3/2) in the lower bound on b(r))  ��f∗  n(reiθ)  �� ⩽  �  �  �  3  2f∗  n(r)e− 1  2 θ2b∗  n(r),  0 ⩽ |θ| ⩽ b∗  n(r)−2/5,  3  2f∗  n(r)e− 1  2 b∗  n(r)1/5,  b∗  n(r)−2/5 ⩽ |θ| ⩽ π,  for n large enough to ensure 8b∗  n(r)−1/5 csc2(ϵ/2) csc(ϵ) ⩽ log(3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We write this briefly as  ��f∗  n(reiθ)  �� ⩽ 3  2f∗  n(r) exp  �  − 1  2 min  �  θ2b∗  n(r), b∗  n(r)1/5��  for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0 where n0 is large enough (just depending on the  parameters and the implied constants in the tameness condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If we multiply this bound  by  |pn(reiθ)| = pn(r)(1 + O(r−ν))  and use b∗  n(r) = r(1 + O(r−µ)) to infer  min  �  θ2b∗  n(r), b∗  n(r)1/5�  = min(θ2r, r1/5) · (1 + O(r−µ)) = min(θ2r, r1/5) + O(r1/5−µ),  we obtain the asserted estimate in the compact form  ��fn(reiθ)  �� ⩽ 2fn(r) exp  �  − 1  2 min  �  θ2r, r1/5��  for n − n3/5 ⩽ r ⩽ n + n3/5 and n ⩾ n0, where n0 is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  37  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Bessel functions of large order in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In the 1950s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Olver  started a systematic and exhaustive study of asymptotic expansions of the Bessel functions  Jν(z) for large order ν and argument z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For the transition region34 z = ν + τν1/3 he obtained  from applying the saddle point method to integral representations of Sommerfeld’s type the  asymptotic expansion [49, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1)] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' also [52, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='19(iii)])  (101a)  Jν(ν + τν1/3) ∼ 21/3  ν1/3 Ai(−21/3τ)  ∞  �  k=0  Ak(τ)  ν2k/3 + 22/3  ν1/3 Ai′(−21/3τ)  ∞  �  k=1  Bk(τ)  ν2k/3  valid when |arg ν| ⩽ π/2 − δ < π with τ being any fixed complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, Ak(τ) and  Bk(τ) are certain rational polynomials of increasing degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' the first few are [49, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='42)]35  A0(τ) = 1,  A1(τ) = −1  5τ,  A2(τ) = − 9  100τ 5 + 3  35τ 2,  (101b)  B0(τ) = 0,  B1(τ) = 3  10τ 2,  B2(τ) = −17  70τ 3 + 1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  (101c)  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The sequence [49, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='18), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='38), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='40)] of formulae in Olver’s  1952 paper gives an actual method36 to calculate Ak(τ) and Bk(τ) (combining reversion and  nesting of power series with recursive formulae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The degrees of Ak are the positive integers  congruent to 0, 1 mod 5 (starting with deg A1 = 1) and the degrees of Bk are the positive  integers congruent to 2, 3 mod 5 (starting with deg B1 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' In both families of polynomials  the coefficients of τ m are zero when m is not congruent mod 3 to the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  As stated in [49, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 422], the expansion (101) can be repeatedly differentiated with respect  to τ, valid under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For a modern account of differentiability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ  and ν, adding uniformity for τ from any compact real set, see the recent work of Sher [58,  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='8] which is based on the (microlocal) theory of so-called polyhomogeneous conormal  joint asymptotic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  The purposes of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3 require to identify a larger region of real τ where the expansion (101)  is uniform as ν → ∞ through positive real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' To this end we use the uniform asymptotic  expansions of Bessel functions for large order ν, pioneered by Olver [50] in 1954 by analyzing  turning points of the Bessel differential equation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [51, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 11] and [69, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' VIII]):  (102a)  Jν(νz) ∼  �  4ζ  1 − z2  �1/4 �  Ai(ν2/3ζ)  ν1/3  ∞  �  k=0  A∗  k(ζ)  ν2k  + Ai′(ν2/3ζ)  ν5/3  ∞  �  k=0  B∗  k(ζ)  ν2k  �  ,  uniformly for z ∈ (0, ∞) as ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, the parameters and coefficients are, for 0 < z < 1,  (102b)  2  3ζ3/2 = log  �  1 +  √  1 − z2  z  �  −  �  1 − z2,  and  A∗  k(ζ) =  2k  �  j=0  �3  2  �j  vjζ−3j/2U2k−j  �  (1 − z2)−1/2�  ,  (102c)  B∗  k(ζ) = −ζ−1/2  2k+1  �  j=0  �3  2  �j  ujζ−3j/2U2k−j+1  �  (1 − z2)−1/2�  ,  (102d)  34Where Jν(ν + τν1/3) changes at about τ ≈ 0 from being superexponentially small (to the left) to being  oscillatory (to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  35Note that we keep the indexing of the polynomials Bk as in [49], which differs from [52, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='19(iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  36By a Mathematica implementation we extended (and reproduced) Olver’s original table [49, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='42)] of  A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , An and B0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , Bn from n = 4 to n = 23 in about 180 hours computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The polynomials A23  and B23 of degree 56 and 57 exhibit rational coefficients that are ratios of integers with up to 65 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  38  FOLKMAR BORNEMANN  where the Uk(x) are recursively defined rational polynomials of degree 3k (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [50, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='19)])  and uk, vk (u0 = v0 = 1) are the rational coefficients of the asymptotic expansions of the  Airy function and its derivative in a sector containing the positive real axis:  (103)  Ai(z) ∼  e−ξ  2√πz1/4  ∞  �  k=0  (−1)k uk  ξk ,  Ai′(z) ∼ −z1/4e−ξ  2√π  ∞  �  k=0  (−1)k vk  ξk ,  ξ = 2  3z3/2,  as z → ∞ within |arg z| ⩽ π − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Note that ζ = ζ(z) can be continued analytically to the  z-plane cut along the negative real axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='37 A∗  k(ζ) and B∗  k(ζ) can be continued accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  As stated in [50, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 342], valid under the same conditions while preserving uniformity, the  expansion can be repeatedly differentiated with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In particular, with 0 < δ < 1 fixed, the power series expansion  (104)  2−1/3ζ = (1 − z) + 3  10(1 − z)2 + 32  175(1 − z)3 + · · ·  converges uniformly for |1 − z| ⩽ 1 − δ (because of the logarithmic singularity at z = 0 the  radius of convergence of this series is exactly 1, so that this range of uniformity cannot be  extended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' If we put  νz = ν + τν1/3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=',  z = 1 + τν−2/3,  plugging the uniformly convergent series ζ(z) into the uniform large ν expansion (102)  recovers the form of the transition region expansion (101) and proves that it holds uniformly  for  |τ| ⩽ (1 − δ)ν2/3  as ν → ∞ through positive real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  At the expense of considerably larger error terms, this result can be extended as follows:  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' For any non-negative integer m and any real τ0 there holds, as ν → ∞ through  positive real values,  (105)  Jν(ν + τν1/3) = 21/3 Ai(−21/3τ)  m  �  k=0  Ak(τ)  ν(2k+1)/3 + 22/3 Ai′(−21/3τ)  m  �  k=1  Bk(τ)  ν(2k+1)/3  + ν−1−2m/3 · O  �  exp(21/3τ)  �  ,  uniformly for −ν2/3 < τ ⩽ τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here, Ak(τ) and Bk(τ) are the rational polynomials in (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Preserving uniformity, the expansion (105) can be repeatedly differentiated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Let us write  Jν(ν + τν1/3) = Em(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ) + Rm(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ),  where Em denotes the sum of the expansion terms in (105) and Rm is the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We  split the range of τ into the two parts  (I): − 3  4ν2/3 ⩽ τ ⩽ τ0,  (II): − ν2/3 < τ ⩽ −3  4ν2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  In part (I), as argued above for δ = 1/4, the expansion (101) is uniformly valid, that is,  Rm(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ) = ν−1−2m/3 · O  �  Am+1(τ) Ai  �  − 21/3τ  ��  + ν−1−2m/3O  �  Bm+1(τ) Ai′ �  − 21/3τ  ��  uniformly for these τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Now, the superexponential decay of the Airy function Ai(x) and its  derivative as x → ∞ through positive values, as displayed in the expansions (103), imply the  asserted uniform bound  Rm(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ) = ν−1−2m/3 · O  �  exp(21/3τ)  �  in part (I) of the range of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  37In particular, for positive real z, the thus defined ζ(z) is a strictly monotonically decreasing real function  with limz→0+ ζ(z) = +∞, ζ(1) = 0 and limz→+∞ ζ(z) = −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' [52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  ASYMPTOTIC EXPANSIONS RELATING TO LONGEST INCREASING SUBSEQUENCES  39  On the other hand, in part (II) of the range of τ, we infer from (103) that for 0 < ϵ < 1/2  Em(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ) = O  �  exp(−(3ν2/3/4)1+ϵ�  = ν−1−2m/3 · O  �  exp(21/3τ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  We now show that also  (106)  Jν  �  ν + τν1/3�  = ν−1−2m/3 · O  �  exp(21/3τ)  �  uniformly in part (II) of the range of τ, so that all terms in (105) are absorbed in the asserted  error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Here we observe  0 < z = 1 + τν2/3 ⩽ 1  4,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='095 · · · ⩽ 2  3ζ3/2 < ∞,  so that the leading order terms in (102) and (103) yield the bound  Jν(νz) ∼ ν−1/2  �  4  1 − z2  �1/4  (ν2/3ζ)1/4 Ai(ν2/3ζ) = ν−1/2 · O  �  exp(− 2  3ζ3/2ν)  �  ,  uniformly for the τ in (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Because of 2  3ζ3/2 ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='095 and −ν ⩽ −21/3ν2/3 < 21/3τ for ν ⩾ 2,  this bound can be relaxed, as required, to  Jν  �  ν + τν1/3�  = Jν(νz) = ν−1−2m/3 · O  �  exp(21/3τ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Finally, the claim about the derivatives follows from the repeated differentiability of the  uniform expansion (102) and the differential equation of the Airy function, Ai′′(x) = x Ai(x)  (so that the general form of the expansions underlying the proof does not change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  □  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The cases m = 0 and m = 1 of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='1 have previously been stated as  [18, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='11)] and [33, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' However, the proofs given there are incomplete: in [18,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2978] the power series (104) is used up to the boundary of its circle of convergence, so  that uniformity becomes an issue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' whereas in [33, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 7] it is claimed that Olver’s transition  expansion (101) would be uniform w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' τ ∈ (−∞, τ0], which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='38  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Supplement: Expansion Terms for m = 3  Put to the extreme, with the help of a CAS such as Mathematica, the methods of the  present paper can be used to calculate the concrete functional form of the expansion terms for  up to m = 7 and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' We refrain, however, from giving any of the computational details39  and just tabulate the results for m = 3 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Auxiliary Transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' The transformation of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='3 supplements (39) by  (107)  ˜F3(t) = 64t  7875F ′(t) − 24t2  875 F ′′(t) − 122  7875F ′′′(t) + 16t  875F (4)(t) −  1  750F (6)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Hard-to-soft edge transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (24) we have  (108)  F3(t) =  � 64t  7875 + 1037t4  7875  �  F ′(t) +  �  − 9t2  175 + 48t5  875  �  F ′′(t)  +  �  − 122  7875 − 8t3  125 + 9t6  2000  �  F ′′′(t) +  � 16t  875 − 9t4  1000  �  F (4)(t) + 3t2  500F (5)(t) −  1  750F (6)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a plot is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  38Besides that the principal branch of Jν(z) (ν ̸∈ Z) is not defined at negative real z, there is a counter-example  for ν = n being a positive integer: choosing τ = 0 in (101) gives to leading order  (−1)nJn(−n) = Jn(n) ∼ 21/3n−1/3 Ai(0)  (n → ∞),  which differs significantly from applying (101) formally to τ = −2n2/3 (for which n + τn1/3 = −n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  39A Mathematica notebook with the results for up to m = 7 comes with the sources of the arXiv version of  the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As for m = 1, 2, we observe also for m = 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' , 7 that the expressions for ˜Fm (and a fortiori  for Fm, F P  m, F D  m , F ∗  m) take the form of a linear combination of higher order derivatives of the Tracy–Widom  distribution F with certain rational polynomials as coefficients: we conjecture that this is generally the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  40  FOLKMAR BORNEMANN  Poissonized length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (47) we have  (109)  F P  3 (t) = −  �  t  1125 +  41t4  283500  �  F ′(t) −  � 11t2  6300 +  t5  47250  �  F ′′(t)  −  �  61  31500+ 19t3  63000+  t6  1296000  �  F ′′′(t)−  � 11t  10500+  t4  72000  �  F (4)(t)−  t2  12000F (5)(t)−  1  6000F (6)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a plot is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  CDF of length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (60) we have  (110)  F D  3 (t) = −  � 562t  7875 +  41t4  283500  �  F ′(t) +  � t2  300 −  t5  47250  �  F ′′(t)  +  � 5137  15750 + 9t3  7000 −  t6  1296000  �  F ′′′(t)+  � 129t  1750 −  t4  12000  �  F (4)(t)− 3t2  1000F (5)(t)− 9  250F (6)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a plot is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  PDF of length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (65) we have  (111)  F ∗  3 (t) = −  � 373  5250 + 41t3  70875  �  F ′(t) −  � 1781t  28000 +  71t4  283500  �  F ′′(t)  +  � 63t2  8000 −  13t5  504000  �  F ′′′(t) +  � 41473  112000 + 13t3  12096 −  t6  1296000  �  F (4)(t)  +  � 131057t  2016000 −  67t4  864000  �  F (5)(t) − 1493t2  576000F (6)(t) − 232319  8064000F (7)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a plot is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Expected Value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (83) we have  (112)  µ3 = 538  7875M1 +  281  4536000M4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a highly accurate numerical value is displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' As a supplement to (85) we have  (113)  ν3 = −1076  7875M2  1 −  281  2268000M1M4 + 893  7875M2 +  1  42000M2M3 +  227  2268000M5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  a highly accurate numerical value is displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' I would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge  (UK), for support and hospitality during the 2022 program “Applicable resurgent asymptotics: towards a  universal theory (ARA2)” where work on the present paper was undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' This work was supported by  EPSRC grant no EP/R014604/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  References  [1] Adler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', van Moerbeke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=': PDEs for the joint distributions of the Dyson, Airy and sine processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' 33(4), 1326–1361 (2005)  [2] Aldous, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=': Probability approximations via the Poisson clumping heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Springer-Verlag, New York  (1989)  [3] Aldous, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=', Diaconis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=': Longest increasing subsequences: from patience sorting to the Baik-Deift-  Johansson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
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+page_content=': An Introduction to Random Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
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+page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNA0T4oBgHgl3EQfEv9v/content/2301.02022v1.pdf'}
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+arXiv:2301.02023v1  [math.AP]  5 Jan 2023
+On a class of mixed local and nonlocal semilinear elliptic
+equation with singular nonlinearity
+Prashanta Garain
+Abstract
+In this article, we consider a combination of local and nonlocal Laplace equation with
+singular nonlinearities. For such mixed problems, we establish existence of at least one
+weak solution for a parameter dependent singular nonlinearity and existence of multiple
+solution for purturbed singular nonlinearity. Our argument is based on the variational
+and approximation approach.
+Keywords: Mixed local and nonlocal equation, singular nonlinearity, existence, regularity.
+2020 Mathematics Subject Classification: 35M10, 35R11, 35B65, 35J75.
+Contents
+1
+Introduction
+1
+1.1
+Functional setting and useful results . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Statement of the main results: . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.3
+Notation and organization of the article . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Preliminaries for the proof of Theorem 1.7
+6
+2.1
+Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+3
+Preliminaries for the proof of Theorem 1.8
+10
+3.1
+Proof of Theorem 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+1
+Introduction
+In this article, we consider the following mixed local and nonlocal semilinear equation with
+singular nonlinearity
+− ∆u + (−∆)su = g(x, u) in Ω,
+u > 0 in Ω,
+u = 0 in Rn \ Ω,
+(1.1)
+where Ω ⊂ Rn is a bounded domain with n ≥ 2. Here −∆ is the classical Laplace operator
+and (−∆)s, s ∈ (0, 1) is the fractional Laplace operator defined by
+(−∆)su = P.V.
+ˆ
+Rn
+u(x) − u(y)
+|x − y|n+2s dy,
+1
+
+Mixed local and nonlocal singular problems
+2
+where P.V. denotes the principal value. We establish existence of at least one weak solution
+of the problem (1.1) for the purely singular nonlinearity g of the form (g1) given by
+(g1)
+g(x, u) = λh(u)u−γ,
+where λ > 0, γ ∈ (0, 1) and
+(h1) h : [0, ∞) → R is a continuous nondecreasing function such that h(0) > 0 and
+(h2)
+lim
+t→0
+h(t)
+tγ
+= ∞,
+lim
+t→∞
+h(t)
+tγ+1 = 0.
+(1.2)
+Further, we establish multiplicity result for the equation (1.1) with the purturbed singular
+nonlinearity g of the form (g2) given by
+(g2)
+g(x, u) = λu−γ + uq,
+where λ > 0, γ ∈ (0, 1) and q ∈ (1, 2∗ − 1) with 2∗ =
+2n
+n−2 if n > 2 and 2∗ = ∞ if n = 2.
+Before proceeding further, we state the functional setting to study the problem (1.1).
+1.1
+Functional setting and useful results
+In this section, we present some known results for the fractional Sobolev space, see [20] for
+more details. Let E ⊂ Rn be a measurable set and |E| denote its Lebesgue measure. Recall
+that the Lebesgue space L2(E), is defined as the space of measurable functions u : E → R
+with the finite norm
+∥u∥L2(E) =
+
+
+ˆ
+E
+|u(x)|2 dx
+
+
+1/2
+.
+Here and in the rest of the paper, it is assumed that Ω ⊂ Rn with n ≥ 2 is a bounded smooth
+domain. The Sobolev space H1(Ω) is defined as the Banach space of locally integrable weakly
+differentiable functions u : Ω → R equipped with the following norm:
+∥u∥H1(Ω) = ∥u∥L2(Ω) + ∥∇u∥L2(Ω).
+The space H1(Rn) is defined analogously. To deal with mixed problems, we use the space
+H1
+0(Ω) = {u ∈ H1(Rn) : u = 0 in Rn \ Ω} under the norm ∥u∥ = ∥∇u∥L2(Ω). It can be shown
+that H1
+0(Ω) is a real separable and reflexive Banach space, see [9, 10, 38].
+The fractional Sobolev space Hs(Ω), 0 < s < 1, is defined by
+Hs(Ω) =
+�
+u ∈ L2(Ω) : |u(x) − u(y)|
+|x − y|
+n
+2 +s
+∈ L2(Ω × Ω)
+�
+,
+which is endowed with the norm
+∥u∥Hs(Ω) =
+�ˆ
+Ω
+|u(x)|2 dx +
+ˆ
+Ω
+ˆ
+Ω
+|u(x) − u(y)|2
+|x − y|n+2s
+dx dy
+� 1
+2
+.
+For the next result, see [20, Proposition 2.2].
+
+Mixed local and nonlocal singular problems
+3
+Lemma 1.1. There exists a constant C = C(n, s) > 0 such that
+∥u∥Hs(Ω) ≤ C∥u∥H1(Ω),
+∀ u ∈ H1(Ω).
+Next, we have the following result from [13, Lemma 2.1].
+Lemma 1.2. There exists a constant C = C(n, s, Ω) such that
+¨
+Rn
+|u(x) − u(y)|2
+|x − y|n+2s
+dx dy ≤ C
+ˆ
+Ω
+|∇u|2 dx,
+∀ u ∈ H1
+0(Ω).
+(1.3)
+For the following Sobolev embedding, see, for example, [21].
+Lemma 1.3. The embedding operators
+H1
+0(Ω) ֒→
+
+
+
+Lt(Ω),
+for t ∈ [1, 2∗], if n > 2,
+Lt(Ω),
+for t ∈ [1, ∞), if n = 2
+are continuous.
+Now we are ready to define the notion of weak solutions for the problem (1.1).
+Definition 1.4. (Weak Solution) Let g be either of the form (g1) or (g2).
+We say that
+u ∈ H1
+0(Ω) is a weak subsolution (or supersolution) of (1.1), if u > 0 in Ω such that for every
+ω ⋐ Ω, there exists a positive constant c(ω) with u ≥ c(ω) > 0 in ω and
+ˆ
+Ω
+∇u∇φ dx +
+¨
+R2n
+(u(x) − u(y))(φ(x) − φ(y))
+|x − y|n+2s
+dxdy ≤ ( or ) ≥ g(x, u)φ dx,
+(1.4)
+for every nonnegative φ ∈ C1
+c (Ω). We say that u ∈ H1
+0(Ω) is a weak solution of (1.1), if the
+equality in (1.4) holds for every φ ∈ C1
+c (Ω) without a sign restriction.
+Remark 1.5. Note that by Lemma 1.1 and Lemma 1.2, it follows that Definition 1.4 is well
+stated.
+Remark 1.6. Let u ∈ H1
+0(Ω) be a weak solution of the problem (1.1) when g is either of the
+form (g1) or (g2). Then following the lines of the proof of [27, Lemma 5.1], it follows that
+the equality in (1.4) holds, for every φ ∈ H1
+0(Ω).
+1.2
+Statement of the main results:
+Our main results in this article reads as follows:
+Theorem 1.7. Let 0 < γ < 1 and g be of the form (g1). Then for every λ > 0, there exists
+a weak solution u ∈ H1
+0(Ω) ∩ L∞(Ω) of the problem (1.1).
+Theorem 1.8. Let 0 < γ < 1 and g be of the form (g2). Then there exists Λ > 0 such that
+for every λ ∈ (0, Λ) the problem (1.1) admits at least two different weak solutions in H1
+0(Ω).
+
+Mixed local and nonlocal singular problems
+4
+To prove our main results stated above, the following result concerning the mixed local
+and nonlocal eigenvalue problem (1.5) will be useful for us.
+− ∆u + (−∆)su = λ|u|p−2u in Ω,
+u = 0 in Rn \ Ω.
+(1.5)
+Lemma 1.9. (i) There exists the least one eigenvalue λ1 > 0 and at least one corresponding
+eigenfunction e1 ∈ H1
+0(Ω) ∩ L∞(Ω) \ {0} which is nonnegative in Ω. (ii) Moreover, for every
+ω ⋐ Ω, there exists a positive constant c(ω) such that e1 ≥ c(ω) > 0 in ω.
+Proof. Part (i) follows from [9, Prop 2.6 and Theorem 2.8]. Part (ii) follows from [24, Theorem
+8.4].
+Singular problems has drawn a great attention over the last three decade. Equations of
+the form
+− α∆u + β(−∆)su = λf(u)u−γ + µur,
+(1.6)
+where α, β, λ, µ, r ≥ 0, γ > 0 are parameters and f is some given function, are studied widely
+in both the local (β = 0) and nonlocal (α = 0) cases separately.
+Here the singularity is
+captured by the parameter γ > 0. Indeed, the quasilinear analouge of the equation (1.6) is
+also investigated in the separate local and nonlocal cases and there is a colossal amount of
+work done for such problems.
+More precisely, in the local case (β = 0), Crandall-Rabinowitz-Tartar [17] proved existence
+of classical solution of (1.6) for λ = 1, µ = 0 and f(u) = 1 for any γ > 0. Further, for a certain
+range of γ, Lazer-McKenna [35] studied the notion of weak solutions. Boccardo-Orsina [12]
+removed this restriction on γ and proved existence of weak solutions for any γ > 0. This study
+has further been investigated in the quasilinear setting by Canino-Sciunzi-Trombetta [15], see
+also De Cave [18] and the references therein. When f(u) ≥ 0 and µ = 0, for 0 < γ < 1
+and a certain range of λ, equation (1.6) is investigated by Ko-Lee-Shivaji in [34]. In the
+purturbed case, we refer to Haitao [31], Hirano-Saccon-Shioji [32], Arcoya-Boccardo-M´erida
+[2, 3], Bal-Garain [5], Giacomoni-Schindler-Tak´aˇc in [30], and the references therein.
+In the nonlocal case (α = 0), equation (1.6) is studied by Fang [22] for µ = 0 and
+further been extended in the quasilinear setting by Canino-Montoro-Sciunzi-Squassina [14].
+The perturbed singular case (µ > 0) is investigated by Barrios-De Bonis-Medina-Peral [6],
+Adimurthi-Giacomoni-Santra [1], Giacomoni-Mukherjee-Sreenadh [28, 29] and generalized by
+Mukherjee-Sreenadh [37] in the quasilinear case and the references therein.
+To the best of our knowledge, singular problems in the mixed local and nonlocal setting is
+very less known. Our main purpose in this article is to contribute in this topic. We believe it
+would be an interesting topic of further investigation. We would like to mention that mixed
+problems are also less known even in the nonsingular case. Using probability theory, Foondun
+[23], Chen-Kim-Song-Vondraˇcek [16] studied regularity results for the equation
+− ∆u + (−∆)su = 0.
+(1.7)
+
+Mixed local and nonlocal singular problems
+5
+Recently based on purely analytic approach, Biagi-Dipierro-Salort-Valdinoci-Vecchi [7, 8, 38]
+studied existence and regularity results for the mixed equation (1.7). Equation (1.7) is also
+studied using analytic approach in the quasilinear case by Garain-Kinnunen [24]. Several
+recent regularity results and other qualitative properties for such problems using analytic
+approach can be found in see [9, 10, 11, 19, 25] and the references therein.
+In the mixed singular case, that is for positive α and β, assuming µ = 0 and f depending
+on x only, the singular equation (1.6) and its quasilinear version is studied recently. In this
+concern, for the quasilinear case, we refer to Garain-Ukhlov [27] for existence, uniqueness,
+regularity and symmetry properties with any γ > 0. Further, associated extremal functions
+are also studied in [27]. Moreover, Arora-Radulescu [4] studied several existence and regularity
+properties (which shows power and exponential type Sobolev regularity depending upon the
+summability of the datum f and the singular exponent γ > 0) for the semilinear equation
+(1.6), where the case γ = 0 is also considered.
+In this article, first we prove the existence and regularity in terms of boundedness for the
+singular problem under the singularity of the form (g1) in Theorem 1.7. In this context, we
+adopt the variational technique introduced in [34] in the mixed case. To this end, we also
+borrow ideas from [31] to prove the sub-supersolution result (Lemma 2.1), where to deal with
+the nonlocal behavior of the equation, we used the technique from [29]. Finally, the eigenvalue
+problem (1.5) and the purely singular problem (2.7) are used to construct subsolution and
+supersolutions, thanks to Lemma 1.9 and Lemma 2.2.
+In the second part of this article, we investigate the multiplicity result for the purturbed
+singularity (g2) in Theorem 1.8.
+Here, we utilise the variational approach introduced in
+Arcoya-Boccardo [2] in combination with the technique from [26] to deal with the nonlocality.
+To this end, we obtain existence multiple solutions of the associated approximate problem
+(3.3). This fact combined with an apriori estimate (Lemma 3.5) gives us the required result.
+1.3
+Notation and organization of the article
+Throughout the rest of the article, by c or C, we mean a positive constant which may vary
+from line to line or even in the same line. The dependency of the constants c or C on the
+parameters r1, r2, . . . , rk is denoted by c(r1, r2, . . . , rk) or C(r1, r2, . . . , rk). For a ∈ R, we
+denote by a+ = max{a, 0} and a− = max{−a, 0}. We use the notation 2∗ =
+2n
+n−2 if n > 2 and
+2∗ = ∞ if n = 2.
+In Section 2, we obtain some preliminary results and prove Theorem 1.7.
+Finally, in
+Section 3, we establish some useful results and prove Theorem 1.8.
+
+Mixed local and nonlocal singular problems
+6
+2
+Preliminaries for the proof of Theorem 1.7
+Throughout this section, we assume g is of the form (g1). First we obtain some useful results.
+Consider the energy functional Jλ : H1
+0(Ω) → R ∪ {±∞} defined by
+Jλ(u) =
+ˆ
+Ω
+G(x, ∇u) +
+¨
+R2n F(x, y, u) dxdy − λ
+ˆ
+Ω
+H(u) dx
+where
+G(x, ∇u) = 1
+2|∇u|2
+F(x, y, u) = |u(x) − u(y)|2
+|x − y|n+2s
+and
+H(t) =
+
+
+
+
+
+ˆ t
+0
+h(τ)τ −γ dτ, if t > 0,
+0, if t ≤ 0.
+Following Haitao [31], we establish the following result in the mixed local and nonlocal setting.
+Lemma 2.1. Suppose that u, u ∈ H1
+0(Ω) ∩ L∞(Ω) are weak subsolution and supersolution of
+(1.1) respectively such that 0 < u ≤ u in Ω and u ≥ c(ω) > 0 for every ω ⋐ Ω, for some
+constant c(ω). Then there exists a weak solution u ∈ H1
+0(Ω) ∩ L∞(Ω) of (1.1) satisfying
+u ≤ u ≤ u in Ω.
+Proof. Let us consider the set
+S = {v ∈ H1
+0(Ω) : u ≤ v ≤ u in Ω}.
+Since u ≤ u in Ω, we have S ̸= ∅. We observe that S is closed and convex. We establish the
+result in the following two Steps.
+Step 1: We claim that Jλ admits a minimizer u over S.
+To this end, we prove that Jλ is weakly sequentially lower semicontinuous over S. Indeed, let
+{vk}k∈N ⊂ S be such that vk ⇀ v weakly in H1
+0(Ω). Then by the hypothesis on h, we have
+H(vk) ≤
+ˆ u
+0
+h(τ)τ −γ dτ ≤ h(∥u∥∞)
+(1 − γ) ∥u∥1−γ
+∞ .
+Therefore by the Lebesgue Dominated Convergence theorem and weak lower semicontinuity of
+norm, the claim follows. Hence, there exists a minimizer u ∈ S of Jλ that is Jλ(u) = inf
+v∈S Jλ(v).
+Step 2: Here, we prove that u is a weak solution of (1.1).
+Let φ ∈ C1
+c (Ω) and ǫ > 0. We define
+ηǫ =
+
+
+
+
+
+
+
+u
+if u + ǫφ ≥ u,
+u + ǫφ
+if u ≤ u + ǫφ ≤ u,
+u
+if u + ǫφ ≤ u.
+
+Mixed local and nonlocal singular problems
+7
+Observe that ηǫ = u + ǫφ − φǫ + φǫ ∈ S, where φǫ = (u + ǫφ − u)+ and φǫ = (u + ǫφ − u)−.
+By Step 1 above, since u is a minimizer of Jλ, we have
+0 ≤ lim
+t→0
+Jλ(u + t(ηǫ − u)) − Jλ(u)
+t
+= I1 + I2 − λJ (say),
+(2.1)
+with
+I1 =
+ˆ
+Ω
+∇u∇(ηǫ − u) dx,
+I2 =
+ˆ
+Q
+(ηǫ − u)(−∆)su dx,
+J =
+ˆ
+Ω
+(ηǫ − u)u−γh(u) dx,
+where we have used the notation Q = R2n \ (CΩ × CΩ), where CΩ := Rn \ Ω. Therefore, we
+have
+0 ≤
+ˆ
+Ω
+∇u∇(ηǫ − u) dx +
+ˆ
+Q
+(ηǫ − u)(−∆)su dx − λ
+ˆ
+Ω
+(ηǫ − u)u−γh(u) dx
+=⇒ 1
+ǫ (Qǫ − Qǫ) ≤
+ˆ
+Ω
+∇u∇φ dx +
+ˆ
+Rn φ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φ dx
+(2.2)
+where
+Qǫ =
+ˆ
+Ω
+∇u∇φǫ dx +
+ˆ
+Rn φǫ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φǫ dx
+and
+Qǫ =
+ˆ
+Ω
+∇u∇φǫ dx +
+ˆ
+Rn φǫ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φǫ dx.
+Estimate of Qǫ: We observe that
+1
+ǫ
+ˆ
+Ω
+∇u∇φǫ dx = 1
+ǫ
+ˆ
+Ω
+∇(u − u)∇φǫ dx ≥
+ˆ
+Ωǫ ∇(u − u)∇φ dx + 1
+ǫ
+ˆ
+Ω
+∇u∇φǫ dx
+�
+≥ o(1) + 1
+ǫ
+ˆ
+Ω
+∇u∇φǫ dx.
+(2.3)
+Further, we notice that
+1
+ǫ
+ˆ
+Rn φǫ(−∆)su dx = 1
+ǫ
+� ˆ
+Rn φǫ(−∆)s(u − u) dx +
+ˆ
+Rn φǫ(−∆)su dx
+�
+≥ o(1) + 1
+ǫ
+ˆ
+Rn φǫ(−∆)su dx,
+(2.4)
+where to estimate the last inequality, we used the the lines of the proof from [29, Page 9].
+
+Mixed local and nonlocal singular problems
+8
+Combining (2.3) and (2.4), we have
+1
+ǫ Qǫ ≥ o(1) + 1
+ǫ
+� ˆ
+Ω
+∇u∇φǫ dx +
+ˆ
+Rn φǫ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φǫ dx
+�
+= o(1) + 1
+ǫ
+� ˆ
+Ω
+∇u∇φǫ dx +
+ˆ
+Rn φǫ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φǫ dx
+�
++ λ
+ǫ
+� ˆ
+Ω
+u−γh(u)φǫ dx −
+ˆ
+Ω
+u−γh(u)φǫ dx
+�
+≥ o(1) + λ
+ǫ
+ˆ
+Ωǫ h(u)(u−γ − u−γ)(u − u) dx
++ λ
+ˆ
+Ωǫ h(u)(u−γ − u−γ)φ dx
+≥ o(1),
+(2.5)
+using that u is a weak supersolution of (1.1), u ≤ u and
+ˆ
+Ωǫ h(u)(u−γ−u−γ)φ dx ≤ 2c(ω)−γh(||u||∞)||φ||∞ <
++∞, where Ωǫ = supp φǫ and ω = supp φ.
+Taking into account that u is a weak subsolution of (1.1), u ≥ u and
+ˆ
+Ωǫ
+h(u)(u−γ −
+u−γ)φ dx ≤ 2c(ω)−γh(∥u∥∞)∥φ∥∞ < +∞, where Ωǫ = supp φǫ and ω = suppφ, in a similar
+way, we obtain
+1
+ǫ Qǫ ≤ o(1).
+(2.6)
+Using the estimates (2.5) and (2.6) in (2.2), we conclude that
+0 ≤
+ˆ
+Ω
+∇u∇φ dx +
+ˆ
+Rn φ(−∆)su dx − λ
+ˆ
+Ω
+u−γh(u)φ dx.
+Since φ ∈ C1
+c (Ω) is arbitrary, our claim follows. This completes the proof.
+Lemma 2.2. Let 0 < γ < 1 and v0 ∈ H1
+0(Ω) be a weak solution of the problem
+− ∆u + (−∆)su = u−γ in Ω,
+u > 0 in Ω,
+u = 0 in Rn \ Ω.
+(2.7)
+Then v0 ∈ L∞(Ω).
+Proof. Let k > 1, then by Remark 1.6 we choose φk = (v0 − k)+ ∈ H1
+0(Ω) as a test function
+in (2.7) and apply H¨older’s along with Young’s inequality with ǫ ∈ (0, 1) to get
+ˆ
+Ω
+|∇φk|2 dx ≤ C(ǫ)|A(k)|
+2
+q′ + ǫ
+ˆ
+Ω
+|∇φk|2 dx,
+where A(k) =
+�
+x ∈ Ω : v0 ≥ k in Ω
+�
+.
+In the above estimate, we have also used that
+H1
+0(Ω) ֒→ Lq(Ω) for some q > 2 from Lemma 1.3. Therefore, fixing ǫ ∈ (0, 1), we obtain
+ˆ
+Ω
+|∇φk|2 dx ≤ C|A(k)|
+2
+q′ ,
+
+Mixed local and nonlocal singular problems
+9
+where C is some positive constant. Let 1 < k < h, then since A(h) ⊂ A(k), we have
+(h − k)p|A(h)|
+2
+q ≤
+� ˆ
+A(h)
+(v0 − k)q dx
+� 2
+q ≤
+� ˆ
+A(k)
+(v0 − k)q dx
+� 2
+q
+≤ C
+ˆ
+Ω
+|∇φk|2 dx ≤ C |A(k)|
+2
+q′ .
+Therefore
+|A(h)| ≤
+C
+(h − k)q |A(k)|q−1.
+Since q > 2, by [33, Lemma B.1], we have ||v0||L∞(Ω) ≤ c, where c is a positive constant.
+Hence the result follows.
+2.1
+Proof of Theorem 1.7
+We construct a pair of weak subsolution and supersolution of (1.1) according to Lemma 2.1.
+By Lemma 1.9, there exists e1 ∈ H1
+0(Ω) ∩ L∞(Ω) such that
+− ∆e1 + (−∆)se1 = λ1e1 in Ω, e1 > 0 in Ω, e1 = 0 in Rn \ Ω
+(2.8)
+and for every ω ⋐ Ω, there exists a positive constant c(ω) with u ≥ c(ω) in ω. By (h2), we
+know that lim
+t→0 t−γh(t) = ∞, so we can choose aλ > 0 sufficiently small such that
+λ1(aλe1) ≤ λ(aλe1)−γh(aλe1).
+(2.9)
+Let u = aλe1, then u ∈ H1
+0(Ω) ∩ L∞(Ω) and by (2.8) and (2.9), we get
+− ∆u + (−∆)su ≤ λ(aλe1)−γh(aλe1) = λu−γh(u) in Ω.
+(2.10)
+By [27, Theorem 2.13] and Lemma 2.2, there exists v0 ∈ H1
+0(Ω) ∩ L∞(Ω) such that for every
+ω ⋐ Ω there exists a positive constant c(ω) satisfying v0 ≥ c(ω) > 0 in ω and
+− ∆v0 + (−∆)sv0 = v−γ
+0
+in Ω, v0 > 0 in Ω, v0 = 0 in Rn \ Ω.
+(2.11)
+By the hypothesis (h2), since lim
+t→∞ t−(γ+1)h(t) = 0, we choose bλ > 0 sufficiently large such
+that
+(bλ∥v0∥∞)−(γ+1)h(bλ∥v0∥∞) ≤
+1
+λ∥v0∥γ+1
+∞
+.
+(2.12)
+We define u := bλv0. Then u ∈ H1
+0(Ω) ∩ L∞(Ω) and using (2.11) and (2.12), we have
+− ∆u + (−∆)su = v−γ
+0 bλ ≥ λ(bλv0)−γh(bλ∥v0∥∞) ≥ λu−γh(u) in Ω,
+(2.13)
+where we have also used the nondecreasing property of h from (h1). Thus, from (2.10) and
+(2.13), it follows that u and u are weak subsolution and supersolution of (1.1) respectively
+and the constants aλ, bλ can be chosen in such a way that u ≤ u. Therefore, by Lemma 2.1,
+the result follows.
+
+Mixed local and nonlocal singular problems
+10
+3
+Preliminaries for the proof of Theorem 1.8
+In this section, we consider the equation (1.1) when g is of the form (g2), which reads as
+−∆u + (−∆)su = λu−γ + uq in Ω,
+u > 0 in Ω,
+u = 0 in Rn \ Ω,
+(3.1)
+where λ > 0, 0 < γ < 1 and q ∈ (1, 2∗ − 1) where 2∗ =
+2n
+n−2 if n > 2 and 2∗ = ∞ if n = 2. To
+this end, we study the functional Iλ : H1
+0(Ω) → R ∪ {±∞} associated with the problem (3.1)
+given by
+Iλ(u) := 1
+2
+ˆ
+Ω
+|∇u|2 dx+ 1
+2
+¨
+R2n
+|u(x) − u(y)|2
+|x − y|n+2s
+dxdy−λ
+ˆ
+Ω
+(u+)1−γ
+1 − γ
+dx−
+1
+q + 1
+ˆ
+Ω
+(u+)q+1 dx.
+(3.2)
+For ǫ > 0, we consider the approximated problem
+−∆u + (−∆)su = λ(u+ + ǫ)−γ + (u+)q in Ω,
+u = 0 in Rn \ Ω.
+(3.3)
+We remark that the energy functional associated with the problem (3.3) is given by
+Iλ,ǫ(u) = 1
+2
+ˆ
+Ω
+|∇u|2 dx + 1
+2
+¨
+R2n
+|u(x) − u(y)|2
+|x − y|n+2s
+dxdy − λ
+ˆ
+Ω
+[(u+ + ǫ)1−γ − ǫ1−γ]
+1 − γ
+dx
+−
+1
+q + 1
+ˆ
+Ω
+(u+)q+1 dx.
+(3.4)
+We observe that Iλ,ǫ ∈ C1�
+H1
+0(Ω), R
+�
+, Iλ,ǫ(0) = 0 and Iλ,ǫ(v) ≤ I0,ǫ(v), for all v ∈ H1
+0(Ω).
+Let us define
+l =
+
+
+
+2∗ =
+2n
+n−2, if n > 2,
+r, if n = 2,
+(3.5)
+where r > 1 is such that 1 < q < r−1 if n = 2. Next we prove that Iλ,ǫ satisfies the Mountain
+Pass Geometry.
+Lemma 3.1. There exists R > 0, ρ > 0 and Λ > 0 depending on R such that
+inf
+∥v∥≤R Iλ,ǫ(v) < 0 and
+inf
+∥v∥=R Iλ,ǫ(v) ≥ ρ, for λ ∈ (0, Λ).
+Moreover, there exists T > R such that Iλ,ǫ(Te1) < −1 for λ ∈ (0, Λ), where e1 is given by
+Lemma 1.9.
+Proof. Recalling the definition of l from (3.5), we define θ = |Ω|
+1
+(
+l
+q+1)
+′
+. By H¨older’s inequality
+and Lemma 1.3, for every v ∈ H1
+0(Ω), we have
+ˆ
+Ω
+(v+)q+1 dx ≤
+�ˆ
+Ω
+|v|l
+� q+1
+l
+|Ω|
+1
+(
+l
+q+1 )′ ≤ Cθ∥v∥q+1,
+(3.6)
+for some positive constant C independent of v. Since
+lim
+t→0
+Iλ,ǫ(te1)
+t
+= −λ
+ˆ
+Ω
+ǫ−γe1 dx < 0,
+
+Mixed local and nonlocal singular problems
+11
+we choose k ∈ (0, 1) sufficiently small and set ∥v∥ = R := k( q+1
+pCθ)
+1
+q−1 such that
+inf
+∥v∥≤R Iλ,ǫ(v) < 0.
+Moreover, using the fact R < ( q+1
+pCθ)
+1
+q−1 and the estimate (3.6), we have
+I0,ǫ(v) ≥ R2
+2 − CθRq+1
+q + 1
+:= 2ρ (say) > 0.
+(3.7)
+We define
+Λ :=
+ρ
+sup
+∥v∥=R
+�
+1
+1 − γ
+ˆ
+Ω
+|v|1−γ dx
+�,
+which is positive. Note that, since ρ, R depends on k, q, |Ω| and C, so does Λ. We observe
+that
+(v+ + ǫ)1−γ − ǫ1−γ ≤ (v+)1−γ.
+(3.8)
+Therefore, we have
+Iλ,ǫ(v) ≥ 1
+p
+ˆ
+Ω
+|∇v|2 dx +
+¨
+R2n
+|v(x) − v(y)|2
+|x − y|n+2s
+dxdy −
+1
+q + 1
+ˆ
+Ω
+(v+)q+1 dx −
+λ
+1 − γ
+ˆ
+Ω
+(v+)1−γ dx
+= I0,ǫ(v) −
+λ
+1 − γ
+ˆ
+Ω
+(v+)1−γ dx.
+Hence, using (3.7), for λ ∈ (0, Λ), we get
+inf
+∥v∥=R Iλ,ǫ(v) ≥
+inf
+∥v∥=R I0,ǫ(v) − λ sup
+∥v∥=R
+�
+1
+1 − γ
+ˆ
+Ω
+|v|1−γ dx
+�
+≥ 2ρ − λ sup
+∥v∥=R
+�
+1
+1 − γ
+ˆ
+Ω
+|v|1−γ dx
+�
+≥ ρ.
+Finally, we observe that I0,ǫ(te1) → −∞, as t → +∞. This gives the existence of T > R such
+that I0,ǫ(Te1) < −1. Therefore,
+Iλ,ǫ(Te1) ≤ I0,ǫ(Te1) < −1,
+which completes the proof.
+Next, we prove that Iλ,ǫ satisfies the Palais Smale (PS)c condition.
+Lemma 3.2. Iλ,ǫ satisfies the (PS)c condition, for any c ∈ R, that is if {uk}k∈N ⊂ H1
+0(Ω) is
+a sequence such that
+Iλ,ǫ(uk) → c and I′
+λ,ǫ(uk) → 0
+(3.9)
+as k → ∞, then {uk}k∈N contains a strongly convergent subsequence in H1
+0(Ω).
+
+Mixed local and nonlocal singular problems
+12
+Proof. We prove the result in two steps below.
+Step 1. First, we claim that if {uk}k∈N ⊂ H1
+0(Ω) satisfies (3.9) then {uk}k∈N is uniformly
+bounded in H1
+0(Ω). To this end, by (3.8), for some positive constant C (independent of k),
+we have
+Iλ,ǫ(uk) −
+1
+q + 1I′
+λ,ǫ(uk)uk =
+�1
+2 −
+1
+q + 1
+� ˆ
+Ω
+|∇uk|2 dx +
+�1
+2 −
+1
+q + 1
+� ¨
+R2n
+|u(x) − u(y)|2
+|x − y|n+2s
+dxdy
+− λ
+ˆ
+Ω
+(u+
+k + ǫ)1−γ − ǫ1−γ
+1 − γ
+dx +
+λ
+q + 1
+ˆ
+Ω
+(u+
+k + ǫ)−γuk dx
+≥
+�1
+2 −
+1
+q + 1
+�
+∥uk∥2 − C∥uk∥1−γ,
+(3.10)
+for some positive constant C (independent of k), where we have also used Lemma 1.3 and
+H¨older’s inequality. Noting q > 1 and using (3.10), we obtain
+Iλ,ǫ(uk) −
+1
+q + 1I′
+λ,ǫ(uk)uk ≥ C1∥uk∥2 − C∥uk∥1−γ,
+(3.11)
+for some positive constants C, C1 (independent of k). Using (3.9), for k large enough, we have
+����Iλ,ǫ(uk) −
+1
+q + 1I′
+λ,ǫ(uk)uk
+���� ≤ C + o(∥uk∥),
+(3.12)
+for some positive constant C (independent of k). Combining (3.11) and (3.12), our claim
+follows.
+Step 2. We claim that up to a subsequence, uk → u0 strongly in H1
+0(Ω) as k → ∞.
+By Step 1, since {uk}k∈N is uniformly bounded in H1
+0(Ω), due to the reflexivity of H1
+0(Ω),
+there exists u0 ∈ H1
+0(Ω) such that up to a subsequence, uk ⇀ u0 weakly in H1
+0(Ω) as k → ∞.
+Again, by (3.9), we have
+lim
+k→∞
+�ˆ
+Ω
+∇uk∇u0 dx +
+¨
+R2n
+(uk(x) − uk(y))(u0(x) − u0(y))
+|x − y|n+2s
+dxdy
+−λ
+ˆ
+Ω
+(u+
+k + ǫ)−γu0 dx −
+ˆ
+Ω
+(u+
+k )qu0 dx
+�
+= 0
+and
+lim
+k→∞
+�ˆ
+Ω
+|∇uk|2 dx+
+¨
+R2n
+|uk(x) − uk(y)|2
+|x − y|n+2s
+dxdy−λ
+ˆ
+Ω
+(u+
+k +ǫ)−γuk dx−
+ˆ
+Ω
+(u+
+k )quk dx
+�
+= 0.
+
+Mixed local and nonlocal singular problems
+13
+The preceding two inequalities give,
+lim
+k→∞
+�ˆ
+Ω
+|∇(uk − u0)|2 dx +
+¨
+R2n
+|(uk(x) − uk(y)) − (u0(x) − u0(y))|2
+|x − y|n+2s
+dxdy
+�
+= lim
+k→∞
+�
+λ
+ˆ
+Ω
+(u+
+k + ǫ)−γuk dx +
+ˆ
+Ω
+(u+
+k )quk dx − λ
+ˆ
+Ω
+(u+
+k + ǫ)−γu0 dx −
+ˆ
+Ω
+(u+
+k )qu0 dx
+�
+− lim
+k→∞
+�ˆ
+Ω
+∇u0∇uk dx −
+ˆ
+Ω
+|∇u0|2 dx
+�
+− lim
+k→∞
+�¨
+R2n
+(u0(x) − u0(y))(uk(x) − uk(y))
+|x − y|n+2s
+dxdy −
+¨
+R2n
+|u0(x) − u0(y)|2
+|x − y|n+2s
+dxdy
+�
+.
+(3.13)
+Since uk ⇀ u0 weakly in H1
+0(Ω) as k → ∞, we observe that
+lim
+k→∞
+�ˆ
+Ω
+∇u0∇uk dx −
+ˆ
+Ω
+|∇u0|2 dx
+�
+= 0.
+(3.14)
+Further, since uk ⇀ u0 weakly in H1
+0(Ω) as k → ∞, it follows that
+lim
+k→∞
+�¨
+R2n
+(u0(x) − u0(y))(uk(x) − uk(y))
+|x − y|n+2s
+dxdy −
+¨
+R2n
+|u0(x) − u0(y)|2
+|x − y|n+2s
+dxdy
+�
+= 0.
+(3.15)
+Indeed, the weak convergence of uk to u0 implies that
+uk(x) − uk(y)
+|x − y|n+2s
+⇀ u0(x) − u0(y)
+|x − y|n+2s
+weakly in
+L2(R2n),
+which combined with the fact that
+u0(x) − u0(y)
+|x − y|
+n+2s
+2
+∈ L2(R2n)
+proves (3.15).
+On the other hand, since
+��(u+
+k + ǫ)−γu0
+�� ≤ ǫ−γu0 and
+ˆ
+Ω
+��ǫ−γu0
+�� dx ≤ ǫ−γ
+ˆ
+Ω
+|u0| dx < +∞,
+by the Lebesgue Dominated convergence theorem, it follows that
+lim
+k→∞
+ˆ
+Ω
+(u+
+k + ǫ)−γu0 dx =
+ˆ
+Ω
+(u+
+0 + ǫ)−γu0 dx.
+(3.16)
+Since uk → u0 pointwise almost everywhere in Ω and for any measurable subset E of Ω,
+ˆ
+E
+|(u+
+k + ǫ)−γuk| dx ≤
+ˆ
+E
+ǫ−γ|uk| dx ≤ ∥ǫ−γ∥L∞(Ω)∥uk∥Ll(Ω)|E|
+l−1
+l
+≤ C(ǫ)|E|
+l−1
+l ,
+using Vitali’s convergence theorem, we have
+lim
+k→∞ λ
+ˆ
+Ω
+(u+
+k + ǫ)−γuk dx = λ
+ˆ
+Ω
+(u+
+0 + ǫ)−γu0 dx.
+(3.17)
+
+Mixed local and nonlocal singular problems
+14
+Since q + 1 < l, we have
+ˆ
+E
+|(u+
+k )qu0| dx ≤ ∥u0∥Ll(Ω)
+�ˆ
+E
+(u+
+k )ql
+′
+dx
+� 1
+l′
+≤ C3|E|α
+and
+ˆ
+E
+|(u+
+k )quk| dx ≤ ∥uk∥Ll(Ω)
+�ˆ
+E
+(u+
+k )ql′ dx
+� 1
+l′
+≤ C4|E|β
+for some positive constants C3, C4, α and β. Again using Vitali’s convergence theorem, we
+get
+lim
+k→∞
+ˆ
+Ω
+(u+
+k )qu0 dx =
+ˆ
+Ω
+(u+
+0 )qu0 dx,
+(3.18)
+and
+lim
+k→∞
+ˆ
+Ω
+(u+
+k )quk dx =
+ˆ
+Ω
+(u+
+0 )qu0 dx.
+(3.19)
+Using (3.14), (3.15), (3.16), (3.17), (3.18) and (3.19) in (3.13), we obtain uk → u0 strongly in
+H1
+0(Ω) as k → ∞ which proves our claim.
+Remark 3.3. Using Lemma 3.1, Lemma 3.2 and the Mountain Pass Lemma, for every
+λ ∈ (0, Λ), there exists ζǫ ∈ H1
+0(Ω) such that I′
+λ,ǫ(ζǫ) = 0 and
+Iλ,ǫ(ζǫ) = inf
+γ∈Γ max
+t∈[0,1] Iλ,ǫ(γ(t)) ≥ ρ > 0,
+where
+Γ =
+�
+γ ∈ C([0, 1], H1
+0(Ω)) : γ(0) = 0, γ(1) = Te1
+�
+.
+Moreover, as a consequence of Lemma 3.1, since for every λ ∈ (0, Λ) we have
+inf
+∥v∥≤R Iλ,ǫ(v) < 0,
+by the weak lower semicontinuity of Iλ,ǫ, there exists a nonzero νǫ ∈ H1
+0(Ω) such that ∥νǫ∥ ≤ R
+and
+inf
+∥v∥≤R Iλ,ǫ(v) = Iλ,ǫ(νǫ) < 0 < ρ ≤ Iλ,ǫ(ζǫ).
+(3.20)
+Thus, ζǫ and νǫ are two different non trivial critical points of Iλ,ǫ, provided λ ∈ (0, Λ).
+Lemma 3.4. The critical points ζǫ and νǫ of Iλ,ǫ are nonnegative in Ω.
+Proof. Let u = ζǫ or νǫ. Therefore, since the integrand λ(u+ + ǫ)−γ + (u+)q is nonnegative in
+Ω, testing (3.3) with v = min{u, 0} and proceeding exactly as in the proof of [27, Pages 11-12,
+Lemma 3.1] (or [4, Page 11, Lemma 3.1]), we get u ≥ 0 in Ω. This completes the proof.
+Lemma 3.5. There exists a constant Θ > 0 (independent of ǫ) such that ∥vǫ∥ ≤ Θ, where
+vǫ = ζǫ or νǫ.
+Proof. We notice that the result trivially holds if vǫ = νǫ. Thus, it is enough to deal with
+the case when vǫ = ζǫ. Recalling the terms from Lemma 3.1 and Remark 3.3, we define
+A = max
+t∈[0,1] I0,ǫ(tTe1) then
+A ≥ max
+t∈[0,1] Iλ,ǫ(tTe1) ≥ inf
+γ∈Γ max
+t∈[0,1] Iλ,ǫ(γ(t)) = Iλ,ǫ(ζǫ) ≥ ρ > 0 > Iλ,ǫ(νǫ).
+
+Mixed local and nonlocal singular problems
+15
+Therefore
+1
+2
+ˆ
+Ω
+|∇ζǫ|2 dx+1
+2
+¨
+R2n
+|ζǫ(x) − ζǫ(y)|2
+|x − y|n+2s
+dxdy−λ
+ˆ
+Ω
+(ζǫ + ǫ)1−γ − ǫ1−γ
+1 − γ
+dx−
+1
+q + 1
+ˆ
+Ω
+ζq+1
+ǫ
+dx ≤ A.
+(3.21)
+Choosing φ = − ζǫ
+2 as a test function in (3.3) we obtain
+−
+1
+q + 1
+ˆ
+Ω
+|∇ζǫ|2 dx−
+1
+q + 1
+¨
+R2n
+|ζǫ(x) − ζǫ(y)|2
+|x − y|n+2s
+dxdy+
+λ
+q + 1
+ˆ
+Ω
+ζǫ
+(ζǫ + ǫ)γ dx+
+1
+q + 1
+ˆ
+Ω
+ζq+1
+ǫ
+dx = 0.
+(3.22)
+Adding (3.21) and (3.22) we have
+�1
+2 −
+1
+q + 1
+�
+∥ζǫ∥2 ≤ λ
+ˆ
+Ω
+(ζǫ + ǫ)1−γ − ǫ1−γ
+1 − γ
+dx −
+λ
+q + 1
+ˆ
+Ω
+ζǫ
+(ζǫ + ǫ)γ dx + A
+≤ C
+ˆ
+Ω
+ζǫ1−γ + A ≤ C∥ζǫ∥1−γ + A,
+for some positive constant C being independent of ǫ, where we have used H¨older’s inequality
+and Lemma 1.3. Thus, since q > 1, the sequence {ζǫ} is uniformly bounded in H1
+0(Ω) with
+respect to ǫ. This completes the proof.
+3.1
+Proof of Theorem 1.8
+By Lemma 3.4 and Lemma 3.5, up to a subsequence, ζǫ ⇀ ζ0 and νǫ ⇀ ν0 weakly in H1
+0(Ω)
+as ǫ → 0+, for some nonnegative ζ0, ν0 ∈ H1
+0(Ω).
+Step 1. Let v0 = ζ0 or ν0. Here, we prove that v0 ∈ H1
+0(Ω) is a weak solution of the problem
+(3.1). Indeed, for any ǫ ∈ (0, 1) and t ≥ 0, we notice that
+λ(t + ǫ)−γ + tq ≥ λ(t + 1)−γ + tq ≥ min
+�
+1, λ
+2
+�
+:= C > 0, say.
+Therefore, recalling that vǫ = ζǫ or νǫ, we have
+− ∆vǫ + (−∆)svǫ = λ(vǫ + ǫ)−γ + vq
+ǫ ≥ C > 0.
+(3.23)
+Using [27, Lemma 3.1] (see also [4, Lemma 3.1]), we get the existence of ξ ∈ H1
+0(Ω) ∩ L∞(Ω)
+satisfying
+−∆ξ + (−∆)sξ = C in Ω, ξ > 0 in Ω, ξ = 0 in Rn \ Ω
+such that for every ω ⋐ Ω, there exists a constant c(ω) > 0 satisfying ξ ≥ c(ω) > 0 in Ω.
+Then, for every nonnegative φ ∈ H1
+0(Ω), we have
+ˆ
+Ω
+∇vǫ∇φ dx +
+¨
+R2n
+(vǫ(x) − vǫ(y))(φ(x) − φ(y))
+|x − y|n+2s
+dxdy =
+ˆ
+Ω
+�
+λ(vǫ + ǫ)−γ + vq
+ǫ
+�
+φ dx ≥
+ˆ
+Ω
+Cφ dx
+=
+ˆ
+Ω
+∇ξ∇φ dx +
+¨
+R2n
+(ξ(x) − ξ(y))(φ(x) − φ(y))
+|x − y|n+2s
+dxdy.
+
+Mixed local and nonlocal singular problems
+16
+Testing with φ = (ξ − vǫ)+ in the above estimate, we obtain
+ˆ
+Ω
+|∇(ξ−vǫ)+|2 dx+
+¨
+R2n
+(ξ(x) − ξ(y) − (vǫ(x) − vǫ(y))((ξ − vǫ)+(x) − (ξ − vǫ)+(y))
+|x − y|n+2s
+dxdy ≤ 0.
+Following the same arguments as in the proof of [36, Lemma 9], the double integral in the
+above estimate become nonnegative.
+Hence, using this fact in the above inequality gives
+vǫ ≥ ξ in Ω. Hence there exists a constant c(ω) > 0 (independent of ǫ) such that
+vǫ ≥ c(ω) > 0, for every ω ⋐ Ω.
+(3.24)
+Using Lemma 3.5 and the fact (3.24) along with the hypothesis on q, we can pass to the limit
+in (3.23) to obtain
+ˆ
+Ω
+∇v0∇φ dx +
+¨
+R2n
+(v0(x) − v0(y) − (vǫ(x) − vǫ(y))((ξ − vǫ)+(x) − (ξ − vǫ)+(y))
+|x − y|n+2s
+dxdy
+= λ
+ˆ
+Ω
+φv−γ
+0 (x) dx +
+ˆ
+Ω
+vq
+0φ dx,
+for every φ ∈ C1
+c (Ω). Hence the claim follows.
+Step 2. Now we establish that ζ0 ̸= ν0. Choosing φ = vǫ ∈ H1
+0(Ω) as a test function in
+(3.3), we get
+ˆ
+Ω
+|∇vǫ|2 dx +
+¨
+R2n
+|vǫ(x) − vǫ(y)|2
+|x − y|n+2s
+dxdy = λ
+ˆ
+Ω
+vǫ(vǫ + ǫ)−γ dx +
+ˆ
+Ω
+vq+1
+ǫ
+dx.
+Since q + 1 < l, using Lemma 1.3 we obtain
+lim
+ǫ→0+
+ˆ
+Ω
+(vǫ)q+1 dx =
+ˆ
+Ω
+vq+1
+0
+dx.
+(3.25)
+Moreover, since
+0 ≤ vǫ(vǫ + ǫ)−γ ≤ v1−γ
+ǫ
+,
+using Vitali’s convergence theorem, it follows that
+λ lim
+ǫ→0+
+ˆ
+Ω
+vǫ(vǫ + ǫ)−γ dx = λ
+ˆ
+Ω
+v1−γ
+0
+dx.
+Therefore, we obtain
+lim
+ǫ→0+
+� ˆ
+Ω
+|∇vǫ|2 dx +
+¨
+R2n
+|vǫ(x) − vǫ(y)|2
+|x − y|n+2s
+dxdy
+�
+= λ
+ˆ
+Ω
+v1−γ
+0
+dx +
+ˆ
+Ω
+vq+1
+0
+dx.
+(3.26)
+By Remark 1.6, choosing φ = v0 as a test function in (3.1) we get
+ˆ
+Ω
+|∇v0|2 dx +
+¨
+R2n
+|v0(x) − v0(y)|2
+|x − y|n+2s
+dxdy = λ
+ˆ
+Ω
+v1−γ
+0
+dx +
+ˆ
+Ω
+vq+1
+0
+dx.
+(3.27)
+
+Mixed local and nonlocal singular problems
+17
+Hence from (3.26) and (3.27), we obtain
+lim
+ǫ→0+
+� ˆ
+Ω
+|∇vǫ|2 dx+
+¨
+R2n
+|vǫ(x) − vǫ(y)|2
+|x − y|n+2s
+dxdy
+�
+=
+ˆ
+Ω
+|∇v0|2 dx+
+¨
+R2n
+|v0(x) − v0(y)|2
+|x − y|n+2s
+dxdy.
+(3.28)
+Using Vitali’s convergence theorem, we have
+lim
+ǫ→0+
+ˆ
+Ω
+[(vǫ + ǫ)1−γ − ǫ1−γ] dx =
+ˆ
+Ω
+v1−γ
+0
+dx.
+(3.29)
+From (3.25), (3.28) and (3.29), we have lim
+ǫ→0+ Iλ,ǫ(vǫ) = Iλ(v0), which along with (3.20) gives
+ζ0 ̸= ν0.
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+Prashanta Garain
+Department of Mathematics,
+Indian Institute of Technology Indore,
+Khandwa Road, Simrol, Indore 453552, India
+e-mail: pgarain92@gmail.com
+
diff --git a/VdA0T4oBgHgl3EQfEv-t/content/tmp_files/load_file.txt b/VdA0T4oBgHgl3EQfEv-t/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf,len=801
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='02023v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='AP]  5 Jan 2023  On a class of mixed local and nonlocal semilinear elliptic  equation with singular nonlinearity  Prashanta Garain  Abstract  In this article, we consider a combination of local and nonlocal Laplace equation with  singular nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' For such mixed problems, we establish existence of at least one  weak solution for a parameter dependent singular nonlinearity and existence of multiple  solution for purturbed singular nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Our argument is based on the variational  and approximation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Keywords: Mixed local and nonlocal equation, singular nonlinearity, existence, regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  2020 Mathematics Subject Classification: 35M10, 35R11, 35B65, 35J75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Contents  1  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1  Functional setting and useful results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  15  1  Introduction  In this article, we consider the following mixed local and nonlocal semilinear equation with  singular nonlinearity  − ∆u + (−∆)su = g(x, u) in Ω,  u > 0 in Ω,  u = 0 in Rn \\ Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1)  where Ω ⊂ Rn is a bounded domain with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Here −∆ is the classical Laplace operator  and (−∆)s, s ∈ (0, 1) is the fractional Laplace operator defined by  (−∆)su = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  ˆ  Rn  u(x) − u(y)  |x − y|n+2s dy,  1  Mixed local and nonlocal singular problems  2  where P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' denotes the principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We establish existence of at least one weak solution  of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) for the purely singular nonlinearity g of the form (g1) given by  (g1)  g(x, u) = λh(u)u−γ,  where λ > 0, γ ∈ (0, 1) and  (h1) h : [0, ∞) → R is a continuous nondecreasing function such that h(0) > 0 and  (h2)  lim  t→0  h(t)  tγ  = ∞,  lim  t→∞  h(t)  tγ+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2)  Further, we establish multiplicity result for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) with the purturbed singular  nonlinearity g of the form (g2) given by  (g2)  g(x, u) = λu−γ + uq,  where λ > 0, γ ∈ (0, 1) and q ∈ (1, 2∗ − 1) with 2∗ =  2n  n−2 if n > 2 and 2∗ = ∞ if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Before proceeding further, we state the functional setting to study the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1  Functional setting and useful results  In this section, we present some known results for the fractional Sobolev space, see [20] for  more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let E ⊂ Rn be a measurable set and |E| denote its Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Recall  that the Lebesgue space L2(E), is defined as the space of measurable functions u : E → R  with the finite norm  ∥u∥L2(E) =  \uf8eb  \uf8ed  ˆ  E  |u(x)|2 dx  \uf8f6  \uf8f8  1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Here and in the rest of the paper, it is assumed that Ω ⊂ Rn with n ≥ 2 is a bounded smooth  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' The Sobolev space H1(Ω) is defined as the Banach space of locally integrable weakly  differentiable functions u : Ω → R equipped with the following norm:  ∥u∥H1(Ω) = ∥u∥L2(Ω) + ∥∇u∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  The space H1(Rn) is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' To deal with mixed problems, we use the space  H1  0(Ω) = {u ∈ H1(Rn) : u = 0 in Rn \\ Ω} under the norm ∥u∥ = ∥∇u∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' It can be shown  that H1  0(Ω) is a real separable and reflexive Banach space, see [9, 10, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  The fractional Sobolev space Hs(Ω), 0 < s < 1, is defined by  Hs(Ω) =  �  u ∈ L2(Ω) : |u(x) − u(y)|  |x − y|  n  2 +s  ∈ L2(Ω × Ω)  �  ,  which is endowed with the norm  ∥u∥Hs(Ω) =  �ˆ  Ω  |u(x)|2 dx +  ˆ  Ω  ˆ  Ω  |u(x) − u(y)|2  |x − y|n+2s  dx dy  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  For the next result, see [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  3  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' There exists a constant C = C(n, s) > 0 such that  ∥u∥Hs(Ω) ≤ C∥u∥H1(Ω),  ∀ u ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Next, we have the following result from [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' There exists a constant C = C(n, s, Ω) such that  ¨  Rn  |u(x) − u(y)|2  |x − y|n+2s  dx dy ≤ C  ˆ  Ω  |∇u|2 dx,  ∀ u ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3)  For the following Sobolev embedding, see, for example, [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' The embedding operators  H1  0(Ω) ֒→  \uf8f1  \uf8f2  \uf8f3  Lt(Ω),  for t ∈ [1, 2∗], if n > 2,  Lt(Ω),  for t ∈ [1, ∞), if n = 2  are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Now we are ready to define the notion of weak solutions for the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' (Weak Solution) Let g be either of the form (g1) or (g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  We say that  u ∈ H1  0(Ω) is a weak subsolution (or supersolution) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1), if u > 0 in Ω such that for every  ω ⋐ Ω, there exists a positive constant c(ω) with u ≥ c(ω) > 0 in ω and  ˆ  Ω  ∇u∇φ dx +  ¨  R2n  (u(x) − u(y))(φ(x) − φ(y))  |x − y|n+2s  dxdy ≤ ( or ) ≥ g(x, u)φ dx,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4)  for every nonnegative φ ∈ C1  c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We say that u ∈ H1  0(Ω) is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1), if the  equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4) holds for every φ ∈ C1  c (Ω) without a sign restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Note that by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2, it follows that Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4 is well  stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let u ∈ H1  0(Ω) be a weak solution of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) when g is either of the  form (g1) or (g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then following the lines of the proof of [27, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1], it follows that  the equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4) holds, for every φ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2  Statement of the main results:  Our main results in this article reads as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let 0 < γ < 1 and g be of the form (g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then for every λ > 0, there exists  a weak solution u ∈ H1  0(Ω) ∩ L∞(Ω) of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let 0 < γ < 1 and g be of the form (g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then there exists Λ > 0 such that  for every λ ∈ (0, Λ) the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) admits at least two different weak solutions in H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  4  To prove our main results stated above, the following result concerning the mixed local  and nonlocal eigenvalue problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5) will be useful for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  − ∆u + (−∆)su = λ|u|p−2u in Ω,  u = 0 in Rn \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5)  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' (i) There exists the least one eigenvalue λ1 > 0 and at least one corresponding  eigenfunction e1 ∈ H1  0(Ω) ∩ L∞(Ω) \\ {0} which is nonnegative in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' (ii) Moreover, for every  ω ⋐ Ω, there exists a positive constant c(ω) such that e1 ≥ c(ω) > 0 in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Part (i) follows from [9, Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Part (ii) follows from [24, Theorem  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Singular problems has drawn a great attention over the last three decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Equations of  the form  − α∆u + β(−∆)su = λf(u)u−γ + µur,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6)  where α, β, λ, µ, r ≥ 0, γ > 0 are parameters and f is some given function, are studied widely  in both the local (β = 0) and nonlocal (α = 0) cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Here the singularity is  captured by the parameter γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Indeed, the quasilinear analouge of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) is  also investigated in the separate local and nonlocal cases and there is a colossal amount of  work done for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  More precisely, in the local case (β = 0), Crandall-Rabinowitz-Tartar [17] proved existence  of classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) for λ = 1, µ = 0 and f(u) = 1 for any γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Further, for a certain  range of γ, Lazer-McKenna [35] studied the notion of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Boccardo-Orsina [12]  removed this restriction on γ and proved existence of weak solutions for any γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This study  has further been investigated in the quasilinear setting by Canino-Sciunzi-Trombetta [15], see  also De Cave [18] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' When f(u) ≥ 0 and µ = 0, for 0 < γ < 1  and a certain range of λ, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) is investigated by Ko-Lee-Shivaji in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' In the  purturbed case, we refer to Haitao [31], Hirano-Saccon-Shioji [32], Arcoya-Boccardo-M´erida  [2, 3], Bal-Garain [5], Giacomoni-Schindler-Tak´aˇc in [30], and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In the nonlocal case (α = 0), equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) is studied by Fang [22] for µ = 0 and  further been extended in the quasilinear setting by Canino-Montoro-Sciunzi-Squassina [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  The perturbed singular case (µ > 0) is investigated by Barrios-De Bonis-Medina-Peral [6],  Adimurthi-Giacomoni-Santra [1], Giacomoni-Mukherjee-Sreenadh [28, 29] and generalized by  Mukherjee-Sreenadh [37] in the quasilinear case and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  To the best of our knowledge, singular problems in the mixed local and nonlocal setting is  very less known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Our main purpose in this article is to contribute in this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We believe it  would be an interesting topic of further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We would like to mention that mixed  problems are also less known even in the nonsingular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Using probability theory, Foondun  [23], Chen-Kim-Song-Vondraˇcek [16] studied regularity results for the equation  − ∆u + (−∆)su = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7)  Mixed local and nonlocal singular problems  5  Recently based on purely analytic approach, Biagi-Dipierro-Salort-Valdinoci-Vecchi [7, 8, 38]  studied existence and regularity results for the mixed equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7) is also  studied using analytic approach in the quasilinear case by Garain-Kinnunen [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Several  recent regularity results and other qualitative properties for such problems using analytic  approach can be found in see [9, 10, 11, 19, 25] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In the mixed singular case, that is for positive α and β, assuming µ = 0 and f depending  on x only, the singular equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) and its quasilinear version is studied recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' In this  concern, for the quasilinear case, we refer to Garain-Ukhlov [27] for existence, uniqueness,  regularity and symmetry properties with any γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Further, associated extremal functions  are also studied in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Moreover, Arora-Radulescu [4] studied several existence and regularity  properties (which shows power and exponential type Sobolev regularity depending upon the  summability of the datum f and the singular exponent γ > 0) for the semilinear equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6), where the case γ = 0 is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In this article, first we prove the existence and regularity in terms of boundedness for the  singular problem under the singularity of the form (g1) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' In this context, we  adopt the variational technique introduced in [34] in the mixed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' To this end, we also  borrow ideas from [31] to prove the sub-supersolution result (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1), where to deal with  the nonlocal behavior of the equation, we used the technique from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Finally, the eigenvalue  problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5) and the purely singular problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7) are used to construct subsolution and  supersolutions, thanks to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In the second part of this article, we investigate the multiplicity result for the purturbed  singularity (g2) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Here, we utilise the variational approach introduced in  Arcoya-Boccardo [2] in combination with the technique from [26] to deal with the nonlocality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  To this end, we obtain existence multiple solutions of the associated approximate problem  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This fact combined with an apriori estimate (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5) gives us the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3  Notation and organization of the article  Throughout the rest of the article, by c or C, we mean a positive constant which may vary  from line to line or even in the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' The dependency of the constants c or C on the  parameters r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' , rk is denoted by c(r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' , rk) or C(r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' , rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' For a ∈ R, we  denote by a+ = max{a, 0} and a− = max{−a, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We use the notation 2∗ =  2n  n−2 if n > 2 and  2∗ = ∞ if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In Section 2, we obtain some preliminary results and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Finally, in  Section 3, we establish some useful results and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  6  2  Preliminaries for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7  Throughout this section, we assume g is of the form (g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' First we obtain some useful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Consider the energy functional Jλ : H1  0(Ω) → R ∪ {±∞} defined by  Jλ(u) =  ˆ  Ω  G(x, ∇u) +  ¨  R2n F(x, y, u) dxdy − λ  ˆ  Ω  H(u) dx  where  G(x, ∇u) = 1  2|∇u|2  F(x, y, u) = |u(x) − u(y)|2  |x − y|n+2s  and  H(t) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ˆ t  0  h(τ)τ −γ dτ, if t > 0,  0, if t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Following Haitao [31], we establish the following result in the mixed local and nonlocal setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Suppose that u, u ∈ H1  0(Ω) ∩ L∞(Ω) are weak subsolution and supersolution of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) respectively such that 0 < u ≤ u in Ω and u ≥ c(ω) > 0 for every ω ⋐ Ω, for some  constant c(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then there exists a weak solution u ∈ H1  0(Ω) ∩ L∞(Ω) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) satisfying  u ≤ u ≤ u in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let us consider the set  S = {v ∈ H1  0(Ω) : u ≤ v ≤ u in Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Since u ≤ u in Ω, we have S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We observe that S is closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We establish the  result in the following two Steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 1: We claim that Jλ admits a minimizer u over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  To this end, we prove that Jλ is weakly sequentially lower semicontinuous over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Indeed, let  {vk}k∈N ⊂ S be such that vk ⇀ v weakly in H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then by the hypothesis on h, we have  H(vk) ≤  ˆ u  0  h(τ)τ −γ dτ ≤ h(∥u∥∞)  (1 − γ) ∥u∥1−γ  ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Therefore by the Lebesgue Dominated Convergence theorem and weak lower semicontinuity of  norm, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Hence, there exists a minimizer u ∈ S of Jλ that is Jλ(u) = inf  v∈S Jλ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 2: Here, we prove that u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Let φ ∈ C1  c (Ω) and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We define  ηǫ =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  u  if u + ǫφ ≥ u,  u + ǫφ  if u ≤ u + ǫφ ≤ u,  u  if u + ǫφ ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  7  Observe that ηǫ = u + ǫφ − φǫ + φǫ ∈ S, where φǫ = (u + ǫφ − u)+ and φǫ = (u + ǫφ − u)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  By Step 1 above, since u is a minimizer of Jλ, we have  0 ≤ lim  t→0  Jλ(u + t(ηǫ − u)) − Jλ(u)  t  = I1 + I2 − λJ (say),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1)  with  I1 =  ˆ  Ω  ∇u∇(ηǫ − u) dx,  I2 =  ˆ  Q  (ηǫ − u)(−∆)su dx,  J =  ˆ  Ω  (ηǫ − u)u−γh(u) dx,  where we have used the notation Q = R2n \\ (CΩ × CΩ), where CΩ := Rn \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Therefore, we  have  0 ≤  ˆ  Ω  ∇u∇(ηǫ − u) dx +  ˆ  Q  (ηǫ − u)(−∆)su dx − λ  ˆ  Ω  (ηǫ − u)u−γh(u) dx  =⇒ 1  ǫ (Qǫ − Qǫ) ≤  ˆ  Ω  ∇u∇φ dx +  ˆ  Rn φ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φ dx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2)  where  Qǫ =  ˆ  Ω  ∇u∇φǫ dx +  ˆ  Rn φǫ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φǫ dx  and  Qǫ =  ˆ  Ω  ∇u∇φǫ dx +  ˆ  Rn φǫ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φǫ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Estimate of Qǫ: We observe that  1  ǫ  ˆ  Ω  ∇u∇φǫ dx = 1  ǫ  ˆ  Ω  ∇(u − u)∇φǫ dx ≥  ˆ  Ωǫ ∇(u − u)∇φ dx + 1  ǫ  ˆ  Ω  ∇u∇φǫ dx  �  ≥ o(1) + 1  ǫ  ˆ  Ω  ∇u∇φǫ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3)  Further, we notice that  1  ǫ  ˆ  Rn φǫ(−∆)su dx = 1  ǫ  � ˆ  Rn φǫ(−∆)s(u − u) dx +  ˆ  Rn φǫ(−∆)su dx  �  ≥ o(1) + 1  ǫ  ˆ  Rn φǫ(−∆)su dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4)  where to estimate the last inequality, we used the the lines of the proof from [29, Page 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  8  Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4), we have  1  ǫ Qǫ ≥ o(1) + 1  ǫ  � ˆ  Ω  ∇u∇φǫ dx +  ˆ  Rn φǫ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φǫ dx  �  = o(1) + 1  ǫ  � ˆ  Ω  ∇u∇φǫ dx +  ˆ  Rn φǫ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φǫ dx  �  + λ  ǫ  � ˆ  Ω  u−γh(u)φǫ dx −  ˆ  Ω  u−γh(u)φǫ dx  �  ≥ o(1) + λ  ǫ  ˆ  Ωǫ h(u)(u−γ − u−γ)(u − u) dx  + λ  ˆ  Ωǫ h(u)(u−γ − u−γ)φ dx  ≥ o(1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5)  using that u is a weak supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1), u ≤ u and  ˆ  Ωǫ h(u)(u−γ−u−γ)φ dx ≤ 2c(ω)−γh(||u||∞)||φ||∞ <  +∞, where Ωǫ = supp φǫ and ω = supp φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Taking into account that u is a weak subsolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1), u ≥ u and  ˆ  Ωǫ  h(u)(u−γ −  u−γ)φ dx ≤ 2c(ω)−γh(∥u∥∞)∥φ∥∞ < +∞, where Ωǫ = supp φǫ and ω = suppφ, in a similar  way, we obtain  1  ǫ Qǫ ≤ o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6)  Using the estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2), we conclude that  0 ≤  ˆ  Ω  ∇u∇φ dx +  ˆ  Rn φ(−∆)su dx − λ  ˆ  Ω  u−γh(u)φ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Since φ ∈ C1  c (Ω) is arbitrary, our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let 0 < γ < 1 and v0 ∈ H1  0(Ω) be a weak solution of the problem  − ∆u + (−∆)su = u−γ in Ω,  u > 0 in Ω,  u = 0 in Rn \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7)  Then v0 ∈ L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let k > 1, then by Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6 we choose φk = (v0 − k)+ ∈ H1  0(Ω) as a test function  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7) and apply H¨older’s along with Young’s inequality with ǫ ∈ (0, 1) to get  ˆ  Ω  |∇φk|2 dx ≤ C(ǫ)|A(k)|  2  q′ + ǫ  ˆ  Ω  |∇φk|2 dx,  where A(k) =  �  x ∈ Ω : v0 ≥ k in Ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  In the above estimate, we have also used that  H1  0(Ω) ֒→ Lq(Ω) for some q > 2 from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Therefore, fixing ǫ ∈ (0, 1), we obtain  ˆ  Ω  |∇φk|2 dx ≤ C|A(k)|  2  q′ ,  Mixed local and nonlocal singular problems  9  where C is some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let 1 < k < h, then since A(h) ⊂ A(k), we have  (h − k)p|A(h)|  2  q ≤  � ˆ  A(h)  (v0 − k)q dx  � 2  q ≤  � ˆ  A(k)  (v0 − k)q dx  � 2  q  ≤ C  ˆ  Ω  |∇φk|2 dx ≤ C |A(k)|  2  q′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Therefore  |A(h)| ≤  C  (h − k)q |A(k)|q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Since q > 2, by [33, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1], we have ||v0||L∞(Ω) ≤ c, where c is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Hence the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7  We construct a pair of weak subsolution and supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9, there exists e1 ∈ H1  0(Ω) ∩ L∞(Ω) such that  − ∆e1 + (−∆)se1 = λ1e1 in Ω, e1 > 0 in Ω, e1 = 0 in Rn \\ Ω  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8)  and for every ω ⋐ Ω, there exists a positive constant c(ω) with u ≥ c(ω) in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' By (h2), we  know that lim  t→0 t−γh(t) = ∞, so we can choose aλ > 0 sufficiently small such that  λ1(aλe1) ≤ λ(aλe1)−γh(aλe1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9)  Let u = aλe1, then u ∈ H1  0(Ω) ∩ L∞(Ω) and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9), we get  − ∆u + (−∆)su ≤ λ(aλe1)−γh(aλe1) = λu−γh(u) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='10)  By [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='13] and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2, there exists v0 ∈ H1  0(Ω) ∩ L∞(Ω) such that for every  ω ⋐ Ω there exists a positive constant c(ω) satisfying v0 ≥ c(ω) > 0 in ω and  − ∆v0 + (−∆)sv0 = v−γ  0  in Ω, v0 > 0 in Ω, v0 = 0 in Rn \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='11)  By the hypothesis (h2), since lim  t→∞ t−(γ+1)h(t) = 0, we choose bλ > 0 sufficiently large such  that  (bλ∥v0∥∞)−(γ+1)h(bλ∥v0∥∞) ≤  1  λ∥v0∥γ+1  ∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='12)  We define u := bλv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Then u ∈ H1  0(Ω) ∩ L∞(Ω) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='12), we have  − ∆u + (−∆)su = v−γ  0 bλ ≥ λ(bλv0)−γh(bλ∥v0∥∞) ≥ λu−γh(u) in Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='13)  where we have also used the nondecreasing property of h from (h1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Thus, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='10) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='13), it follows that u and u are weak subsolution and supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) respectively  and the constants aλ, bλ can be chosen in such a way that u ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1,  the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  10  3  Preliminaries for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8  In this section, we consider the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) when g is of the form (g2), which reads as  −∆u + (−∆)su = λu−γ + uq in Ω,  u > 0 in Ω,  u = 0 in Rn \\ Ω,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1)  where λ > 0, 0 < γ < 1 and q ∈ (1, 2∗ − 1) where 2∗ =  2n  n−2 if n > 2 and 2∗ = ∞ if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' To  this end, we study the functional Iλ : H1  0(Ω) → R ∪ {±∞} associated with the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1)  given by  Iλ(u) := 1  2  ˆ  Ω  |∇u|2 dx+ 1  2  ¨  R2n  |u(x) − u(y)|2  |x − y|n+2s  dxdy−λ  ˆ  Ω  (u+)1−γ  1 − γ  dx−  1  q + 1  ˆ  Ω  (u+)q+1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2)  For ǫ > 0, we consider the approximated problem  −∆u + (−∆)su = λ(u+ + ǫ)−γ + (u+)q in Ω,  u = 0 in Rn \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3)  We remark that the energy functional associated with the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3) is given by  Iλ,ǫ(u) = 1  2  ˆ  Ω  |∇u|2 dx + 1  2  ¨  R2n  |u(x) − u(y)|2  |x − y|n+2s  dxdy − λ  ˆ  Ω  [(u+ + ǫ)1−γ − ǫ1−γ]  1 − γ  dx  −  1  q + 1  ˆ  Ω  (u+)q+1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4)  We observe that Iλ,ǫ ∈ C1�  H1  0(Ω), R  �  , Iλ,ǫ(0) = 0 and Iλ,ǫ(v) ≤ I0,ǫ(v), for all v ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Let us define  l =  \uf8f1  \uf8f2  \uf8f3  2∗ =  2n  n−2, if n > 2,  r, if n = 2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5)  where r > 1 is such that 1 < q < r−1 if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Next we prove that Iλ,ǫ satisfies the Mountain  Pass Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' There exists R > 0, ρ > 0 and Λ > 0 depending on R such that  inf  ∥v∥≤R Iλ,ǫ(v) < 0 and  inf  ∥v∥=R Iλ,ǫ(v) ≥ ρ, for λ ∈ (0, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Moreover, there exists T > R such that Iλ,ǫ(Te1) < −1 for λ ∈ (0, Λ), where e1 is given by  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Recalling the definition of l from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5), we define θ = |Ω|  1  (  l  q+1)  ′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' By H¨older’s inequality  and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3, for every v ∈ H1  0(Ω), we have  ˆ  Ω  (v+)q+1 dx ≤  �ˆ  Ω  |v|l  � q+1  l  |Ω|  1  (  l  q+1 )′ ≤ Cθ∥v∥q+1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6)  for some positive constant C independent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Since  lim  t→0  Iλ,ǫ(te1)  t  = −λ  ˆ  Ω  ǫ−γe1 dx < 0,  Mixed local and nonlocal singular problems  11  we choose k ∈ (0, 1) sufficiently small and set ∥v∥ = R := k( q+1  pCθ)  1  q−1 such that  inf  ∥v∥≤R Iλ,ǫ(v) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Moreover, using the fact R < ( q+1  pCθ)  1  q−1 and the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6), we have  I0,ǫ(v) ≥ R2  2 − CθRq+1  q + 1  := 2ρ (say) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7)  We define  Λ :=  ρ  sup  ∥v∥=R  �  1  1 − γ  ˆ  Ω  |v|1−γ dx  �,  which is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Note that, since ρ, R depends on k, q, |Ω| and C, so does Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We observe  that  (v+ + ǫ)1−γ − ǫ1−γ ≤ (v+)1−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8)  Therefore, we have  Iλ,ǫ(v) ≥ 1  p  ˆ  Ω  |∇v|2 dx +  ¨  R2n  |v(x) − v(y)|2  |x − y|n+2s  dxdy −  1  q + 1  ˆ  Ω  (v+)q+1 dx −  λ  1 − γ  ˆ  Ω  (v+)1−γ dx  = I0,ǫ(v) −  λ  1 − γ  ˆ  Ω  (v+)1−γ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Hence, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='7), for λ ∈ (0, Λ), we get  inf  ∥v∥=R Iλ,ǫ(v) ≥  inf  ∥v∥=R I0,ǫ(v) − λ sup  ∥v∥=R  �  1  1 − γ  ˆ  Ω  |v|1−γ dx  �  ≥ 2ρ − λ sup  ∥v∥=R  �  1  1 − γ  ˆ  Ω  |v|1−γ dx  �  ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Finally, we observe that I0,ǫ(te1) → −∞, as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This gives the existence of T > R such  that I0,ǫ(Te1) < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Therefore,  Iλ,ǫ(Te1) ≤ I0,ǫ(Te1) < −1,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Next, we prove that Iλ,ǫ satisfies the Palais Smale (PS)c condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Iλ,ǫ satisfies the (PS)c condition, for any c ∈ R, that is if {uk}k∈N ⊂ H1  0(Ω) is  a sequence such that  Iλ,ǫ(uk) → c and I′  λ,ǫ(uk) → 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9)  as k → ∞, then {uk}k∈N contains a strongly convergent subsequence in H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  12  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We prove the result in two steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' First, we claim that if {uk}k∈N ⊂ H1  0(Ω) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9) then {uk}k∈N is uniformly  bounded in H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' To this end, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8), for some positive constant C (independent of k),  we have  Iλ,ǫ(uk) −  1  q + 1I′  λ,ǫ(uk)uk =  �1  2 −  1  q + 1  � ˆ  Ω  |∇uk|2 dx +  �1  2 −  1  q + 1  � ¨  R2n  |u(x) − u(y)|2  |x − y|n+2s  dxdy  − λ  ˆ  Ω  (u+  k + ǫ)1−γ − ǫ1−γ  1 − γ  dx +  λ  q + 1  ˆ  Ω  (u+  k + ǫ)−γuk dx  ≥  �1  2 −  1  q + 1  �  ∥uk∥2 − C∥uk∥1−γ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='10)  for some positive constant C (independent of k), where we have also used Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3 and  H¨older’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Noting q > 1 and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='10), we obtain  Iλ,ǫ(uk) −  1  q + 1I′  λ,ǫ(uk)uk ≥ C1∥uk∥2 − C∥uk∥1−γ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='11)  for some positive constants C, C1 (independent of k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9), for k large enough, we have  ����Iλ,ǫ(uk) −  1  q + 1I′  λ,ǫ(uk)uk  ���� ≤ C + o(∥uk∥),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='12)  for some positive constant C (independent of k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='12), our claim  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We claim that up to a subsequence, uk → u0 strongly in H1  0(Ω) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  By Step 1, since {uk}k∈N is uniformly bounded in H1  0(Ω), due to the reflexivity of H1  0(Ω),  there exists u0 ∈ H1  0(Ω) such that up to a subsequence, uk ⇀ u0 weakly in H1  0(Ω) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Again, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='9), we have  lim  k→∞  �ˆ  Ω  ∇uk∇u0 dx +  ¨  R2n  (uk(x) − uk(y))(u0(x) − u0(y))  |x − y|n+2s  dxdy  −λ  ˆ  Ω  (u+  k + ǫ)−γu0 dx −  ˆ  Ω  (u+  k )qu0 dx  �  = 0  and  lim  k→∞  �ˆ  Ω  |∇uk|2 dx+  ¨  R2n  |uk(x) − uk(y)|2  |x − y|n+2s  dxdy−λ  ˆ  Ω  (u+  k +ǫ)−γuk dx−  ˆ  Ω  (u+  k )quk dx  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  13  The preceding two inequalities give,  lim  k→∞  �ˆ  Ω  |∇(uk − u0)|2 dx +  ¨  R2n  |(uk(x) − uk(y)) − (u0(x) − u0(y))|2  |x − y|n+2s  dxdy  �  = lim  k→∞  �  λ  ˆ  Ω  (u+  k + ǫ)−γuk dx +  ˆ  Ω  (u+  k )quk dx − λ  ˆ  Ω  (u+  k + ǫ)−γu0 dx −  ˆ  Ω  (u+  k )qu0 dx  �  − lim  k→∞  �ˆ  Ω  ∇u0∇uk dx −  ˆ  Ω  |∇u0|2 dx  �  − lim  k→∞  �¨  R2n  (u0(x) − u0(y))(uk(x) − uk(y))  |x − y|n+2s  dxdy −  ¨  R2n  |u0(x) − u0(y)|2  |x − y|n+2s  dxdy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='13)  Since uk ⇀ u0 weakly in H1  0(Ω) as k → ∞, we observe that  lim  k→∞  �ˆ  Ω  ∇u0∇uk dx −  ˆ  Ω  |∇u0|2 dx  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='14)  Further, since uk ⇀ u0 weakly in H1  0(Ω) as k → ∞, it follows that  lim  k→∞  �¨  R2n  (u0(x) − u0(y))(uk(x) − uk(y))  |x − y|n+2s  dxdy −  ¨  R2n  |u0(x) − u0(y)|2  |x − y|n+2s  dxdy  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='15)  Indeed, the weak convergence of uk to u0 implies that  uk(x) − uk(y)  |x − y|n+2s  ⇀ u0(x) − u0(y)  |x − y|n+2s  weakly in  L2(R2n),  which combined with the fact that  u0(x) − u0(y)  |x − y|  n+2s  2  ∈ L2(R2n)  proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  On the other hand, since  ��(u+  k + ǫ)−γu0  �� ≤ ǫ−γu0 and  ˆ  Ω  ��ǫ−γu0  �� dx ≤ ǫ−γ  ˆ  Ω  |u0| dx < +∞,  by the Lebesgue Dominated convergence theorem, it follows that  lim  k→∞  ˆ  Ω  (u+  k + ǫ)−γu0 dx =  ˆ  Ω  (u+  0 + ǫ)−γu0 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='16)  Since uk → u0 pointwise almost everywhere in Ω and for any measurable subset E of Ω,  ˆ  E  |(u+  k + ǫ)−γuk| dx ≤  ˆ  E  ǫ−γ|uk| dx ≤ ∥ǫ−γ∥L∞(Ω)∥uk∥Ll(Ω)|E|  l−1  l  ≤ C(ǫ)|E|  l−1  l ,  using Vitali’s convergence theorem, we have  lim  k→∞ λ  ˆ  Ω  (u+  k + ǫ)−γuk dx = λ  ˆ  Ω  (u+  0 + ǫ)−γu0 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='17)  Mixed local and nonlocal singular problems  14  Since q + 1 < l, we have  ˆ  E  |(u+  k )qu0| dx ≤ ∥u0∥Ll(Ω)  �ˆ  E  (u+  k )ql  ′  dx  � 1  l′  ≤ C3|E|α  and  ˆ  E  |(u+  k )quk| dx ≤ ∥uk∥Ll(Ω)  �ˆ  E  (u+  k )ql′ dx  � 1  l′  ≤ C4|E|β  for some positive constants C3, C4, α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Again using Vitali’s convergence theorem, we  get  lim  k→∞  ˆ  Ω  (u+  k )qu0 dx =  ˆ  Ω  (u+  0 )qu0 dx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='18)  and  lim  k→∞  ˆ  Ω  (u+  k )quk dx =  ˆ  Ω  (u+  0 )qu0 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='19)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='14), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='19) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='13), we obtain uk → u0 strongly in  H1  0(Ω) as k → ∞ which proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='2 and the Mountain Pass Lemma, for every  λ ∈ (0, Λ), there exists ζǫ ∈ H1  0(Ω) such that I′  λ,ǫ(ζǫ) = 0 and  Iλ,ǫ(ζǫ) = inf  γ∈Γ max  t∈[0,1] Iλ,ǫ(γ(t)) ≥ ρ > 0,  where  Γ =  �  γ ∈ C([0, 1], H1  0(Ω)) : γ(0) = 0, γ(1) = Te1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Moreover, as a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1, since for every λ ∈ (0, Λ) we have  inf  ∥v∥≤R Iλ,ǫ(v) < 0,  by the weak lower semicontinuity of Iλ,ǫ, there exists a nonzero νǫ ∈ H1  0(Ω) such that ∥νǫ∥ ≤ R  and  inf  ∥v∥≤R Iλ,ǫ(v) = Iλ,ǫ(νǫ) < 0 < ρ ≤ Iλ,ǫ(ζǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='20)  Thus, ζǫ and νǫ are two different non trivial critical points of Iλ,ǫ, provided λ ∈ (0, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' The critical points ζǫ and νǫ of Iλ,ǫ are nonnegative in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let u = ζǫ or νǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Therefore, since the integrand λ(u+ + ǫ)−γ + (u+)q is nonnegative in  Ω, testing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3) with v = min{u, 0} and proceeding exactly as in the proof of [27, Pages 11-12,  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1] (or [4, Page 11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1]), we get u ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' There exists a constant Θ > 0 (independent of ǫ) such that ∥vǫ∥ ≤ Θ, where  vǫ = ζǫ or νǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' We notice that the result trivially holds if vǫ = νǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Thus, it is enough to deal with  the case when vǫ = ζǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Recalling the terms from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3, we define  A = max  t∈[0,1] I0,ǫ(tTe1) then  A ≥ max  t∈[0,1] Iλ,ǫ(tTe1) ≥ inf  γ∈Γ max  t∈[0,1] Iλ,ǫ(γ(t)) = Iλ,ǫ(ζǫ) ≥ ρ > 0 > Iλ,ǫ(νǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  15  Therefore  1  2  ˆ  Ω  |∇ζǫ|2 dx+1  2  ¨  R2n  |ζǫ(x) − ζǫ(y)|2  |x − y|n+2s  dxdy−λ  ˆ  Ω  (ζǫ + ǫ)1−γ − ǫ1−γ  1 − γ  dx−  1  q + 1  ˆ  Ω  ζq+1  ǫ  dx ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='21)  Choosing φ = − ζǫ  2 as a test function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3) we obtain  −  1  q + 1  ˆ  Ω  |∇ζǫ|2 dx−  1  q + 1  ¨  R2n  |ζǫ(x) − ζǫ(y)|2  |x − y|n+2s  dxdy+  λ  q + 1  ˆ  Ω  ζǫ  (ζǫ + ǫ)γ dx+  1  q + 1  ˆ  Ω  ζq+1  ǫ  dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='22)  Adding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='22) we have  �1  2 −  1  q + 1  �  ∥ζǫ∥2 ≤ λ  ˆ  Ω  (ζǫ + ǫ)1−γ − ǫ1−γ  1 − γ  dx −  λ  q + 1  ˆ  Ω  ζǫ  (ζǫ + ǫ)γ dx + A  ≤ C  ˆ  Ω  ζǫ1−γ + A ≤ C∥ζǫ∥1−γ + A,  for some positive constant C being independent of ǫ, where we have used H¨older’s inequality  and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Thus, since q > 1, the sequence {ζǫ} is uniformly bounded in H1  0(Ω) with  respect to ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='8  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5, up to a subsequence, ζǫ ⇀ ζ0 and νǫ ⇀ ν0 weakly in H1  0(Ω)  as ǫ → 0+, for some nonnegative ζ0, ν0 ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Let v0 = ζ0 or ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Here, we prove that v0 ∈ H1  0(Ω) is a weak solution of the problem  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Indeed, for any ǫ ∈ (0, 1) and t ≥ 0, we notice that  λ(t + ǫ)−γ + tq ≥ λ(t + 1)−γ + tq ≥ min  �  1, λ  2  �  := C > 0, say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Therefore, recalling that vǫ = ζǫ or νǫ, we have  − ∆vǫ + (−∆)svǫ = λ(vǫ + ǫ)−γ + vq  ǫ ≥ C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='23)  Using [27, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1] (see also [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1]), we get the existence of ξ ∈ H1  0(Ω) ∩ L∞(Ω)  satisfying  −∆ξ + (−∆)sξ = C in Ω, ξ > 0 in Ω, ξ = 0 in Rn \\ Ω  such that for every ω ⋐ Ω, there exists a constant c(ω) > 0 satisfying ξ ≥ c(ω) > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Then, for every nonnegative φ ∈ H1  0(Ω), we have  ˆ  Ω  ∇vǫ∇φ dx +  ¨  R2n  (vǫ(x) − vǫ(y))(φ(x) − φ(y))  |x − y|n+2s  dxdy =  ˆ  Ω  �  λ(vǫ + ǫ)−γ + vq  ǫ  �  φ dx ≥  ˆ  Ω  Cφ dx  =  ˆ  Ω  ∇ξ∇φ dx +  ¨  R2n  (ξ(x) − ξ(y))(φ(x) − φ(y))  |x − y|n+2s  dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Mixed local and nonlocal singular problems  16  Testing with φ = (ξ − vǫ)+ in the above estimate, we obtain  ˆ  Ω  |∇(ξ−vǫ)+|2 dx+  ¨  R2n  (ξ(x) − ξ(y) − (vǫ(x) − vǫ(y))((ξ − vǫ)+(x) − (ξ − vǫ)+(y))  |x − y|n+2s  dxdy ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Following the same arguments as in the proof of [36, Lemma 9], the double integral in the  above estimate become nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Hence, using this fact in the above inequality gives  vǫ ≥ ξ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Hence there exists a constant c(ω) > 0 (independent of ǫ) such that  vǫ ≥ c(ω) > 0, for every ω ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='24)  Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='5 and the fact (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='24) along with the hypothesis on q, we can pass to the limit  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='23) to obtain  ˆ  Ω  ∇v0∇φ dx +  ¨  R2n  (v0(x) − v0(y) − (vǫ(x) − vǫ(y))((ξ − vǫ)+(x) − (ξ − vǫ)+(y))  |x − y|n+2s  dxdy  = λ  ˆ  Ω  φv−γ  0 (x) dx +  ˆ  Ω  vq  0φ dx,  for every φ ∈ C1  c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Hence the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Now we establish that ζ0 ̸= ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Choosing φ = vǫ ∈ H1  0(Ω) as a test function in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3), we get  ˆ  Ω  |∇vǫ|2 dx +  ¨  R2n  |vǫ(x) − vǫ(y)|2  |x − y|n+2s  dxdy = λ  ˆ  Ω  vǫ(vǫ + ǫ)−γ dx +  ˆ  Ω  vq+1  ǫ  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Since q + 1 < l, using Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='3 we obtain  lim  ǫ→0+  ˆ  Ω  (vǫ)q+1 dx =  ˆ  Ω  vq+1  0  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='25)  Moreover, since  0 ≤ vǫ(vǫ + ǫ)−γ ≤ v1−γ  ǫ  ,  using Vitali’s convergence theorem, it follows that  λ lim  ǫ→0+  ˆ  Ω  vǫ(vǫ + ǫ)−γ dx = λ  ˆ  Ω  v1−γ  0  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Therefore, we obtain  lim  ǫ→0+  � ˆ  Ω  |∇vǫ|2 dx +  ¨  R2n  |vǫ(x) − vǫ(y)|2  |x − y|n+2s  dxdy  �  = λ  ˆ  Ω  v1−γ  0  dx +  ˆ  Ω  vq+1  0  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='26)  By Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='6, choosing φ = v0 as a test function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='1) we get  ˆ  Ω  |∇v0|2 dx +  ¨  R2n  |v0(x) − v0(y)|2  |x − y|n+2s  dxdy = λ  ˆ  Ω  v1−γ  0  dx +  ˆ  Ω  vq+1  0  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='27)  Mixed local and nonlocal singular problems  17  Hence from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='27), we obtain  lim  ǫ→0+  � ˆ  Ω  |∇vǫ|2 dx+  ¨  R2n  |vǫ(x) − vǫ(y)|2  |x − y|n+2s  dxdy  �  =  ˆ  Ω  |∇v0|2 dx+  ¨  R2n  |v0(x) − v0(y)|2  |x − y|n+2s  dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='28)  Using Vitali’s convergence theorem, we have  lim  ǫ→0+  ˆ  Ω  [(vǫ + ǫ)1−γ − ǫ1−γ] dx =  ˆ  Ω  v1−γ  0  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='29)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='29), we have lim  ǫ→0+ Iλ,ǫ(vǫ) = Iλ(v0), which along with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='20) gives  ζ0 ̸= ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  References  [1] Adimurthi, Jacques Giacomoni, and Sanjiban Santra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Radulescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Combined effects in mixed local-nonlocal  stationary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' arXiv e-prints, page arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='06701, November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Multiplicity of solution for a quasilinear equation  with singular nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Mediterr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Open Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' A Faber-Krahn  inequality for mixed local and nonlocal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' arXiv e-prints, page arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='00830,  April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Mixed local and  nonlocal elliptic operators: regularity and maximum principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Partial Differ-  ential Equations, 47(3):585–629, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' A Hong-Krahn-  Szeg¨o inequality for mixed local and nonlocal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=', 5(1):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' 014,  25, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' A Brezis-Oswald approach for mixed  local and nonlocal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' arXiv e-prints, page arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Semilinear elliptic  equations involving mixed local and nonlocal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' da Silva, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content='  Asymptot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' NoDEA Nonlinear Differ-  ential Equations Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Crandall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Rabinowitz, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+page_content=' Salort and Eugenio Vecchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' On the mixed local-nonlocal H´enon equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content=' Dif-  ferential Integral Equations, 35(11-12):795–818, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='  Prashanta Garain  Department of Mathematics,  Indian Institute of Technology Indore,  Khandwa Road, Simrol, Indore 453552, India  e-mail: pgarain92@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdA0T4oBgHgl3EQfEv-t/content/2301.02023v1.pdf'}
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+Traveling wave enantio-selective electron paramagnetic resonance
+M. Donaire,1, ∗ N. Bruyant,2 and G.L.J.A. Rikken2, †
+1Departamento de F´ısica Te´orica, At´omica y ´Optica and IMUVA,
+Universidad de Valladolid, Paseo Bel´en 7, 47011 Valladolid, Spain
+2Laboratoire National des Champs Magn´etiques Intenses UPR3228
+CNRS/EMFL/INSA/UGA/UPS, Toulouse & Grenoble,France
+(Dated: January 13, 2023)
+We propose a novel method for enantio-selective electron paramagnetic resonance spectroscopy
+based on magneto-chiral anisotropy. We calculate the strength of this effect and propose a dedicated
+interferometer setup for its observation.
+Introduction
+Electron paramagnetic resonance (EPR) spectroscopy is
+a powerful technique to study the local environment and
+the dynamics of spin-carrying entities, like transition
+metal ion complexes and organic radicals [1]. Also, those
+systems that do not intrinsically carry a spin can still
+be studied by EPR through spin-labelling, i.e., by se-
+lectively adding-on a spin carrying probe [2]. Many of
+the systems studied by EPR are chiral, i.e., they exist
+in two non-superimposable forms (enantiomers) that are
+each other’s mirror image, particularly in biochemistry
+where enzymes, metalloproteins, membranes, etc., are
+chiral subjects of intense EPR activity [3]. However, EPR
+is universally believed to be blind to chirality. Here we
+present the paradigm shift that EPR in the proper con-
+figuration is intrinsically sensitive to chirality because of
+magneto-chiral anisotropy (MChA).
+MChA corresponds to an entire class of effects in chiral
+media under an external magnetic field, which show an
+enantio-selective difference in the propagation of any un-
+polarized flux that propagates parallel or anti-parallel to
+the magnetic field. This difference has its origin in the si-
+multaneous breaking of parity and time-reversal symme-
+tries as a result of the chirality of the media and the mag-
+netization induced by the external magnetic field, respec-
+tively. Generally, such a difference manifests itself in the
+velocity or the attenuation of the flux. MChA has been
+predicted since 1962 in the optical properties of chiral
+systems in magnetic fields [4–8], and was finally observed
+in the 1990’s [9–11]. Nowadays it is observed across the
+entire electromagnetic spectrum, from microwaves [12] to
+X-rays [13]. The existence of MChA was further gener-
+alized to electrical transport [14] (in carbon nano tubes
+[15], organic conductors [16], metals [17–19] and semicon-
+ductors [20]), to sound propagation [21] and to dielectric
+properties [22].
+EPR is basically a strongly resonant form of magnetic
+circular dichroism and magnetic circular birefringence
+[23], effects well known in the optical wavelength range,
+where they however only represent small perturbations
+∗Electronic address: manuel.donaire@uva.es
+†Electronic address: geert.rikken@lncmi.cnrs.fr
+of the optical properties of the medium. By analogy, one
+should expect that MChA can manifest itself also in EPR
+of chiral media. This expectation can be formalized by
+the observation that the EPR transition probability P
+induced by a propagating electromagnetic field between
+the spin levels of a chiral medium in a magnetic field,
+is allowed by parity and time-reversal symmetry to have
+the form
+P D/L(ω, ˆk, B0) = P0(ω, B0)[1 + γD/L(ω)ˆk · B0].
+(1)
+In this equation, B0 is an external and constant magnetic
+field, P0 is the leading order transition probability be-
+tween the Zeeman levels, common to both enantiomers,
+the handedness of the medium is represented by D− right
+and L− left, with γD = −γL, and ˆk is a unitary vector
+in the direction of the wave vector of the electromagnetic
+field driving the transition whose frequency ω is of the or-
+der of µBB0/ℏ. This shows that the EPR transition prob-
+ability is enantioselectively modified when probed by an
+electromagnetic wave travelling parallel or anti-parallel
+to the magnetic field, an effect that we shall call travel-
+ing wave enantioselective EPR (TWEEPR). TWEEPR
+is quantified by the anisotropy factor gD/L
+T
+, which repre-
+sents the relative difference between the transition prob-
+abilities of both enantiomers,
+gD/L
+T
+≡ [P D/L(ω, �k, B0) − P D/L(ω, �k, −B0)]
+[P D/L(ω, �k, B0) + P D/L(ω, �k, −B0)]
+= γD/Lˆk·B0.
+(2)
+As spin is related to the absence of time-reversal sym-
+metry, and chirality is related to the absence of parity
+symmetry, one might expect that the two are decou-
+pled and that gD/L
+T
+is vanishingly small, thereby reducing
+TWEEPR to an academic curiosity. However, below we
+will show through a model calculation that, because of
+the ubiquitous spin-orbit coupling, TWEEPR represents
+a significant and measurable fraction of the EPR transi-
+tion probability for realistic chiral systems and that its
+anisotropy factor is not much smaller than that of optical
+MChA. Lastly, we will describe a dedicated TWEEPR
+setup.
+The model
+As for the spin system of our model calculation of
+TWEEPR, without loss of generality, we have chosen
+arXiv:2301.04755v1  [physics.chem-ph]  11 Jan 2023
+
+2
+a crystalline quasi-octahedral Cu(II) chiral complex be-
+cause this ion is one of the most extensively studied
+systems by EPR, it has the largest spin-orbit coupling
+among the first row transition metals, and it has the
+simplest energy diagram. Its electromagnetic response is
+attributed to a single unpaired electron that, in the 3d9
+configuration of the Cu(II) complex, behaves as a hole of
+positive charge +e. We model the binding potential of
+the hole by that of an isotropic harmonic oscillator that
+represents the rest of the ion, and is perturbed by the chi-
+ral potential V D/L
+C
+that results from its interaction with
+the chiral environment of the crystal lattice, and by the
+spin-orbit coupling. In turn, as we will show, this model
+allows us to find analytic expressions for both the optical
+and the EPR magnetochiral anisotropy parameters, gD/L
+O
+and gD/L
+T
+, respectively, in terms of the parameters of the
+model, both being proportional to the chiral coupling.
+Our model can thus relate gD/L
+T
+to its optical analogue
+gD/L
+O
+. The latter is experimentally determined for several
+systems. In particular, for CsCuCl3 both MChD [24] and
+EPR [25] have been reported. This approach thereby re-
+sults in a generic analytical expression for gD/L
+T
+in terms
+of the parameters of our model, and in a semi-empirical
+and quantitative prediction for gD/L
+T
+for this particular
+material in terms of its experimental optical MChD. The
+latter can be extended to any material for which optical
+MChD has been determined. Below we detail our model,
+which is a variant of Condon’s model for optical activity
+[26, 27], and its extension to optical magnetochiral bire-
+fringence [28].
+The Hamiltonian describing the system is given by H =
+H0 + V D/L
+C
++ VSO, with
+H0 =
+p2
+2me
++ meω2
+0r2
+2
+− µB(L + gS) · B0,
+(3)
+V D/L
+C
+= CD/Lxyz,
+VSO = λL · S,
+(4)
+where r = (x, y, z) and p are the position and kinetic
+momentum vectors of the harmonic oscillator, ω0 is its
+natural frequency, L and S are their orbital and spin an-
+gular momentum operators, respectively, CD = −CL is
+the right/left-handed chiral coupling, g ≃ 2 is the Land´e
+factor, λ ≃ −0.1 eV is the spin-orbit (SO) coupling pa-
+rameter, and B0 ≡ B0ˆz is the external magnetic field.
+The interaction with an electromagnetic plane-wave of
+frequency ω, propagating along B0, is given in a multi-
+pole expansion by
+W = −er · Eω(t)/2 − µB(L + gS) · Bω(t)/2 + h.c., (5)
+where Eω(t) = iωAωe−iωt and Bω(t) = i¯nk ∧ Aωe−iωt
+are the complex-valued electric and magnetic fields in
+terms of the electromagnetic vector potential, Aω, eval-
+uated at the center of mass of the ion. Note that the
+field incident on a molecule of the complex is the effec-
+tive field which propagates throughout the medium with
+an effective index of refraction ¯n. Hence it is the effective
+wavevector ¯nk that appears.
+FIG. 1: Energy levels of Cu(II) in a chiral quasi-octahedral
+configuration. Approximate experimental values are ∆0 ≃ 1.5
+eV, ∆1 ≃ 0.5 eV, ∆2 ≃≈ 0.23 eV.
+In our model, the 3d orbitals are represented by linear
+combinations of the n = 2, l = 2 states of the isotropic
+harmonic oscillator –see Appendix A. Essential to the
+original Condon model was the anisotropy of the har-
+monic oscillator, which removes all axis and planes of
+symmetry. In our model, such an anisotropy is provided
+by the interaction of the ion with the surrounding lig-
+ands of the complex, which in the case of CsCuCl3 form
+an quasi-octahedral structure. In the first place, that in-
+teraction causes the elongation of the 3d orbitals which
+lie along the z-axis, opening an optical gap ∆0. Also, in
+conjunction with the Jahn-Teller distortion and the he-
+lical configuration of the Cu(II) ions, it removes the de-
+generacy between the orbitals lying on the xy plane and
+generates a small energy gap δ between the states dzx and
+dyz, with λ ≫ δ. The ground state of the Cu(II) ion in
+the octahedral configuration Ψ is, at finite temperature
+and subject to a magnetic field, a linear combination of
+the doublet dx2−y2 ⊗ {↑, ↓},
+|Ψ⟩ = |dx2−y2⟩ ⊗ (cos θ/2 ↑ + sin θ/2 ↓),
+(6)
+where θ, being a function of B0 and the temperature,
+is the angle between the magnetization of the sample
+and B0. For EPR, spin-flip takes place at a resonance
+frequency Ω = gµBB0/ℏ when the up ↑ component of
+Ψ turns into |Φ⟩ = |dx2−y2⟩⊗ ↓ , with probability pro-
+portional to cos2 θ/2, and the down ↓ component turns
+into |Φ′⟩ = |dx2−y2⟩⊗ ↑ with probability proportional to
+sin2 θ/2. The net absorption probability is thus propor-
+tional to cos2 θ/2 − sin2 θ/2 = cos θ and hence to the
+degree of magnetization along B0.
+At
+B0 = 1T, Ω
+corresponds to an energy 150 µeV. In contrast, optical
+
+3dz
+3dzx
+3dx-y
+3dxy
+3dyz
+Chiralground state
+zx
+t2g.
+Φ4
+yz
+△2
+x
+△1
+3d9
+Ao
+eg
+crystalfield splitting
+Jahn-Tellereffect
+SO,chiral, and
+(octahedralsymmetry)(tetragonaldistortion)
+Zeeman splitting3
+absorption happens at an energy ∆0 ≃ 1.5 eV towards
+the quadruplet {dzx, dyz} ⊗ {↑, ↓}.
+Applying standard
+perturbation theory with the spin-orbit and the Zeeman
+potentials upon this quasidegenerate quadruplet , we end
+up with the four states φi, i = 1, .., 4, as appear in the
+energy diagram represented in Fig.1 –a brief description
+can be found in the Appendix A. It is of note that these
+states play a crucial role in the E1M1 transitions of both
+EPR and its optical analogue.
+Results
+Using up to fourth order time-dependent perturbation
+theory on VSO, VC and W, in the adiabatic regime, our
+model allows us to calculate the standard EPR and op-
+tical transition probabilities, as well as the MChA cor-
+rections to both of them, with the latter two being both
+proportional to CD/L. As for gD/L
+T
+, the probability dif-
+ference in the denominator of Eq.(2) is an enantioselec-
+tive E1M1 transition, whereas the denominator equals in
+good approximation the leading order M1M1 transition,
+gD/L
+T
+= P D/L
+E1M1/PM1M1|ω≈Ω, with
+PM1M1|ω≈Ω = ℏ−2���
+� T
+0
+dte−i(T −t)(Ω/2−iΓ/2)e−it(ω−Ω/2)⟨Φ| − gµBS · Bω|Ψ⟩
+���
+2
+− ℏ−2���
+� T
+0
+dte−i(T −t)(2ω−Ω/2−iΓ/2)
+× e−it(ω+Ω/2)⟨Φ′| − gµBS · Bω|Ψ⟩
+���
+2
+,
+P D/L
+E1M1|ω≈Ω = −2ℏ−2Re
+� T
+0
+dte−i(T −t)(Ω/2−iΓ/2)⟨˜Φ| − er · (¯n2 + 2)Eω/3|˜Ψ⟩e−it(ω−Ω/2)
+� T
+0
+dτ ei(T −τ)(Ω/2+iΓ/2)
+× ⟨Ψ| − gµBS · B∗
+ω|Φ⟩eiτ(ω−Ω/2) + 2ℏ−2Re
+� T
+0
+dt e−i(T −t)(2ω−Ω/2)⟨˜Φ′| − er · (¯n2 + 2)Eω/3|˜Ψ⟩
+× e−it(ω+Ω/2−iΓ/2)
+� T
+0
+dτ ei(T −τ)(2ω−Ω/2)⟨Ψ| − gµBS · B∗
+ω|Φ′⟩eiτ(ω+Ω/2+iΓ/2),
+ΓT ≫ 1,
+(7)
+where Γ is the linewidth of EPR absorption, ΓT ≫ 1
+implies the adiabatic approximation, and the states ˜Ψ,
+˜Φ, and ˜Φ′ are dressed with the states φi, i = 1, .., 4, on
+account of the spin-orbit and chiral interactions. Using
+a linearly polarized microwave probe field in Eq.(7), the
+resultant expression for the TWEEPR anisotropy factor
+reads
+gD/L
+T
+≃ c CD/Lℏ Ω δ
+meω3
+0∆2
+0
+¯n2 + 2
+3¯n
+,
+(8)
+where the second factor on the right hand side describes
+the effect of the refractive index on the local electric field
+and the wavevector. It is worth noting that the aforemen-
+tioned dependence on magnetization, ∼ cos θ, cancels out
+in the ratio between probabilities. For further details, see
+Appendix B.
+The values for the unknown parameters in Eq.(8) can
+be deduced comparing the predictions of the model with
+the experimental results for optical MChD [24] and EPR
+[25] in CsCuCl3.
+In particular, we can estimate gD/L
+T
+from the data on the non-reciprocal absorption coeffi-
+cient in optical MChD, αA = α(B0 ↿↾ k) − α(B0 ⇃↾ k).
+The calculation goes as follows. In terms of the E1M1
+absorption probability at resonance, ω = ∆0/ℏ, αA reads
+αA = 4cµ0ρ∆0Γ′
+|Eω|2
+P D/L
+E1M1|ω=∆0/ℏ,
+(9)
+where Γ′ is the linewidth of optical absorption, and ρ
+is the molecular number density of the complex. Using
+our model, a calculation analogous to that for P D/L,EP R
+E1M1
+but for its optical counterpart, P D/L,O
+E1M1 – Appendices B,
+C and D-, allows as to express gD/L
+T
+in Eq.(8) in terms
+of αA,
+gD/L
+T
+=
+c ℏ3Γ′Ω ˜∆αA
+2∆3
+0µ0µ2
+Bρ cos θ,
+(10)
+where ˜∆−1 = ∆−1
+0
++ ∆−1
+2
+− 3∆−1
+1
+is the inverse of an
+effective energy interval which takes account of the opti-
+cal transitions to intermediate states –see Fig.1. It is of
+note that, whereas the magnetic transition is driven in
+EPR by the spin operator [Eq.(7)], it is driven by the or-
+bital angular momentum in the optical case. In turn, this
+causes MChD to be stronger in the optical case and pro-
+portional to the degree of magnetization cos θ, which can
+be approximated by cos θ ≈ µ0B0/kBT [31]. The optical
+MChA parameter, gD/L
+0
+, has an analogous expression to
+that in Eq.(2) with ℏω ≈ ∆0, being proportional to αA.
+Hence, our model allows us to estimate its upper bound,
+gD/L
+0
+≤ (cCD/Lδ cos θ)/(meω3
+0 ˜∆) – see Appendices C
+and D, from which gD/L
+T
+/gD/L
+0
+≳ (ℏΩ ˜∆)/(∆2
+0 cos θ).
+Note that, since both Ω and cos θ are proportional to
+B0, the ratio between EPR and optical MChA factors is
+independent of the field strength.
+Finally,
+substituting
+the
+experimental
+values
+for
+CsCuCl3 of all the variables in Eq.(10), for B0 = 14 T
+at a temperature of 4.2 K, we obtain gD/L
+T
+≈ 1.5 · 10−2,
+
+4
+which is small but not beyond the resolution of high field
+EPR spectrometers. For an X band EPR spectrometer
+(B = 0, 35 T), this means gD/L
+T
+≈ 3 · 10−4 which will
+require a different approach, as we discuss below.
+Implementation
+In commercial EPR spectrometers, resonant standing
+wave cavities are used to enhance sensitivity.
+Such a
+cavity can be regarded as containing equal amounts of
+traveling waves with k and −k. The MChA γD/L term
+in Eq.(1) can therefore not give a net contribution to
+the resonance in such a configuration. For this term to
+be observed, a traveling wave configuration should be
+used. Such configurations are not unknown in EPR; sev-
+eral reported home-built EPR spectrometers have used
+one-pass transmission configurations [32] [33]. Sensitiv-
+ity for such a travelling wave configuration can be en-
+hanced by means of a Mach-Zehnder interferometer [34]
+or a unidirectional ring resonator [35]. In such a configu-
+FIG. 2:
+Schematic setup of the TWEEPR interferometer.
+The waves counterpropagating through the sample S are de-
+picted in red and blue.
+ration, MChA can be obtained as the difference between
+the microwave transmissions for the two opposing mag-
+netic field directions, similar to what was realized in the
+optical case [11].
+As the EPR lines can be quite nar-
+row, the two oppositely oriented magnetic fields should
+have the same magnitude with high precision, which re-
+quires a tight control of this field, possibly with another
+EPR or NMR feedback circuit. Stabilizing a field this
+way can be quite time-consuming, and TWEEPR being
+a small difference on the already small EPR absorption,
+the extensive signal-averaging through field alternations
+that would be required to obtain a good signal-to-noise-
+ratio, makes such an approach impractical. We there-
+fore propose another approach in the form of an X band
+microwave interferometer that removes the normal EPR
+contribution from the output signal, through destructive
+interference between counter-propagating waves through
+the sample at a fixed magnetic field, as illustrated in
+Figure 2.
+This leaves ideally only the TWEEPR con-
+tribution. By applying an additional small modulation
+field and using phase sensitive detection (PSD) sufficient
+sensitivity is obtained to resolve this small contribution.
+When tuned to total destructive interference at zero field,
+the interferometer output as given by the PSD is propor-
+tional to the TWEEPR response d[T(B0 ↿↾ k)−T(B0 ⇃↾
+k)]/dB0 = γD/L(ω). The sensitivity of the interferome-
+ter can be further improved by inserting the sample in a
+unidirectional resonant ring resonator. Q factors above
+103 have been reported for such configurations [36] and
+would bring a corresponding increase in sensitivity. It
+seems therefore quite feasible that TWEEPR can evolve
+into a standard characterization technique in the form of
+standalone dedicated TWEEPR spectrometers. An al-
+ternative to this configuration could be the microwave
+equivalent of the first observation of optical MChA in
+luminescence [9], using pulsed EPR echo techniques [1]
+with a similar interferometer setup.
+Discussion
+In general, the non-local response of a chiral system of
+size a to an electromagnetic wave with wave vector k is
+of the order ka, so one could have expected gD/L
+T
+/gD/L
+O
+to be of the order ℏΩ/∆0, the relevant spatial length
+scale for both TWEEPR and optical MChD being the
+orbital size.
+This ratio is of the order of 10−4, which
+would have put TWEEPR beyond experimental reach.
+However, in contrast to the optical absorption, which to
+zeroth order is independent of the magnetic field, the
+normal EPR absorption scales with the magnetization of
+the spin system. Since the MChA corrections are propor-
+tional to the magnetization in both EPR and the optical
+case, the cancellation of the factor cos θ ≪ 1 applies to
+gD/L
+T
+only, and it appears thereby in the denominator of
+gD/L
+T
+/gD/L
+O
+, resulting in Eq.(10). For room temperature
+X-band EPR of Cu(II), this results in gD/L
+T
+/gD/L
+O
+of the
+order of 10−1, which makes TWEEPR experimentally
+feasible under those conditions. As a consequence, and
+in contrast to many other magnetic resonance techniques,
+going to low temperatures is not necessarily favorable for
+TWEEPR. Going to higher magnetic field does not af-
+fect gD/L
+T
+/gD/L
+O
+, the increase in Ω being compensated by
+the concomitant increase of cos θ because of the higher
+resonance field.
+The main results of our model are an analytic expres-
+sion for the TWEEPR anisotropy factor [Eq.(8)] and an
+expression for its relationship with the optical anisotropy
+absorption coefficient [Eq.(10)]. The expression in Eq.(8)
+shows that gD/L
+T
+has a linear dependence on the magnetic
+field strength (through Ω) and on the chirality (through
+CD/L), as predicted by symmetry arguments. The de-
+pendence on the spin-orbit coupling does not appear ex-
+plicitly, because we have considered the case for Cu(II),
+where the level splitting δ is much smaller than the SO
+coupling λ. In the inverse case, gD/L
+T
+would be propor-
+tional to λ instead. Adapting the calculation to other chi-
+
+Det
+sig
+atn
+mod
+PSD
+moo
+ref
+vout
+uw5
+ral transition metal complexes is conceptually straight-
+forward and should result in an expression similar to
+Eq.(8), apart from numerical factors of order unity. A
+rather different case is represented by chiral organic rad-
+icals, where the unpaired electron is delocalized on one
+or more interatomic bonds and a different microscopic
+model should be used for the calculation of gD/L
+T
+. One
+might however expect that such differences apply also to
+the calculation of gD/L
+O
+for such radicals, preserving a
+relationship similar to that in Eq.(10).
+Acknowledgements
+This work was supported by the Agence Nationale de
+la Recherche (SECRETS, (ANR PRC 20-CE06-0023-
+01) and the Laboratory of Excellence NanoX (ANR-17-
+EURE-0009)). We gratefully acknowledge helpful discus-
+sions with Anne-Laure Barra.
+In the Appendices we describe the theoretical model used in our calculations, we offer explicit expressions for the
+transition probabilities that enter the anisotropy factors in EPR and optical MChD, and comment on the limitations
+of our model.
+Appendix A: Fundamentals of the model
+As outlined in the article, in order to estimate the MChA factors of a chiral Cu(II) complex, we consider a variant
+of the one-electron model proposed by Condon for the study of natural optical activity in chiral compounds [26, 27].
+The total Hamiltonian of our model is H = H0 + V D/L
+C
++ VSO, where H0 =
+p2
+2me + meω2
+0r2
+2
++ VZ is the unperturbed
+Hamiltonian, with VZ = −µB(L + gS) · B0 being the Zeeman potential; and V D/L
+C
+= CD/Lxyz, VSO = λL · S being
+the chiral potential and the spin-orbit coupling, respectively. We stick to the nomenclature used in the article. The
+chiral Hamiltonian, V D/L
+C
+, results from the electrostatic interaction of the ion with the chiral configuration of the
+ligands in the complex, and produces the necessary parity asymmetry which is at the origin of natural optical activity.
+The orbital contribution of the Zeeman potential was added in Ref.[28] to the original Condon’s model to estimate
+the magneto-chiral birefringence of diamagnetic chiral compounds. In order to account for magnetochiral dichroism
+(MChD) in a paramagnetic complex, we introduce here the spin contribution to the Zeeman potential as well as the
+spin-orbit coupling. In contrast to the approach in Ref.[28] and for simplicity, we consider an isotropic harmonic
+oscillator, whereas the anisotropy caused by the crystal field is introduced in an effective manner through the energy
+intervals between the 3d orbitals, as depicted in Fig.1 in the article.
+The eigenstates of H0 are labeled with the eigenvalues of the orbital angular momentum and spin operators,
+{|nL, nR, nz⟩} ⊗ {↑, ↓} [29], upon which V D/L
+C
+and VSO act perturbatively. In a Cu(II) complex, the chromophoric
+charge is the unpaired electron of the 3d9 electronic configuration which behaves as a hole of positive charge. In the
+absence of ligands, the 3d orbitals of the ion can be represented approximately by the n = 2, l = 2 states of the
+harmonic oscillator of our model. However, the ligands’ fields affect the electronic configuration of the ion, removing
+the degeneracy of the d-states. In particular, for octahedral coordination geometries around the ion, the set of d-
+orbitals splits into doubly degenerate eg orbitals, dx2−y2 and dz2, and triply degenerate t2g orbitals, dxy, dyz and
+dzx. The energy interval between eg and t2g states, ∆0, lies in the visible region of the spectrum, ∆0 ≃ 1.5 eV. As a
+result, the eg orbitals become the ground states, and can be approximated by linear combinations of l = 2, ml = 0, ±2
+eigenstates of the harmonic oscillator. The fact that the chromophoric charge in the eg states cannot rotate into any
+other orbital leads to an effective quenching of the orbital angular momentum of the ground state. Below a certain
+temperature, an additional Jahn-Teller (JT) distortion takes place when the ligands along one of the axes, say the
+z-axis, move away from the ion in order to minimize the electronic repulsion, giving rise to the complete removal of
+the degeneracy in the eg level, and to a partial lifting of the degeneracy in the t2g orbitals. The isotropy of the system
+is thus broken and the ground state becomes unique, up to spin degeneracy. For the particular case of the CsCuCl3
+crystal, the bonds along the z-axis get elongated and the ground state is the dx2−y2 orbital. Fig.1 in the article depicts
+the energy splitting of the distorted d-orbitals, including the approximate values of the energy intervals. Lastly, the
+JT distortion in conjuntion with the helical deformation of the crystal along the c-axis, of coordiates [1,1,1] in the
+local axis basis, removes the degeneracy between the orbitals lying on the xy plane in a small ammount δ. Below,
+we write the approximate expression of the 3d orbitals in terms of the harmonic oscillator eigenstates, {|nL, nR, nz⟩},
+
+6
+together with their corresponding energies,
+|dzx⟩ = (|0, 1, 1⟩ − |1, 0, 1⟩)/
+√
+2,
+E = ∆0,
+|dyz⟩ = i(|0, 1, 1⟩ + |1, 0, 1⟩)/
+√
+2,
+E = ∆0 − δ,
+|dxy⟩ = i(|0, 2, 0⟩ − |2, 0, 0⟩)/
+√
+2,
+E = ∆0 − ∆2,
+|dz2⟩ = (|1, 1, 0⟩ −
+√
+2|0, 0, 2⟩)/
+√
+3,
+E = ∆0 − ∆1,
+|dx2−y2⟩ = (|0, 2, 0⟩ + |2, 0, 0⟩)/
+√
+2,
+E = 0.
+(A1)
+Altogether, the crystal field combined with the JT distortion and the helical deformation turns the crystalline structure
+into a chiral one. In accord with Condon’s model, the potential V D/L
+C
+reproduces the electrostatic interaction of the
+chromophoric charge with the surrounding chiral structure, removing all axes and planes of symmetry from the system.
+It is through the chiral potential that E1 transitions between the 3d orbitals take place in our model. In addition
+to the above interactions, MChD in EPR requires necessarily the coupling between the spin and the orbital angular
+momentum of the unpaired electron hole through the potential VSO, where the coupling constant is λ ≈ −0.1 eV. In
+particular, the SO interaction together with the Zeeman potential break the quasi-degeneracy between the four states
+{|dzx⟩, |dyz⟩} ⊗ {↑, ↓}, providing the following eigenstates for λ ≫ δ,
+|Φ1⟩ ≈ |1, 0, 1⟩⊗ ↓ + δ
+2λ|0, 1, 1⟩⊗ ↓,
+E ≃ ∆0 − λ/2 + ℏΩ,
+|Φ2⟩ ≈ |0, 1, 1⟩⊗ ↑ + δ
+2λ|1, 0, 1⟩⊗ ↑,
+E ≃ ∆0 − λ/2 − ℏΩ,
+|Φ3⟩ ≈ |0, 1, 1⟩⊗ ↓ − δ
+2λ|1, 0, 1⟩⊗ ↓,
+E ≃ ∆0 + λ/2 + ℏΩ + δ2
+4λ2 (λ + ℏΩ),
+|Φ4⟩ ≈ |1, 0, 1⟩⊗ ↑ − δ
+2λ|0, 1, 1⟩⊗ ↑,
+E ≃ ∆0 + λ/2 − ℏΩ + δ2
+4λ2 (λ − ℏΩ).
+(A2)
+{Φ1, Φ2, Φ3, Φ4} are indeed the eigenstates of the Hamiltonian VZ+VSO restricted to the subspace {|dzx⟩, |dyz⟩}⊗{↑, ↓}.
+They constitute the intermediate states of the transition processes in EPR mediated by the interaction of the spin
+with the chiral structure of the surrounding charges.
+In the following, we apply to our system time-dependent quantum perturbation techniques to compute first the
+MChA factor in EPR, gD/L
+T
+. Next, in order to estimate the value of the unknowns of our model, we compute the
+anisotropy factor in optical MChD for the same system. Finally, making use of the experimental values available for
+CsCuCl3 in the literature [24, 25], we estimate the strength of TWEEPR.
+Appendix B: MChD in EPR
+Let us consider a CsCuCl3 complex, initially prepared in its ground state, and partially polarized along a uniform
+magnetic field B = B0ˆz directed along the z-axis,
+|Ψ⟩ = |dx2−y2⟩ ⊗ (cos θ/2 ↑ + sin θ/2 ↓) ≈
+1
+√
+2(|0, 2, 0⟩ + |2, 0, 0⟩) ⊗ (cos θ/2 ↑ + sin θ/2 ↓),
+(B1)
+where we have approximated the actual ground state with the corresponding state of our harmonic oscillator model
+in the basis {|nL, nR, nz⟩} ⊗ {↑, ↓}, and θ is the angle between the magnetic moment of the complex and the z-axis,
+cos θ = ℏ−1⟨Ψ|2S|Ψ⟩ · ˆz. At temperature T, cos θ ≈ µ0B0/kBT [31]. Under the action of an incident electromagnetic
+field of frequency ω close to the transition frequency, Ω = gµBB0/ℏ, and wave vector k parallel to B0, the complex
+gets partially excited towards the state
+|Φ⟩ = |dx2−y2⟩⊗ ↓≈
+1
+√
+2(|0, 2, 0⟩ + |2, 0, 0⟩)⊗ ↓,
+(B2)
+
+7
+with probability proportional to cos2 θ/2; and partially de-excited (through stimulated emission) towards the state
+|Φ′⟩ = |dx2−y2⟩⊗ ↑≈
+1
+√
+2(|0, 2, 0⟩ + |2, 0, 0⟩)⊗ ↑,
+(B3)
+with probability proportional to sin2 θ/2. Since the rest of probability factors are equivalent, the net absorption
+probability in EPR is proportional to cos2 θ/2 − sin2 θ/2 = cos θ, and thus proportional to the magnetization of the
+complex.
+As mentioned in the article, from symmetry considerations and in leading order, the numerator and the denom-
+inator in the ratio gD/L
+T
+= [P D/L(ω, ˆk, B0) − P D/L(ω, ˆk, −B0)]/[P D/L(ω, ˆk, B0) + P D/L(ω, ˆk, −B0)] for ω ≈ Ω are
+dominated, respectively, by the electric-magnetic dipole (E1M1) and the magnetic-magnetic dipole (M1M1) transi-
+tion probabilities, the magnetic transition being driven by the spin operator only. That leads to the approximate
+expression,
+gD/L
+T
+≃ P D/L
+E1M1(ω, ˆk, B0)
+PM1M1(ω, ˆk, B0)
+���
+ω≈Ω.
+(B4)
+In what follows, we compute the transition probabilities PM1M1 and P D/L
+E1M1 for ω ≈ Ω using time-dependent
+perturbation theory in the adiabatic regime. This regime is the suitable one for a probe field whose duration is
+much longer than the typical lifetime for excitation or de-excitation.
+As in the article, the Hamiltonian of the
+interaction of our system with the microwave probe field reads, in the electric and magnetic dipole approximation,
+W = −er·Eω(t)/2−µB(L+2S)·Bω(t)/2+h.c.. In this equation, Eω(t) = Eωe−iωt = iωAωe−iωt, Bω(t) = Bωe−iωt =
+i¯nk ∧ Aωe−iωt, are the complex-valued electric and magnetic fields, respectively, with Aω being the complex-valued
+amplitude of the plane-wave electromagnetic vector potential of frequency ω ≈ Ω, evaluated at the center of mass of
+the Cu(II) ion, and ¯n being the effective refractive index of the sample. The local depolarization changes the local
+electric field incident on each Cu(II) ion to Eω(¯n2 + 2)/3. Under the action of W, with k along B0, the expressions
+for PM1M1 and P D/L
+E1M1 read, respectively, at leading order in the coupling constants of the interaction potentials,
+PM1M1|ω≈Ω = ℏ−2
+�����
+� T
+0
+dte−i(T −t)(Ω/2−iΓ/2)e−it(ω−Ω/2)⟨Φ| − gµBS · Bω|Ψ⟩
+�����
+2
+− ℏ−2
+�����
+� T
+0
+dte−i(T −t)(2ω−Ω/2−iΓ/2)e−it(ω+Ω/2)⟨Φ′| − gµBS · Bω|Ψ⟩
+�����
+2
+,
+(B5)
+
+8
+P D/L
+E1M1|ω≈Ω = 2Re(−i)3ℏ−4 �
+p,q̸=Ψ
+� T
+0
+dte−i(T −t)(Ω/2−iΓ/2)⟨Φ| − er · (¯n2 + 2)Eω/3|p⟩
+� t
+−∞
+dt′eηt′e−i(t−t′)(Ep+ω)
+× ⟨p|V D/L
+C
+|q⟩
+� t′
+−∞
+dt′′eηt′′e−i(t′−t′′)(Eq+ω)⟨q|VSO|Ψ⟩e−it′′(ω−Ω/2)i
+� T
+0
+dτ ei(T −τ)(Ω/2+iΓ/2)
+× ⟨Ψ| − gµBS · B∗
+ω|Φ⟩eiτ(ω−Ω/2) + 2Re(−i)3ℏ−4 �
+p,q̸=Φ
+� T
+−∞
+dt eηte−i(T −t)(Ω/2−iΓ/2)⟨Φ|VSO|p⟩
+×
+� t
+−∞
+dt′eηt′e−i(t−t′)Ep⟨p|V D/L
+C
+|q⟩
+� t′
+0
+dt′′e−i(t′−t′′)Eq⟨q| − er · (¯n2 + 2)Eω/3|Ψ⟩e−it′′(ω−Ω/2)
+× i
+� T
+0
+dτ ei(T −τ)(Ω/2+iΓ/2)⟨Ψ| − gµBS · B∗
+ω|Φ⟩eiτ(ω−Ω/2)
+− 2Re(−i)3ℏ−4 �
+p,q̸=Ψ
+� T
+0
+dt e−i(T −t)(2ω−Ω/2)⟨Φ′| − er · (¯n2 + 2)Eω/3|p⟩
+� t
+−∞
+dt′eηt′e−i(t−t′)(Ep+ω)
+× ⟨p|V D/L
+C
+|q⟩
+� t′
+−∞
+dt′′eηt′′e−i(t′−t′′)(Eq+ω)⟨q|VSO|Ψ⟩e−it′′(ω+Ω/2−iΓ/2)i
+� T
+0
+dτ ei(T −τ)(2ω−Ω/2)
+× ⟨Ψ| − gµBS · B∗
+ω|Φ′⟩eiτ(ω+Ω/2+iΓ/2) − 2Re(−i)3ℏ−4 �
+p,q̸=Φ′
+� T
+−∞
+dt eηte−i(T −t)(2ω−Ω/2)
+× ⟨Φ′|VSO|p⟩
+� t
+−∞
+dt′eηt′e−i(t−t′)(2ω+Ep)⟨p|V D/L
+C
+|q⟩
+� t′
+0
+dt′′e−i(t′−t′′)(2ω+Eq)
+× ⟨q| − er · (¯n2 + 2)Eω/3|Ψ⟩e−it′′(ω+Ω/2−iΓ/2)i
+� T
+0
+dτ ei(T −τ)(2ω−Ω/2)⟨Ψ| − gµBS · B∗
+ω|Φ′⟩
+× eiτ(ω+Ω/2+iΓ/2),
+η → 0+, ΓT ≫ 1.
+(B6)
+In these equations the states p and q stand for the excited states of the 3d9 configuration together with other eigenstates
+of H0 with n ̸= 2. The quasi-stationary condition η → 0+ accounts for the stationarity of the chiral and the spin-orbit
+interactions; whereas the adiabatic limit ΓT ≫ 1 takes into account the long duration of the probe field with respect
+to the lifetime Γ−1, with Γ being the linewidth of absorption and T the observation time.
+The diagrammatical
+representation of the processes involved in the above equation is given in Fig.3. In the article, the contributions of
+the quasi-stationary processes were incorporated into the dressed states ˜Ψ, ˜Φ, ˜Φ′. More specifically, the bare states
+are dressed with the quadruplet {Φ1, .., Φ4} through VSO, and with harmonic states with n ̸= 2 by VC. In terms of
+the eigenstates of the harmonic oscillator, they read
+|˜Φ⟩ =
+�
+(|020⟩ + |200⟩)/
+√
+2 +
+λ
+√
+2∆0
+(1 + ∆2/∆0)(|020⟩ − |200⟩) + iλCD/LK3/2
+2ℏω0∆0
+(1 + ∆2/∆0)
+× (|001⟩ − 2|111⟩)
+�
+↓ +
+�
+λ
+√
+2∆0
+(1 + 3ℏΩ/∆0)|011⟩ +
+δ
+√
+2∆2
+0
+(ℏΩ + λ/2)|101⟩
++ −iCD/LK3/2λ
+2ℏω0∆0
+(1 + 3ℏΩ/∆0)(|210⟩ −
+√
+3|030⟩ −
+√
+2|100⟩) + iCD/LK3/2δ
+2ℏω0∆2
+0
+× (ℏΩ + λ/2)(|120⟩ −
+√
+3|300⟩ −
+√
+2|010⟩)
+�
+↑
+|˜Φ′⟩ =
+�
+(|020⟩ + |200⟩)/
+√
+2 +
+−λ
+√
+2∆0
+(1 + ∆2/∆0)(|020⟩ − |200⟩) + −iλCD/LK3/2
+2ℏω0∆0
+(1 + ∆2/∆0)
+× (|001⟩ − 2|111⟩)
+�
+↑ +
+� −λ
+√
+2∆0
+(1 − 3ℏΩ/∆0)|101⟩ +
+δ
+√
+2∆2
+0
+(ℏΩ − λ/2)|011⟩
++ −iCD/LK3/2λ
+2ℏω0∆0
+(1 − 3ℏΩ/∆0)(|120⟩ −
+√
+3|300⟩ −
+√
+2|010⟩) + −iCD/LK3/2δ
+2ℏω0∆2
+0
+× (ℏΩ − λ/2)(|210⟩ −
+√
+3|030⟩ −
+√
+2|100⟩)
+�
+↓
+|˜Ψ⟩ = cos θ/2|˜Φ′⟩ + sin θ/2|˜Φ⟩,
+K = ℏ/(2meω0).
+(B7)
+
+9
+FIG. 3: Diagrammatic representation of the processes which contribute to PM1M1 and P D/L
+E1M1 for ω ≈ Ω at leading order in
+the perturbative interactions, i.e., at second order and fourth order, respectively. Time runs along the vertical direction from
+0 to the observation time T , where the probability is computed. Intermediate atomic states are labeled as p and q. Diagrams
+with two-photon states account for stimulated emission.
+Using a linearly polarized incident field and averaging in orientations around the ˆz-axis, we obtain, for λ ≫ δ,
+PM1M1|ω≈Ω ≃
+ℏ−2µ2
+B|Bω|2
+4[(ω − Ω)2 + Γ2/4] cos θ,
+(B8)
+P D/L
+E1M1|ω≈Ω ≃ (¯n2 + 2)
+3
+CD/LΩδ
+meω3
+0∆2
+0
+ℏ−1µ2
+B|Bω||Eω|
+4[(ω − Ω)2 + Γ2/4] cos θ,
+(B9)
+gD/L
+T
+≃ (¯n2 + 2)
+3¯n
+c CD/LℏΩδ
+meω3
+0∆2
+0
++ O(δ/λ, λ/∆0).
+(B10)
+Lastly, it is worth mentioning that for the case δ > λ, i.e., when anisotropy dominates over the spin-orbit coupling,
+gD/L
+T
+scales as (cℏCD/LΩδλ)/(meω3
+0∆3
+0) instead. This scenario will be addressed in a separate publication [30].
+Appendix C: Optical MChD
+Optical MChD involves transitions of frequency ∆0 from the ground state |Ψ⟩ to the quasi-degenerate quadruplet
+{|dzx⟩, |dyz⟩} ⊗ {↑, ↓} which, in account of the Zeeman and spin-orbit interactions, for δ ≪ λ, corresponds to the
+set of states {Φ1, ..., Φ4} of Eq.(A2). In contrast to EPR, the absorption probability in the denominator of the ratio
+gD/L
+O
+= [P D/L(ω, ˆk, B0)−P D/L(ω, ˆk, −B0)]/[P D/L(ω, ˆk, B0)+P D/L(ω, ˆk, −B0)] for ω ≈ ∆0/ℏ may not be dominated
+by the magnetic-magnetic dipole absorption probability. This might be so because the d-orbitals of the Cu(II) ion
+
+0
+-er:(n*+2)Eo/3
+P
+D
+-gsS·Ba
+M1M1
+Vso
+tlo
+PDIL
+2 Re
+D
+E1M1
+D
+p
+910
+hybridize generally with the σ and π orbitals of the ligands, allowing for additional electric-electric dipole (E1E1)
+transitions. For the sake of simplicity, we will neglect the latter in our calculations, which implies that our preliminar
+estimate for gD/L
+O
+must be intended as an approximate upper bound. As for the case of EPR, the numerator of
+the ratio in gD/L
+O
+is again dominated by the electric-magnetic dipole absorption probability, and the non-vanishing
+terms come from magnetic transitions driven by the spin angular momentum –Eq.(C2) below. However, in contrast
+to EPR, the magnetic transitions in the denominator are mainly driven by the orbital angular momentum operator
+–see Eq.(C1) below. In turn, this causes the E1M1 transition probability to depend on the spin polarization of the
+complex, whereas neither the M1M1 nor the E1E1 probabilities do. Note also that stimulated emission from the state
+|Ψ⟩ is absent in optical MChD. All in all, this implies that gD/L
+O
+is proportional to the magnetization of the sample,
+which is itself proportional to the degree of spin-polarization along B0, cos θ, in agreement with experiments. In Fig.4
+FIG. 4: Diagrammatic representation of PM1M1 and P D/L
+E1M1 for ω ≈ ∆0/ℏ at leading order in the perturbative interactions,
+i.e., at second and up to fifth order, respectively. Intermediate atomic states are labeled as p, q, r, s.
+we depict some of the diagrams which contribute to PM1M1 and P D/L
+E1M1 in optical MChD. Following a perturbative
+approach analogous to that in EPR, for an incident electromagnetic plane wave with k ∥ B0 and assuming δ ≪ λ,
+one arrives at
+PM1M1|ω≈∆0/ℏ ≃
+ℏ−2µ2
+B|Bω|2
+4[(ω − ∆0/ℏ)2 + Γ
+′2/4],
+(C1)
+P D/L
+E1M1|ω≈∆0/ℏ ≃ (¯n2 + 2)
+3
+CD/Lδ
+2meω3
+0 ˜∆
+ℏ−2µ2
+B|Bω||Eω|
+4[(ω − ∆0/ℏ)2 + Γ
+′2/4] cos θ,
+(C2)
+gD/L
+O
+≲ P D/L,O
+E1M1
+P O
+M1M1
+���
+ω≈∆0/ℏ ≃ (¯n2 + 2)
+3¯n
+c CD/Lδ cos θ
+2 meω3
+0 ˜∆
+,
+(C3)
+
+-er:(n+2) Eo/3
+P
+Di-
+wo -μ(L+gS), Ba
+M1M1
+Vso
+Vc
+tlo
+Di --
+PDIL
+= 2 Re
+4
+E1M1
+Di -.
++
+r
+r
+011
+01111
+where ˜∆−1 = ∆−1
+0
++ ∆−1
+2
+− 3∆−1
+1 , and Γ′ is the linewidth of optical absorption. As anticipated, the fact that the
+magnetic dipole transition in P D/L
+E1M1 is dominated by the orbital angular momentum operator causes its leading
+order term to depend on the magnetization ∼ cos θ. Hence, time-reversal invariance happens to be broken by the
+spin-polarization of the complex.
+Appendix D: Estimate of gD/L
+T
+In the first place, we work out the relationship between gD/L
+T
+and gD/L
+O
+. Comparing Eq.(B9) with Eq.(C2) at
+resonance, and taking into account Eqs.(B10) and (C3), we arrive at the following relationships,
+P D/L
+E1M1|ω=Ω
+P D/L
+E1M1|ω=∆0/ℏ
+≃ 2ℏΩ ˜∆Γ
+′2
+∆2
+0Γ2
+,
+gD/L
+T
+gD/L
+O
+≳
+2ℏΩ ˜∆
+∆2
+0 cos θ.
+(D1)
+Next, considering the experimental data obtained in Ref.[24] for gD/L
+O
+and applying the relationship in Eq.(D1), we
+can estimate a lower bound for gD/L
+T
+. That is, substituting into Eq.(D1) the experimental values gD/L
+O
+≈ 0.025,
+cos θ ≈ 0.4, for B0 = 14T at a temperature of 4.2 K, we obtain gD/L
+T
+≳ 10−4.
+Alternatively, we can estimate gD/L
+T
+using the experimental data of Ref.[24] for the non-reciprocal absorption
+coefficient of optical MChD, αA = α(B0 ↿↾ k) − α(B0 ⇃↾ k). In order to do so, we first write down αA as a function
+of P D/L,O
+E1M1 at resonance,
+αA = 4cµ0ρΓ′∆0
+|Eω|2
+P D/L
+E1M1|ω=∆0/ℏ,
+(D2)
+where ρ is the molecular density of the CsCuCl3 complex (mass density 3.5g/cm3). Substituting the expression for
+P D/L,O
+E1M1 (ω = ∆0/ℏ) in the above equation and using Eq.(B10) we arrive at the equalities,
+CD/Lδ =
+3ℏ2meω3
+0 ˜∆Γ′αA
+2(¯n2 + 2)ρµ0µ2
+B∆0 cos θ,
+gD/L
+T
+=
+c ℏ3Γ′Ω ˜∆αA
+2∆3
+0µ0µ2
+Bρ cos θ.
+(D3)
+Substituting the experimental values for all the variables in Eq.(D3), for B0 = 14 T at a temperature of 4.2 K, with
+Γ′ ≈ 0.1eV and ¯n ≈ 1.5, we obtain gD/L
+T
+≈ 1.5 · 10−2, in agreement with our previous lower bound estimate.
+Appendix E: Further comments on the Hamiltonian model
+Despite the success of our model to derive analytical estimates for the MChA factors, there is still room for
+improvement. In the first place, concerning the chiral Hamiltonian VC, it was written in terms of the local axis of the
+octahedral structure, x, y, z, while it should be adapted to the crystal axis to account for the helical distribution of the
+active ions along the c-axis. In fact, the experimental data on αA taken from the literature to estimate gD/L
+T
+consider
+B0 along the c-axis. Also, the harmonic oscillator model, which is considered only distorted in the n = 2, l = 2 level,
+may not be accurate enough to account for the intermediate transitions induced by the chiral potential to levels with
+n ̸= 2. Hence, a more accurate confining potential model, though less generic, can be obtained using a more detailed
+formulation of the crystal field and the JT distortion for the particular case of CsCuCl3–see, eg., Ref.[37]. Finally,
+our estimate of the unknown combination CD/Lδ in terms of αA [Eq.(D3)], involves ¯n-dependent factors [Eq.(B10],
+which account for effective incident fields, as well as ρ-dependent factors. For high densities and ¯n ≈ 1.5 those factors
+are likely to depend on near field terms and spatial correlations when evaluated at the absorption frequency [38].
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+Phys. Rev. Lett. 113, 235501 (2014). Y. Okamura, F. Ka-
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+[13] M. Ceol´ın, S. Goberna-Ferr´on and J. R. Gal´an-Mascar´os,
+Adv. Mat. 2012, DOI: 10.1002/adma.201200786, R. Ses-
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+F.Wilhelm and A. Rogalev, Nat. Phys. 11, 69 (2015).
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+Lett. 87, 236602 (2001).
+[15] V. Krsti´c, S. Roth, M. Burghard, K. Kern and G.L.J.A.
+Rikken, J. Chem. Phys. 117, 11315 (2002).
+[16] F. Pop, P. Auban-Senzier, E. Canadell, G. L. J. A.
+Rikken and N. Avarvari, Nat. Comm. 5, 3757 (2014).
+[17] T. Yokouchi, N. Kanazawa, A. Kikkawa, D. Morikawa, K.
+Shibata, T. Arima, Y. Taguchi, F. Kagawa, Y. Tokura,
+Nat. Comm. 8, 866 (2017).
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+Mattmann, S. Pan´e, B. J. Nelson, and P. Gambardella,
+Appl. Phys. Lett. 112, 242401 (2018).
+[19] R. Aoki,1 Y. Kousaka and Y. Togawa, Phys. Rev. Lett.
+122, 057206 (2019).
+[20] G.L.J.A. Rikken and N. Avarvari, Phys. Rev. B (2019).
+[21] T. Nomura, X.-X. Zhang, S. Zherlitsyn, J. Wosnitza, Y.
+Tokura, N. Nagaosa, and S. Seki, Phys. Rev. Lett. 122,
+145901 (2019).
+[22] G.L.J.A. Rikken and N. Avarvari, Nat. Comm. 13, 3564
+(2022).
+[23] W. Roy Mason, A practical guide to magnetic circular
+dichroism, Wiley 2008.
+[24] N. Nakagawa et al, Phys. Rev. B 96, 121102(R) (2017).
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+Japan 61, 1344 (1992).
+[26] E.U. Condon, Rev. Mod. Phys. 9, 432 (1937).
+[27] E. U. Condon, William Altar, and Henry Eyring, J.
+Chem. Phys. 5, 753 (1937).
+[28] M. Donaire, G. L.J.A. Rikken, and B. A. van Tiggelen,
+Eur. Phys. J. D 68, 33 (2014).
+[29] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Me-
+chanics, Wiley-VCH (1992).
+[30] M. Donaire and G. L.J.A. Rikken, in preparation.
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+Lett. 115, 267207 (2015); A. Sera, Y. Kousaka, J. Akim-
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+
diff --git a/Z9E3T4oBgHgl3EQf2gtA/content/tmp_files/load_file.txt b/Z9E3T4oBgHgl3EQf2gtA/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0dc7f250f7cdcea358cf9000333303bbffdc13ff
--- /dev/null
+++ b/Z9E3T4oBgHgl3EQf2gtA/content/tmp_files/load_file.txt
@@ -0,0 +1,783 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf,len=782
+page_content='Traveling wave enantio-selective electron paramagnetic resonance  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Donaire,1, ∗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Bruyant,2 and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rikken2, †  1Departamento de F´ısica Te´orica, At´omica y ´Optica and IMUVA,  Universidad de Valladolid, Paseo Bel´en 7, 47011 Valladolid, Spain  2Laboratoire National des Champs Magn´etiques Intenses UPR3228  CNRS/EMFL/INSA/UGA/UPS, Toulouse & Grenoble,France  (Dated: January 13, 2023)  We propose a novel method for enantio-selective electron paramagnetic resonance spectroscopy  based on magneto-chiral anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' We calculate the strength of this effect and propose a dedicated  interferometer setup for its observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Introduction  Electron paramagnetic resonance (EPR) spectroscopy is  a powerful technique to study the local environment and  the dynamics of spin-carrying entities, like transition  metal ion complexes and organic radicals [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Also, those  systems that do not intrinsically carry a spin can still  be studied by EPR through spin-labelling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', by se-  lectively adding-on a spin carrying probe [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Many of  the systems studied by EPR are chiral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', they exist  in two non-superimposable forms (enantiomers) that are  each other’s mirror image, particularly in biochemistry  where enzymes, metalloproteins, membranes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', are  chiral subjects of intense EPR activity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' However, EPR  is universally believed to be blind to chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Here we  present the paradigm shift that EPR in the proper con-  figuration is intrinsically sensitive to chirality because of  magneto-chiral anisotropy (MChA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  MChA corresponds to an entire class of effects in chiral  media under an external magnetic field, which show an  enantio-selective difference in the propagation of any un-  polarized flux that propagates parallel or anti-parallel to  the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This difference has its origin in the si-  multaneous breaking of parity and time-reversal symme-  tries as a result of the chirality of the media and the mag-  netization induced by the external magnetic field, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Generally, such a difference manifests itself in the  velocity or the attenuation of the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' MChA has been  predicted since 1962 in the optical properties of chiral  systems in magnetic fields [4–8], and was finally observed  in the 1990’s [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Nowadays it is observed across the  entire electromagnetic spectrum, from microwaves [12] to  X-rays [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The existence of MChA was further gener-  alized to electrical transport [14] (in carbon nano tubes  [15], organic conductors [16], metals [17–19] and semicon-  ductors [20]), to sound propagation [21] and to dielectric  properties [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  EPR is basically a strongly resonant form of magnetic  circular dichroism and magnetic circular birefringence  [23], effects well known in the optical wavelength range,  where they however only represent small perturbations  ∗Electronic address: manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='donaire@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='es  †Electronic address: geert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='rikken@lncmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='fr  of the optical properties of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' By analogy, one  should expect that MChA can manifest itself also in EPR  of chiral media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This expectation can be formalized by  the observation that the EPR transition probability P  induced by a propagating electromagnetic field between  the spin levels of a chiral medium in a magnetic field,  is allowed by parity and time-reversal symmetry to have  the form  P D/L(ω, ˆk, B0) = P0(ω, B0)[1 + γD/L(ω)ˆk · B0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (1)  In this equation, B0 is an external and constant magnetic  field, P0 is the leading order transition probability be-  tween the Zeeman levels, common to both enantiomers,  the handedness of the medium is represented by D− right  and L− left, with γD = −γL, and ˆk is a unitary vector  in the direction of the wave vector of the electromagnetic  field driving the transition whose frequency ω is of the or-  der of µBB0/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This shows that the EPR transition prob-  ability is enantioselectively modified when probed by an  electromagnetic wave travelling parallel or anti-parallel  to the magnetic field, an effect that we shall call travel-  ing wave enantioselective EPR (TWEEPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' TWEEPR  is quantified by the anisotropy factor gD/L  T  , which repre-  sents the relative difference between the transition prob-  abilities of both enantiomers,  gD/L  T  ≡ [P D/L(ω, �k, B0) − P D/L(ω, �k, −B0)]  [P D/L(ω, �k, B0) + P D/L(ω, �k, −B0)]  = γD/Lˆk·B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (2)  As spin is related to the absence of time-reversal sym-  metry, and chirality is related to the absence of parity  symmetry, one might expect that the two are decou-  pled and that gD/L  T  is vanishingly small, thereby reducing  TWEEPR to an academic curiosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' However, below we  will show through a model calculation that, because of  the ubiquitous spin-orbit coupling, TWEEPR represents  a significant and measurable fraction of the EPR transi-  tion probability for realistic chiral systems and that its  anisotropy factor is not much smaller than that of optical  MChA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Lastly, we will describe a dedicated TWEEPR  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The model  As for the spin system of our model calculation of  TWEEPR, without loss of generality, we have chosen  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='04755v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='chem-ph]  11 Jan 2023  2  a crystalline quasi-octahedral Cu(II) chiral complex be-  cause this ion is one of the most extensively studied  systems by EPR, it has the largest spin-orbit coupling  among the first row transition metals, and it has the  simplest energy diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Its electromagnetic response is  attributed to a single unpaired electron that, in the 3d9  configuration of the Cu(II) complex, behaves as a hole of  positive charge +e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' We model the binding potential of  the hole by that of an isotropic harmonic oscillator that  represents the rest of the ion, and is perturbed by the chi-  ral potential V D/L  C  that results from its interaction with  the chiral environment of the crystal lattice, and by the  spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In turn, as we will show, this model  allows us to find analytic expressions for both the optical  and the EPR magnetochiral anisotropy parameters, gD/L  O  and gD/L  T  , respectively, in terms of the parameters of the  model, both being proportional to the chiral coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Our model can thus relate gD/L  T  to its optical analogue  gD/L  O  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The latter is experimentally determined for several  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In particular, for CsCuCl3 both MChD [24] and  EPR [25] have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This approach thereby re-  sults in a generic analytical expression for gD/L  T  in terms  of the parameters of our model, and in a semi-empirical  and quantitative prediction for gD/L  T  for this particular  material in terms of its experimental optical MChD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The  latter can be extended to any material for which optical  MChD has been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Below we detail our model,  which is a variant of Condon’s model for optical activity  [26, 27], and its extension to optical magnetochiral bire-  fringence [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The Hamiltonian describing the system is given by H =  H0 + V D/L  C  + VSO, with  H0 =  p2  2me  + meω2  0r2  2  − µB(L + gS) · B0,  (3)  V D/L  C  = CD/Lxyz,  VSO = λL · S,  (4)  where r = (x, y, z) and p are the position and kinetic  momentum vectors of the harmonic oscillator, ω0 is its  natural frequency, L and S are their orbital and spin an-  gular momentum operators, respectively, CD = −CL is  the right/left-handed chiral coupling, g ≃ 2 is the Land´e  factor, λ ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1 eV is the spin-orbit (SO) coupling pa-  rameter, and B0 ≡ B0ˆz is the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The interaction with an electromagnetic plane-wave of  frequency ω, propagating along B0, is given in a multi-  pole expansion by  W = −er · Eω(t)/2 − µB(L + gS) · Bω(t)/2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', (5)  where Eω(t) = iωAωe−iωt and Bω(t) = i¯nk ∧ Aωe−iωt  are the complex-valued electric and magnetic fields in  terms of the electromagnetic vector potential, Aω, eval-  uated at the center of mass of the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Note that the  field incident on a molecule of the complex is the effec-  tive field which propagates throughout the medium with  an effective index of refraction ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Hence it is the effective  wavevector ¯nk that appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 1: Energy levels of Cu(II) in a chiral quasi-octahedral  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Approximate experimental values are ∆0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5  eV, ∆1 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 eV, ∆2 ≃≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='23 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  In our model, the 3d orbitals are represented by linear  combinations of the n = 2, l = 2 states of the isotropic  harmonic oscillator –see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Essential to the  original Condon model was the anisotropy of the har-  monic oscillator, which removes all axis and planes of  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In our model, such an anisotropy is provided  by the interaction of the ion with the surrounding lig-  ands of the complex, which in the case of CsCuCl3 form  an quasi-octahedral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In the first place, that in-  teraction causes the elongation of the 3d orbitals which  lie along the z-axis, opening an optical gap ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Also, in  conjunction with the Jahn-Teller distortion and the he-  lical configuration of the Cu(II) ions, it removes the de-  generacy between the orbitals lying on the xy plane and  generates a small energy gap δ between the states dzx and  dyz, with λ ≫ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The ground state of the Cu(II) ion in  the octahedral configuration Ψ is, at finite temperature  and subject to a magnetic field, a linear combination of  the doublet dx2−y2 ⊗ {↑, ↓},  |Ψ⟩ = |dx2−y2⟩ ⊗ (cos θ/2 ↑ + sin θ/2 ↓),  (6)  where θ, being a function of B0 and the temperature,  is the angle between the magnetization of the sample  and B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For EPR, spin-flip takes place at a resonance  frequency Ω = gµBB0/ℏ when the up ↑ component of  Ψ turns into |Φ⟩ = |dx2−y2⟩⊗ ↓ , with probability pro-  portional to cos2 θ/2, and the down ↓ component turns  into |Φ′⟩ = |dx2−y2⟩⊗ ↑ with probability proportional to  sin2 θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The net absorption probability is thus propor-  tional to cos2 θ/2 − sin2 θ/2 = cos θ and hence to the  degree of magnetization along B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  At  B0 = 1T, Ω  corresponds to an energy 150 µeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In contrast, optical  3dz  3dzx  3dx-y  3dxy  3dyz  Chiralground state  zx  t2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Φ4  yz  △2  x  △1  3d9  Ao  eg  crystalfield splitting  Jahn-Tellereffect  SO,chiral, and  (octahedralsymmetry)(tetragonaldistortion)  Zeeman splitting3  absorption happens at an energy ∆0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 eV towards  the quadruplet {dzx, dyz} ⊗ {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Applying standard  perturbation theory with the spin-orbit and the Zeeman  potentials upon this quasidegenerate quadruplet , we end  up with the four states φi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='., 4, as appear in the  energy diagram represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1 –a brief description  can be found in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' It is of note that these  states play a crucial role in the E1M1 transitions of both  EPR and its optical analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Results  Using up to fourth order time-dependent perturbation  theory on VSO, VC and W, in the adiabatic regime, our  model allows us to calculate the standard EPR and op-  tical transition probabilities, as well as the MChA cor-  rections to both of them, with the latter two being both  proportional to CD/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' As for gD/L  T  , the probability dif-  ference in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (2) is an enantioselec-  tive E1M1 transition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' whereas the denominator equals in  good approximation the leading order M1M1 transition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  gD/L  T  = P D/L  E1M1/PM1M1|ω≈Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' with  PM1M1|ω≈Ω = ℏ−2���  � T  0  dte−i(T −t)(Ω/2−iΓ/2)e−it(ω−Ω/2)⟨Φ| − gµBS · Bω|Ψ⟩  ���  2  − ℏ−2���  � T  0  dte−i(T −t)(2ω−Ω/2−iΓ/2)  × e−it(ω+Ω/2)⟨Φ′| − gµBS · Bω|Ψ⟩  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  P D/L  E1M1|ω≈Ω = −2ℏ−2Re  � T  0  dte−i(T −t)(Ω/2−iΓ/2)⟨˜Φ| − er · (¯n2 + 2)Eω/3|˜Ψ⟩e−it(ω−Ω/2)  � T  0  dτ ei(T −τ)(Ω/2+iΓ/2)  × ⟨Ψ| − gµBS · B∗  ω|Φ⟩eiτ(ω−Ω/2) + 2ℏ−2Re  � T  0  dt e−i(T −t)(2ω−Ω/2)⟨˜Φ′| − er · (¯n2 + 2)Eω/3|˜Ψ⟩  × e−it(ω+Ω/2−iΓ/2)  � T  0  dτ ei(T −τ)(2ω−Ω/2)⟨Ψ| − gµBS · B∗  ω|Φ′⟩eiτ(ω+Ω/2+iΓ/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  ΓT ≫ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (7)  where Γ is the linewidth of EPR absorption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' ΓT ≫ 1  implies the adiabatic approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and the states ˜Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  ˜Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and ˜Φ′ are dressed with the states φi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='., 4, on  account of the spin-orbit and chiral interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Using  a linearly polarized microwave probe field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (7), the  resultant expression for the TWEEPR anisotropy factor  reads  gD/L  T  ≃ c CD/Lℏ Ω δ  meω3  0∆2  0  ¯n2 + 2  3¯n  ,  (8)  where the second factor on the right hand side describes  the effect of the refractive index on the local electric field  and the wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' It is worth noting that the aforemen-  tioned dependence on magnetization, ∼ cos θ, cancels out  in the ratio between probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For further details, see  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The values for the unknown parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (8) can  be deduced comparing the predictions of the model with  the experimental results for optical MChD [24] and EPR  [25] in CsCuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  In particular, we can estimate gD/L  T  from the data on the non-reciprocal absorption coeffi-  cient in optical MChD, αA = α(B0 ↿↾ k) − α(B0 ⇃↾ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The calculation goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In terms of the E1M1  absorption probability at resonance, ω = ∆0/ℏ, αA reads  αA = 4cµ0ρ∆0Γ′  |Eω|2  P D/L  E1M1|ω=∆0/ℏ,  (9)  where Γ′ is the linewidth of optical absorption, and ρ  is the molecular number density of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Using  our model, a calculation analogous to that for P D/L,EP R  E1M1  but for its optical counterpart, P D/L,O  E1M1 – Appendices B,  C and D-, allows as to express gD/L  T  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (8) in terms  of αA,  gD/L  T  =  c ℏ3Γ′Ω ˜∆αA  2∆3  0µ0µ2  Bρ cos θ,  (10)  where ˜∆−1 = ∆−1  0  + ∆−1  2  − 3∆−1  1  is the inverse of an  effective energy interval which takes account of the opti-  cal transitions to intermediate states –see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' It is of  note that, whereas the magnetic transition is driven in  EPR by the spin operator [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (7)], it is driven by the or-  bital angular momentum in the optical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In turn, this  causes MChD to be stronger in the optical case and pro-  portional to the degree of magnetization cos θ, which can  be approximated by cos θ ≈ µ0B0/kBT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The optical  MChA parameter, gD/L  0  , has an analogous expression to  that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (2) with ℏω ≈ ∆0, being proportional to αA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Hence, our model allows us to estimate its upper bound,  gD/L  0  ≤ (cCD/Lδ cos θ)/(meω3  0 ˜∆) – see Appendices C  and D, from which gD/L  T  /gD/L  0  ≳ (ℏΩ ˜∆)/(∆2  0 cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Note that, since both Ω and cos θ are proportional to  B0, the ratio between EPR and optical MChA factors is  independent of the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Finally,  substituting  the  experimental  values  for  CsCuCl3 of all the variables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (10), for B0 = 14 T  at a temperature of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2 K, we obtain gD/L  T  ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 · 10−2,  4  which is small but not beyond the resolution of high field  EPR spectrometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For an X band EPR spectrometer  (B = 0, 35 T), this means gD/L  T  ≈ 3 · 10−4 which will  require a different approach, as we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Implementation  In commercial EPR spectrometers, resonant standing  wave cavities are used to enhance sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Such a  cavity can be regarded as containing equal amounts of  traveling waves with k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The MChA γD/L term  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (1) can therefore not give a net contribution to  the resonance in such a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For this term to  be observed, a traveling wave configuration should be  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Such configurations are not unknown in EPR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' sev-  eral reported home-built EPR spectrometers have used  one-pass transmission configurations [32] [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Sensitiv-  ity for such a travelling wave configuration can be en-  hanced by means of a Mach-Zehnder interferometer [34]  or a unidirectional ring resonator [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In such a configu-  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 2:  Schematic setup of the TWEEPR interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The waves counterpropagating through the sample S are de-  picted in red and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  ration, MChA can be obtained as the difference between  the microwave transmissions for the two opposing mag-  netic field directions, similar to what was realized in the  optical case [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  As the EPR lines can be quite nar-  row, the two oppositely oriented magnetic fields should  have the same magnitude with high precision, which re-  quires a tight control of this field, possibly with another  EPR or NMR feedback circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Stabilizing a field this  way can be quite time-consuming, and TWEEPR being  a small difference on the already small EPR absorption,  the extensive signal-averaging through field alternations  that would be required to obtain a good signal-to-noise-  ratio, makes such an approach impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' We there-  fore propose another approach in the form of an X band  microwave interferometer that removes the normal EPR  contribution from the output signal, through destructive  interference between counter-propagating waves through  the sample at a fixed magnetic field, as illustrated in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  This leaves ideally only the TWEEPR con-  tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' By applying an additional small modulation  field and using phase sensitive detection (PSD) sufficient  sensitivity is obtained to resolve this small contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  When tuned to total destructive interference at zero field,  the interferometer output as given by the PSD is propor-  tional to the TWEEPR response d[T(B0 ↿↾ k)−T(B0 ⇃↾  k)]/dB0 = γD/L(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The sensitivity of the interferome-  ter can be further improved by inserting the sample in a  unidirectional resonant ring resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Q factors above  103 have been reported for such configurations [36] and  would bring a corresponding increase in sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' It  seems therefore quite feasible that TWEEPR can evolve  into a standard characterization technique in the form of  standalone dedicated TWEEPR spectrometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' An al-  ternative to this configuration could be the microwave  equivalent of the first observation of optical MChA in  luminescence [9], using pulsed EPR echo techniques [1]  with a similar interferometer setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Discussion  In general, the non-local response of a chiral system of  size a to an electromagnetic wave with wave vector k is  of the order ka, so one could have expected gD/L  T  /gD/L  O  to be of the order ℏΩ/∆0, the relevant spatial length  scale for both TWEEPR and optical MChD being the  orbital size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  This ratio is of the order of 10−4, which  would have put TWEEPR beyond experimental reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  However, in contrast to the optical absorption, which to  zeroth order is independent of the magnetic field, the  normal EPR absorption scales with the magnetization of  the spin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Since the MChA corrections are propor-  tional to the magnetization in both EPR and the optical  case, the cancellation of the factor cos θ ≪ 1 applies to  gD/L  T  only, and it appears thereby in the denominator of  gD/L  T  /gD/L  O  , resulting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For room temperature  X-band EPR of Cu(II), this results in gD/L  T  /gD/L  O  of the  order of 10−1, which makes TWEEPR experimentally  feasible under those conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' As a consequence, and  in contrast to many other magnetic resonance techniques,  going to low temperatures is not necessarily favorable for  TWEEPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Going to higher magnetic field does not af-  fect gD/L  T  /gD/L  O  , the increase in Ω being compensated by  the concomitant increase of cos θ because of the higher  resonance field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The main results of our model are an analytic expres-  sion for the TWEEPR anisotropy factor [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (8)] and an  expression for its relationship with the optical anisotropy  absorption coefficient [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (8)  shows that gD/L  T  has a linear dependence on the magnetic  field strength (through Ω) and on the chirality (through  CD/L), as predicted by symmetry arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The de-  pendence on the spin-orbit coupling does not appear ex-  plicitly, because we have considered the case for Cu(II),  where the level splitting δ is much smaller than the SO  coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In the inverse case, gD/L  T  would be propor-  tional to λ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Adapting the calculation to other chi-  Det  sig  atn  mod  PSD  moo  ref  vout  uw5  ral transition metal complexes is conceptually straight-  forward and should result in an expression similar to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (8), apart from numerical factors of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' A  rather different case is represented by chiral organic rad-  icals, where the unpaired electron is delocalized on one  or more interatomic bonds and a different microscopic  model should be used for the calculation of gD/L  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' One  might however expect that such differences apply also to  the calculation of gD/L  O  for such radicals, preserving a  relationship similar to that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Acknowledgements  This work was supported by the Agence Nationale de  la Recherche (SECRETS, (ANR PRC 20-CE06-0023-  01) and the Laboratory of Excellence NanoX (ANR-17-  EURE-0009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' We gratefully acknowledge helpful discus-  sions with Anne-Laure Barra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  In the Appendices we describe the theoretical model used in our calculations, we offer explicit expressions for the  transition probabilities that enter the anisotropy factors in EPR and optical MChD, and comment on the limitations  of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Appendix A: Fundamentals of the model  As outlined in the article, in order to estimate the MChA factors of a chiral Cu(II) complex, we consider a variant  of the one-electron model proposed by Condon for the study of natural optical activity in chiral compounds [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The total Hamiltonian of our model is H = H0 + V D/L  C  + VSO, where H0 =  p2  2me + meω2  0r2  2  + VZ is the unperturbed  Hamiltonian, with VZ = −µB(L + gS) · B0 being the Zeeman potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and V D/L  C  = CD/Lxyz, VSO = λL · S being  the chiral potential and the spin-orbit coupling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' We stick to the nomenclature used in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The  chiral Hamiltonian, V D/L  C  , results from the electrostatic interaction of the ion with the chiral configuration of the  ligands in the complex, and produces the necessary parity asymmetry which is at the origin of natural optical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The orbital contribution of the Zeeman potential was added in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' [28] to the original Condon’s model to estimate  the magneto-chiral birefringence of diamagnetic chiral compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In order to account for magnetochiral dichroism  (MChD) in a paramagnetic complex, we introduce here the spin contribution to the Zeeman potential as well as the  spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In contrast to the approach in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' [28] and for simplicity, we consider an isotropic harmonic  oscillator, whereas the anisotropy caused by the crystal field is introduced in an effective manner through the energy  intervals between the 3d orbitals, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1 in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The eigenstates of H0 are labeled with the eigenvalues of the orbital angular momentum and spin operators,  {|nL, nR, nz⟩} ⊗ {↑, ↓} [29], upon which V D/L  C  and VSO act perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In a Cu(II) complex, the chromophoric  charge is the unpaired electron of the 3d9 electronic configuration which behaves as a hole of positive charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In the  absence of ligands, the 3d orbitals of the ion can be represented approximately by the n = 2, l = 2 states of the  harmonic oscillator of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' However, the ligands’ fields affect the electronic configuration of the ion, removing  the degeneracy of the d-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In particular, for octahedral coordination geometries around the ion, the set of d-  orbitals splits into doubly degenerate eg orbitals, dx2−y2 and dz2, and triply degenerate t2g orbitals, dxy, dyz and  dzx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The energy interval between eg and t2g states, ∆0, lies in the visible region of the spectrum, ∆0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' As a  result, the eg orbitals become the ground states, and can be approximated by linear combinations of l = 2, ml = 0, ±2  eigenstates of the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The fact that the chromophoric charge in the eg states cannot rotate into any  other orbital leads to an effective quenching of the orbital angular momentum of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Below a certain  temperature, an additional Jahn-Teller (JT) distortion takes place when the ligands along one of the axes, say the  z-axis, move away from the ion in order to minimize the electronic repulsion, giving rise to the complete removal of  the degeneracy in the eg level, and to a partial lifting of the degeneracy in the t2g orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The isotropy of the system  is thus broken and the ground state becomes unique, up to spin degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For the particular case of the CsCuCl3  crystal, the bonds along the z-axis get elongated and the ground state is the dx2−y2 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1 in the article depicts  the energy splitting of the distorted d-orbitals, including the approximate values of the energy intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Lastly, the  JT distortion in conjuntion with the helical deformation of the crystal along the c-axis, of coordiates [1,1,1] in the  local axis basis, removes the degeneracy between the orbitals lying on the xy plane in a small ammount δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Below,  we write the approximate expression of the 3d orbitals in terms of the harmonic oscillator eigenstates, {|nL, nR, nz⟩},  6  together with their corresponding energies,  |dzx⟩ = (|0, 1, 1⟩ − |1, 0, 1⟩)/  √  2,  E = ∆0,  |dyz⟩ = i(|0, 1, 1⟩ + |1, 0, 1⟩)/  √  2,  E = ∆0 − δ,  |dxy⟩ = i(|0, 2, 0⟩ − |2, 0, 0⟩)/  √  2,  E = ∆0 − ∆2,  |dz2⟩ = (|1, 1, 0⟩ −  √  2|0, 0, 2⟩)/  √  3,  E = ∆0 − ∆1,  |dx2−y2⟩ = (|0, 2, 0⟩ + |2, 0, 0⟩)/  √  2,  E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (A1)  Altogether, the crystal field combined with the JT distortion and the helical deformation turns the crystalline structure  into a chiral one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In accord with Condon’s model, the potential V D/L  C  reproduces the electrostatic interaction of the  chromophoric charge with the surrounding chiral structure, removing all axes and planes of symmetry from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  It is through the chiral potential that E1 transitions between the 3d orbitals take place in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In addition  to the above interactions, MChD in EPR requires necessarily the coupling between the spin and the orbital angular  momentum of the unpaired electron hole through the potential VSO, where the coupling constant is λ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In  particular, the SO interaction together with the Zeeman potential break the quasi-degeneracy between the four states  {|dzx⟩, |dyz⟩} ⊗ {↑, ↓}, providing the following eigenstates for λ ≫ δ,  |Φ1⟩ ≈ |1, 0, 1⟩⊗ ↓ + δ  2λ|0, 1, 1⟩⊗ ↓,  E ≃ ∆0 − λ/2 + ℏΩ,  |Φ2⟩ ≈ |0, 1, 1⟩⊗ ↑ + δ  2λ|1, 0, 1⟩⊗ ↑,  E ≃ ∆0 − λ/2 − ℏΩ,  |Φ3⟩ ≈ |0, 1, 1⟩⊗ ↓ − δ  2λ|1, 0, 1⟩⊗ ↓,  E ≃ ∆0 + λ/2 + ℏΩ + δ2  4λ2 (λ + ℏΩ),  |Φ4⟩ ≈ |1, 0, 1⟩⊗ ↑ − δ  2λ|0, 1, 1⟩⊗ ↑,  E ≃ ∆0 + λ/2 − ℏΩ + δ2  4λ2 (λ − ℏΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (A2)  {Φ1, Φ2, Φ3, Φ4} are indeed the eigenstates of the Hamiltonian VZ+VSO restricted to the subspace {|dzx⟩, |dyz⟩}⊗{↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  They constitute the intermediate states of the transition processes in EPR mediated by the interaction of the spin  with the chiral structure of the surrounding charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  In the following, we apply to our system time-dependent quantum perturbation techniques to compute first the  MChA factor in EPR, gD/L  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Next, in order to estimate the value of the unknowns of our model, we compute the  anisotropy factor in optical MChD for the same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Finally, making use of the experimental values available for  CsCuCl3 in the literature [24, 25], we estimate the strength of TWEEPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Appendix B: MChD in EPR  Let us consider a CsCuCl3 complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' initially prepared in its ground state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and partially polarized along a uniform  magnetic field B = B0ˆz directed along the z-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  |Ψ⟩ = |dx2−y2⟩ ⊗ (cos θ/2 ↑ + sin θ/2 ↓) ≈  1  √  2(|0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 0⟩ + |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 0⟩) ⊗ (cos θ/2 ↑ + sin θ/2 ↓),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B1)  where we have approximated the actual ground state with the corresponding state of our harmonic oscillator model  in the basis {|nL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' nR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' nz⟩} ⊗ {↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' ↓},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and θ is the angle between the magnetic moment of the complex and the z-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  cos θ = ℏ−1⟨Ψ|2S|Ψ⟩ · ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' At temperature T, cos θ ≈ µ0B0/kBT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Under the action of an incident electromagnetic  field of frequency ω close to the transition frequency, Ω = gµBB0/ℏ, and wave vector k parallel to B0, the complex  gets partially excited towards the state  |Φ⟩ = |dx2−y2⟩⊗ ↓≈  1  √  2(|0, 2, 0⟩ + |2, 0, 0⟩)⊗ ↓,  (B2)  7  with probability proportional to cos2 θ/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' and partially de-excited (through stimulated emission) towards the state  |Φ′⟩ = |dx2−y2⟩⊗ ↑≈  1  √  2(|0, 2, 0⟩ + |2, 0, 0⟩)⊗ ↑,  (B3)  with probability proportional to sin2 θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Since the rest of probability factors are equivalent, the net absorption  probability in EPR is proportional to cos2 θ/2 − sin2 θ/2 = cos θ, and thus proportional to the magnetization of the  complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  As mentioned in the article, from symmetry considerations and in leading order, the numerator and the denom-  inator in the ratio gD/L  T  = [P D/L(ω, ˆk, B0) − P D/L(ω, ˆk, −B0)]/[P D/L(ω, ˆk, B0) + P D/L(ω, ˆk, −B0)] for ω ≈ Ω are  dominated, respectively, by the electric-magnetic dipole (E1M1) and the magnetic-magnetic dipole (M1M1) transi-  tion probabilities, the magnetic transition being driven by the spin operator only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' That leads to the approximate  expression,  gD/L  T  ≃ P D/L  E1M1(ω, ˆk, B0)  PM1M1(ω, ˆk, B0)  ���  ω≈Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B4)  In what follows, we compute the transition probabilities PM1M1 and P D/L  E1M1 for ω ≈ Ω using time-dependent  perturbation theory in the adiabatic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This regime is the suitable one for a probe field whose duration is  much longer than the typical lifetime for excitation or de-excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  As in the article, the Hamiltonian of the  interaction of our system with the microwave probe field reads, in the electric and magnetic dipole approximation,  W = −er·Eω(t)/2−µB(L+2S)·Bω(t)/2+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='. In this equation, Eω(t) = Eωe−iωt = iωAωe−iωt, Bω(t) = Bωe−iωt =  i¯nk ∧ Aωe−iωt, are the complex-valued electric and magnetic fields, respectively, with Aω being the complex-valued  amplitude of the plane-wave electromagnetic vector potential of frequency ω ≈ Ω, evaluated at the center of mass of  the Cu(II) ion, and ¯n being the effective refractive index of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The local depolarization changes the local  electric field incident on each Cu(II) ion to Eω(¯n2 + 2)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Under the action of W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' with k along B0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' the expressions  for PM1M1 and P D/L  E1M1 read,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' at leading order in the coupling constants of the interaction potentials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  PM1M1|ω≈Ω = ℏ−2  �����  � T  0  dte−i(T −t)(Ω/2−iΓ/2)e−it(ω−Ω/2)⟨Φ| − gµBS · Bω|Ψ⟩  �����  2  − ℏ−2  �����  � T  0  dte−i(T −t)(2ω−Ω/2−iΓ/2)e−it(ω+Ω/2)⟨Φ′| − gµBS · Bω|Ψ⟩  �����  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B5)  8  P D/L  E1M1|ω≈Ω = 2Re(−i)3ℏ−4 �  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='q̸=Ψ  � T  0  dte−i(T −t)(Ω/2−iΓ/2)⟨Φ| − er · (¯n2 + 2)Eω/3|p⟩  � t  −∞  dt′eηt′e−i(t−t′)(Ep+ω)  × ⟨p|V D/L  C  |q⟩  � t′  −∞  dt′′eηt′′e−i(t′−t′′)(Eq+ω)⟨q|VSO|Ψ⟩e−it′′(ω−Ω/2)i  � T  0  dτ ei(T −τ)(Ω/2+iΓ/2)  × ⟨Ψ| − gµBS · B∗  ω|Φ⟩eiτ(ω−Ω/2) + 2Re(−i)3ℏ−4 �  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='q̸=Φ  � T  −∞  dt eηte−i(T −t)(Ω/2−iΓ/2)⟨Φ|VSO|p⟩  ×  � t  −∞  dt′eηt′e−i(t−t′)Ep⟨p|V D/L  C  |q⟩  � t′  0  dt′′e−i(t′−t′′)Eq⟨q| − er · (¯n2 + 2)Eω/3|Ψ⟩e−it′′(ω−Ω/2)  × i  � T  0  dτ ei(T −τ)(Ω/2+iΓ/2)⟨Ψ| − gµBS · B∗  ω|Φ⟩eiτ(ω−Ω/2)  − 2Re(−i)3ℏ−4 �  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='q̸=Ψ  � T  0  dt e−i(T −t)(2ω−Ω/2)⟨Φ′| − er · (¯n2 + 2)Eω/3|p⟩  � t  −∞  dt′eηt′e−i(t−t′)(Ep+ω)  × ⟨p|V D/L  C  |q⟩  � t′  −∞  dt′′eηt′′e−i(t′−t′′)(Eq+ω)⟨q|VSO|Ψ⟩e−it′′(ω+Ω/2−iΓ/2)i  � T  0  dτ ei(T −τ)(2ω−Ω/2)  × ⟨Ψ| − gµBS · B∗  ω|Φ′⟩eiτ(ω+Ω/2+iΓ/2) − 2Re(−i)3ℏ−4 �  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='q̸=Φ′  � T  −∞  dt eηte−i(T −t)(2ω−Ω/2)  × ⟨Φ′|VSO|p⟩  � t  −∞  dt′eηt′e−i(t−t′)(2ω+Ep)⟨p|V D/L  C  |q⟩  � t′  0  dt′′e−i(t′−t′′)(2ω+Eq)  × ⟨q| − er · (¯n2 + 2)Eω/3|Ψ⟩e−it′′(ω+Ω/2−iΓ/2)i  � T  0  dτ ei(T −τ)(2ω−Ω/2)⟨Ψ| − gµBS · B∗  ω|Φ′⟩  × eiτ(ω+Ω/2+iΓ/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  η → 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' ΓT ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B6)  In these equations the states p and q stand for the excited states of the 3d9 configuration together with other eigenstates  of H0 with n ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' The quasi-stationary condition η → 0+ accounts for the stationarity of the chiral and the spin-orbit  interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' whereas the adiabatic limit ΓT ≫ 1 takes into account the long duration of the probe field with respect  to the lifetime Γ−1, with Γ being the linewidth of absorption and T the observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  The diagrammatical  representation of the processes involved in the above equation is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In the article, the contributions of  the quasi-stationary processes were incorporated into the dressed states ˜Ψ, ˜Φ, ˜Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' More specifically, the bare states  are dressed with the quadruplet {Φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='., Φ4} through VSO, and with harmonic states with n ̸= 2 by VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In terms of  the eigenstates of the harmonic oscillator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' they read  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='|˜Φ⟩ =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(|020⟩ + |200⟩)/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + ∆2/∆0)(|020⟩ − |200⟩) + iλCD/LK3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + ∆2/∆0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='× (|001⟩ − 2|111⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='↓ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + 3ℏΩ/∆0)|011⟩ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(ℏΩ + λ/2)|101⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='+ −iCD/LK3/2λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + 3ℏΩ/∆0)(|210⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='3|030⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2|100⟩) + iCD/LK3/2δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='× (ℏΩ + λ/2)(|120⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='3|300⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2|010⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='↑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='|˜Φ′⟩ =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(|020⟩ + |200⟩)/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='−λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + ∆2/∆0)(|020⟩ − |200⟩) + −iλCD/LK3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 + ∆2/∆0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='× (|001⟩ − 2|111⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='↑ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='� −λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 − 3ℏΩ/∆0)|101⟩ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(ℏΩ − λ/2)|011⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='+ −iCD/LK3/2λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(1 − 3ℏΩ/∆0)(|120⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='3|300⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2|010⟩) + −iCD/LK3/2δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2ℏω0∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='× (ℏΩ − λ/2)(|210⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='3|030⟩ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2|100⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='↓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='|˜Ψ⟩ = cos θ/2|˜Φ′⟩ + sin θ/2|˜Φ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  K = ℏ/(2meω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B7)  9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 3: Diagrammatic representation of the processes which contribute to PM1M1 and P D/L  E1M1 for ω ≈ Ω at leading order in  the perturbative interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', at second order and fourth order, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Time runs along the vertical direction from  0 to the observation time T , where the probability is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Intermediate atomic states are labeled as p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Diagrams  with two-photon states account for stimulated emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Using a linearly polarized incident field and averaging in orientations around the ˆz-axis, we obtain, for λ ≫ δ,  PM1M1|ω≈Ω ≃  ℏ−2µ2  B|Bω|2  4[(ω − Ω)2 + Γ2/4] cos θ,  (B8)  P D/L  E1M1|ω≈Ω ≃ (¯n2 + 2)  3  CD/LΩδ  meω3  0∆2  0  ℏ−1µ2  B|Bω||Eω|  4[(ω − Ω)2 + Γ2/4] cos θ,  (B9)  gD/L  T  ≃ (¯n2 + 2)  3¯n  c CD/LℏΩδ  meω3  0∆2  0  + O(δ/λ, λ/∆0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (B10)  Lastly, it is worth mentioning that for the case δ > λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', when anisotropy dominates over the spin-orbit coupling,  gD/L  T  scales as (cℏCD/LΩδλ)/(meω3  0∆3  0) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This scenario will be addressed in a separate publication [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Appendix C: Optical MChD  Optical MChD involves transitions of frequency ∆0 from the ground state |Ψ⟩ to the quasi-degenerate quadruplet  {|dzx⟩, |dyz⟩} ⊗ {↑, ↓} which, in account of the Zeeman and spin-orbit interactions, for δ ≪ λ, corresponds to the  set of states {Φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', Φ4} of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='(A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In contrast to EPR, the absorption probability in the denominator of the ratio  gD/L  O  = [P D/L(ω, ˆk, B0)−P D/L(ω, ˆk, −B0)]/[P D/L(ω, ˆk, B0)+P D/L(ω, ˆk, −B0)] for ω ≈ ∆0/ℏ may not be dominated  by the magnetic-magnetic dipole absorption probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' This might be so because the d-orbitals of the Cu(II) ion  0  er:(n*+2)Eo/3  P  D  gsS·Ba  M1M1  Vso  tlo  PDIL  2 Re  D  E1M1  D  p  910  hybridize generally with the σ and π orbitals of the ligands, allowing for additional electric-electric dipole (E1E1)  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For the sake of simplicity, we will neglect the latter in our calculations, which implies that our preliminar  estimate for gD/L  O  must be intended as an approximate upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' As for the case of EPR, the numerator of  the ratio in gD/L  O  is again dominated by the electric-magnetic dipole absorption probability, and the non-vanishing  terms come from magnetic transitions driven by the spin angular momentum –Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (C2) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' However, in contrast  to EPR, the magnetic transitions in the denominator are mainly driven by the orbital angular momentum operator  –see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (C1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In turn, this causes the E1M1 transition probability to depend on the spin polarization of the  complex, whereas neither the M1M1 nor the E1E1 probabilities do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Note also that stimulated emission from the state  |Ψ⟩ is absent in optical MChD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' All in all, this implies that gD/L  O  is proportional to the magnetization of the sample,  which is itself proportional to the degree of spin-polarization along B0, cos θ, in agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 4: Diagrammatic representation of PM1M1 and P D/L  E1M1 for ω ≈ ∆0/ℏ at leading order in the perturbative interactions,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', at second and up to fifth order, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Intermediate atomic states are labeled as p, q, r, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  we depict some of the diagrams which contribute to PM1M1 and P D/L  E1M1 in optical MChD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Following a perturbative  approach analogous to that in EPR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' for an incident electromagnetic plane wave with k ∥ B0 and assuming δ ≪ λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  one arrives at  PM1M1|ω≈∆0/ℏ ≃  ℏ−2µ2  B|Bω|2  4[(ω − ∆0/ℏ)2 + Γ  ′2/4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (C1)  P D/L  E1M1|ω≈∆0/ℏ ≃ (¯n2 + 2)  3  CD/Lδ  2meω3  0 ˜∆  ℏ−2µ2  B|Bω||Eω|  4[(ω − ∆0/ℏ)2 + Γ  ′2/4] cos θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (C2)  gD/L  O  ≲ P D/L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='O  E1M1  P O  M1M1  ���  ω≈∆0/ℏ ≃ (¯n2 + 2)  3¯n  c CD/Lδ cos θ  2 meω3  0 ˜∆  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (C3)  er:(n+2) Eo/3  P  Di-  wo -μ(L+gS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Ba  M1M1  Vso  Vc  tlo  Di --  PDIL  = 2 Re  4  E1M1  Di -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  +  r  r  011  01111  where ˜∆−1 = ∆−1  0  + ∆−1  2  − 3∆−1  1 , and Γ′ is the linewidth of optical absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' As anticipated, the fact that the  magnetic dipole transition in P D/L  E1M1 is dominated by the orbital angular momentum operator causes its leading  order term to depend on the magnetization ∼ cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Hence, time-reversal invariance happens to be broken by the  spin-polarization of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Appendix D: Estimate of gD/L  T  In the first place, we work out the relationship between gD/L  T  and gD/L  O  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (B9) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (C2) at  resonance, and taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (B10) and (C3), we arrive at the following relationships,  P D/L  E1M1|ω=Ω  P D/L  E1M1|ω=∆0/ℏ  ≃ 2ℏΩ ˜∆Γ  ′2  ∆2  0Γ2  ,  gD/L  T  gD/L  O  ≳  2ℏΩ ˜∆  ∆2  0 cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (D1)  Next, considering the experimental data obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' [24] for gD/L  O  and applying the relationship in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (D1), we  can estimate a lower bound for gD/L  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' That is, substituting into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (D1) the experimental values gD/L  O  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='025,  cos θ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='4, for B0 = 14T at a temperature of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2 K, we obtain gD/L  T  ≳ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Alternatively, we can estimate gD/L  T  using the experimental data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' [24] for the non-reciprocal absorption  coefficient of optical MChD, αA = α(B0 ↿↾ k) − α(B0 ⇃↾ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In order to do so, we first write down αA as a function  of P D/L,O  E1M1 at resonance,  αA = 4cµ0ρΓ′∆0  |Eω|2  P D/L  E1M1|ω=∆0/ℏ,  (D2)  where ρ is the molecular density of the CsCuCl3 complex (mass density 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5g/cm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Substituting the expression for  P D/L,O  E1M1 (ω = ∆0/ℏ) in the above equation and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (B10) we arrive at the equalities,  CD/Lδ =  3ℏ2meω3  0 ˜∆Γ′αA  2(¯n2 + 2)ρµ0µ2  B∆0 cos θ,  gD/L  T  =  c ℏ3Γ′Ω ˜∆αA  2∆3  0µ0µ2  Bρ cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  (D3)  Substituting the experimental values for all the variables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (D3), for B0 = 14 T at a temperature of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='2 K, with  Γ′ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1eV and ¯n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5, we obtain gD/L  T  ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 · 10−2, in agreement with our previous lower bound estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Appendix E: Further comments on the Hamiltonian model  Despite the success of our model to derive analytical estimates for the MChA factors, there is still room for  improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In the first place, concerning the chiral Hamiltonian VC, it was written in terms of the local axis of the  octahedral structure, x, y, z, while it should be adapted to the crystal axis to account for the helical distribution of the  active ions along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' In fact, the experimental data on αA taken from the literature to estimate gD/L  T  consider  B0 along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Also, the harmonic oscillator model, which is considered only distorted in the n = 2, l = 2 level,  may not be accurate enough to account for the intermediate transitions induced by the chiral potential to levels with  n ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Hence, a more accurate confining potential model, though less generic, can be obtained using a more detailed  formulation of the crystal field and the JT distortion for the particular case of CsCuCl3–see, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Finally,  our estimate of the unknown combination CD/Lδ in terms of αA [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (D3)], involves ¯n-dependent factors [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' (B10],  which account for effective incident fields, as well as ρ-dependent factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' For high densities and ¯n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='5 those factors  are likely to depend on near field terms and spatial correlations when evaluated at the absorption frequency [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Goldfarb and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Stoll, Wiley Chichester  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  [2] Spin labeling, Biological Magnetic Resonance vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 14 ed  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Berliner, Kluwer, New York 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 2012, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='1002/adma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='201200786, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Ses-  soli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Boulon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Caneschi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Mannini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Poggini,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='Wilhelm and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rogalev, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 11, 69 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rikken, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' F¨olling and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Wyder, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Krsti´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Roth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Burghard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Kern and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Rikken, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Auban-Senzier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Canadell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Rikken and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Avarvari, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Kanazawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Kikkawa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Morikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Shibata, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Arima, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Taguchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Kagawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Tokura,  Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Pan´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Gambardella,  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Kousaka and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Togawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Condon, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 9, 432 (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Condon, William Altar, and Henry Eyring, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' 5, 753 (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Donaire, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rikken, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' van Tiggelen,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' D 68, 33 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Diu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Laloe, Quantum Me-  chanics, Wiley-VCH (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Abe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Kimura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Matsuda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' No-  mura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Kousaka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Sera, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Kawamata, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Koike, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Inoue,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Townsend, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Weissman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Sierins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' Syms, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=', Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
+page_content='  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQf2gtA/content/2301.04755v1.pdf'}
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+Publications of the Astronomical Society of Australia (2020), 1–8
+doi:
+RESEARCH PAPER
+Very long baseline interferometry observations of the high-redshif
+blazar candidate J0141–5427
+K. É. Gabányi,1,2,3 S. Belladitta,4,5,6 S. Frey,2,7,8 G. Orosz,9,10 L. I. Gurvits,9,11,22 K. Rozgonyi,33,44,55 T. An,66,77 H. Cao,88
+Z. Paragi,9 and K. Perger2,7
+1Department of Astronomy, Institute of Geography and Earth Sciences, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
+2Konkoly Observatory, ELKH Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary
+3ELKH-ELTE Extragalactic Astrophysics Research Group, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
+4INAF, Osservatorio Astronomico di Brera, Via Brera 28, 20121 Milano, Italy
+5DiSAT, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy
+6Max-Planck-Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
+7CSFK, MTA Centre of Excellence, Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary
+8Institute of Physics, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
+9Joint Institute for VLBI ERIC, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
+10School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
+11Faculty of Aerospace Engineering, Delf University of Technology, Kluyverweg 1, 2629 HS, Delf, The Netherlands
+22CSIRO Astronomy and Space Science, PO Box 76, Epping, NSW 1710, Australia
+33University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany
+44International Centre for Radio Astronomy Research, The University of Western Australia, Crawley, WA 6009, Australia
+55Australian Research Council, Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO 3D), Canberra, ACT 2611, Australia
+66Shanghai Astronomical Observatory, CAS, Nandan Road 80, Shanghai 200030, PR China
+77Peng Cheng Laboratory, Shenzhen 518066, PR China
+88School of Physics and Electronic Information, Huanggang Normal University, 146 Xingang 2nd Road, Huanggang, Hubei 438000, PR China
+Author for correspondence: K. É. Gabányi, Email: k.gabanyi@astro.elte.hu.
+(Received dd Mmm 2022; revised dd Mmm YYYY; accepted dd Mmm YYYY; first published online dd Mmm YYYY)
+Abstract
+Active galactic nuclei (AGN) have been observed as far as redshift z ∼ 7. They are crucial in investigating the early Universe as well as the growth
+of supermassive black holes at their centres. Radio-loud AGN with their jets seen at a small viewing angle are called blazars and show relativistic
+boosting of their emission. Thus, their apparently brighter jets are easier to detect in the high-redshift Universe. DES J014132.4–542749.9 is a
+radio-luminous but X-ray weak blazar candidate at z = 5. We conducted high-resolution radio interferometric observations of this source with
+the Australian Long Baseline Array at 1.7 and 8.5 GHz. A single, compact radio emitting feature was detected at both frequencies with a flat
+radio spectrum. We derived the milliarcsecond-level accurate position of the object. The frequency dependence of its brightness temperature is
+similar to that of blazar sources observed at lower redshifts. Based on our observations, we can confirm its blazar nature. We compared its radio
+properties with those of two other similarly X-ray-weak and radio-bright AGN, and found that they show very different relativistic boosting
+characteristics.
+Keywords: active galactic nuclei – very long baseline interferometry – galaxies: high-redshift
+1.
+Introduction
+Active galactic nuclei (AGN) are the most luminous persistent
+astronomical objects, and they are invaluable probes for inves-
+tigating the high-redshift Universe. Roughly ten per cent of
+AGN are radio-loud, jetted sources (e.g., Ivezić et al., 2002).
+In them, the radio emission originates from the synchrotron
+emission of the jets. When the jets are seen at a small angles
+to the line of sight, e.g., ≲ 10◦ (Urry & Padovani, 1995), rela-
+tivistic beaming causes significant flux density enhancement
+of the advancing jet. Thus, these beamed sources called blazars
+can be preferentially detected even at high redshifts (z ≳ 4) in
+radio bands.
+Blazars can be identified using high-resolution very long
+baseline interferometry (VLBI) radio observations. They are
+characterized by a bright feature that is compact at milliarcsec
+(mas) scale, the jet base, which usually has a flat radio spectrum
+at GHz frequencies (e.g., Hovatta et al., 2014). The apparent
+brightness temperature of this dominant component exceeds
+the equipartition limit, Teq
+B ≈ 5 × 1010 K (Readhead, 1994),
+and sometimes even the inverse Compton limit (∼ 1012 K,
+Kellermann & Pauliny-Toth, 1969), indicating the potential
+prevalence of relativistic beaming. Also, apparent superlumi-
+nal motion of components can often be observed in blazar
+jets. However, in the case of high-redshift sources, the steep-
+spectrum jet components are harder to detect, because the
+observed frequencies correspond to (1 + z) times higher emit-
+ted frequencies in the source’s rest frame, thus the extended
+regions of jets are often undetectably faint (Gurvits et al., 2015).
+Blazars can also be classified via their broad-band spectral
+energy distribution (SED) featuring non-thermal emission
+over the electromagnetic spectrum and exhibiting relativistic
+beaming effect (e.g., Massaro et al., 2009).
+arXiv:2301.04533v1  [astro-ph.GA]  11 Jan 2023
+
+2
+K. É. Gabányi et al.
+Belladitta et al. (2019) reported the discovery of DES J014132.4–542749.9
+(hereafter J0141–5427), a radio-bright but X-ray-weak AGN
+at z = 5.00 ± 0.01. The source, according to archival data and
+newly obtained X-ray observations of the authors, is an order
+of magnitude fainter in X-rays than other blazars with simi-
+lar radio luminosities. Belladitta et al. (2019) showed that the
+SED of J0141–5427 can be best described with a relativistically
+beamed blazar SED if a very high magnetic field strength of
+∼ 9 G is assumed.
+We initiated VLBI observations of J0141–5427 with the
+Australian Long Baseline Array (LBA) at 1.7 and 8.5 GHz to
+ascertain its blazar nature.
+Hereafter we use the flat ΛCDM cosmological model
+with parameters of H0 = 70 km s–1 Mpc–1, Ωm = 0.27, and
+ΩΛ = 0.73. At the redshift of J0141–5427, 1 mas angular size
+corresponds to a projected linear length of ∼ 6.5 pc, and the lu-
+minosity distance of the object is DL = 48273.2 Mpc (Wright,
+2006).
+2.
+Observations and data reduction
+Observations of J0141–5427 with the LBA were conducted in
+2020, under the project code v591 (PI: K. É. Gabányi) in phase-
+referenced mode (Beasley & Conway, 1995). In this observing
+mode, the pointing directions of the telescopes change regu-
+larly between the target source and a nearby phase-reference
+calibrator within the atmospheric coherence time permitted
+by the radio propagation media. The delay, and delay rate solu-
+tions can be then transferred (interpolated) from the calibrator
+to the target source. The nodding cycles in both the 1.7 and
+8.5 GHz observations were 5 min long, with 3.5 min spent on
+the target and 1.5 min on the calibrator. The phase-reference
+calibrator was ICRF J015649.7–543948 in both experiments.
+Additional calibrator sources were also observed to facilitate
+amplitude calibration, and to monitor the stability of the array.
+The 1.7 GHz observation took place on 2020 June 26 and
+27, the participating antennas were Ceduna (CD), Hobart
+(HO), Mopra (MP), Parkes (PA), the tied array of the Australia
+Telescope Compact Array (ATCA) in Australia, and Harte-
+beesthoek (HH) in South Africa. The observation lasted for
+10 h, the on-target time was 4.5 h. The 8.5 GHz observation
+took place on 2020 July 14, with the following participating
+antennas: CD, HO, MP, PA, Katherine (KE), Yarragadee (YG),
+the tied array of ATCA in Australia, the 12-m Warkworth an-
+tenna (WW) in New Zealand, and HH in South Africa. The
+observation lasted for 10.25 h, with an on-target time of 4.9 h.
+In both observations, the total bandwidth of 128 MHz was
+divided into 8 intermediate frequency bands (IF) of 32 chan-
+nels each. The correlator integration time was set to 2 s. The
+correlation was done at the Pawsey Supercomputing Centre in
+Perth, on a DiFX software correlator (Deller et al., 2011). The
+longest baselines of the arrays (providing the finest angular
+resolution) were those to HH. At 8.5 GHz, HH could only
+participate in the last 18 min of the observation.
+Data reduction was done using the National Radio Astron-
+omy Observatory (NRAO) Astronomical Image Processing
+System (AIPS, Greisen 1990) following standard procedures of
+ionospheric and parallactic angle corrections, manual phase
+calibrations and fringe-fitting of the calibrator sources, and
+following the LBA guide on amplitude calibrationa. The nec-
+essary files for amplitude calibration were created from the
+system temperature measurements and gain curves provided
+by the participating stations or system equivalent flux densities
+listed in the LBA amplitude calibration user’s guide. In the
+absence of system temperature measurements (at the antennas
+CD and HO at 1.7 GHz and HO, KE, and WW at 8.5 GHz),
+nominal system temperature values were used.
+The fringe-fitting was performed for all calibrator sources.
+Solutions were found for ∼ 86% and ≳ 98% of the data at
+1.7 GHz and 8.5 GHz, respectively.
+At the AT, MP, and PA antennas, wider filters were used,
+resulting in clearly lower amplitude values in the channel-
+averaged data at those IFs corresponding to the edges of the
+bands (IFs 1, 4, 5, 8) compared to the ones at the middle (IFs 2,
+3, 6, 7). Therefore, the edge IFs were scaled up by a constant
+factor of 1.169 to bring them closer to the values measured
+in the middle of the band at these antennas in the case of 1.7-
+GHz observation, before channel averaging. At the 8.5-GHz
+observation, instead of such scaling, we flagged the first 10
+channels for IFs 1 and 5 and the last 10 channels for IFs 4 and
+8, for the three antennas using wider filters (AT, MP, and PA).
+After the fringe-fitting performed on the calibrator sources,
+and the application of the above described amplitude scaling for
+the 1.7-GHz amplitudes, the channel-averaged data of the cali-
+brator sources were imported into the Caltech DIFMAP package
+(Shepherd, 1997) for hybrid mapping. The hybrid-mapping
+procedure involves subsequent steps of CLEANing (Högbom,
+1974) and phase self-calibration of the data. As the last step, am-
+plitude self-calibration was done. The gain correction factors
+obtained for different calibrator sources were in good agree-
+ment for the same antennas and IFs. The flux density values
+of the phase-reference calibrator obtained this way at both fre-
+quencies were in good agreement with the ones measured by
+the ATCA closest in time and at similar frequencies according
+to the ATCA Calibrator Databaseb with the VLBI-measured
+flux densities ∼ 15 % and ∼ 3 % lower than the ones measured
+by ATCA at 1.7 GHz and at 8.5 GHz, respectively. The dif-
+ference is most probably caused by resolution effect, the LBA
+observations resolved out the large-scale emission detected by
+ATCA. Thus, we accepted the gain correction factors obtained
+in DIFMAP for the phase-reference calibrator, and we adjusted
+the antenna gains accordingly in AIPS to further improve the
+amplitude calibration.
+To improve the delay and rate solutions, the phase-reference
+calibrator was fringe-fitted again using the CLEAN component
+model of its brightness distribution derived from the hybrid
+mapping, to take the source structure into account. The ob-
+tained solutions were applied to the phase-reference calibrator
+as well as to the target source, and subsequently both were
+ahttps://www.atnf.csiro.au/vlbi/dokuwiki/doku.php/lbaops/
+lbacalibrationnotes (accessed 2022.09.06)
+bhttps://www.narrabri.atnf.csiro.au/calibrators/calibrator_database.html
+(accessed 2022.09.06)
+
+Publications of the Astronomical Society of Australia
+3
+imaged in DIFMAP.
+In the case of the phase-reference calibrator, the amplitude
+self-calibration performed after this second hybrid mapping
+showed that the gain correction factors were mostly ≲ 10%
+for the 1.7-GHz data and mostly ≲ 5% for the 8.5-GHz data,
+except for single IFs of AT and CD, and a few discrepant IFs of
+YG. Additionally, it seemed that amplitude self-calibration of
+HH was not constrained at 1.7-GHz, and it could not correct
+the amplitudes. Thus, we conservatively assume the amplitude
+calibration of these LBA data is reliable at 10% level.
+Due to an unfortunate typing mistake made by the PI
+at the time of scheduling, the observations and subsequent
+correlations were done at a target source position with 4′′
+offset in declination from the previously known position. A
+significant offset from the phase centre may cause reduction of
+the peak intensity and distortion of the obtained image through
+bandwidth smearing and time-average smearing effects (Bridle
+& Schwab, 1999).
+The bandwidth smearing effect would have been substan-
+tial (intensity reduction of a point source by ∼ 80–90 %) if the
+data were averaged over all the channels within an IF (Bridle
+& Schwab, 1999; Wrobel, 1995). Therefore, the hybrid map-
+ping of the target source was performed on the unaveraged
+data. We disregarded the first and last 5 channels of all 8 IFs
+to account for bandpass effects.
+At both frequencies, time averaging was done for 2 s at
+the correlator. However, because of the different resolutions,
+time-average smearing affects the two data sets differently. At
+1.7 GHz, this effect is negligible, the peak intensity reduction
+of a point source is less than 1% at 4′′ from the pointing centre.
+At 8.5 GHz, if calculated for the highest achievable resolution
+obtained with the longest baseline, between HH and YG,
+time-average smearing would cause an average peak intensity
+reduction of a point source by 15%. Excluding the baselines
+to HH, the average amplitude reduction of a point source is
+∼ 5% at 4′′ distance. Since HH could only participate in the
+last 18 min of the 8.5 GHz observation, we excluded the data on
+the baselines to HH. Therefore the effects of the unintentional
+pointing offset introduced in the target source position could
+be mitigated satisfactorily.
+The target source J0141–5427 turned out to be bright
+enough for attempting a direct fringe-fitting. Before that, the
+visibility data set was shifted by 4′′ in declination direction
+to its a priori known correct position using the task CLCOR
+in AIPS. At 1.7 GHz, fringes with a signal-to-noise level ex-
+ceeding 6σ were found for 69% of data, including the longest
+baselines to HH. We continued imaging both the fringe-fitted
+and the phase-referenced 1.7-GHz data of the target, and the
+results were in good agreement. The peak intensity was less
+by ∼ 10 mJy beam–1 (∼ 13 %) in the phase-referenced image
+compared to the one obtained after fringe-fitting the data due
+to the coherence loss (Martí-Vidal et al., 2010). At 8.5 GHz,
+at the same signal-to-noise level, fringes were found for only
+24% of data, and no fringes were found on the baselines to
+HH. Therefore, we did not use the fringe-fitted data of the
+target for the higher frequency observation.
+At both frequencies, phase self-calibration and amplitude
+self-calibration were performed with subsequently shorter time
+intervals during the hybrid mapping of J0141–5427. However,
+only the best-behaving, least noisy antennas were used in
+the self-calibration processes. Thus, HH and CD were kept
+fixed for the 1.7-GHz observation. In the case of the 8.5-
+GHz observation, originally all antennas were used in phase
+self-calibration (except for HH which was not used in the
+hybrid mapping), but for the shortest time intervals, and in the
+amplitude self-calibration, only ATCA, MP, CD, and PA were
+included, while the gains of the remaining antennas were kept
+fixed.
+3.
+Results
+At both frequencies, a single radio-emitting feature was de-
+tected (Figs. 1 and 2).
+Figure 1. 1.7-GHz naturally-weighted LBA map of the fringe-fitted data of
+J0141–5427. The peak intensity is 76.7 mJy beam–1. The lowest contours
+are at ±1.7 mJy beam–1, corresponding to 4σ image noise level. Further
+positive contours increase by a factor of 2. The elliptical Gaussian restor-
+ing beam size is 25.7 mas × 6.2 mas at a major axis position angle of –8.9◦,
+and it is shown in the lower lef corner of the image.
+We derived the coordinates of the brightest pixel at both
+frequencies using the AIPS verb MAXFIT. At 8.5 GHz, the
+right ascension and declination are RA = 1h41m32.44937s
+and Dec = –54◦27′49.9705′′, respectively. We estimate that
+these coordinates are accurate within 0.8 mas. The most dom-
+inant sources of the uncertainty are the positional accuracy
+of the phase reference calibrator (0.37 mas in right ascension
+and 0.33 mas in declination direction, according to the most
+recent version of the Radio Fundamental Catalogc) and the
+crfc_2022b, http://astrogeo.org/sol/rfc/rfc_2022b/rfc_2022b_cat.txt (ac-
+cessed 2022.09.06)
+
+4
+K. É. Gabányi et al.
+Figure 2.
+8.5-GHz naturally-weighted phase-referenced LBA map of
+J0141–5427. The peak intensity is 28.5 mJy beam–1. The lowest contours
+are at ±0.7 mJy beam–1, corresponding to 4σ image noise level. Further
+positive contours increase by a factor of 2. The elliptical Gaussian restor-
+ing beam size is 3.9 mas × 2.8 mas at a major axis position angle of 5.3◦, and
+it is shown in the lower lef corner of the image.
+astrometric errors strongly depending on the phase-reference
+calibrator–target angular separation. For the latter, we conser-
+vatively assumed the value derived for observations taken at
+5 GHz by Chatterjee et al. (2004). The coordinates derived
+from the phase-referenced 1.7-GHz observation agree with
+the 8.5-GHz values within the uncertainties. Additionally,
+they agree within the uncertainty with the optical position
+provided in the Dark Energy Survey 2nd data release (Ab-
+bott et al., 2021). These newly derived radio coordinates of
+J0141–5427 are much more accurate than those previously
+obtained from lower-resolution radio observations, e.g. the
+AT20 survey with ∼ 1′′ positional accuracy (Murphy et al.,
+2010).
+To quantitatively describe the brightness distribution of
+the source, we fitted the visibility data with Gaussian model
+components. At 1.7 GHz, a single circular Gaussian compo-
+nent with a flux density of (80.3 ± 8.4) mJy and a full-width
+at half-maximum (FWHM) size of ∼ 2.2 mas can adequately
+describe the data. However, according to Lister et al. (2021),
+the typical uncertainty of a single isolated Gaussian bright-
+ness distribution component diameter is 20% of the restoring
+beam FWHM size. As such, the size of the component is
+not well-constrained. The highly elongated restoring beam
+of the 1.7-GHz experiment, major axis 25.7 mas in roughly
+north–south direction and minor axis 6.2 mas in the perpen-
+dicular direction, would result in an asymmetric source size
+uncertainty in the two perpendicular orientations. In the finer
+resolution east–west direction, the FWHM size of the emit-
+ting feature is (2.2 ± 1.5) mas, while it is not constrained in the
+perpendicular direction, (2.2±5.1) mas. Nevertheless, the com-
+pactness of the radio emission is further supported by the high
+percentage of fringe solutions found on the longest baselines
+to HH.
+At 8.5 GHz, an elliptical Gaussian component with a flux
+density of (40.8 ± 4.1) mJy, a major and a minor axis FWHM
+sizes of (3.1 ± 0.8) mas and (1.4 ± 0.6) mas, respectively, and
+a major axis position angle of –13.7◦ was needed to fit the
+datad. The 8.5-GHz observation is somewhat affected by time
+smearing effect as described in Section 2. While the peak in-
+tensity reduction of a point source may not be significant, time-
+average smearing can cause distortion of the image. Therefore,
+we also analysed the data set by excluding the longer baselines
+where the smearing effect is expected to be more pronounced.
+We only retained the antennas of MP, PA, ATCA, HO, and
+CD. We obtained the same parameters within the errors for
+the fitted Gaussian brightness distribution model, suggesting
+that the modeling results are robust.
+Assuming the same amount of coherence loss we seen at
+1.7 GHz (∼ 15 %), the flux density of the detected feature is
+(46.9 ± 4.7) mJy at 8.5 GHz.
+4.
+Discussion
+4.1
+Brightness temperature
+The brightness temperature of the source in the rest-frame of
+the source can be calculated with the following equation (e.g.,
+Veres et al., 2010; Hovatta et al., 2014):
+Tb = 1.22 × 1012
+S
+θmajθminν2o
+(1 + z),
+(1)
+where S is the flux density in units of Jy, νo is the observing
+frequency in unit of GHz, and θmaj and θmin are the major and
+minor axes (FWHM) of the Gaussian radio-emitting feature
+in units of mas. The brightness temperature of the modeled
+feature measured at an observing frequency of 8.5 GHz is
+Tb, νo=8.5 = (1.1 ± 0.9) × 109 K. At 1.7 GHz observing fre-
+quency, due to the poorly constrained component size, the
+brightness temperature has much larger error, Tb, νo=1.7 =
+(4.2 ± 3.1) × 1010 K. Despite the large uncertainty, Tb, νo=1.7
+exceeds Tb, νo=8.5, which would contradict the naive expec-
+tations of detecting more compact, thus of higher brightness
+temperature, emitting feature in higher-resolution VLBI ob-
+servation.
+However, the 8.5 GHz and 1.7 GHz observing frequencies
+correspond to ∼ 51.0 and ∼ 10.2 GHz rest-frame frequencies,
+respectively, at the redshift of the source (z = 5). Cheng
+et al. (2020) studied a large sample (more than 800 objects) of
+compact, bright radio-loud AGN (mostly blazars), and showed
+that Tb at 43 GHz and at 86 GHz rest-frame frequencies are
+below the values obtained at lower rest-frame frequencies,
+between 2 GHz and 22 GHz, due to synchrotron opacity effect.
+By analysing data from large multi-frequency VLBI sur-
+veys, Cheng et al. (2020) found that the frequency-dependence
+dPosition angles are measured from north through east.
+
+Publications of the Astronomical Society of Australia
+5
+of the core brightness temperature can be well described with
+a broken power law (up until 240 GHz), with the maximum
+brightness temperature reached at the break frequency of
+∼ 6.8 GHz. Using their best fit parameters for the shape
+of the curve, we obtain a brightness temperature value of
+(6.4 ± 2.0) × 1010 K at the break frequency of 6.8 GHz (rest-
+frame, corresponding to 1.1 GHz observing frequency). If,
+instead, we fit for both the break frequency, νj and the bright-
+ness temperature at νj, we obtain (20.5 ± 8.2) × 1010 K at a
+rest-frame frequency νj = (3.6 ± 0.4) GHz (corresponding to
+an observing frequency of 0.6 GHz).
+The brightness temperature values of J0141–5427 obtained
+at rest-frame frequencies of 51 and 10.7 GHz clearly indi-
+cate that the radio emission is related to the activity of the
+central supermassive black hole in an AGN, and cannot be
+explained by star formation in the host galaxy (Condon, 1992).
+At face value, they do not exceed the theoretical equipartition
+limit of ∼ 5 × 1010 K of Readhead (1994) and the empiri-
+cally found median intrinsic brightness temperature of blazar
+sources, 4.1 × 1010 K (Homan et al., 2021). Thus, Doppler
+boosting is not crucially needed to explain the brightness tem-
+perature values, however it cannot be ruled out. On the other
+hand, taking the decrease in brightness temperature at high
+rest-frame frequencies well above the break frequency into
+account, the measured brightness temperatures are compatible
+with those of a slightly Doppler boosted blazar source.
+Brightness temperature values significantly below the equipar-
+tition limit usually correspond to physical processes in evolved
+plasma regions and not in the compact regions of blazar jets.
+However, such low brightness temperatures can also be mea-
+sured in blazars due to insufficient resolution, when the core
+and a close jet component cannot be resolved and thus the
+fitted size is larger, resulting in lower Tb value. Hovatta et al.
+(2014) studied 190 blazar jets of the Monitoring Of Jets in
+Active galactic nuclei with VLBA Experiments (MOJAVE, Lis-
+ter et al., 2019) survey, and showed that in sources at higher
+redshiftse the derived core parameters are more likely con-
+taminated by a neighbouring jet component due to the lower
+effective linear resolution.
+Interestingly, high brightness temperature values, close to
+the equipartion limit are rarely observed in other z > 5 blazars
+(Coppejans et al., 2016; Zhang et al., 2022), with the notable
+exception of J0906+6930 (An et al., 2020).
+4.2
+Flux density and spectral index
+Compared to lower-resolution radio observations of J0141–5427,
+there is a significant difference in the recovered flux density.
+According to the AT20G survey, the source had a flux density
+of (70.0 ± 4.0) mJy at 8.6 GHz, measured between 2004 and
+2008 with the ATCA (Chhetri et al., 2013). This discrepancy
+can be due to resolution effect, i.e. the LBA observation re-
+solving out a significant fraction of extended radio emission,
+and/or source flux density variability in time.
+eThe highest-redshift objects of this sample are at z ≈ 3.3 (Hovatta et al.,
+2014).
+We can derive the spectral index (α) of the compact radio-
+emitting feature between 1.7 GHz and 8.5 GHz observing
+frequencies using the flux densities obtained from the Gaussian
+model fitting to our LBA visibility data. The spectral index is
+defined as S ∝ να. For J0141–5427, α = –0.33 ± 0.13, thus it
+has a flat radio spectrum. This is a typical spectral index value
+for the core of jetted AGN (e.g., Hovatta et al., 2014) between
+8 and 15 GHz.
+0.1
+1
+10
+100
+Observed frequency (GHz)
+1
+10
+100
+1000
+Flux density (mJy)
+0.6
+6
+60
+600
+Rest-frame frequency (GHz)
+Figure 3. Radio spectrum of J0141–5427. Black circles are low-resolution
+archival measurements (for references, see Belladitta et al., 2019). Orange
+circles are from the RACS DR1 (McConnell et al., 2020; Hale et al., 2021), and
+from the SPT-SZ survey (Everett et al., 2020). Red squares are our LBA flux
+densities. The brown line represents a power-law fit to the low-resolution
+data (black and orange symbols).
+The source shows a similarly flat radio spectrum between
+76 MHz and 20 GHz in archival low-resolution radio observa-
+tions as reported by Belladitta et al. (2019). Since that publica-
+tion, the first data release of the Australian Square Kilometre
+Array (SKA) Pathfinder (ASKAP), the Rapid ASKAP Contin-
+uum Survey (RACS, McConnell et al., 2020; Hale et al., 2021)
+has become public. In addition, we included high-frequency
+radio flux density measurements obtained with the South
+Pole Telescope (SPT) within the framework of SPT Sunyaev–
+Zeldovich survey (SPT-SZ, Everett et al., 2020). The most
+complete radio spectrum of J0141–5427 is shown in Fig. 3.
+J0141–5427 has been detected in RACS as a single-component
+source with a flux density of (174.0 ± 13.0) mJy at 888 MHz
+(Fig. 4). This value agrees within the errors with the closest-
+frequency measurements taken at 843 MHz by the Sydney
+University Molonglo Survey (SUMSS, Mauch et al., 2003) in
+2002. At higher frequencies, J0141–5427 was detected in all
+three bands of the SPT-SZ, at 95 GHz, 150 GHz, and 220 GHz
+(however, at the highest frequency only with a signal-to-noise
+ratio of 2.9) with flux densities of S95 = (19.1 ± 2.2) mJy,
+S150 = (11.2 ± 1.2) mJy, and S220 = (10.1 ± 4.2) mJy, respec-
+tively. These measurements indicate a possible steepening of
+the radio spectrum at high frequencies. However, the broad-
+band radio spectrum is still flat with α220
+0.076 = –0.39 ± 0.02.
+The observing frequencies of SPT correspond to rest-frame
+
+6
+K. É. Gabányi et al.
+Declination (J2000)
+Right Ascension (J2000)
+01 41 40
+38
+36
+34
+32
+30
+28
+26
+24
+-54 27 00
+15
+30
+45
+28 00
+15
+30
+45
+Figure 4. ASKAP image of J0141–5427 at 888 MHz from RACS (McConnell
+et al., 2020; Hale et al., 2021). Peak brightness is 162.6 mJy beam–1. The
+lowest contours are drawn at ±0.68 mJy beam–1 corresponding to an image
+noise level of 3σ, further positive contour levels increase by a factor of two.
+The restoring beam is 17.84′′ × 11.28′′ at a major axis position angle of
+–46.7◦, as shown in the lower lef corner of the image.
+frequencies of 570 GHz, 900 GHz, and 1320 GHz, where the
+emission from the dust in the host galaxy may have a growing
+contribution to the measured flux density (Planck Collabora-
+tion et al., 2016; Massardi et al., 2022).
+There is no sign of spectral turnover of the radio spectrum
+at observed frequencies around 0.6 GHz and 1.0 GHz corre-
+sponding to the rest-frame turnover values estimated from the
+brightness temperatures (see Sect. 4.1). There is a hint of spec-
+tral flattening at around a few hundred MHz measured by the
+GaLactic and Extragalactic All-sky MWA Survey (GLEAM)
+(Belladitta et al., 2019), which is followed by a steepening at
+lower observed frequencies, below ∼ 130 MHz. However, this
+apparent rise of the flux density with decreasing frequency
+(and thus decreasing angular resolution) could be caused by
+source confusion; according to Franzen et al. (2019), confusion
+is the limiting noise factor at ≲ 100 MHz in the GLEAM data.
+The effect of confusion, the target source being blended with
+its neighbours, has also been seen in lower frequency GLEAM
+data by An et al. (2022, submitted).
+4.3
+Magnetic field strength
+The magnetic field strength of a compact synchrotron self-
+absorbed source can be estimated if the frequency of the spec-
+tral turnover from the optically thick to the optically thin re-
+gion, and the flux density (Sj) and the angular size of the emit-
+ting region at the turnover point (θj) are known (Marscher,
+1983),
+B = 10–5b(α)θ4
+j ν5
+j S–2
+j
+δ
+1 + z,
+(2)
+where δ is the relativistic Doppler boosting factor, and b(α) is
+a numerical factor depending on the spectral index tabulated
+in Marscher (1983).
+Using νj = 6.8 GHz from Cheng et al. (2020) and assum-
+ing that α does not change till the turnover, we can calcu-
+late the expected flux density at this (rest-frame) frequency,
+Sj = 92.1 mJy. The size of the emitting region can be derived
+from the fitted brightness temperature as θj = 3.0 mas. Thus,
+the magnetic field strength can be given as B = 1.6δ G. Alter-
+natively, using the fitted turnover (rest-frame) frequency value
+of, νj = 3.6 GHz, one can obtain a much lower magnetic field
+strength of B = 0.083δ G.
+Since there is no indication of substantial relativistic boost-
+ing in the source, the Doppler factor is expected to have a
+value below 10, the above estimated magnetic field strength
+remains well below the one obtained by Belladitta et al. (2019),
+B ≈ 9 G. However, the value derived by Belladitta et al. (2019)
+characterizes the magnetic field strength at close proximity
+(fraction of a parsec) to the black hole, while the one esti-
+mated from the radio jet is much farther away from the central
+engine.
+Additionally, the above calculation of the magnetic field
+strength relies on the brightness temperature and size estima-
+tions, which may only be limiting values (upper limit on the
+actual source size, thus lower limit on the brightness tempera-
+ture) due to the resolution. Therefore, this can also hinder the
+comparison of the magnetic field strength derived from the
+X-ray observations and from radio data.
+4.4
+Radio power
+We can use the derived spectral index and flux densities to
+calculate the monochromatic radio powers (Hogg et al., 2002):
+Pν = 4πD2
+LSν(1 + z)–α–1
+(3)
+The obtained radio power values are P1.7 = (7.9 ± 0.8) ×
+1027 W Hz–1 and P8.5 = (4.6±1.4)×1027 W Hz–1. Compared
+to other high-redshift radio-loud AGN, J0141–5427 is among
+the most powerful ones in the radio regime (Coppejans et al.,
+2016; Sotnikova et al., 2021; Krezinger et al., 2022).
+4.5
+J0141–5427 as a potential VLBI reference source
+The sky density of known compact bright extragalactic radio
+sources suitable as VLBI calibrators at declinations below about
+–40◦ is significantly lower than at higher declinations (e.g.,
+Charlot et al., 2020). This is because most VLBI networks
+operate on the northern hemisphere. While J0141–5427 with
+its 8.5-GHz flux density of ∼ 47 mJy (Sect. 3) is not bright
+enough for the inclusion in the regular geodetic VLBI obser-
+vational programmes (e.g., Plank et al., 2017), it may serve as
+a phase-reference source for observing weaker nearby targets
+for high-resolution imaging or relative astrometric position-
+ing. This is especially true at lower frequencies, as indicated
+by the high rate of fringe-fit solutions found for J0141–5427
+in our experiment at 1.7 GHz. So far, VLBI imaging surveys
+of low-declination southern radio AGN have mainly concen-
+trated on bright sources with at least ∼ 100 mJy flux densities
+(e.g., Shen et al., 1997, 1998; Ojha et al., 2004, 2005, 2010;
+Müller et al., 2018).
+
+Publications of the Astronomical Society of Australia
+7
+5.
+Other X-ray weak blazar candidates
+Since J0141–5427 is the only known blazar candidate at high
+redshift with an intense radio but with a very weak X-ray emis-
+sion, Belladitta et al. (2019) searched for similar X-ray weak
+radio-bright blazar candidates in the local Universe using the
+5th edition of the Roma-BZCAT multifrequency catalogue of
+blazars (Massaro et al., 2009). They selected flat-spectrum radio
+sources with flux densities measured at 1.4 GHz or 843 MHz
+exceeding 1.5 Jy. All these sources have X-ray detections. The
+authors focused only on sources with 1.4-GHz radio power
+similar to that of J0141–5427. They found only two objects
+(2 % of their sample) with as low X-ray-to-radio luminosity
+ratio as for J0141–5427.
+5BZQ J2206–1835 is a quasar at redshift z = 0.619 (Morton
+& Tritton, 1982). It was observed in the prelaunch survey of
+the VLBI Space Observatory Programme by Fomalont et al.
+(2000) at 5 GHz. It was detected only at the shortest baselines
+of the Very Long Baseline Array (VLBA). In a 22 GHz VLBA
+survey, Moellenbrock et al. (1996) did not detect the source.
+Thus these high-resolution observations did not confirm the
+blazar nature of 5BZQ J2206–1835, as they failed to reveal any
+bright compact radio-emitting feature at mas scale.
+5BZQ J2038+5119, also known as 3C 418, is a quasar at
+a redshift of z = 1.686 (Spinrad et al., 1985). It was observed
+within the framework of the MOJAVE (Lister et al., 2019)
+survey at 15 GHz. It has a one-sided jet structure with ap-
+parent superluminal motion exceeding 6c (Lister et al., 2019).
+The brightness temperature of the core component is between
+4.6 × 1011 K and 5.2 × 1012 K (according to the brightness dis-
+tribution model of the jet obtained given in Lister et al., 2019),
+thus it exceeds the equipartition limit and implies Doppler
+boosting. The object was also detected in γ-rays by the Large
+Area Telescope onboard the Fermi satellite (Abdollahi et al.,
+2020).
+Thus, the three similarly weak at X-ray radio-loud AGN
+exhibit very different radio characteristics, forming a heteroge-
+neous group. One of them is a genuine relativistically boosted
+blazar, another one is not a blazar according to its VLBI ob-
+servations, and J0141–5427 has a modest measured brightness
+temperature, however, it is compact enough to be detected on
+intercontinental radio interferometric baselines.
+6.
+Summary
+Belladitta et al. (2019) reported the discovery of a radio-loud
+AGN at a redshift of z = 5, which they identified as a possible
+blazar. Contrary to the expectations, the X-ray emission of
+this source, J0141–5427, is very weak.
+We performed mas-scale resolution radio imaging obser-
+vations of J0141–5427 using the Australian LBA at 1.7 and
+8.5 GHz. We detected a single bright, compact feature at
+both frequencies. This and the flat radio spectrum of the mas-
+scale feature strengthen its blazar classification. The estimated
+brightness temperature values clearly indicate the AGN origin
+of the radio emission.
+The relatively low brightness temperature value measured
+at the rest-frame frequency of ∼ 50 GHz is in accordance with
+the findings of Cheng et al. (2020). Thus, it still allows for
+moderate relativistic Doppler boosting that could be directly
+observable at a lower frequency, in support of the blazar nature
+of the source. High-resolution VLBI imaging at observed
+frequencies below 1 GHz can sample the assumed turn-over
+region in the brightness temperature values and provide a
+Doppler factor for J0141–5427. However, such low-frequency
+(≲ 1 GHz), high-resolution observations are currently not
+achievable.
+We investigated the radio properties of two other blazar
+candidates which have similarly low X-ray-to-radio luminos-
+ity ratios as J0141–5427. We found that while one of them
+(J2038+5119) clearly shows relativistically boosted radio emis-
+sion, the other one (J2206–1835) is certainly not a blazar.
+J0141–5427 was detected in X-ray so far in only one obser-
+vation in 2005, while remained undetected in 2018 (Belladitta
+et al., 2019). Since blazars are known to show significant
+variability, a new X-ray observation may provide a better
+constraint on the high-energy properties of this source.
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+Acknowledgement
+We thank the referee for his useful feedback that have improved
+this manuscript. The Long Baseline Array is part of the Aus-
+tralia Telescope National Facility (https://ror.org/05qajvd42,
+accessed 2022.09.10) which is funded by the Australian Gov-
+ernment for operation as a National Facility managed by
+CSIRO. This work was supported by resources provided by
+the Pawsey Supercomputing Centre with funding from the
+Australian Government and the Government of Western Aus-
+tralia. The ASKAP radio telescope is part of the Australia
+Telescope National Facility which is managed by Australia’s
+national science agency, CSIRO. Operation of ASKAP is
+funded by the Australian Government with support from
+the National Collaborative Research Infrastructure Strategy.
+ASKAP uses the resources of the Pawsey Supercomputing
+Research Centre. Establishment of ASKAP, the Murchison
+Radio-astronomy Observatory and the Pawsey Supercomput-
+ing Research Centre are initiatives of the Australian Govern-
+ment, with support from the Government of Western Australia
+and the Science and Industry Endowment Fund. We acknowl-
+edge the Wajarri Yamatji people as the traditional owners of
+the Observatory site. This paper includes archived data ob-
+tained through the CSIRO ASKAP Science Data Archive,
+CASDA (https://data.csiro.au). LIG acknowledges support by
+the CSIRO Distinguished Visitor Programme. HC acknowl-
+edges support from the Hebei Natural Science Foundation
+of China (Grant No. A2022408002), and the National Natu-
+ral Science Foundation of China (Grants No. U2031116 and
+U1731103). This research was supported by the Australian Re-
+search Council Centre of Excellence for All Sky Astrophysics
+in three Dimensions (ASTRO-3D), through project num-
+ber CE170100013. KR acknowledges support from the Bun-
+desministerium für Bildung und Forschung (BMBF) award
+05A20WM4. This research was supported by the Hungar-
+ian National Research, Development and Innovation Office
+(NKFIH), grant number OTKA K134213.
+
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+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Gabányi, Email: k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='gabanyi@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  (Received dd Mmm 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' revised dd Mmm YYYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' accepted dd Mmm YYYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' first published online dd Mmm YYYY)  Abstract  Active galactic nuclei (AGN) have been observed as far as redshift z ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' They are crucial in investigating the early Universe as well as the growth  of supermassive black holes at their centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Radio-loud AGN with their jets seen at a small viewing angle are called blazars and show relativistic  boosting of their emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, their apparently brighter jets are easier to detect in the high-redshift Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' DES J014132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4–542749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9 is a  radio-luminous but X-ray weak blazar candidate at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We conducted high-resolution radio interferometric observations of this source with  the Australian Long Baseline Array at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' A single, compact radio emitting feature was detected at both frequencies with a flat  radio spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We derived the milliarcsecond-level accurate position of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The frequency dependence of its brightness temperature is  similar to that of blazar sources observed at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Based on our observations, we can confirm its blazar nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We compared its radio  properties with those of two other similarly X-ray-weak and radio-bright AGN, and found that they show very different relativistic boosting  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Keywords: active galactic nuclei – very long baseline interferometry – galaxies: high-redshift  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Introduction  Active galactic nuclei (AGN) are the most luminous persistent  astronomical objects, and they are invaluable probes for inves-  tigating the high-redshift Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Roughly ten per cent of  AGN are radio-loud, jetted sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  In them, the radio emission originates from the synchrotron  emission of the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' When the jets are seen at a small angles  to the line of sight, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', ≲ 10◦ (Urry & Padovani, 1995), rela-  tivistic beaming causes significant flux density enhancement  of the advancing jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, these beamed sources called blazars  can be preferentially detected even at high redshifts (z ≳ 4) in  radio bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Blazars can be identified using high-resolution very long  baseline interferometry (VLBI) radio observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' They are  characterized by a bright feature that is compact at milliarcsec  (mas) scale, the jet base, which usually has a flat radio spectrum  at GHz frequencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The apparent  brightness temperature of this dominant component exceeds  the equipartition limit, Teq  B ≈ 5 × 1010 K (Readhead, 1994),  and sometimes even the inverse Compton limit (∼ 1012 K,  Kellermann & Pauliny-Toth, 1969), indicating the potential  prevalence of relativistic beaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Also, apparent superlumi-  nal motion of components can often be observed in blazar  jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, in the case of high-redshift sources, the steep-  spectrum jet components are harder to detect, because the  observed frequencies correspond to (1 + z) times higher emit-  ted frequencies in the source’s rest frame, thus the extended  regions of jets are often undetectably faint (Gurvits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Blazars can also be classified via their broad-band spectral  energy distribution (SED) featuring non-thermal emission  over the electromagnetic spectrum and exhibiting relativistic  beaming effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Massaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='04533v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='GA]  11 Jan 2023  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Gabányi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019) reported the discovery of DES J014132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4–542749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9  (hereafter J0141–5427), a radio-bright but X-ray-weak AGN  at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The source, according to archival data and  newly obtained X-ray observations of the authors, is an order  of magnitude fainter in X-rays than other blazars with simi-  lar radio luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019) showed that the  SED of J0141–5427 can be best described with a relativistically  beamed blazar SED if a very high magnetic field strength of  ∼ 9 G is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We initiated VLBI observations of J0141–5427 with the  Australian Long Baseline Array (LBA) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz to  ascertain its blazar nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Hereafter we use the flat ΛCDM cosmological model  with parameters of H0 = 70 km s–1 Mpc–1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='27, and  ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At the redshift of J0141–5427, 1 mas angular size  corresponds to a projected linear length of ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 pc, and the lu-  minosity distance of the object is DL = 48273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 Mpc (Wright,  2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Observations and data reduction  Observations of J0141–5427 with the LBA were conducted in  2020, under the project code v591 (PI: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Gabányi) in phase-  referenced mode (Beasley & Conway, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In this observing  mode, the pointing directions of the telescopes change regu-  larly between the target source and a nearby phase-reference  calibrator within the atmospheric coherence time permitted  by the radio propagation media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The delay, and delay rate solu-  tions can be then transferred (interpolated) from the calibrator  to the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The nodding cycles in both the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz observations were 5 min long, with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 min spent on  the target and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 min on the calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The phase-reference  calibrator was ICRF J015649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7–543948 in both experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Additional calibrator sources were also observed to facilitate  amplitude calibration, and to monitor the stability of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz observation took place on 2020 June 26 and  27, the participating antennas were Ceduna (CD), Hobart  (HO), Mopra (MP), Parkes (PA), the tied array of the Australia  Telescope Compact Array (ATCA) in Australia, and Harte-  beesthoek (HH) in South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The observation lasted for  10 h, the on-target time was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz observation  took place on 2020 July 14, with the following participating  antennas: CD, HO, MP, PA, Katherine (KE), Yarragadee (YG),  the tied array of ATCA in Australia, the 12-m Warkworth an-  tenna (WW) in New Zealand, and HH in South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The  observation lasted for 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='25 h, with an on-target time of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  In both observations, the total bandwidth of 128 MHz was  divided into 8 intermediate frequency bands (IF) of 32 chan-  nels each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The correlator integration time was set to 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The  correlation was done at the Pawsey Supercomputing Centre in  Perth, on a DiFX software correlator (Deller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The  longest baselines of the arrays (providing the finest angular  resolution) were those to HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, HH could only  participate in the last 18 min of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Data reduction was done using the National Radio Astron-  omy Observatory (NRAO) Astronomical Image Processing  System (AIPS, Greisen 1990) following standard procedures of  ionospheric and parallactic angle corrections, manual phase  calibrations and fringe-fitting of the calibrator sources, and  following the LBA guide on amplitude calibrationa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The nec-  essary files for amplitude calibration were created from the  system temperature measurements and gain curves provided  by the participating stations or system equivalent flux densities  listed in the LBA amplitude calibration user’s guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In the  absence of system temperature measurements (at the antennas  CD and HO at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz and HO, KE, and WW at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz),  nominal system temperature values were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The fringe-fitting was performed for all calibrator sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Solutions were found for ∼ 86% and ≳ 98% of the data at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At the AT, MP, and PA antennas, wider filters were used,  resulting in clearly lower amplitude values in the channel-  averaged data at those IFs corresponding to the edges of the  bands (IFs 1, 4, 5, 8) compared to the ones at the middle (IFs 2,  3, 6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore, the edge IFs were scaled up by a constant  factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='169 to bring them closer to the values measured  in the middle of the band at these antennas in the case of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-  GHz observation, before channel averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz  observation, instead of such scaling, we flagged the first 10  channels for IFs 1 and 5 and the last 10 channels for IFs 4 and  8, for the three antennas using wider filters (AT, MP, and PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  After the fringe-fitting performed on the calibrator sources,  and the application of the above described amplitude scaling for  the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz amplitudes, the channel-averaged data of the cali-  brator sources were imported into the Caltech DIFMAP package  (Shepherd, 1997) for hybrid mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The hybrid-mapping  procedure involves subsequent steps of CLEANing (Högbom,  1974) and phase self-calibration of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' As the last step, am-  plitude self-calibration was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The gain correction factors  obtained for different calibrator sources were in good agree-  ment for the same antennas and IFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The flux density values  of the phase-reference calibrator obtained this way at both fre-  quencies were in good agreement with the ones measured by  the ATCA closest in time and at similar frequencies according  to the ATCA Calibrator Databaseb with the VLBI-measured  flux densities ∼ 15 % and ∼ 3 % lower than the ones measured  by ATCA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz and at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The dif-  ference is most probably caused by resolution effect, the LBA  observations resolved out the large-scale emission detected by  ATCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, we accepted the gain correction factors obtained  in DIFMAP for the phase-reference calibrator, and we adjusted  the antenna gains accordingly in AIPS to further improve the  amplitude calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  To improve the delay and rate solutions, the phase-reference  calibrator was fringe-fitted again using the CLEAN component  model of its brightness distribution derived from the hybrid  mapping, to take the source structure into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The ob-  tained solutions were applied to the phase-reference calibrator  as well as to the target source, and subsequently both were  ahttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='atnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='au/vlbi/dokuwiki/doku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='php/lbaops/  lbacalibrationnotes (accessed 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='06)  bhttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='narrabri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='atnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='au/calibrators/calibrator_database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='html  (accessed 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='06)  Publications of the Astronomical Society of Australia  3  imaged in DIFMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  In the case of the phase-reference calibrator, the amplitude  self-calibration performed after this second hybrid mapping  showed that the gain correction factors were mostly ≲ 10%  for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz data and mostly ≲ 5% for the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz data,  except for single IFs of AT and CD, and a few discrepant IFs of  YG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Additionally, it seemed that amplitude self-calibration of  HH was not constrained at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz, and it could not correct  the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, we conservatively assume the amplitude  calibration of these LBA data is reliable at 10% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Due to an unfortunate typing mistake made by the PI  at the time of scheduling, the observations and subsequent  correlations were done at a target source position with 4′′  offset in declination from the previously known position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' A  significant offset from the phase centre may cause reduction of  the peak intensity and distortion of the obtained image through  bandwidth smearing and time-average smearing effects (Bridle  & Schwab, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The bandwidth smearing effect would have been substan-  tial (intensity reduction of a point source by ∼ 80–90 %) if the  data were averaged over all the channels within an IF (Bridle  & Schwab, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Wrobel, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore, the hybrid map-  ping of the target source was performed on the unaveraged  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We disregarded the first and last 5 channels of all 8 IFs  to account for bandpass effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At both frequencies, time averaging was done for 2 s at  the correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, because of the different resolutions,  time-average smearing affects the two data sets differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz, this effect is negligible, the peak intensity reduction  of a point source is less than 1% at 4′′ from the pointing centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, if calculated for the highest achievable resolution  obtained with the longest baseline, between HH and YG,  time-average smearing would cause an average peak intensity  reduction of a point source by 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Excluding the baselines  to HH, the average amplitude reduction of a point source is  ∼ 5% at 4′′ distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Since HH could only participate in the  last 18 min of the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz observation, we excluded the data on  the baselines to HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore the effects of the unintentional  pointing offset introduced in the target source position could  be mitigated satisfactorily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The target source J0141–5427 turned out to be bright  enough for attempting a direct fringe-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Before that, the  visibility data set was shifted by 4′′ in declination direction  to its a priori known correct position using the task CLCOR  in AIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz, fringes with a signal-to-noise level ex-  ceeding 6σ were found for 69% of data, including the longest  baselines to HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We continued imaging both the fringe-fitted  and the phase-referenced 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz data of the target, and the  results were in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The peak intensity was less  by ∼ 10 mJy beam–1 (∼ 13 %) in the phase-referenced image  compared to the one obtained after fringe-fitting the data due  to the coherence loss (Martí-Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz,  at the same signal-to-noise level, fringes were found for only  24% of data, and no fringes were found on the baselines to  HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore, we did not use the fringe-fitted data of the  target for the higher frequency observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At both frequencies, phase self-calibration and amplitude  self-calibration were performed with subsequently shorter time  intervals during the hybrid mapping of J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However,  only the best-behaving, least noisy antennas were used in  the self-calibration processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, HH and CD were kept  fixed for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In the case of the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-  GHz observation, originally all antennas were used in phase  self-calibration (except for HH which was not used in the  hybrid mapping), but for the shortest time intervals, and in the  amplitude self-calibration, only ATCA, MP, CD, and PA were  included, while the gains of the remaining antennas were kept  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Results  At both frequencies, a single radio-emitting feature was de-  tected (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz naturally-weighted LBA map of the fringe-fitted data of  J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The peak intensity is 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 mJy beam–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The lowest contours  are at ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 mJy beam–1, corresponding to 4σ image noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Further  positive contours increase by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The elliptical Gaussian restor-  ing beam size is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 mas × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 mas at a major axis position angle of –8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9◦,  and it is shown in the lower lef corner of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We derived the coordinates of the brightest pixel at both  frequencies using the AIPS verb MAXFIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, the  right ascension and declination are RA = 1h41m32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='44937s  and Dec = –54◦27′49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9705′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We estimate that  these coordinates are accurate within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The most dom-  inant sources of the uncertainty are the positional accuracy  of the phase reference calibrator (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='37 mas in right ascension  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='33 mas in declination direction, according to the most  recent version of the Radio Fundamental Catalogc) and the  crfc_2022b, http://astrogeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='org/sol/rfc/rfc_2022b/rfc_2022b_cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='txt (ac-  cessed 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='06)  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Gabányi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz naturally-weighted phase-referenced LBA map of  J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The peak intensity is 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 mJy beam–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The lowest contours  are at ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 mJy beam–1, corresponding to 4σ image noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Further  positive contours increase by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The elliptical Gaussian restor-  ing beam size is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9 mas × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 mas at a major axis position angle of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='3◦, and  it is shown in the lower lef corner of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  astrometric errors strongly depending on the phase-reference  calibrator–target angular separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' For the latter, we conser-  vatively assumed the value derived for observations taken at  5 GHz by Chatterjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The coordinates derived  from the phase-referenced 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz observation agree with  the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz values within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Additionally,  they agree within the uncertainty with the optical position  provided in the Dark Energy Survey 2nd data release (Ab-  bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' These newly derived radio coordinates of  J0141–5427 are much more accurate than those previously  obtained from lower-resolution radio observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' the  AT20 survey with ∼ 1′′ positional accuracy (Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  To quantitatively describe the brightness distribution of  the source, we fitted the visibility data with Gaussian model  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz, a single circular Gaussian compo-  nent with a flux density of (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4) mJy and a full-width  at half-maximum (FWHM) size of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 mas can adequately  describe the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, according to Lister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2021),  the typical uncertainty of a single isolated Gaussian bright-  ness distribution component diameter is 20% of the restoring  beam FWHM size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' As such, the size of the component is  not well-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The highly elongated restoring beam  of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7-GHz experiment, major axis 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 mas in roughly  north–south direction and minor axis 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 mas in the perpen-  dicular direction, would result in an asymmetric source size  uncertainty in the two perpendicular orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In the finer  resolution east–west direction, the FWHM size of the emit-  ting feature is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5) mas, while it is not constrained in the  perpendicular direction, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1) mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Nevertheless, the com-  pactness of the radio emission is further supported by the high  percentage of fringe solutions found on the longest baselines  to HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz, an elliptical Gaussian component with a flux  density of (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1) mJy, a major and a minor axis FWHM  sizes of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8) mas and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6) mas, respectively, and  a major axis position angle of –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7◦ was needed to fit the  datad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz observation is somewhat affected by time  smearing effect as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' While the peak in-  tensity reduction of a point source may not be significant, time-  average smearing can cause distortion of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore,  we also analysed the data set by excluding the longer baselines  where the smearing effect is expected to be more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We only retained the antennas of MP, PA, ATCA, HO, and  CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We obtained the same parameters within the errors for  the fitted Gaussian brightness distribution model, suggesting  that the modeling results are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Assuming the same amount of coherence loss we seen at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz (∼ 15 %), the flux density of the detected feature is  (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7) mJy at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Discussion  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1  Brightness temperature  The brightness temperature of the source in the rest-frame of  the source can be calculated with the following equation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  Veres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2014):  Tb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='22 × 1012  S  θmajθminν2o  (1 + z),  (1)  where S is the flux density in units of Jy, νo is the observing  frequency in unit of GHz, and θmaj and θmin are the major and  minor axes (FWHM) of the Gaussian radio-emitting feature  in units of mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The brightness temperature of the modeled  feature measured at an observing frequency of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz is  Tb, νo=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9) × 109 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz observing fre-  quency, due to the poorly constrained component size, the  brightness temperature has much larger error, Tb, νo=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 =  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1) × 1010 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Despite the large uncertainty, Tb, νo=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7  exceeds Tb, νo=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5, which would contradict the naive expec-  tations of detecting more compact, thus of higher brightness  temperature, emitting feature in higher-resolution VLBI ob-  servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  However, the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz observing frequencies  correspond to ∼ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0 and ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 GHz rest-frame frequencies,  respectively, at the redshift of the source (z = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Cheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2020) studied a large sample (more than 800 objects) of  compact, bright radio-loud AGN (mostly blazars), and showed  that Tb at 43 GHz and at 86 GHz rest-frame frequencies are  below the values obtained at lower rest-frame frequencies,  between 2 GHz and 22 GHz, due to synchrotron opacity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  By analysing data from large multi-frequency VLBI sur-  veys, Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2020) found that the frequency-dependence  dPosition angles are measured from north through east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Publications of the Astronomical Society of Australia  5  of the core brightness temperature can be well described with  a broken power law (up until 240 GHz), with the maximum  brightness temperature reached at the break frequency of  ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Using their best fit parameters for the shape  of the curve, we obtain a brightness temperature value of  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0) × 1010 K at the break frequency of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 GHz (rest-  frame, corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 GHz observing frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' If,  instead, we fit for both the break frequency, νj and the bright-  ness temperature at νj, we obtain (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2) × 1010 K at a  rest-frame frequency νj = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4) GHz (corresponding to  an observing frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The brightness temperature values of J0141–5427 obtained  at rest-frame frequencies of 51 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz clearly indi-  cate that the radio emission is related to the activity of the  central supermassive black hole in an AGN, and cannot be  explained by star formation in the host galaxy (Condon, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  At face value, they do not exceed the theoretical equipartition  limit of ∼ 5 × 1010 K of Readhead (1994) and the empiri-  cally found median intrinsic brightness temperature of blazar  sources, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 × 1010 K (Homan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, Doppler  boosting is not crucially needed to explain the brightness tem-  perature values, however it cannot be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' On the other  hand, taking the decrease in brightness temperature at high  rest-frame frequencies well above the break frequency into  account, the measured brightness temperatures are compatible  with those of a slightly Doppler boosted blazar source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Brightness temperature values significantly below the equipar-  tition limit usually correspond to physical processes in evolved  plasma regions and not in the compact regions of blazar jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  However, such low brightness temperatures can also be mea-  sured in blazars due to insufficient resolution, when the core  and a close jet component cannot be resolved and thus the  fitted size is larger, resulting in lower Tb value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  (2014) studied 190 blazar jets of the Monitoring Of Jets in  Active galactic nuclei with VLBA Experiments (MOJAVE, Lis-  ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019) survey, and showed that in sources at higher  redshiftse the derived core parameters are more likely con-  taminated by a neighbouring jet component due to the lower  effective linear resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Interestingly, high brightness temperature values, close to  the equipartion limit are rarely observed in other z > 5 blazars  (Coppejans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2022), with the notable  exception of J0906+6930 (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2  Flux density and spectral index  Compared to lower-resolution radio observations of J0141–5427,  there is a significant difference in the recovered flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  According to the AT20G survey, the source had a flux density  of (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0) mJy at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 GHz, measured between 2004 and  2008 with the ATCA (Chhetri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This discrepancy  can be due to resolution effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' the LBA observation re-  solving out a significant fraction of extended radio emission,  and/or source flux density variability in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  eThe highest-redshift objects of this sample are at z ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='3 (Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We can derive the spectral index (α) of the compact radio-  emitting feature between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz observing  frequencies using the flux densities obtained from the Gaussian  model fitting to our LBA visibility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The spectral index is  defined as S ∝ να.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' For J0141–5427, α = –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='13, thus it  has a flat radio spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This is a typical spectral index value  for the core of jetted AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2014) between  8 and 15 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1  1  10  100  Observed frequency (GHz)  1  10  100  1000  Flux density (mJy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6  6  60  600  Rest-frame frequency (GHz)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Radio spectrum of J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Black circles are low-resolution  archival measurements (for references, see Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Orange  circles are from the RACS DR1 (McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021), and  from the SPT-SZ survey (Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Red squares are our LBA flux  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The brown line represents a power-law fit to the low-resolution  data (black and orange symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The source shows a similarly flat radio spectrum between  76 MHz and 20 GHz in archival low-resolution radio observa-  tions as reported by Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Since that publica-  tion, the first data release of the Australian Square Kilometre  Array (SKA) Pathfinder (ASKAP), the Rapid ASKAP Contin-  uum Survey (RACS, McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021)  has become public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In addition, we included high-frequency  radio flux density measurements obtained with the South  Pole Telescope (SPT) within the framework of SPT Sunyaev–  Zeldovich survey (SPT-SZ, Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The most  complete radio spectrum of J0141–5427 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  J0141–5427 has been detected in RACS as a single-component  source with a flux density of (174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0) mJy at 888 MHz  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This value agrees within the errors with the closest-  frequency measurements taken at 843 MHz by the Sydney  University Molonglo Survey (SUMSS, Mauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2003) in  2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' At higher frequencies, J0141–5427 was detected in all  three bands of the SPT-SZ, at 95 GHz, 150 GHz, and 220 GHz  (however, at the highest frequency only with a signal-to-noise  ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9) with flux densities of S95 = (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2) mJy,  S150 = (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2) mJy, and S220 = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2) mJy, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' These measurements indicate a possible steepening of  the radio spectrum at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, the broad-  band radio spectrum is still flat with α220  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='076 = –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The observing frequencies of SPT correspond to rest-frame  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Gabányi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Declination (J2000)  Right Ascension (J2000)  01 41 40  38  36  34  32  30  28  26  24  54 27 00  15  30  45  28 00  15  30  45  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' ASKAP image of J0141–5427 at 888 MHz from RACS (McConnell  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Peak brightness is 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 mJy beam–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The  lowest contours are drawn at ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='68 mJy beam–1 corresponding to an image  noise level of 3σ, further positive contour levels increase by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The restoring beam is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='84′′ × 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='28′′ at a major axis position angle of  –46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7◦, as shown in the lower lef corner of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  frequencies of 570 GHz, 900 GHz, and 1320 GHz, where the  emission from the dust in the host galaxy may have a growing  contribution to the measured flux density (Planck Collabora-  tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Massardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  There is no sign of spectral turnover of the radio spectrum  at observed frequencies around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 GHz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0 GHz corre-  sponding to the rest-frame turnover values estimated from the  brightness temperatures (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' There is a hint of spec-  tral flattening at around a few hundred MHz measured by the  GaLactic and Extragalactic All-sky MWA Survey (GLEAM)  (Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019), which is followed by a steepening at  lower observed frequencies, below ∼ 130 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, this  apparent rise of the flux density with decreasing frequency  (and thus decreasing angular resolution) could be caused by  source confusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' according to Franzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019), confusion  is the limiting noise factor at ≲ 100 MHz in the GLEAM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The effect of confusion, the target source being blended with  its neighbours, has also been seen in lower frequency GLEAM  data by An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2022, submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='3  Magnetic field strength  The magnetic field strength of a compact synchrotron self-  absorbed source can be estimated if the frequency of the spec-  tral turnover from the optically thick to the optically thin re-  gion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' and the flux density (Sj) and the angular size of the emit-  ting region at the turnover point (θj) are known (Marscher,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  1983),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  B = 10–5b(α)θ4  j ν5  j S–2  j  δ  1 + z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  (2)  where δ is the relativistic Doppler boosting factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' and b(α) is  a numerical factor depending on the spectral index tabulated  in Marscher (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Using νj = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8 GHz from Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2020) and assum-  ing that α does not change till the turnover, we can calcu-  late the expected flux density at this (rest-frame) frequency,  Sj = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='1 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The size of the emitting region can be derived  from the fitted brightness temperature as θj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='0 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus,  the magnetic field strength can be given as B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6δ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Alter-  natively, using the fitted turnover (rest-frame) frequency value  of, νj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 GHz, one can obtain a much lower magnetic field  strength of B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='083δ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Since there is no indication of substantial relativistic boost-  ing in the source, the Doppler factor is expected to have a  value below 10, the above estimated magnetic field strength  remains well below the one obtained by Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019),  B ≈ 9 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, the value derived by Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019)  characterizes the magnetic field strength at close proximity  (fraction of a parsec) to the black hole, while the one esti-  mated from the radio jet is much farther away from the central  engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Additionally, the above calculation of the magnetic field  strength relies on the brightness temperature and size estima-  tions, which may only be limiting values (upper limit on the  actual source size, thus lower limit on the brightness tempera-  ture) due to the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Therefore, this can also hinder the  comparison of the magnetic field strength derived from the  X-ray observations and from radio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4  Radio power  We can use the derived spectral index and flux densities to  calculate the monochromatic radio powers (Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2002):  Pν = 4πD2  LSν(1 + z)–α–1  (3)  The obtained radio power values are P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='8) ×  1027 W Hz–1 and P8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4)×1027 W Hz–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Compared  to other high-redshift radio-loud AGN, J0141–5427 is among  the most powerful ones in the radio regime (Coppejans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Sotnikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Krezinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5  J0141–5427 as a potential VLBI reference source  The sky density of known compact bright extragalactic radio  sources suitable as VLBI calibrators at declinations below about  –40◦ is significantly lower than at higher declinations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  Charlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This is because most VLBI networks  operate on the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' While J0141–5427 with  its 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5-GHz flux density of ∼ 47 mJy (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' 3) is not bright  enough for the inclusion in the regular geodetic VLBI obser-  vational programmes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Plank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2017), it may serve as  a phase-reference source for observing weaker nearby targets  for high-resolution imaging or relative astrometric position-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This is especially true at lower frequencies, as indicated  by the high rate of fringe-fit solutions found for J0141–5427  in our experiment at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' So far, VLBI imaging surveys  of low-declination southern radio AGN have mainly concen-  trated on bright sources with at least ∼ 100 mJy flux densities  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 1997, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Ojha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2004, 2005, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Publications of the Astronomical Society of Australia  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Other X-ray weak blazar candidates  Since J0141–5427 is the only known blazar candidate at high  redshift with an intense radio but with a very weak X-ray emis-  sion, Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019) searched for similar X-ray weak  radio-bright blazar candidates in the local Universe using the  5th edition of the Roma-BZCAT multifrequency catalogue of  blazars (Massaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' They selected flat-spectrum radio  sources with flux densities measured at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4 GHz or 843 MHz  exceeding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' All these sources have X-ray detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The  authors focused only on sources with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='4-GHz radio power  similar to that of J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' They found only two objects  (2 % of their sample) with as low X-ray-to-radio luminosity  ratio as for J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  5BZQ J2206–1835 is a quasar at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='619 (Morton  & Tritton, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' It was observed in the prelaunch survey of  the VLBI Space Observatory Programme by Fomalont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  (2000) at 5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' It was detected only at the shortest baselines  of the Very Long Baseline Array (VLBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' In a 22 GHz VLBA  survey, Moellenbrock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (1996) did not detect the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Thus these high-resolution observations did not confirm the  blazar nature of 5BZQ J2206–1835, as they failed to reveal any  bright compact radio-emitting feature at mas scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  5BZQ J2038+5119, also known as 3C 418, is a quasar at  a redshift of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='686 (Spinrad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' It was observed  within the framework of the MOJAVE (Lister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019)  survey at 15 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' It has a one-sided jet structure with ap-  parent superluminal motion exceeding 6c (Lister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The brightness temperature of the core component is between  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='6 × 1011 K and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='2 × 1012 K (according to the brightness dis-  tribution model of the jet obtained given in Lister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019),  thus it exceeds the equipartition limit and implies Doppler  boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The object was also detected in γ-rays by the Large  Area Telescope onboard the Fermi satellite (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Thus, the three similarly weak at X-ray radio-loud AGN  exhibit very different radio characteristics, forming a heteroge-  neous group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' One of them is a genuine relativistically boosted  blazar, another one is not a blazar according to its VLBI ob-  servations, and J0141–5427 has a modest measured brightness  temperature, however, it is compact enough to be detected on  intercontinental radio interferometric baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  Summary  Belladitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2019) reported the discovery of a radio-loud  AGN at a redshift of z = 5, which they identified as a possible  blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Contrary to the expectations, the X-ray emission of  this source, J0141–5427, is very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We performed mas-scale resolution radio imaging obser-  vations of J0141–5427 using the Australian LBA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='7 and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We detected a single bright, compact feature at  both frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This and the flat radio spectrum of the mas-  scale feature strengthen its blazar classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The estimated  brightness temperature values clearly indicate the AGN origin  of the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  The relatively low brightness temperature value measured  at the rest-frame frequency of ∼ 50 GHz is in accordance with  the findings of Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Thus, it still allows for  moderate relativistic Doppler boosting that could be directly  observable at a lower frequency, in support of the blazar nature  of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' High-resolution VLBI imaging at observed  frequencies below 1 GHz can sample the assumed turn-over  region in the brightness temperature values and provide a  Doppler factor for J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' However, such low-frequency  (≲ 1 GHz), high-resolution observations are currently not  achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  We investigated the radio properties of two other blazar  candidates which have similarly low X-ray-to-radio luminos-  ity ratios as J0141–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We found that while one of them  (J2038+5119) clearly shows relativistically boosted radio emis-  sion, the other one (J2206–1835) is certainly not a blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  J0141–5427 was detected in X-ray so far in only one obser-  vation in 2005, while remained undetected in 2018 (Belladitta  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Since blazars are known to show significant  variability, a new X-ray observation may provide a better  constraint on the high-energy properties of this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
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+page_content=' 2022, A&A, 662, L2  Acknowledgement  We thank the referee for his useful feedback that have improved  this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The Long Baseline Array is part of the Aus-  tralia Telescope National Facility (https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='org/05qajvd42,  accessed 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='10) which is funded by the Australian Gov-  ernment for operation as a National Facility managed by  CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This work was supported by resources provided by  the Pawsey Supercomputing Centre with funding from the  Australian Government and the Government of Western Aus-  tralia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' The ASKAP radio telescope is part of the Australia  Telescope National Facility which is managed by Australia’s  national science agency, CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Operation of ASKAP is  funded by the Australian Government with support from  the National Collaborative Research Infrastructure Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='  ASKAP uses the resources of the Pawsey Supercomputing  Research Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' Establishment of ASKAP, the Murchison  Radio-astronomy Observatory and the Pawsey Supercomput-  ing Research Centre are initiatives of the Australian Govern-  ment, with support from the Government of Western Australia  and the Science and Industry Endowment Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' We acknowl-  edge the Wajarri Yamatji people as the traditional owners of  the Observatory site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This paper includes archived data ob-  tained through the CSIRO ASKAP Science Data Archive,  CASDA (https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' LIG acknowledges support by  the CSIRO Distinguished Visitor Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' HC acknowl-  edges support from the Hebei Natural Science Foundation  of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' A2022408002), and the National Natu-  ral Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' U2031116 and  U1731103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This research was supported by the Australian Re-  search Council Centre of Excellence for All Sky Astrophysics  in three Dimensions (ASTRO-3D), through project num-  ber CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' KR acknowledges support from the Bun-  desministerium für Bildung und Forschung (BMBF) award  05A20WM4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
+page_content=' This research was supported by the Hungar-  ian National Research, Development and Innovation Office  (NKFIH), grant number OTKA K134213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E3T4oBgHgl3EQfdQqQ/content/2301.04533v1.pdf'}
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+Estimating Distributions with Low-dimensional Structures
+Using Mixtures of Generative Models
+Rong Tang and Yun Yang
+University of Illinois Urbana Champaign
+Abstract
+There has been a growing interest in statistical inference from data satisfying the so-called manifold
+hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a
+submanifold of much lower dimension. In machine learning, encoder-decoder pair based generative
+modelling approaches have been successful in learning complicated high-dimensional distributions
+such as those over images and texts by explicitly imposing the low-dimensional manifold structure.
+In this work, we introduce a new approach for estimating distributions on unknown submanifolds via
+mixtures of generative models. We show that conventional generative modeling approaches using a
+single encoder-decoder pair are generally unable to capture data distributions under the manifold
+hypothesis, unless the underlying manifold admits a global parametrization; however, this issue can
+be solved by using a collection of encoder-decoder pairs for learning different local patches of the
+data supporting manifold. A rigorous theoretical analysis is developed to demonstrate that the
+proposed estimator attains the minimax-optimal rate of convergence for the implicit estimation of
+data distributions with manifold structures. Our experiments show that, by utilizing parameter
+sharing, the proposed method can significantly improve the performance of conventional auto-encoder
+based generative modelling approaches with minimal additional computational efforts.
+Keywords: Autoencoder; distribution estimation; generative model; manifold; minimax-rate.
+1
+Introduction
+Modelling and estimating complicated high-dimensional distributions with low-dimensional structures
+remains one of the major challenges in modern statistical learning. Suppose we observe n i.i.d. samples
+{X1, X2, . . . , Xn} living in an ambient Euclidean space RD according to some unknown distribution
+µ∗. We wish to estimate µ∗ based on the samples for conducting statistical inference and generating
+new samples. One of the most popular nonparametric methods for distribution estimation is kernel
+density estimation (KDE). It has been shown that when µ∗ admits a density function relative to the
+Lebesgue measure of RD, and the density function is β-smooth, then KDE can achieve the optimal rate
+n−
+β
+2β+D for recovering the density value at any point in RD (Silverman, 2018; Tsybakov, 2009). However,
+the non-parametric rate n−
+β
+2β+D suffers from the curse of dimensionality as the ambient dimension D
+appears in the rate exponent and can be enormous in machine learning applications involving images and
+texts (Brock et al., 2018; Oord et al., 2016). In order to avoid this exponential blow-up of the dimension,
+a common practice is to assume some additional structure in the data so that the effective dimension of
+the data space is relatively low.
+One such structure that has attracted much attention recently is the so-called manifold hypothesis,
+which assumes the date to live on a d-dimensional submanifold M embedded in the possibly high-
+dimensional ambient space RD. Although submanifolds have more complicated geometry than the
+1
+arXiv:2301.00890v1  [stat.ME]  2 Jan 2023
+
+conventional Euclidean spaces, the manifold hypothesis is a natural assumption to make in a number of
+areas of science and technology. For example, in computer vision and medical imaging, data are usually
+images represented as vectorized pixel intensities. Although images may contain millions of pixels, it is
+usually determined by a comparatively smaller set of global characteristics such as camera projection,
+lighting condition, texture, object position and orientation. Other examples of high-dimensional complex
+data with low-dimensional manifold structures appear in natural language processing (Luo et al., 2020;
+Ling et al., 2017), protein-protein interaction detection (You et al., 2010; Terradot et al., 2004), and
+astronomy and shape analysis (Mardia, 1999; Jupp & Mardia, 2009).
+Statistical theory and methodology for modeling manifold valued data have been developed in various
+contexts (Lin et al., 2020, 2017; Zhang et al., 2022; Lan et al., 2021; Divol, 2022; Tang & Yang, 2022;
+Berenfeld et al., 2022).
+Specifically, the problem of estimating a probability measure lying on an
+unknown low-dimensional Riemannian submanifold has been studied in a number of recent works. For
+example, Divol (2022) consider a kernel density type estimator based on a preliminary step of estimating
+the volume measure of the submanifold using local polynomial estimation techniques. They prove that
+the developed estimator can achieve the minimax-optimal error bound under the Wasserstein loss. Tang &
+Yang (2022) construct a two-step estimator: the first step estimates the data supporting submanifold; and
+the second step recovers the distribution on the estimated submanifold based on wavelet type estimators.
+They also show that such an estimation is minimax-optimal with respect to certain adversarial loss
+functions.
+Berenfeld et al. (2022) develop a Bayesian procedure based on location-scale mixtures of
+Gaussians for estimating the density of data living close to an unknown submanifold with theoretical
+guarantees. However, although these existing methods are theoretically appealing, they usually have poor
+computational scalability with the ambient dimensionality and the sample size, making them costly to
+implement for modeling massive and high-dimensional real data, such as images and texts.
+Auto-encoder based deep generative modeling approaches in the machine learning literature, such as
+variational auto-encoder (VAE) (Kingma & Welling, 2013; Rezende et al., 2014; Kingma et al., 2016),
+Wasserstein auto-encoder (WAE) (Tolstikhin et al., 2019), InfoVAE (Zhao et al., 2019) and inferential
+Wasserstein generative adversarial networks (iWGAN) (Chen et al., 2022), have achieved great successes
+in generating synthetic realistic-looking images and texts, and are usually very efficient to implement.
+However, despite their empirical successes, a general theoretical framework explaining whether and how
+these generative modelling approaches benefit from the low-dimensional manifold structure is lacking,
+and it is also not clear whether these existing methods are theoretically optimal in the minimax sense.
+For example, the key step in the auto-encoder is the extraction of d (d ≪ D) latent features (via an
+encoder Q : RD → Rd) that can be used for accurately reconstructing the original data (via a decoder
+G : Rd → RD). In other words, these auto-encoder based methods implicitly assume data X to have a
+low-dimensional structure so that they can be accurately reconstructed in the sense that X ≈ G(Q(X)).
+Moreover, it is often the case that real-world data falls on a manifold that does not admit a global
+parametrization. For example, when the data space is a boundaryless manifold such as a sphere, or
+disconnected (Khayatkhoei et al., 2018). This lack of global parametrization makes conventional auto-
+encoder methods equipped with a single encoder/decoder pair incapable of recovering the entire data
+space without incurring distortions. Our empirical results (c.f. Fig. 1) also suggest that conventional
+auto-encoder based generative modelling approaches tend to generate off real-manifold samples with
+unrealistic appearances.
+In this article, we propose a new generative modelling approach for learning manifold-supported
+distributions that is theoretically minimax-optimal, computationally efficient, and empirically promising
+in generating complicated yet realistic-looking data. Unlike most existing generative modelling procedures
+that rely on the strong assumption of the existence of a global parametrization of the data space, we
+employ multiple encoder/decoder pairs, where each pair corresponds to the parametrization of a local
+2
+
+patch of the data supporting manifold. Moreover, we utilize the partition of unity technique for gluing
+local probability measures estimated in the patches to form a global estimation of the probability measure
+on the manifold. In addition, most existing methods simply plug in the data empirical distribution in
+constructing the objective function for defining a GAN (Goodfellow et al., 2014) type estimator, which
+may lead to theoretical deficiency due to the failure of taking the smoothness of the target distribution
+into account. We instead propose to plug in a smoothness-regularized version that provably improves the
+estimation accuracy. Concretely, we show that when the target distribution is α-smooth and lies in a
+β-smooth d-dimensional submanifold in RD, then the corresponding estimator �µ based on n data points
+achieves a non-asymptotic error bound of order O
+� log n
+√n ∨ n−
+α∧(β−1)+1
+2(α∧(β−1))+d �
+(here a ∧ b denotes min{a, b})
+under the 1-Wasserstein distance, which corresponds to the minimax rate modulo a logarithmic factor
+when α ≤ β − 1. The implied rate of convergence does not suffer from the “curse of dimensionality” and
+only depends on the intrinsic dimensionality d of the data. Our numerical results also show that the
+proposed method tends to be more accurate than conventional auto-encoder based generative modelling
+approaches and classic kernel density estimators for learning target distributions with low-intrinsic
+dimensional structures.
+1.1
+Notation
+We summarize some necessary notations and definitions here. For any positive integer k, we use the
+shorthand [k] := {1, 2, · · · , k}. We use ∥ · ∥p to denote the usual vector ℓp norm, and reserve ∥ · ∥ for
+the ℓ2 norm. We use Sd
+1 = {x ∈ Rd+1 : ∥x∥ = 1} to denote the d-dimensional unit sphere in Rd+1. For
+a probability measure µ, we use supp(µ) to denote its support. For any measure ν and map G, the
+push-forward measure µ = G#ν is defined as the unique measure such that µ(A) = ν(G−1(A)) holds
+for any measurable set A. For two probability measures µ, ν, the 1-Wasserstein distance between µ and
+ν is defined as W1(µ, ν) = inf
+� �
+∥x − T(x)∥1 dµ(x) : T#µ = ν
+�
+= sup
+� �
+f(x)d(µ − ν) : Lip(f) ≤ 1
+�
+,
+where Lip(f) denotes the minimal Lipschitz constant for f. When no ambiguity arises, for an absolutely
+continuous probability measure ν, we may also use ν to refer to its density function. We use P(Ω) to
+denote the set of probability measures on Ω. We use Br(x) to denote the closed ball centered at x
+with radius r under the ℓ2 distance. We use Cα
+r (Ω) to denote the set of all α-Hölder smooth functions
+with Hölder norm ∥ · ∥Cα(Ω) being bounded by r (see for example, Evans (2010a)). Similarly, we use
+Cα
+r (Ω; RD) =
+�
+f = (f1, . . . , fD) : Ω → RD �� ∀ j ∈ [D], fj ∈ Cα
+r (Ω)
+�
+to denote the vector valued function
+space counterpart.
+1.2
+Organization
+The rest of the paper is organized as follows. In Section 2, we give a brief introduction to the auto-encoder
+based generative modelling approaches and define smooth distributions on manifolds. Our proposed
+model is introduced in Section 3, and its implementation and theoretical properties are described in
+Section 4 and Section 5, respectively. Simulations and a real data application are provided in Section 6
+and Section 7.
+2
+Background
+2.1
+Auto-encoder based generative modelling approaches
+Assume i.i.d. data X(n) = {X1, X2, · · · , Xn} sampled from an unknown target distribution µ∗ over
+data space X ⊂ RD are available. In the literature of generative modelling, the target distribution µ∗
+is implicitly specified by its sampling scheme, represented by a generative model. Mathematically, a
+3
+
+generative model is defined as a pair (ν0, G), where ν0 is a distribution on a low-dimensional latent space
+Z ⊂ Rd, called generative distribution, that is easy to sample from; and G : Z → X is a map from Z to
+X, called generative map, so that if Z ∼ ν0, then G(Z) ∼ µ∗. The goal of generative modelling is to fit a
+generative model that specifies a stochastic process whose simulated data look indistinguishable from real
+data. In particular, auto-encoder based generative modelling approaches introduce a family of encoders
+Q that send the data X to the (low-dimensional) latent variables Z, and a family of decoders G that
+reconstruct the data from the latent variables, so that they jointly minimizes the following objective:
+1
+n
+n
+�
+i=1
+c
+�
+Xi, G(Q(Xi))
+�
++ Penalty term,
+where c(·, ·) is a cost function. The first component n−1 �n
+i=1 c(Xi, G(Q(Xi))) of the objective function
+corresponds to the reconstruction cost: a common choice is the squared loss c(x, y) = ∥x − y∥2
+2. This
+reconstruction cost enforces the push-forward measure of (G ◦ Q)#µ∗ to be as close to µ∗ as possible.
+On the other hand, the second component involves a penalty term for regularizing the encoder/decoder
+pair. For example, in VAE (Kingma & Welling, 2013; Rezende et al., 2014; Kingma et al., 2016), the
+penalty term is chosen to be an averaged Kullback-Leibler (KL) divergence between the latent variable
+distribution induced by the (probabilistic) encoder and a prior distribution ν0; in iWGAN (Chen et al.,
+2022), the penalty term is chosen to be an approximation to the 1-Wasserstein distance between the
+reconstructed data distribution (G ◦ Q)#µ∗ and induced distribution from the generative model G#ν0
+using prior ν0. Other approaches choose penalty terms for directly matching the encoder-induced latent
+variable distribution and a given prior in the latent space (for example, WAE, Tolstikhin et al. (2019);
+infoVAE, Zhao et al. (2019); Sliced WAE, Kolouri et al. (2018)), which leads to the following training
+objective:
+1
+n
+n
+�
+i=1
+c
+�
+Xi, G(Q(Xi))
+�
++ λ · D(Q#�µem, ν0),
+(1)
+where �µem : = n−1 �n
+i=1 δXi denotes the discrete empirical distribution of the data X(n), and D is a
+generic discrepancy metrics characterizing closeness between distributions over the latent space. We will
+call any method whose training objective takes the form as (1) a latent distribution matched auto-encoder
+(LDMAE) for future reference. Ideally, the learned decoder �G from minimizing (1) has the property
+that �µ = �G#ν0 ≈ µ∗. Comparing with approaches directly dealing with distributions on the ambient
+space (Goodfellow et al., 2014; Arjovsky et al., 2017; Khayatkhoei et al., 2018), the latent distribution
+matching schemes bring several computational benefits.
+First, the choice of D for quantifying the
+discrepancy between distributions in the latent space Z is more flexible since these distributions usually
+admit a density function relative to the Lebesgue measure over Z; in contrast, many commonly used
+discrepancies metrics such as the total variation distance, the Hellinger distance and the KL divergence
+are known to be unsuitable for characterizing closeness between nearly singular measures over the ambient
+space (Li et al., 2017; Xu et al., 2018). Second, computing a discrepancy between distributions in the
+relatively low-dimensional latent space is much more efficient and does not suffer from the curse of
+dimensionality, make the training process more stable and less time-consuming. In addition, the extra
+flexibility of D allows one to employ those metrics that have simple and explicit computational formulas,
+such as the squared maximum mean discrepancy (MMD) (Tolstikhin et al., 2019; Zhao et al., 2019) and
+the sliced Wasserstein distance (Kolouri et al., 2018).
+At the end of this subsection, we describe two important limitations of LDMAE, which motivate our
+proposed method to be introduced in Section 3. Concretely, based on the aforementioned decomposition
+perspective of the training objective (1), the estimation error of �µ from the target distribution µ∗ depends
+on two terms: (1) the “distance” between µ∗ and ( �G ◦ �Q)#µ∗; (2) the “distance” between �Q#µ∗ and ν0,
+4
+
+where �Q denotes the learned encoder from minimizing (1). Let M denote the support of the true data
+generating distribution µ∗ as a d-dimensional submanifold embedded in RD. To control the first distance,
+one needs M to have a global parametrization, that is, we can find some continuous maps G∗ : Rd → RD
+and Q∗ : RD → Rd so that for any x ∈ M, G∗(Q∗(x)) = x. This global-parametrization condition does
+not hold for many common manifolds, such as disconnected manifolds and boundaryless manifolds like
+spheres and torus. The second distance depends on how well the empirical data distribution �µem induced
+empirical latent distribution �Q#�µem can approximate the population level distribution �Q#µ∗. However,
+even though we assume that manifold M admits a global parameterization such that µ∗ = G∗
+#ν0 and
+Q∗ = (G∗)−1 for some G∗, Q∗ in the decoder and encoder families, the discrete empirical distribution
+�µem may suffer from statistical deficiency for approximating a smooth measure (Liang, 2020; Tang &
+Yang, 2022). As a result, simply plugging-in Q#�µem in the penalty term may lead to overfitting.
+2.2
+Partition of unity and distributions on manifolds
+Intuitively speaking, a manifold is a topological space that locally resembles the Euclidean space. Formally,
+we have the following mathematical definition of a manifold.
+Definition 1. A d-dimensional manifold M is defined as a topological space satisfying:(1) There exists
+an atlas on M consisting of a collection of d-dimensional charts A = {(Uλ, ϕλ)}λ∈Λ covering M, that
+is, M = �
+λ∈Λ Uλ. (2) Each chart 1(U, ϕ) in atlas A consists of a homeomorphism ϕ : U → �U, called
+coordinate map, from an open set U ⊂ M to an open set �U ⊂ Rd.
+We call a manifold M a (β-smooth) submanifold embedded on RD if M ⊂ RD, and the coordinate
+map ϕ and its inverse ϕ−1 in each chart are β-smooth maps when identified as functions defined on
+subsets of Euclidean spaces. Another useful notion related to the manifold is partition of unity.
+Definition 2. A partition of unity of a manifold M is a collection of functions {ρλ}λ∈Λ satisfying
+1. 0 ≤ ρλ ≤ 1 for all λ ∈ Λ, and �
+λ∈Λ ρλ(x) = 1 for all x ∈ M.
+2. Each point x ∈ M has a neighborhood which intersects supp(ρλ) for only finitely many λ ∈ Λ.
+Using the partition of unity, one can glue constructions in the local charts to form a global construction
+on the manifold. A partition of unity can be constructed from any open cover {Uλ}λ∈Λ of the manifold
+in a way where the partition {ρλ}λ∈Λ is indexed over the same set and supp(ρλ) ⊂ Uλ for any λ ∈ Λ.
+Such a partition of unity is said to be subordinate to the open cover {Uλ}λ∈Λ.
+For a manifold M with atlas A = {(Uλ, ϕλ)}λ∈Λ, suppose Λ is finite and we write it as Λ = [K].
+Given a partition of unity {ρk}k∈[K] subordinate to the open cover {Uk}k∈[K], one can decompose any
+distribution µ∗ on M as
+µ∗ =
+�
+k∈[K]
+ρkµ∗ =
+�
+k∈[K]
+(ϕ−1
+k )#
+�
+(ϕk)#(ρkµ∗)
+�
+,
+(2)
+where the first inequality uses �
+k∈[K] ρk(x) = 1 for all x ∈ M, and the second inequality uses the fact
+that supp(ρk) ⊂ Uk and ϕk is a homeomorphism on Uk → ϕk(Uk).2 Then if we write ν∗
+k = (ϕk)#
+�
+ρkµ∗
+Eµ∗[ρk]
+�
+,
+µ∗ can be expressed as the following (mixture of) generative models:
+µ∗ =
+�
+k∈[K]
+Eµ∗[ρk] · (ϕ−1
+k )#ν∗
+k,
+(3)
+1Subscript λ is suppressed for the simplicity of notation.
+2If (ϕk)#(ρkµ∗) admits an α-smooth density function for α ∈ [0, β − 1] relative to the Lebesgue measure on Rd for each
+k ∈ [K], then µ∗ is said to be an α-smooth distribution on M.
+5
+
+Decomposition (3) suggests that any distribution µ∗ lying on a d-dimensional submanifold embedded on
+RD whose atlas composed of at most K-number of charts belongs to the following mixture of generative
+models class:
+S∗ =
+�
+µ =
+�
+k∈[K]
+pk · (Gk)#νk
+��� Gk : Rd → RD, νk ∈ P(Rd), 0 ≤ pk ≤ 1,
+�
+k∈[K]
+pk = 1
+�
+.
+This space forms our model space representing distributions on manifolds.
+3
+Mixture of latent distribution matched auto-encoder
+From discussions in Section 2, we see that conventional auto-encoder based generative modelling approaches
+may suffer from low representation power when the target distribution to be estimated lies on a general
+submanifold without global parametrization.
+However, the property that any manifold-supported
+distribution can be expressed in the form of a mixture of generative models (c.f. decomposition (3))
+motivates us to employ multiple encoder/decoder pairs, and to use the partition of unity to glue them
+together with proper weights.
+Recall that we have a set of n i.i.d observations X(n) = {X1, . . . , Xn} sampled from the target
+distribution µ∗ lying on a d-dimensional submanifold M embedded in RD with d ≤ D. Let {Sk}k∈[K]
+be a suitably chosen open cover to M = supp(µ∗) ⊂ RD, fix a partition of unity {ρk}k∈[K] subordinate
+to {Sk}k∈[K],3 which can be chosen without the knowledge of M (c.f. Section 4). For any generic
+approximation family G consists of (G, Q, v), where G = (G1, G2, · · · , GK) with Gk : Rd → RD,
+Q = (Q1, Q2, · · · , QK) with Qk : RD → Rd, and v = (ν1, ν2, · · · , νK) with νk ∈ P(Rd), we define the
+following estimator, which we call mixture of latent distribution matched auto-encoder (MLDMAE)
+estimator:
+�µ =
+�
+k∈[K]
+�pk · ( �Gk)#�νk,
+with
+�pk = 1
+n
+n
+�
+i=1
+ρk(Xi)
+and
+( �G, �Q, �v) = arg min
+(G,Q,v)∈G
+K
+�
+k=1
+� 1
+n
+n
+�
+i=1
+c
+�
+Xi, Gk(Qk(Xi))
+�
+· ρk(Xi) + λk · D
+�
+�νk,Qk, νk
+��
+,
+(4)
+where recall that c(·, ·) is the cost function, �νk,Qk is a (smoothness-regularized) estimator to the density
+of (Qk)#( µ∗·ρk
+Eµ∗[ρk]) (the precise definition is available in Appendix C), and D(·, ·) is a generic discrepancy
+measure between distributions on the latent space. Different from conventional LDMAE estimators,
+MLDMAE can also use empirical Bayes method to select data-dependent prior distributions for local
+latent variables, which adds extra flexibility in the modeling and may potentially reduce the approximation
+error. We show in Theorem 1 that for some carefully chosen approximation family G, cost function c,
+discrepancy metric D, and smoothness-regularized estimator �νk,Qk, the resulting estimator �µ attains the
+minimax rate of convergence under the 1-Wasserstein distance as W1(�µ, µ∗) ≤ C n−
+α∧(β−1)+1
+2(α∧(β−1))+d ∨ log n
+√n
+when µ∗ is an α-smooth distribution on an unknown β-smooth d-dimensional submanifold.
+The objective function of MLDMAE can be decomposed into two parts: the reconstruction cost
+n−1 �K
+k=1
+�n
+i=1 c
+�
+Xi, Gk(Qk(Xi))
+�
+· ρk(Xi) and the penalty �K
+k=1 λk · D
+�
+�νk,Qk, νk
+�
+. We may also allow
+the latent dimension d to be different across encoder/decoder pairs over k ∈ [K]. The reconstruction cost
+aims to learn local parametrizations of the supporting manifold of µ∗ by enforcing the encoder/decoder
+pair ( �Qk, �Gk) to represent some coordinate system (ϕk, ϕ−1
+k ) of local patch Uk = M ∩ Sk of M in
+decomposition (3). Therefore, it corresponds to the support recovery of µ∗. By employing multiple
+3Here we may consider any function ρk : RD → [0, 1] so that supp(ρk) ⊂ Sk ⊂ RD and �
+k∈[K] ρk(x) = 1 for any
+x ∈ ∪k∈[K]Sk. Note that {ρk|Sk∩M}k∈[K] would form a partition of unity to M.
+6
+
+encoder/decoder pairs, the MLDMAE estimator avoids the restrictive global-parametrization assumption
+that is implicitly assumed in conventional LDMAE estimators. As a result, MLDMAE is suitable for
+a wider range of problems (see Fig. 1 for an illustration). On the other hand, the penalty term aims
+to enforce the reweighted local latent distribution ( �Qk)#
+� µ∗·ρk
+Eµ∗[ρk]
+�
+to match some member �νk in the
+pre-specified prior family, so that �νk is close to the ν∗
+k in decomposition (3).
+(a) Real data
+(b) LDMAE
+(c) MLDMAE
+Figure 1: Comparison between LDMAE and MLDMAE when the target distribution is the uniform
+distribution on a sphere. Figure (a) plots the real data, and Figures (b), (c) plot the randomly generated
+samples from the MLDMAE and LDMAE estimators respectively, based on 10000 training samples. The
+partition of unity chosen in MLDMAE is the smooth partition of unity described in Section 4 with K = 10
+and γ = 10. The discrepancy metric D(·, ·) is chosen to be the MMD with Gaussian kernel. We can
+see that LDMAE fails to capture the correct shape of a sphere. The reason is that the sphere (or any
+boundaryless manifold) requires at least two covering charts in its describing atlas. The LDMAE model
+uses a single pair of encoder/decoder, and thus it returns a curve that has start/end points. On the
+contrary, our estimator is able to learn general manifolds that can not be globally parametrized.
+In practice, instead of selecting the best data-dependent priors, we can also fix the prior as a simple
+distribution ν0 such as an isotropic Gaussian. Moreover, D(·, ·) can be chosen as certain squared maximum
+mean discrepancy (MMD) loss4 that can be efficiently computed in a closed-form formula. The k-th
+smoothness-regularized distribution �νk,Qk in (4) can be constructed by applying kernel-smoothing to its
+(weighted) empirical counterpart (Qk)#�µk
+n with �µk
+n = (n�pk)−1 �n
+i=1 ρk(Xi) δXi, leading to �νk,Qk(z) =
+(n�pk)−1 �n
+i=1 �k(z, Qk(Xi))ρk(Xi) for a suitable kernel �k.
+Note that when kernel �k is the Gaussian
+kernel �k(x, y) = (2πh)− d
+2 exp(− ∥x−y∥2
+2h
+) with bandwidth parameter h, then �νk,Qk corresponds to the
+Gaussian-smoothed distribution ( �Qk)#�µk
+n, where �Qk is the randomly perturbed encoder defined by
+�Qk(X) = Qk(X) +
+√
+h · N(0, Id).5 Employing such a smoothness-regularized distribution can be viewed
+as applying a randomized data augmentation to increase the variability of the encoded training samples,
+which mitigates potential overfitting to data and improves the generalization ability of the resulting
+estimator.
+Introducing the encoder-decoder structure as in our estimator brings several benefits. Computationally,
+the encoders turn the high-dimensional data into low-dimensional latent variables so that we only need to
+compute a penalty term over low-dimensional distributions. Therefore, the MLDMAE framework brings
+less computational burden compared with generative modelling approaches (e.g. iWGAN) that directly
+deal with distributions in the ambient space. Theoretically, when d ≪ D, the data distribution µ∗ becomes
+a singular measure in RD. As a consequence, with the information about the supporting manifold of µ∗,
+which is explicitly induced by the encoder-decoder pairs, it is possible to utilize classical techniques of
+nonparametric density estimation, such as wavelet truncation, to construct a minimax-optimal estimator.
+Specifically, the underlying true latent variable distribution (Qk)#( µ∗·ρk
+Eµ∗[ρk]) defined in the mixture of
+generative models (3) is, with high probability, absolutely continuous with respect to the Lebesgue
+measure on Rd (c.f. Lemma 4 in Appendix C), which enables us to develop smoothness-regularized
+4For a positive-definite reproducing kernel k, the MMD loss is defined as MMD2(µ1, µ2) = EX,X′∈µ1[k(X, X′)] +
+EY,Y ′∈µ2[k(Y, Y ′)] − 2EX∈µ1,Y ∈µ2[k(X, Y )]
+5Here for randomized map Q, the push forward measure Q#µ is defined as the measure so that for any measureable
+function f,
+�
+f(x) d[Q#µ] = E
+� �
+f(Q(x)) dµ
+�
+where the expectation is with respect to the randomness of Q.
+7
+
+1.00
+0.75
+0.50 
+0.25
+0.00 
+0.25
+0.50
+0.75
+1.00
+1.00
+0.75
+-0.50
+0.25
+0.00
+0.25
+0.50 
+0.75
+1.001.0
+0.5
+0.0 
+0.5
+1.0
+1.00
+0.750.500.25
+0.00
+0.25
+0.50 
+0.75
+1.001.00
+0.75
+0.50 
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.00
+0-0.75
+0.500.25
+0.00
+0.25
+0.50
+0.75
+1.00estimators by borrowing techniques from Liang (2020) and Singh et al. (2018). Indeed, as suggested
+by Liang (2020) and Tang & Yang (2022), the rate O
+�
+n− α+1
+2α+d ∨ log n
+√n
+�
+achieved by the MLDMAE estimator
+when α ≤ β − 1 is minimax-optimal up to logarithmic factor relative to the 1-Wasserstein distance.
+4
+Computation
+In this section, we discuss some important computational aspects of the proposed method.
+Choice of partition of unity: One issue we need to address is how to choose a reasonable partition of
+unity {ρk}k∈[K]. To do this, we can first run a clustering algorithm such as (mini-batch) K-means to
+the data using a sufficiently large cluster number K. Based on the clustering result, one straightforward
+choice of ρk is the indicator function 1(x ∈ k-th cluster). We can also choose a smooth partition of
+unity by the following: firstly we record the centroid of the k-th cluster as ak and the smallest radius rk
+so that data points in the k-th cluster are included in Brk(ak). Then we can construct an open cover
+{Sk = Brk+ε(ak)◦}k∈[K] where ε is a small positive number so that {Sk}k∈[K] can cover the unknown
+support of µ∗ with high probability. Given the open cover, for each k ∈ [K], we define a local partition
+function as �ρk(x) = ((rk + ε)2 − ∥x − ak∥2)γ · 1(x ∈ Sk), where γ > 1 is a tuning parameter. The resulting
+{ρk}k∈[K] forms a partition of unity for M with ρk = �ρk/
+� �K
+k′=1 �ρk′�
+for k ∈ [K]. Note that �ρk tends to
+give less weight to points away from the centroid ak for large γ.
+Choice of penalty terms: We choose the smoothness-regularized distribution �νk,Qk as the Gaussian
+kernel-smoothed version of (Qk)#�µk
+n as described in Section 3. This �νk,Qk relates to the commonly-used
+Gaussian encoder in VAE, and thus we can utilize the reparametrization trick in VAE to optimize the
+desired objective function. For the priors v, we can consider a simple distribution ν0 such as standard
+Gaussian N(0, Id) as is usually done in conventional generative modelling approaches. Moreover, to
+ensure the smoothness of the learned manifold, we can consider data-driven priors described in Remark 1
+in Section 5 below, so that νk and �νk, �
+Qk can be ensured to have matching tails. For the discrepancy metric
+D(·, ·), to prevent instability in the adversarial training, we can: (1) choose D(·, ·) to be the (squared)
+MMD characterized by a positive-definite kernel k, such as the inverse multiquadratics (IMQ) kernel
+k(x, y) = Cim/(Cim + ∥x − y∥2
+2) and the RBF kernel k(x, y) = exp(−∥x − y∥2/C); (2) when the intrinsic
+(latent) dimension is of order O(1), we can choose D(·, ·) as the 1-Wasserstein distance computed by the
+“POT” package (Flamary et al., 2021), which returns the optimal transport map between two discrete
+measures using network simplex algorithm (Bonneel et al., 2011).
+Construction of decoders and encoders: The decoders G and encoders Q can be realized through
+neural networks. However, for a large K, we will have a large number of parameters to train if we see
+each decoder/encoder as an independent neural network. To address this issue, we enable parameter
+sharing inside the set of decoders G and the set of encoders Q. Specifically, for the set of encoders, we set
+the last (output) layer to be free among the encoders {Qk}k∈[K] while other layers to be tied. Moreover,
+since the decoder-encoder structure aims to reconstruct the data, we want the decoder Gk to be close
+to the inverse of the encoder Qk. To achieve this, for the set of decoders, we oppositely set the first
+(input) layer to be free among the decoders {Gk}k∈[K] and tie other layers. The rationale behind our
+parameter sharing scheme is that in the encoder, the first (convolutional) layer focuses on “low-level”
+features extraction and other layers extract “high-level” features. Therefore, the described parameter
+sharing scheme enables the networks to leverage the common “low-level” information among different
+clusters of the data, hence improves the model training efficiency.
+Optimization of objective function: Recall that �νk,Qk = �Qk(�µk
+n), where �Qk is a randomized perturbed
+encoder �Qk(X) = Qk(X) +
+√
+h · N(0, Id) and �µk
+n is the re-weighted empirical measure �µk
+n = �µn·ρk
+�pk . Given
+a discrepancy metric D(·, ·) and cost function c(·, ·) defining the penalty and reconstruction cost, we can
+8
+
+rewrite the objective function as
+K
+�
+k=1
+�
+�pk
+�
+c
+�
+x, Gk(Qk(x))
+�
+d�µk
+n + λk · D
+�
+νk, ( �Qk)#�µk
+n
+��
+.
+To approximate the gradient of this objective function, we sample from the measure �µk
+n by introducing
+auxiliary random variables u ∈ Unif(0, 1). Specifically, we include data Xi into the empirical measure �µk
+n
+if u < ρk(Xi) and otherwise we exclude the data from �µk
+n. Moreover, the penalty term can be estimated
+using finite samples from νk and ( �Qk)#�µm
+n .
+Algorithm 1: Algorithm for implementing MLDMAE
+Input: Regularization coefficient {λk}k∈[K], partition of unity {ρk}k∈[K], discrepancy metric
+D(·, ·) and cost function c(·, ·), priors {νk}k∈[K], latent (intrinsic) dimension d;
+Data:X(n) = {X1, X2 · · · , Xn};
+repeat
+Sample a mini-batch dataset D from X(n);
+for k ← 1 to K do
+Initialize an empty dataset �
+Dk;
+for X ∈ D do
+Generate random variable u form Unif(0, 1);
+if u ≤ ρk(X) then
+Add X to dataset �
+Dk;
+Generate dataset Lk from prior νk;
+Generate dataset Ek from N(Qφ,k(X), h Id) for X uniformly picked in �
+Dk.
+Update φ and θ by one step first-order method (e.g., Adam, Kingma & Ba (2014)) with
+objective function:
+K
+�
+k=1
+� �pk
+| �
+Dk|
+�
+X∈ �
+Dk
+c
+�
+X, Gθ,k(Qφ,k(X))
+�
++ λk · D
+�
+1
+|Lk|
+�
+z∈Lk
+δz,
+1
+|Ek|
+�
+z∈Ek
+δz
+��
+,
+where δz is the Dirac measure on point z.
+until (φ, θ) converges;
+Based on the above discussion, we can now develop the algorithm as described in Algorithm 1
+for implementing the MLDMAE estimation, where we use Gθ = {Gθ,1, Gθ,2, · · · , Gθ,M} and Qφ =
+{Qφ,1, Qφ,2, · · · , Qφ,M} to denote the decoders and encoders parametrized by θ and φ respectively.
+5
+Theoretical Analysis
+In this section, we derive the finite sample error of the MLDMAE estimator. We first state the following
+assumptions on the target distribution µ∗ and the approximation family used in defining the estimator (4).
+Assumption A (Target distribution): The target distribution µ∗ on manifold M satisfies that: (1)
+M ⊂ ∪k∈[K]Sk; (2) for any k ∈ [K], there exists G∗
+k ∈ Cβ
+L(Rd; RD) and Q∗
+k ∈ Cβ
+L(RD; Rd) so that
+x = G∗
+k(Q∗
+k(x)) holds for any x ∈ M ∩ Sk; (3) for any m ∈ [K], let pk = Eµ∗[ρk], then pk > 0 and
+ν∗
+k = (Q∗
+k)#( µ∗·ρk
+pk ) ∈ Cα
+L(Rd); let Ωk = Q∗
+k(M ∩ Sk), there exists a function gk : R+ → R+, so that for
+any r > 0 and z ∈ Ωk, there exists z′ ∈ Ωk such that z ∈ Br(z′) and ν∗
+k(z′) ≥ gk(r).
+Assumption B (Approximation family): The approximation family G satisfies that (1) (G∗, Q∗, v∗) ∈
+G with G∗ = (G∗
+1, G∗
+2, · · · , G∗
+K), Q∗ = (Q∗
+1, Q∗
+2, · · · , Q∗
+K) and v∗ = (ν∗
+1, ν∗
+2, · · · , ν∗
+K); (2) for any
+9
+
+(G, Q, v) ∈ G, it holds that Gk ∈ Cβ
+L(Rd; RD) and Qk ∈ Cβ
+L(RD; Rd) for any k ∈ [K].
+Example (Manifold-supported distributions): For any α-smooth distribution µ∗ on a β-smooth
+d-dimensional boundaryless compact submanifold embedded in RD and with a positive density, we
+can find a suitable open cover {Sk}k∈[K] and partition of unity {ρk}k∈[K] so that Assumption A holds.
+Moreover, the (mixture of) generative model class induced by the approximation family G =
+�
+(G, Q, v) :
+∀k ∈ [K], Gk ∈ Cβ
+L(Rd; RD), Qk ∈ Cβ
+L(RD; Rd), νk = (Vk#ν0)·ρk(Gk(z))
+Eν0[ρk(Gk(Vk(z)))], Vk ∈ Cα+1
+L
+(Bd
+1; Bd
+1)
+�
+with ν0 being
+any fixed α-smooth distribution on Bd
+1 whose density value bounded away from zero (e.g., uniform
+distribution), suffices to model the manifold-supported distributions µ∗ (i.e., Assumption B holds for the
+approximation family G). In particular, when α + 1 = β, we can consider the compositions Gk ◦ Vk as
+β-smooth encoders for preventing the estimation of priors, that is, we can use the approximation family
+G =
+�
+(G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ
+L(Rd; RD), Qk ∈ Cβ
+L(RD; Rd), νk =
+ν0·ρk(Gk(z))
+Eν0[ρk(Gk(z))]
+�
+. Further details are
+available in Appendix B.
+Remark 1. The choice of the approximation family suggests that, for learning a manifold-supported
+distribution, instead of taking the priors νk to be some fixed simple distribution ν0, we may rescale ν0 by
+the weight ρk(Gk(·)) so that the resulting distribution has a matching tail as Qk#(ρkµ∗/Eµ∗[ρk]) after con-
+vergence. However, incorporating such a prior family in MLDMAE may lead to an unstable training due to
+the high irregularity of functions ρk. To address this issue, we can consider “data-driven” priors as follows:
+we first fix νk to be some simple fixed distribution ν0, such as uniform distribution or truncated normal,
+then we run the MLDMAE algorithm to obtain estimators of encoder/decoder pairs {( �G[1]
+k , �Q[1]
+k )}k∈[K].
+Now we fix the priors νk to be ν0 rescaled by ρk(G[1]
+k (·)), and run the MLDMAE algorithm with initializa-
+tion {( �G[1]
+k , �Q[1]
+k )}k∈[K] to obtain estimators of encoder/decoder pairs {( �G[2]
+k , �Q[2]
+k )}k∈[K]. The above steps
+can continue by fixing the priors νk to be ν0 · ρk(G[l]
+k (·)) and obtaining estimators {( �G[l+1]
+k
+, �Q[l+1]
+k
+)}k∈[K],
+and stop until no improvement in validation error is seen. Using such a data-driven prior can largely
+improve the performance of MLDMAE at the intersections of the support of different partition functions,
+see Fig. 2 for an illustration.
+(a) MLDMAE: truncated normal prior (b) MLDMAE: data-driven prior (once
+update)
+(c)
+MLDMAE:
+data-driven
+prior
+(twice updates)
+Figure 2: Performance of MLDMAE with different choices of priors when the target measure is the
+uniform distribution on a sphere. Figure (a) plots the generated samples from MLDMAE estimator when
+the priors are truncated normal. Figures (b) and (c) plot the generated samples from MLDMAE estimator
+with data-driven priors described in Remark 1 under once and twice updates respectively, where ν0 is the
+truncated normal as in Figure (a). We can see with a simple truncated normal prior, the generated plot
+tends to be non-smooth at the intersection of different partition functions. While once update of priors
+using the strategy described in Remark 1 can lead to much better performance.
+Example (Distributions with clustering structures): Another example is a distribution induced
+by the mixture of generative models µ∗ = �K
+k=1 pk · (G∗
+k)#ν∗
+k, where supports of generative models are
+disjoint. In this case, the supporting manifold of µ∗ is a disconnected manifold, and we can simply
+10
+
+10
+0.5
+ 0°0
+0.5
+1.0
+1.00
+00.750.50
+-0.25
+0.00
+0.25
+0.50
+0.75
+1.001.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.00
+0.75
+-0.50
+0.25
+0.00
+0.25
+0.50 
+0.75
+1.001.00
+0.75
+0.50 
+0.25 
+0.00 
+0.25
+0.50
+0.75
+1.00
+1.00
+0.75
+0.50 0.25
+0.00
+0.25
+0.50
+0.75
+1.00choose {Sk}k∈[K] to be any disjoint sets that can cover each support of the generative model (i.e.,
+supp((G∗
+k)#ν∗
+k) ⊂ Sk for k ∈ [K]), and take ρk to be the indicator function 1(x ∈ Sk). With such choices,
+Assumption A holds if for each k ∈ [K], ν∗
+k is α-smooth with a compact support, and G∗
+k is β-smooth
+with a β-smooth inverse.
+Theorem 1. Fix α ≥ 0, β ≥ 1 and a partition of unity {ρk}k∈[K] subordinate to the open cover
+{Sk}k∈[K]. Suppose the target distribution µ∗ satisfies Assumption A and the approximation family G
+satisfies Assumption B. If we choose the discrepancy metric D(·, ·) to be the 1-Wasserstein distance W1
+and cost function c(·, ·) to be the squared ℓ2 loss, then there exists a choice of regularization coefficients
+{λk}k∈[K] so that for large enough n,
+1. if α = 0, then by choosing the re-weighted empirical measure �νk,Qk =
+1
+�pkn
+�n
+i=1 δQ(Xi)ρk(Xi) with
+�pk = 1
+n
+�n
+i=1 ρk(Xi) as the plug-in, the resulting estimator �µ satisfies with probability at least 1−n−1
+that
+W1(�µ, µ∗) ≤ C n− 1
+d ∨ log n
+√n .
+(5)
+2. if α > 0, β > 1, then there exists a smoothness-regularized empirical measure �νk,Qk as the plug-in,
+so that the resulting estimator �µ satisfies with probability at least 1 − n−1 that
+W1(�µ, µ∗) ≤ C n− (α∧(β−1))+1
+2(α∧(β−1))+d ∨ log n
+√n .
+(6)
+The smoothness-regularized empirical measure �νk,Qk adopted in the proof of Theorem 1 can either be
+based on wavelet truncation or kernel density estimator of the measure Qk#( µ∗·ρk
+pk ). Based on the minimax
+lower bound developed in Liang (2020) and Tang & Yang (2022), the convergence rate in Theorem 1 is
+minimax-optimal relative to the 1-Wasserstein distance when α ≤ β −1. If Assumption A holds for K = 1,
+then statement (5) can provide a theoretical guarantee to the LDMAE estimator. The ambient dimension
+D does not appear in the exponent of the developed rate; thus Theorem 1 shows the adaptiveness of
+MLDMAE to low-dimensional submanifold structures since the bound does not suffer from the “curse of
+dimensionality”. Moreover, the MLDMAE estimator can take advantage of the smoothness of the target
+measure to further enhance the estimation accuracy by regularizing the empirical measure in the penalty.
+6
+Simulation
+In this section, we present some visual results of the MLDMAE approach when apply to common manifolds:
+2D-spiral and torus. The precise data generating distributions are given in Appendix A. The penalty
+term D(·, ·) is chosen to be the 1-Wasserstein distance computed by the “POT” package (Flamary et al.,
+2021) and the cost function c(·, ·) is the squared ℓ2 loss. As benchmarks, we also consider (1) the classic
+kernel density estimator (KDE) with Gaussian kernel that is commonly employed in statistics literature
+for density estimation; (2) the LDMAE estimator with the same kind of cost function and discrepancy
+metric as MLDMAE. The generated samples from the generators learned by different approaches are
+given in Fig. 3. The 1-Wasserstein distance between the estimated distribution and the true distribution
+are given in Table 1. We can see that employing multiple encoders and decoders can lead to much better
+performance than employing a single pair of encoder and decoder. In particular, we can see even though
+we increase the complexity of the encoder/decoder family, LDMAE still can not capture the correct shape
+of these standard manifolds in statistics. On the other hand, by allowing multiple encoders/decoders
+in MLDMAE, the manifold structure can be correctly learned with a relatively simple encoder/decoder
+structure and a smaller number of training parameters. Moreover, MLDMAE can beat the classic KDE
+in both examples.
+11
+
+Training sample
+KDE
+MLDMAE (1-NN)
+LDMAE (1-NN)
+LDMAE (2-NN)
+Figure 3: The figure illustrates the performance of MLDMAE, LDMAE and KDE when the target
+measures lying on a spiral and torus. We observe n = 1000 and n = 3000 training points for the example
+of spiral (Top row) and torus (Bottom row) respectively. The first column plots the training samples. The
+second column plots the generated samples from the classic kernel density estimator, which corresponds
+to adding Gaussian noises to the original training samples. The third column plots the generated samples
+from our proposed MLDMAE estimator (K = 10 for spiral and K = 15 for torus), the encoders and
+decoders are parameterized by one-hidden layer neural networks with hidden layer size being 128, the
+partition of unity is chosen to be the smooth partition of unity described in Section 4 with γ = 10, the
+priors are selected as described in Remark 1 with ν0 being a truncated normal. The fourth and fifth
+columns plot the generated samples from the LDMAE estimator where the encoder-decoder pair are
+parameterized by one-hidden layer and two-hidden layer neural networks respectively.
+KDE
+MLDMAE
+LDMAE (1-NN)
+LDMAE (2-NN)
+Spiral
+W1 distance
+0.2119
+0.1936
+0.6315
+0.4666
+Number of training parameters
+/
+4492
+1027
+34051
+Torus
+W1 distance
+0.2837
+0.2335
+0.9101
+0.6193
+Number of training parameters
+/
+10529
+1412
+34565
+Table 1: The table gives the 1-Wasserstein distance between the target measure and the distribution
+estimators of different approaches for the spiral and torus examples. We also provide numbers of training
+parameters for different methods.
+7
+Real Data Application
+In this section, we empirically evaluate the proposed MLDMAE approach using three real-world datasets:
+MNIST handwritten digit (LeCun et al., 1995), Fashion-MNIST (Xiao et al., 2017) and CelebA 64 ×
+64 (Krizhevsky et al., 2009). For comparison, we consider LDMAE approach and variational auto-enocoder
+(VAE) (Kingma & Welling, 2013; Rezende et al., 2014; Kingma et al., 2016), which are commonly used auto-
+encoder based generative modelling approaches. In all reported experiments, we set the reconstruction cost
+to be the squared ℓ2 loss, and fix the priors νk to be a standard Gaussian N(0, Id). The partition of unity
+is chosen to be the indicator functions described in Section 4. The encoders and decoders are modelled by
+convolutional deep neural networks with parameter sharing as described in Section 4, and further details
+are available in Appendix A. We consider two kinds of discrepancy metric D(·, ·) in the penalty terms
+of LDMAE and MLDMAE, one is the 1-Wasserstein distance computed through “POT” package, and
+the other one is MMD with the inverse multiquadratics (IMQ) kernel k(x, y) = 2d/(2d + ∥x − y∥2
+2). As
+described in Tolstikhin et al. (2019), the inverse multiquadratics kernel has a much heavier tail than the
+12
+
+0
+-2
+-4
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+10
+1.51.5
+Z Label
+_4
+4
+3
+2
+0
+X Label
+1
+0
+-1
+2
+-2
+-3
+m
+41.5
+Z Label
+4
+3
+0
+2
+1
+2
+X Label
+0
+1
+-2
+4
+3
+-3
+-44
+2
+0
+-2
+-4
+-6
+-2 5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.51.5
+Z Label
+0.9
+4
+3
+0
+2
+1
+X Label
+0
+1
+-2
+-3
+4
+3
+-44
+2
+0
+-2
+-4
+-6
+-2.5
+2.0
+1.5
+1.0
+0.5
+0'0
+0.5 
+10
+1.5Z Label
+-4
+4
+0
+2
+X Label
+0
+2
+Y Label
+2
+-42
+0
+-2 
+-4
+-6 
+-8
+-10
+-2
+0
+61.5
+Z Label
+0.5
+4
+六人
+3
+3
+2
+1
+0
+X Label
+1
+-1
+2
+-2
+w
+-3
+-4
+44
+2
+0
+-2
+-4
+-6
+-2
+-1
+0
+1
+2
+3conventional RBF kernel k(x, y) = exp(−∥x − y∥2/C), so it can provide more meaningful gradients for
+outliers. The numbers K of clusters for MLDMAE are selected so that the number of free parameters
+is around 1
+5 ∼ 1
+2 of the number of sharing parameters, and it turns out K = 5 works well for all the
+examples. Another important factor is the latent dimension, which is not explicit for the real dataset.
+Selecting a too small latent dimension would lead to large reconstruction errors and thus result in noisy
+generated samples. On the contrary, selecting a too large latent dimension would lead to the singularity
+of the encoded distribution and thus result in numerical instabilities. We use d = 4 for Fashion-MNIST,
+d = 8 for MNIST handwritten digit and d = 64 for CelebA, which seems to work reasonably well, and
+trainings of MLDMAE are stable and robust to initialization.
+To quantitatively assess the MLDMAE estimator, for the dataset of MNIST handwritten digit and
+Fashion-MNIST, we consider two kinds of evaluation metrics: one is the test log-likelihood (Test LL) used
+in Goodfellow et al. (2014) by fitting a Gaussian Parzen window to the generated samples and reporting
+the log-likelihood evaluated at the test samples, and the other one is the 1-Wasserstein distance (W1)
+between the test samples and generated samples. The generated samples are shown in Fig. 4 and the Test
+LL and W1 distance are provided in Table 2. We can see for the Fashion-MNIST dataset, MLDMAE
+with W1 or MMD penalty obviously outperforms VAE and LDMAE. For the MNIST handwritten digit
+dataset, MLDMAE with MMD penalty outperforms the Wasserstein penalty in the W1 metric, and it
+may attribute to the fact that the MNIST digit dataset has a relatively larger latent dimension d = 8, so
+we need a very large batch size for accurately estimating the Wasserstein penalty, which will reduce the
+number of gradient updating. In addition, Fig. 5 gives the trends of the Test LL and W1 distance as the
+cluster number K increases for the MNIST digit dataset. We can see the trends of both metrics become
+smooth when M ≥ 10, this is consistent with the underlying fact that the MNIST digit dataset contains
+10 digits (clusters). Furthermore, we can see an obvious improvement in both metrics when increasing K
+in the range of [1, 10], while the total number of training parameters only increases by 7% when cluster
+number K increases by 1.
+Table 2: MNIST handwritten digit and Fashion-MNIST dataset:
+Test LL and W1 distance for different approaches, “+MMD” and
+“+W1” represent the choices of the discrepancy metric D(·, ·) in
+the penalty terms.
+Fashion-MNIST
+MNIST
+Test LL ↑
+W1 ↓
+Test LL ↑
+W1 ↓
+VAE
+533
+77.7
+393
+63.2
+LDMAE+MMD
+549
+75.5
+381
+63.0
+LDMAE+W1
+562
+74.6
+400
+68.1
+MLDMAE+MMD
+557
+66.0
+443
+56.6
+MLDMAE+W1
+562
+65.8
+456
+63.1
+Table 3: CelebA dataset: FID and
+KID for different approaches.
+The
+penalty term in LDMAE and MLD-
+MAE are chosen to be MMD penalties.
+FID ↓
+KID ↓
+VAE
+63.0
+0.063
+LDMAE+MMD
+55.5
+0.057
+LDMAE+SW1
+62.8
+0.064
+MLDMAE+MMD
+51.1
+0.051
+MLDMAE+SW1
+52.0
+0.052
+13
+
+VAE
+LDMAE+MMD
+LDMAE+W1
+MLDMAE+MMD
+MLDMAE+W1
+Figure 4: The generated samples from different approaches for Fashion-MNIST (Top row) and MNIST
+handwritten digit dataset (Bottom row). The first column corresponds to the VAE estimator; the second
+and third columns correspond to the LMDAE estimator with MMD and W1 penalty respectively; the
+fourth and firth columns correspond to the MLMDAE estimator with MMD and W1 penalty respectively.
+Figure 5: Negative test log-likelihood (red) and W1 distance (black) of MLDMAE with MMD penalty
+and different cluster numbers K for the MNIST handwritten digit dataset.
+For the celebA dataset, since the ambient dimension D = 64 × 64 × 3 is extremely large. Instead
+of using test log-likelihood or W1 distance, which are evaluated in the ambient space; we consider
+two commonly-used metrics for color image data: FID (Heusel et al., 2017) and KID (Bińkowski
+et al., 2018), in which the original high-dimensional data is fed into an ImageNet-pretrained inception
+network to obtain 2048-dimensional inception (feature) representations, and the FID and KID are the
+fréchet distance and the squared MMD between inception representations of generated samples and test
+samples, respectively. Moreover, as described previously, the W1 penalty is unsuitable for large intrinsic
+dimensions, for avoiding the curse of dimensionality, we consider the so-called sliced Wasserstein distance:
+SW1(µ, ν) := Eθ∼Unif(Sd−1
+1
+)
+�
+W1(Projθ#µ, Projθ#ν)
+�
+, where Projθ denotes the projection function to the
+direction θ and Unif(Sd−1
+1
+) denotes the uniform distribution on Sd−1
+1
+. The expectation over Unif(Sd−1
+1
+)
+can be estimated by Monte Carlo method. The sliced-Wasserstein distance slices high-dimensional
+probability densities into sets of one-dimensional marginal distributions and compare these marginal
+distributions via the Wasserstein distance, it has similar qualitative properties to the Wasserstein distance,
+but is much easier to compute. The generated samples and FID, KID are given in Fig. 6 and Table 3.
+14
+
+9
+Test LL
+W1
+460
+Wasserstein distance
+Test Log-likelihood
+440
+420
+8
+400
+6
+5
+1
+2
+4
+6
+8
+10
+12
+14
+Cluster number50
+100
+150
+200
+250
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+9
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+50
+100
+150
+200
+25050 
+3
+100
+150
+200
+250
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+d5@337280)
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+0
+50
+100
+150
+200
+25050
+100
+150
+200
+250
+50
+100
+150
+200
+250We can see that the MLDMAE estimator can achieve the best performance under all evaluation metrics.
+The MLDMAE with MMD penalty performs slightly better than the SW1 penalty, while it also requires
+more computation time (the computation time for MLDMAE+MMD is 150s per epoch using NVIDIA
+A100-SXM4-40GB GPU, while that is 120s per epoch for MLDMAE+SW1).
+(a) VAE
+(b) LDMAE+MMD
+(c) LDMAE+SW1
+(d) MLDMAE+MMD
+(e) MLDMAE+SW1
+Figure 6: The generated samples from different approaches (first column: VAE; second and third columns:
+LDMAE with MMD and SW1 penalty respectively; fourth and fifth columns: MLDMAE with MMD and
+SW1 penalty respectively) for the CelebA dataset.
+8
+Conclusion
+In this work, we proposed a new approach, mixture of latent distribution matched auto-encoder (MLD-
+MAE), to improve the conventional auto-encoder based generative modelling approaches for learning
+manifold-supported distributions. We showed theoretically that the proposed estimator can learn manifold-
+supported distributions with a minimax-optimal convergence rate. Moreover, we conducted experiments
+to show that by employing multiple encoder/decoder pairs, the estimators derived from MLDMAE can
+substantially boost the target distribution estimation accuracy. In our theoretical analysis, we consider
+the case where the penalty term is chosen to be the Wasserstein distance. We leave the theoretical
+analysis for some other adversarial losses, such as the MMD distance considered in our experiments, to
+future work.
+15
+
+100
+200
+300
+400
+500
+100
+200
+300
+400
+5000 
+100
+200
+300
+400
+500
+100
+200
+OOE
+400
+500100
+200
+300
+400
+500
+100
+200
+300
+400
+5000 
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500100
+200
+300
+400
+500
+D
+100
+200
+OOE
+400
+500References
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+18
+
+Appendix
+Notations: We adopt the notations in the manuscript, and further introduce the following additional
+notations for technical proofs. We use N(F, �d, ϵ) to denote the ϵ-covering number of function space
+F with respect to pseudo-metric �d. We use Br(x) to denote the closed ball centered at x with radius
+r under the ℓ2 distance; in particular, we use Bd
+r denote Br(0d) when no ambiguity may arise. We
+denote Sd−1
+1
+= {x ∈ Rd : ∥x∥ = 1}.
+For a function f : Ω → Rd, we use Jf(x) to denote the
+d × m Jacobian matrix of f at x ∈ Ω. For a function f : Rd → R, we use f (a) to denote its mixed
+partial derivative ∂|a|f/∂xa1
+1 · · · ∂xad
+d . We define the α-smooth Hölder (function) class (see e.g., Evans
+(2010b)) with radius r > 0 over Ω as Cα
+r (Ω) :=
+�
+f : Ω → R
+�� ∥f∥Cα(Ω) = �
+|a|≤⌊α⌋ maxx∈Ω |f (a)(x)| +
+�
+|a|=⌊α⌋ maxx,y∈Ω, x̸=y
+��f (a)(x) − f (a)(y)
+�� /∥x − y∥˜α−⌊α⌋ ≤ r
+�
+. Similarly, we use Cα
+r (Ω; RD) =
+�
+f =
+(f1, . . . , fD) : Ω → RD �� ∀ j ∈ [D], fj ∈ Cα
+r (Ω)
+�
+to denote the vector valued function space counterpart.
+For an f ∈ Cα
+r (Ω; RD) and a multi-index a ∈ Nd
+0, we denote f (a) as the D dimensional vector whose j-th
+component is the mixed partial derivative [fj](a) of fj for j ∈ [D]. Throughout, C, c, C0, c0, C1, c1,
+C2, c2,. . . are generically used to denote positive constants whose values might change from one line to
+another, but are independent from everything else.
+A
+Remaining implementation details
+A.1
+Simulation
+The training points D in spiral are generated via the following steps: (1) generate φ0 ∼ N(0, 1); (2) set
+φ = 3πφ0; (3) generate data point X though X = [ cos(φ+2)·φ
+π
+, 2 sin(φ+2)·φ
+π
+]. The training points D in torus
+are generated via the following steps: (1) generate φ0, φ1 ∼ N(0, 1); (2) set φ = 2πφ0 and θ = 2πφ1; (3)
+generate data point X through X = [(3 + cos(θ)) cos(φ), (3 + cos(θ)) sin(φ), sin(θ)]. The cluster number
+M in MLDMAE is M = 10 for the dataset of spiral and M = 15 for the dataset of torus. The partition of
+unity is given by ρk = �ρk/
+� �K
+k′=1 �ρk′�
+with �ρk = (r2
+k − ∥x − ak∥2)10 · 1(x ∈ Sk), where {ak}k∈[K] are the
+centers returned by the K-means algorithm, rk = sup{∥x−ak∥ : x ∈ D; ∀k1 ∈ [K], ∥x−ak∥ ≤ ∥x−ak1∥},
+and Sk = Brk(ak).
+A.2
+Real data application
+The specification of our models trained on MNIST handwritten digit, Fashion-MNIST and CelebA are
+described in Table 4, 5, and 6. “Shared” is short for parameter sharing among encoders or among decoders.
+All models are optimized using Adam optimization with learning rate 0.001, β1 = 0.9, and β2 = 0.999. The
+partition of unity for all datasets is chosen as the indicator function ρk(x) = 1(x ∈ cluster m) for k ∈ [K].
+The codes for reproducing the experiments are available in https://github.com/rtang1997/MLDMAE.
+19
+
+Operation
+Kernel
+Strides
+Feature maps
+Activation
+Shared?
+Decoder Gk(z) : k ∈ [K], z ∈ N(0, Id)
+8
+Fully connected
+3 × 3 × 128
+ReLU
+No
+Transposed convolution
+3 × 3
+2 × 2
+7 × 7 × 64
+ReLU
+Yes
+Transposed convolution
+3 × 3
+2 × 2
+14 × 14 × 32
+ReLU
+Yes
+Transposed convolution
+3 × 3
+2 × 2
+28 × 28 × 1
+ReLU
+Yes
+Encoder Qk(x) : k ∈ [K]
+28 × 28 × 1
+Convolution
+3 × 3
+2 × 2
+26 × 26 × 3
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+12 × 12 × 32
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+5 × 5 × 64
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+2 × 2 × 128
+LeakyReLU
+Yes
+Fully connected
+8
+No
+Cluster number K for MLDMAE
+5
+Batch size
+256 for MLDMAE, and 128 for WAE and VAE
+Number of epochs
+50
+Leaky ReLU slope
+0.1
+Regularization coefficients (λk)
+100 for MMD penalty and 10 for W1 penalty
+Bandwidth (h) for MLDMAE
+0.01
+Number of training samples
+60k
+Table 4: Encoder/decoder Network architecture and hyperparameters for the MNIST handwritten digit
+dataset.
+Operation
+Kernel
+Strides
+Feature maps
+Activation
+Shared?
+Decoder Gk(z) : k ∈ [K], z ∈ N(0, Id)
+4
+Fully connected
+3 × 3 × 128
+ReLU
+No
+Transposed convolution
+3 × 3
+2 × 2
+7 × 7 × 64
+ReLU
+Yes
+Transposed convolution
+3 × 3
+2 × 2
+14 × 14 × 32
+ReLU
+Yes
+Transposed convolution
+3 × 3
+2 × 2
+28 × 28 × 1
+ReLU
+Yes
+Encoder Qk(x) : k ∈ [K]
+28 × 28 × 1
+Convolution
+3 × 3
+2 × 2
+26 × 26 × 3
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+12 × 12 × 32
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+5 × 5 × 64
+LeakyReLU
+Yes
+Convolution
+3 × 3
+2 × 2
+2 × 2 × 128
+LeakyReLU
+Yes
+Fully connected
+4
+No
+Cluster number K for MLDMAE
+5
+Batch size
+256 for MLDMAE, and 128 for WAE and VAE
+Number of epochs
+50
+Leaky ReLU slope
+0.1
+Regularization coefficients (λk)
+100 for MMD penalty and 10 for W1 penalty
+Bandwidth (h) for MLDMAE
+0.01
+Number of training samples
+60k
+Table 5: Encoder/decoder architecture and hyperparameters for the Fashion-MNIST dataset.
+20
+
+Operation
+Kernel
+Strides
+Feature maps
+Activation
+Shared?
+Decoder Gk(z) : k ∈ [K], z ∈ N(0, Id)
+64
+Fully connected
+8 × 8 × 1024
+ReLU
+No
+Transposed convolution
+5 × 5
+2 × 2
+16 × 16 × 512
+ReLU
+Yes
+Batch normalization
+Transposed convolution
+5 × 5
+2 × 2
+32 × 32 × 256
+ReLU
+Yes
+Batch normalization
+Transposed convolution
+5 × 5
+2 × 2
+64 × 64 × 128
+ReLU
+Yes
+Batch normalization
+Transposed convolution
+3 × 3
+1 × 1
+64 × 64 × 3
+Tanh
+Yes
+Encoder Qk(x) : k ∈ [K]
+64 × 64 × 3
+Convolution
+5 × 5
+2 × 2
+32 × 32 × 128
+ReLU
+Yes
+Batch normalization
+Convolution
+5 × 5
+2 × 2
+16 × 16 × 256
+ReLU
+Yes
+Batch normalization
+Convolution
+5 × 5
+2 × 2
+8 × 8 × 512
+ReLU
+Yes
+Batch normalization
+Convolution
+5 × 5
+2 × 2
+4 × 4 × 1023
+ReLU
+Yes
+Batch normalization
+Fully connected
+64
+No
+Cluster number K for MLDMAE
+5
+Batch size
+256 for MLDMAE, and 128 for WAE and VAE
+Number of epochs
+50
+Regularization coefficients (λk)
+100
+Bandwidth (h) for SWAE and MLDMAE
+0.01
+Number of training samples
+180k
+Table 6: Encoder/decoder architecture and hyperparameters for the CelebA dataset.
+B
+Generative modelling of Distributions on submanifolds
+A submanifold in the ambient space RD can be viewed as a nonlinear “subspace”. Borrow the definition
+in Tang & Yang (2022), we define the family of smooth distributions on d-dimensional smooth compact
+submanifolds without boundaries on RD as the set P∗ = P∗(d, D, α, β, L∗) with d ≤ D, β > 1 and
+α ∈ (0, β − 1] composed of all probability measures µ ∈ P(RD) satisfying:
+1. µ is an α-smooth distribution on a β-smooth d-dimensional compact submanifold M embedded in
+RD.
+2. The density µ relative to the volume measure of M is uniformly bounded from below by 1/L∗ on
+M.
+3. M is covered by an atlas A = {(Uλ, φλ)}λ∈Λ on M such that: a) each chart (U, φ) in atlas A
+satisfies ∥φ−1∥Cβ(φ(U)) ≤ L∗ and ∥µ ◦ φ−1∥Cα(φ(U)) ≤ L∗; b) for any z ∈ φ(U), the Jacobian of φ−1(z)
+is full rank and all its singular values are lower bounded by 1/L∗ in absolute values. Moreover, for any
+x ∈ M, there exists a λ ∈ Λ such that Uλ and φλ(Uλ) covers B1/L∗(x)∩M and B1/L∗(φλ(x)) respectively.
+We have the following lemma describing the mixture of generative model classes that can model the
+submanifold-supported distributions.
+21
+
+Lemma 1. Consider OK = {Sk = B◦
+rk(ak)}K
+m=1, for k ∈ [K], choosing ρk(x) =
+�ρk(x)
+�
+k∈[K] �ρk(x) with
+�ρk(x) = (r2
+k − ∥x − ak∥2)γ · 1(x ∈ Sk) for γ ≥ α + 1. There exists a constant r∗ that only depends
+on (L∗, β, α, d, D) so that for any µ∗ ∈ P∗(d, D, α, β, L∗), if (1) supp(µ∗) ⊂ ∪k∈[K]Sk; (2) for any
+k ∈ [K], rk ≤ r∗ and ak ∈ M; (3) there exists some positive constants L∗
+1 so that mink∈[K] rk ≥ L∗
+1 and
+infx∈M
+�
+k∈[K] �ρk(x) ≥ L∗
+1. Then:
+1.
+there exist some universal constants (L, c) that only depend on (L∗, L∗
+1, β, α, d, D, γ) so that
+Assumption A holds for µ∗ with upper bound L and function gk(r) = c (rγ ∧ 1) for any k ∈ [K];
+2. consider ν0 ∈ P(Bd
+1) whose density being α-smooth and bounded below from zero, and approximation
+families
+(a)
+G1 =
+�
+(G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ
+L(Rd; RD), Qk ∈ Cβ
+L(RD; Rd), νk ∈ P(Bd
+1) with νk ∈
+Cα
+L(Bd
+1)
+�
+;
+(b).
+G2 =
+�
+(G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ
+L(Rd; RD), Qk ∈ Cβ
+L(RD; Rd), vk = (Vm#ν0)·ρk(Gk(z))
+Eν0[ρk(Gk(Vk(z)))] , Vk ∈
+Cα+1
+L
+(Bd
+1; Bd
+1)
+�
+;
+(c).
+G3 =
+�
+(G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ
+L(Rd; RD), Qk ∈ Cβ
+L(RD; Rd), vk =
+ν0·ρk(Gk(z))
+Eν0[ρk(Gk(z))]
+�
+;
+then for sufficiently large L, we have Assumption B holds for G1 or G2, that is, the approximation families
+G1 and G2 are both sufficient to model distributions inside P∗(d, D, α, β, L∗). Moreover, if α = β − 1,
+then Assumption B holds for G3.
+The next lemma shows that OK satisfying conditions of Lemma 1 can be found based on a small portion
+of data.
+Lemma 2. Consider any µ∗ ∈ P∗(d, D, α, β, L∗) with support M, fix r∗ being an arbitrary positive
+constant and let n1 ≤ n be a positive integer. Then for any positive constant c, there exist constants C, c1
+that only depend on (d, D, β, L∗, r∗, c) so that when C ≤ n1 ≤ n, let I1 be any subset of [n] with |I1| = n1,
+it holds with probability larger than 1 − n−c
+1
+that
+(1). M ⊂ �
+i∈I1 Bc1( log n1
+n1
+)
+1
+d (xi);
+(2). there exists a constant K that only depends on (d, D, β, L∗, r∗) and a subset {ak}K
+k=1 ⊂ {Xi}i∈I1
+so that
+(a). �
+i∈I1 Bc1( log n1
+n1
+)
+1
+d (xi) ⊂ �K
+k=1 Br∗(ak).
+(b). infx∈M
+�
+k∈[K] �ρk(x) > ( r∗
+√
+2)2γ, where �ρk(x) = ((r∗)2 − ∥x − ak∥2)γ · 1(x ∈ Br∗(ak)).
+C
+Proof of Theorem 5.1
+We first consider the case α > 0 and β > 1. To simplify the notation, we write ˜α = α ∧ (β − 1). We
+consider two kinds of smoothness-regularized empirical measure �νk,Qk, one is based on kernel density
+estimator and one is based on wavelet estimator.
+Kernel density estimator: Define
+�νk,Qk(y) =
+1
+n�pkhd
+n
+�
+i=1
+�k
+�y − Qk(Xi)
+h
+�
+ρk(Xi),
+�pk = 1
+n
+n
+�
+i=1
+ρk(Xi),
+(7)
+with h = n−1/(2�α+d) and �k : Rd → R satisfies that
+1.�k(·) is ⌈˜α⌉ ∨ ⌈ d
+2⌉ smooth in Rd and has support contained in [−1, 1]d;
+2.
+�
+Rd �k(z) dz = 1 and for any j ∈ Nd
+0 with 1 ≤ |j| ≤ ⌊α⌋ + 1,
+� �k(z) · zj dz = 0;
+22
+
+3. for any z ∈ Rd, �k(z) = �k(−z).
+Wavelet estimator: Define �νk,Qk(y) as
+�νk,Qk(y) = 1
+�pk
+� �
+m∈S
+�aQk
+m φm(y) +
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+�θQk
+ljmψljm(y)
+�
+,
+(8)
+with
+�pk = 1
+n
+n
+�
+i=1
+ρk(Xi);
+S = {m ∈ Zd | supp(φm) ∩ [−L, L]d ̸= ∅};
+Slj = {m ∈ Zd | supp(ψljm) ∩ [−L, L]d ̸= ∅};
+�aQk
+m = 1
+n
+n
+�
+i=1
+φm(Qk(Xi))ρk(Xi);
+�θQk
+ljm = 1
+n
+n
+�
+i=1
+ψljm(Qk(Xi))ρk(Xi),
+where 2dJ ≍ n
+d
+2α+d and {φm, ψljm : l = 1, · · · , 2d − 1, j ∈ N, m ∈ Zd} is the orthonormal wavelet
+basis for Besov space on Rd defined as φm(y) = φ(y − m) and ψljm(y) = 2
+jd
+2 ψl(2jy − m), and it holds
+that φ(·) and ψl(·) are compactly supported and have bounded ⌈α ∨ ( d
+2 − α)⌉ order derivatives for any
+1 ≤ l ≤ 2d − 1 (Bouzebda & Didi, 2017).
+We will show both choices of �νk,Qk can lead to the desired result. By Assumption A of µ∗, for any k ∈ [K],
+there exist G∗
+k ∈ Cβ
+L(Rd; RD) and Q∗
+k ∈ Cβ
+L(RD; Rd) so that and for any x ∈ M ∩ Sk, G∗
+k(Q∗
+k(x)) = x. By
+the optimality of �G, �Q and �v for the training objective, we can get that
+K
+�
+k=1
+� 1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) + λk ·
+sup
+f∈Lip1(Rd)
+� �
+f(z)�νk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+��
+≤
+K
+�
+k=1
+λk ·
+sup
+f∈Lip1(Rd)
+� �
+f(z)ν∗
+k(z) dz −
+�
+f(z)�νk,Q∗
+k(z) dz
+�
+,
+(9)
+where recall ν∗
+k = (Q∗
+k)#( µ∗ρk
+pk ). Then we have the following lemma.
+Lemma 3. For any fixed c1 and c2, define
+�Qk = {Q ∈ Cβ
+L(RD; Rd) : the density ν∗
+k,Q of Q#
+�µ∗ · ρk
+pk
+�
+exists and ν∗
+k,Q ∈ C ˜α
+c2(Rd)}.
+Then there exists a constant c3 such that it holds with probability larger than 1 −
+1
+n2 that for any k ∈ [K],
+sup
+Q∈ �
+Qk
+sup
+f∈Lip1(Rd)
+� �
+f(z)ν∗
+k,Q(z) dz −
+�
+f(z)�νk,Q(z) dz
+�
+≤ c4
+�
+n− ˜
+α+1
+2 ˜
+α+d + log n
+√n
+�
+,
+where �νk,Q can either be the kernel density estimator in (7) or wavelet estimator in (8).
+So we choose λk = λ =
+�
+n− ˜
+α+1
+2 ˜
+α+d + log n
+√n
+�−1 · n− 2β
+d −1 for any k ∈ [K], then by the second statement of
+23
+
+Lemma 3 and Q∗
+k ∈ �Qk, it holds with probability larger than 1 − M n−2 that,
+K
+�
+k=1
+� 1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) + λ ·
+sup
+f∈Lip1(Rd)
+� �
+f(z)�νk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+��
+≤ C λ ·
+�
+n− ˜
+α+1
+2 ˜
+α+d + log n
+√n
+�
+.
+So it holds with probability larger than 1 − M n−2 that for any k ∈ [K],
+1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ C n− 2β
+d −1,
+(10)
+and
+sup
+f∈Lip1(Rd)
+� �
+f(z)�νk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+�
+≤ C
+�
+n− ˜
+α+1
+2 ˜
+α+d + log n
+√n
+�
+.
+(11)
+Then we use the following lemma for bounding the population level reconstruction error.
+Lemma 4. For the estimator �G and �Q, there exist positive constants N, c1, c2 and c3 such that when
+n ≥ N, for any k ∈ [K],
+(1). it holds with probability larger than 1−c1 n−3 that E[∥X− �Gk( �Qk(X))∥2·ρk(X)] ≤ c2 n− β
+d ∨ log n
+√n ;
+(2).
+if ˜α > 0, then it holds with probability larger than 1 − c1 n−3 that the density ν∗
+k, �
+Qk of
+( �Qk)#
+� µ∗·ρk
+pk
+�
+exists and belongs to C ˜α
+c3(Rd).
+Then by Lemma 3 and second statement of Lemma 4, there exist constants c, c1 such that it holds
+with probability larger than 1 − c n−2 that
+sup
+f∈Lip1(Rd)
+� �
+f(z)ν∗
+k, �
+Qk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+�
+≤ c1
+�
+n− ˜
+α+1
+2 ˜
+α+d + log n
+√n
+�
+.
+(12)
+24
+
+So combined with equation (10), (11) and (12), it holds with probability larger than 1 − 1
+n that
+W1(�µ, µ∗) =
+sup
+f∈Lip1(RD)
+� K
+�
+k=1
+�
+pk · f(X) d
+�µ∗ · ρk
+pk
+�
+−
+K
+�
+k=1
+�
+�pk · f(X) d( �Gk)#�νk
+�
+(i)
+≤ C
+�
+log n
+n
++
+sup
+f∈Lip1(RD)
+� K
+�
+k=1
+�
+�pk · f(X) d
+�µ∗ · ρk
+pk
+�
+−
+K
+�
+k=1
+�
+�pk · f(X) d( �Gk)#�νk
+�
+≤ C
+�
+log n
+n
++
+K
+�
+k=1
+sup
+f∈Lip1(RD)
+� �
+�pk · f(X) d
+�µ∗ · ρk
+pk
+�
+−
+�
+�pk · f(x) d( �Gk)#�νk
+�
+≤ C
+�
+log n
+n
++
+K
+�
+k=1
+sup
+f∈Lip1(RD)
+� � �pk
+pk
+· f(X)ρk(X)dµ∗ −
+� �pk
+pk
+· f( �Gk( �Qk(X))ρk(X)dµ∗
++
+� �pk
+pk
+· f( �Gk( �Qk(X))ρk(X)dµ∗ −
+�
+�pk · f(x) d( �Gk)#�νk
+�
+≤ C
+�
+log n
+n
++ 2
+K
+�
+k=1
+Eµ∗
+�
+∥X − �Gk( �Qk(X))∥ · ρk(X)
+�
++
+K
+�
+k=1
+sup
+f∈Lip1(RD)
+� �
+f( �Gk(z))ν∗
+k, �
+Qk(z) dz −
+�
+f( �Gk(z)) �νk(z) dz
+�
+≤ C1 n− β
+d ∨ log n
+√n +
+K
+�
+k=1
+�
+sup
+f∈Lip1(RD)
+� �
+f( �Gk(z))ν∗
+k, �
+Qk(z) dz −
+�
+f( �Gk(z))�νk, �
+Qk(z) dz
+�
++
+sup
+f∈Lip1(RD)
+� �
+f( �Gk(z))�νk, �
+Qk(z) dz −
+�
+f( �Gk(z)) �νk(z) dz
+��
+(ii)
+≤ C1 n− β
+d ∨ log n
+√n + C2
+K
+�
+k=1
+�
+sup
+f∈Lip1(Rd)
+� �
+f(z)ν∗
+k, �
+Qk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+�
++
+sup
+f∈Lip1(Rd)
+� �
+f(z)�νk, �
+Qk(z) dz −
+�
+f(z) �νk(z) dz
+��
+≤ C2 n− ˜
+α+1
+2 ˜
+α+d ∨ log n
+√n ,
+(13)
+where (i) uses Bernstein’s inequality to obtain that
+���pk − pk
+�� ≤ C
+�
+log n
+n
+holds with probability at least
+1 − n−3, and (ii) uses the fact that β ≥ 1.
+Then we consider the case α = 0. Define �νk,Q =
+1
+�pkn
+�n
+i=1 δQ(Xi)ρk(Xi). For any f ∈ Lip1(Rd), we
+have
+�
+f(z) d�νk,Q =
+1
+�pkn
+n
+�
+i=1
+f(Q(Xi))ρk(Xi).
+Then we have the following lemma.
+Lemma 5. There exists a constant c so that it holds with probability larger than 1 −
+1
+n2 that
+sup
+Q∈Cβ
+L(RD;Rd)
+sup
+f∈Lip1(Rd)
+� 1
+pk
+�
+f(Q(x))ρk(x) dµ∗ −
+�
+f(z)�νk,Q(z) dz
+�
+≤ c
+�log n
+√n + n− 1
+d
+�
+.
+Then by Lemma 5 and equation (9), choose λk = λ =
+�
+n− 1
+d + log n
+√n
+�−1 · n− 2β
+d −1, we have that
+statement (10) holds and
+sup
+f∈Lip1(Rd)
+� �
+f(z)�νk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+�
+≤ C
+�
+n− 1
+d + log n
+√n
+�
+.
+25
+
+Then by Lemma 4, we have
+E[∥X − �Gk( �Qk(X))∥2ρk(X)] ≤ c2n− β
+d ∨ log n
+√n .
+So combined with Lemma 5, following equation (13), we have
+W1(�µ, µ∗) ≤ C
+�
+log n
+n
++ 2
+K
+�
+k=1
+Eµ∗
+�
+∥X − �Gk( �Qk(X))∥ · ρk(X)
+�
++
+K
+�
+k=1
+sup
+f∈Lip1(RD)
+� �
+f( �Gk(z))ν∗
+k, �
+Qk(z) dz −
+�
+f( �Gk(z)) �νk(z) dz
+�
+≤ C1 n− β
+d ∨ log n
+√n + C2
+K
+�
+k=1
+�
+sup
+f∈Lip1(Rd)
+� �
+f(z)ν∗
+k, �
+Qk(z) dz −
+�
+f(z)�νk, �
+Qk(z) dz
+�
++
+sup
+f∈Lip1(RD)
+� �
+f(z)�νk, �
+Qk(z) dz −
+�
+f(z) �νk(z) dz
+��
+≤ C2 n− 1
+d ∨ log n
+√n .
+C.1
+Proof of Lemma 3: kernel density estimator
+Fix an arbitrary k ∈ [K]. Since �Qk ⊆ Cβ
+L(RD; Rd), it holds that for any Q ∈ �Qk, ν∗
+k,Q ∈ C ˜α
+L(Rd) and
+supp(ν∗
+k,Q) ∈ [−L, L]d, where ν∗
+k,Q is the density of the push-forward measure of µ∗·ρk
+pk
+by map Q. Recall
+that
+�pk · �νk,Q(y) =
+1
+nhd
+n
+�
+i=1
+�k(y − Q(Xi)
+h
+)ρk(Xi).
+Since ν∗
+k,Q and �νk,Q are both compactly supported, there exists a constant C so that for any Q ∈ �Qk,
+sup
+f∈Lip1(Rd)
+� �
+f(y)ν∗
+k,Q(y) dy −
+�
+f(y)�νk,Q(y) dy
+�
+≤ C
+sup
+f∈C1
+1(Rd)
+� �
+f(y)ν∗
+k,Q(y) dy −
+�
+f(y)�νk,Q(y) dy
+�
+,
+where C1
+1(Rd) =
+�
+f : Rd → R | supz∈Rd �
+j∈Nd
+0,|j|≤1 |f (j)(z)| ≤ 1
+�
+. Then we consider f ∈ C1
+1(Rd), we can
+get
+�
+f(y)ν∗
+k,Q(y) dy −
+�
+f(y)�νk,Q(y) dy
+= 1
+�pk
+� �
+pk · f(y)ν∗
+k,Q(y) dy −
+�
+�pk · f(y)�νk,Q(y) dy
+�
++
+�
+f(y)ν∗
+k,Q(y) dy ·
+�
+1 − pk
+�pk
+�
+≤ 1
+�pk
+���
+�
+f(y) · pk · ν∗
+k,Q(y) dy −
+�
+f(y) · EX(n)
+�
+�pk · �νk,Q(y)
+�
+dy
+���
+�
+��
+�
+(A)
++ 1
+�pk
+���
+�
+f(y) · EX(n)
+�
+�pk · �νk,Q(y)
+�
+dy −
+�
+f(y) · �pk · �νk,Q(y) dy
+���
+�
+��
+�
+(B)
++
+��1 − pk
+�pk
+��
+�
+��
+�
+(C)
+.
+(14)
+First for term (C),by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤
+C
+�
+log n
+n , then by pk > 0, for large enough n, we have
+��1 − pk
+�pk
+�� ≤ C
+�
+log n
+n . For bounding the term (A),
+we use a similar strategy as in the proof of Lemma 4.3 of Divol (2022). Recall ν∗
+k,Q = Q#[ µ∗·ρk
+pk ], we can
+26
+
+write
+�
+f(y) · EX(n)
+�
+�pk · �νk,Q(y)
+�
+dy =
+�
+f(y) · 1
+hd · Eµ∗��k(y − Q(X)
+h
+) · ρk(X)
+�
+dy
+=
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · pk · ν∗
+k,Q(z) dz dy
+Denote υ(·) = pk · ν∗
+k,Q(·), we can obtain
+���
+�
+f(y) · pk · ν∗
+k,Q(y) dy −
+�
+f(y) · EX(n)
+�
+�pk · �νk,Q(y)
+�
+dy
+���
+=
+���
+�
+f(y) · υ(y) dy −
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · υ(z) dz dy
+���
+=
+���
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · (υ(z) − υ(y)) dz dy
+���.
+When ⌊˜α⌋ is even, denote s = ⌊˜α⌋; when ⌊˜α⌋ is odd, denote s = ⌊˜α⌋ − 1. Then using Taylor’s theorem,
+we can decompose
+υ(z) − υ(y) =
+�
+j∈Nd
+0
+1≤|j|<s
+υ(j)(y)
+j!
+· (z − y)j +
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1υ(j)(y + t(z − y)) dt · (z − y)j.
+Using the fact that for any j ∈ Nd
+0 with 1 ≤ |j| ≤ ⌊˜α⌋,
+�
+Rd �k(z) · zj dz = 0 and �k(·) = �k(−·), we can obtain
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) ·
+�
+j∈Nd
+0
+|j|<s
+υ(j)(y)
+j!
+· (z − y)j dz dy
+=
+� �
+f(y) · �k(z) ·
+�
+j∈Nd
+0
+|j|<s
+υ(j)(y)
+j!
+· zj · h|j| dz dy = 0,
+and
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1υ(j)(y) · (z − y)j dt dz dy
+=
+� �
+f(y) · �k(z) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1υ(j)(y) · zj · hs dt dz dy = 0.
+Therefore, we have
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · (υ(z) − υ(y)) dz dy
+=
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(y + t(z − y)) − υ(j)(y)
+�
+· (z − y)j dt dz dy
+(i)
+=
+� �
+f(y) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y)
+�
+·
+�x − y
+t
+�j dt dx dy,
+27
+
+where (i) uses the change of variable x = y + t(z − y) and �k(·) = �k(−·). Then by switching the variable x
+and y, using the facts that s is a even number and k is an even function, we have
+� �
+f(y) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y)
+�
+·
+�x − y
+t
+�j dt dx dy
+=
+� �
+f(x) ·
+1
+tdhd · �k(y − x
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(y) − υ(j)(x)
+�
+·
+�y − x
+t
+�j dt dx dy
+= −
+� �
+f(x) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y)
+�
+·
+�x − y
+t
+�j dt dx dy.
+Therefore, when ⌊˜α⌋ is even, we can obtain
+���
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · (υ(z) − υ(y)) dz dy
+���
+=
+���1
+2
+� �
+(f(y) − f(x)) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y)
+�
+·
+�x − y
+t
+�j dt dx dy
+���
+=
+���1
+2
+� �
+(f(y) − f(y + wth)) · �k(w) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(y + wth) − υ(j)(y)
+�
+·
+�
+wh
+�j dt dw dy
+���
+(ii)
+≲ h1+˜α ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+�
+[−L−h,L+h]d
+�
+[−1,1]d ∥w∥1+˜α · �k(w) ·
+� 1
+0
+(1 − t)s−1 dt dw dy
+≲ n− 1+ ˜
+α
+2 ˜
+α+d ,
+where (ii) uses υ(·) = pk · ν∗
+k,Q(·) has support contained in [−L, L]d and �k(·) has support contained in
+[−1, 1]d. On the other hand, when ⌊˜α⌋ is odd, recall s = ⌊˜α⌋ − 1 is even, we have
+���
+� �
+f(y) · 1
+hd · �k(y − z
+h
+) · (υ(z) − υ(y)) dz dy
+���
+=
+���1
+2
+� �
+(f(x) − f(y)) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y)
+�
+·
+�x − y
+t
+�j dt dx dy
+���
+≤
+����
+1
+2
+� �
+(f(x) − f(y)) ·
+1
+tdhd · �k(x − y
+th
+)
+·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1�
+υ(j)(x) − υ(j)(y) −
+�
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) · (x − y)q�
+·
+�x − y
+t
+�j dt dx dy
+����
++
+����
+1
+2
+� �
+(f(x) − f(y)) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) · (x − y)q ·
+�x − y
+t
+�j dt dx dy
+����
+≤ C h1+˜α
++
+����
+1
+2
+� �
+(f(x) − f(y)) ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) · (x − y)q ·
+�x − y
+t
+�j dt dx dy
+����
+�
+��
+�
+(D)
+.
+28
+
+Then for the term (D), using Taylor’s theorem and f ∈ C1
+1(Rd), we can write
+f(x) − f(y) =
+�
+j∈Nd
+0
+|j|=1
+� 1
+0
+f (j)(y + t(x − y)) dt · (x − y)j.
+(15)
+So using the fact that for any j ∈ Nd
+0 with 1 ≤ |j| ≤ ⌊˜α⌋ + 1,
+�
+Rd �k(z) · zj dz = 0, we can obtain
+� � �
+l∈Nd
+0
+|l|=1
+f (l)(y) · (x − y)l ·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) · (x − y)q ·
+�x − y
+t
+�j dt dx dy
+=
+�
+l∈Nd
+0
+|l|=1
+�
+j∈Nd
+0
+|j|=s
+�
+q∈Nd
+0
+|q|=1
+� 1
+0
+1
+tdhd · s
+j! ·
+� �
+f (l)(y) · (x − y)l · �k(x − y
+th
+)(1 − t)s−1υ(j+q)(y) · (x − y)q ·
+�x − y
+t
+�j dx dy dt
+= 0.
+Therefore,
+(D) =
+����
+1
+2
+� � �
+l∈Nd
+0
+|l|=1
+� 1
+0
+�
+f (l)(y + ω(x − y)) − f (l)(y)
+�
+dω · (x − y)l
+·
+1
+tdhd · �k(x − y
+th
+) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!
+� 1
+0
+(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) · (x − y)q ·
+�x − y
+t
+�j dt dx dy
+����
+(iii)
+=
+����
+1
+2
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(θ) − f (l)(y)
+�
+·
+�θ − y
+ω
+�l
+·
+1
+ωdtdhd · �k(θ − y
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) ·
+�θ − y
+ω
+�q ·
+�θ − y
+ωt
+�j dω dt dθ dy
+����,
+where (iii) uses the change of variable y + ω(x − y) = θ. By switching the variable θ and y, we have
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(θ) − f (l)(y)
+�
+·
+�θ − y
+ω
+�l
+·
+1
+ωdtdhd · �k(θ − y
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) ·
+�θ − y
+ω
+�q ·
+�θ − y
+ωt
+�j dω dt dθ dy
+=
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(y) − f (l)(θ)
+�
+·
+�y − θ
+ω
+�l
+·
+1
+ωdtdhd · �k(y − θ
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(θ) ·
+�y − θ
+ω
+�q ·
+�y − θ
+ωt
+�j dω dt dθ dy
+= −
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(θ) − f (l)(y)
+�
+·
+�θ − y
+ω
+�l
+·
+1
+ωdtdhd · �k(θ − y
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(θ) ·
+�θ − y
+ω
+�q ·
+�θ − y
+ωt
+�j dω dt dθ dy,
+29
+
+which leads to
+����
+1
+2
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(θ) − f (l)(y)
+�
+·
+�θ − y
+ω
+�l
+·
+1
+ωdtdhd · �k(θ − y
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+υ(j+q)(y) ·
+�θ − y
+ω
+�q ·
+�θ − y
+ωt
+�j dω dt dθ dy
+����,
+=
+����
+1
+4
+� � � 1
+0
+� 1
+0
+�
+l∈Nd
+0
+|l|=1
+�
+f (l)(θ) − f (l)(y)
+�
+·
+�θ − y
+ω
+�l
+·
+1
+ωdtdhd · �k(θ − y
+ωth ) ·
+�
+j∈Nd
+0
+|j|=s
+s
+j!(1 − t)s−1 �
+q∈Nd
+0
+|q|=1
+�
+υ(j+q)(y) − υ(j+q)(θ)
+�
+·
+�θ − y
+ω
+�q ·
+�θ − y
+ωt
+�j dω dt dθ dy
+����
+≲ h1+˜α,
+where the last inequality uses the fact that s = ⌊˜α⌋ − 1 and υ is ˜α is smooth. We can then obtain that
+(A) ≲ n− ˜
+α+1
+2 ˜
+α+d .
+It remains to bound term (B), which is
+(B) =
+���
+�
+f(y) · EX(n)
+�
+�pk · �νk,Q(y)
+�
+dy −
+�
+f(y) · �pk · �νk,Q(y) dy
+���
+=
+����
+1
+n
+n
+�
+i=1
+�
+f(y) · 1
+hd · �k
+�y − Q(Xi)
+h
+�
+· ρk(Xi) dy − Eµ∗
+� �
+f(y) · 1
+hd · �k
+�y − Q(X)
+h
+�
+· ρk(X) dy
+�����.
+By standard symmetrization, we can get
+E
+�
+sup
+f∈C1
+1(Rd)
+Q∈ �
+Qk
+����
+1
+n
+n
+�
+i=1
+�
+f(y) · 1
+hd · �k
+�y − Q(Xi)
+h
+�
+· ρk(Xi) dy − E
+� �
+f(y) · 1
+hd · k
+�y − Q(X)
+h
+�
+· ρk(X) dy
+�����
+�
+≤ 2E
+�
+sup
+f∈C1
+1(Rd)
+Q∈ �
+Qk
+����
+1
+n
+n
+�
+i=1
+εi
+�
+f(y) · 1
+hd · �k
+�y − Q(Xi)
+h
+�
+· ρk(Xi) dy
+����
+�
+,
+where {εi}n
+i=1 are n i.i.d. copies from Rademacher distribution, i.e. P(εi = 1) = P(εi = −1) = 0.5.
+Define function set
+F =
+�
+f : M → R | f(x) =
+�
+g(y) · 1
+hd · �k
+�y − Q(x)
+h
+�
+· ρk(x) dy; g ∈ C1
+1(Rd); Q ∈ �Qk
+�
+.
+Since there exists a constant L so that supp(Q) = [−L, L]d, we can first consider the function set
+F1 =
+�
+f : [−L, L]d → R | f(z) =
+�
+g(y) · 1
+hd · �k
+�y − z
+h
+�
+dy; g ∈ C1
+1(Rd)
+�
+.
+For any f ∈ F1 and j ∈ Nd
+0 with 1 ≤ |j| ≤ ⌊ d
+2⌋, we have
+f (j)(z) =
+�
+g(y)
+1
+hd+|j| �k(j)(z − y
+h
+) dy
+=
+1
+h|j|
+�
+g(z − ht)�k(j)(t) dt.
+30
+
+Since |j| ≥ 1, there exists j1 ∈ Nd
+0 with |j1| = 1 so that every element in j − j1 is non-negative. Then we
+have
+|f (j)(z)| =
+���
+1
+h|j|−1
+�
+g(j1)(z − ht)�k(j−j1)(t) dt
+��� ≲
+1
+h|j|−1 ,
+where the last inequality uses g ∈ C1
+1(Rd). Moreover, when d ≥ 2, for any j ∈ Nd
+0 with |j| = ⌊ d
+2⌋ and
+z1, z2 ∈ Rd it holds that
+|f (j)(z1) − f (j)(z2)| =
+��
+1
+h⌊ d
+2 ⌋−1
+�
+g(j1)(z1 − ht)�k(j−j1)(t) dt −
+1
+h⌊ d
+2 ⌋−1
+�
+g(j1)(z2 − ht)�k(j−j1)(t) dt
+��
+=
+���
+1
+h⌊ d
+2 ⌋−1
+�
+1
+hd g(j1)(y)
+��k(j−j1)(z1 − y
+h
+) − �k(j−j1)(z2 − y
+h
+)
+�
+dy
+���
+If ∥z1 − z2∥ ≤ h, then
+|f (j)(z1) − f (j)(z2)| =
+���
+1
+h⌊ d
+2 ⌋−1
+�
+1
+hd g(j1)(y)
+��k(j−j1)(z1 − y
+h
+) − �k(j−j1)(z2 − y
+h
+)
+�
+dy
+���
+≤ C
+1
+h⌊ d
+2 ⌋ ∥z1 − z2∥ ≤ C
+1
+h
+d
+2 −1 ∥z1 − z2∥
+d
+2 −⌊ d
+2 ⌋;
+If ∥z1 − z2∥ ≥ h, then
+|f (j)(z1) − f (j)(z2)| ≤ |f (j)(z1)| + |f (j)(z2)| ≤ C
+1
+h⌊ d
+2 ⌋−1 ≤ C
+1
+h
+d
+2 −1 ∥z1 − z2∥
+d
+2 −⌊ d
+2 ⌋.
+So we can get when d ≥ 2, there exists a constant C such that for any f ∈ F1, it holds that
+C h
+d
+2 −1f ∈ C
+d
+2
+1 ([−L, L]d).
+Similarly, when d = 1, we have for any z1, z2 ∈ Rd it holds that
+|f(z1) − f(z2)| =
+��
+�
+(g(z1 − ht) − g(z2 − ht)) · �k
+�
+t
+�
+dy
+�� ≲ ∥z1 − z2∥.
+Therefore, we can conclude that
+F1 ∈ C
+d
+2
+L1 h−( d
+2 −1)+ ([−L, L]d),
+for a constant L1, where (a)+ = max(a, 0). So for any ϵ > 0, we can find a set N f
+ϵ ⊆ F1 such that
+�
+log |N f
+ϵ | ≲ h−( d
+2 −1)+
+ϵ
+and for any f ∈ F1, there exists �f ∈ N f
+ϵ such that
+sup
+y∈[−L,L]d|f(y) − �f(y)| ≤ ϵ.
+Then, to derive an ϵ-covering number for F, we introduce the following lemma.
+Lemma 6. (Lemma 12 of Tang & Yang (2022)) Let XG =
+�
+x ∈ RD : x = G(z), z ∈ Bd
+1
+�
+be a d-
+dimensional submanifold induced by a Lipschitz continuous map G : Rd → RD, then it holds for any
+�γ > 0 that
+log N
+�
+C�γ
+1 (RD), ∥ · ∥L∞(XG), ϵ
+�
+≤ C ϵ− d
+�γ ,
+∀ϵ > 0,
+where N(F, �d, ϵ) denotes the ϵ-covering number of function space F with respect to pseudo-metric �d, and
+∥f∥L∞(XG) = sup
+x∈XG
+��f(x)
+�� denotes the functional supreme norm constrained on set XG.
+Then since Ωk = Q∗
+k(M ∩ Sk) is compactly supported and M ∩ Sk = G∗
+k(Ωk), for any ϵ > 0, we can
+31
+
+find a function set N Q
+ϵ such that
+�
+log |N Q
+ϵ | ≲ ( 1
+ϵ )
+d
+2β and for any Q ∈ �Qk, there exists �Q ∈ N Q
+ϵ such that
+sup
+x∈M∩Sk
+∥ �Q(x) − Q(x)∥ ≤ ϵ.
+Then for any Q ∈ �Qk and f ∈ F1, there exists �Q ∈ N Q
+ϵ , �f ∈ N f
+ϵ and a constant c such that
+���f(Q(x))ρk(x) − �f( �Q(x))ρk(x)
+���
+≤
+���f(Q(x))ρk(x) − �f(Q(x))ρk(x)
+��� +
+��� �f(Q(x))ρk(x) − �f( �Q(x))ρk(x)
+���
+≤ cϵ.
+So we can get
+log N(F, ϵ, ∥ · ∥∞) ≲
+�1
+ϵ
+� d
+β
++
+�
+h−( d
+2 −1)+
+ϵ
+�2
+.
+Choose δ =
+� 1
+n
+� ˜
+α+1
+2 ˜
+α+d ∨
+� 1
+n
+� β
+d , we can get
+1
+√n
+� 1
+δ
+��1
+ϵ
+� d
+2β
++ h1− d
+2 ∨ 1
+ϵ
+�
+dϵ
+≲ log n
+√n +
+� 1
+n
+� ˜
+α+1
+2 ˜
+α+d +
+� 1
+n
+� β
+d .
+By Dudley’s entropy integral bound (see for example, Theorem 5.22 of Wainwright (2019)), it holds that
+E
+�
+sup
+f∈C1
+1(Rd)
+Q∈ �
+Qk
+����
+1
+n
+n
+�
+i=1
+εi
+�
+f(y) · 1
+hd · �k
+�y − Q(Xi)
+h
+�
+· ρk(Xi) dy
+����
+�
+≲ log n
+√n +
+� 1
+n
+� ˜
+α+1
+2 ˜
+α+d +
+� 1
+n
+� β
+d .
+The statement is then followed by Talagrand concentration inequality (see for example, Theorem 3.27 of
+Wainwright (2019)) and the fact that ˜α + 1 ≤ β.
+C.2
+Proof of Lemma 3: Wavelet estimator
+Fix an arbitrary k ∈ [K]. Since �Qk ⊆ Cβ
+L(RD; Rd), it holds that for any Q ∈ �Qk, supp(ν∗
+k,Q) ⊆ [−L, L]d,
+where ν∗
+k,Q is the density of the push-forward measure of µ∗·ρk
+pk
+by map Q. Moreover, by ν∗
+k,Q ∈ Cα
+L(Rd)
+with support contained in [−L, L]d and 0 < pk = Eµ∗[ρk] ≤ 1, we can write pk · ν∗
+k,Q(y) as
+pk · ν∗
+k,Q(y) =
+�
+m∈S
+aQ
+mφm(y) +
+2d−1
+�
+l=1
++∞
+�
+j=0
+�
+m∈Slj
+θQ
+ljmψljm(y),
+where {φm, ψljm : l = 1, · · · , 2d − 1, j ∈ N, m ∈ Zd} is the orthonormal wavelet basis for Besov space on
+Rd defined as φm(y) = φ(y − m) and ψljm(y) = 2
+jd
+2 ψl(2jy − m), and it holds that φ(·) and ψl(·) for any
+1 ≤ l ≤ 2d − 1 are compactly supported and have bounded β order derivatives (Bouzebda & Didi, 2017).
+Then there exists a constant C such that |θQ
+ljm| ≤ C(2−dj)
+α
+d + 1
+2 and aQ
+m ≤ C. Recall that
+�pk · �νk,Q(y) =
+�
+m∈S
+�aQ
+mφm(y) +
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+�θQ
+ljmψljm(y),
+32
+
+with
+�aQ
+m = 1
+n
+n
+�
+i=1
+φm(Q(Xi))ρk(Xi);
+�θQ
+ljm = 1
+n
+n
+�
+i=1
+ψljm(Q(Xi))ρk(Xi).
+We have
+E[�aQ
+m] = pk ·
+�
+φm(y)ν∗
+k,Q(y)dy = aQ
+m,
+E[�θQ
+ljm] = pk ·
+�
+ψljm(y)ν∗
+k,Q(y)dy = θQ
+ljm.
+Moreover, by the fact that φ(·) and ψl(·) are compactly supported, we can get that there exists a constant
+C such that for any 1 ≤ l ≤ 2d − 1 and j ∈ N, it holds that |Slj| ≤ C2dj and |S| ≤ C. Since ν∗
+k,Q and
+�νk,Q are both compactly supported. There exists a constant C so that for any Q ∈ �Qk,
+sup
+f∈Lip1(Rd)
+� �
+f(y)ν∗
+k,Q(y) dy −
+�
+f(y)�νk,Q(y) dy
+�
+≤ C
+sup
+f∈C1
+1(Rd)
+� �
+f(y)ν∗
+k,Q(y) dy −
+�
+f(y)�νk,Q(y) dy
+�
+.
+Then we consider f ∈ C1
+1(Rd), similarly, we can rewrite
+f(y) =
+�
+m∈Zd
+bmφm(y) +
+2d−1
+�
+l=1
++∞
+�
+j=0
+�
+m∈Zd
+fljmψljm(y)
+where |fljm| ≤ C1(2−dj)
+1
+d + 1
+2 and |bm| ≤ C1. So we can get
+�
+f(y)ν∗
+k,Q(y)dy −
+�
+f(y)�νk,Q(y)dy
+= 1
+�pk
+� �
+f(y)pk · ν∗
+k,Q(y)dy −
+�
+�pk · f(y)�νk,Q(y)dy
+�
++
+�
+f(y)ν∗
+k,Q(y)dy ·
+�
+1 − pk
+�pk
+�
+= 1
+�pk
+�
+f(y)
+�
+��
+m∈S
+(�aQ
+m − E�aQ
+m)φm(y) +
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+(�θQ
+ljm − E�θQ
+ljm)ψljm(y)
+�
+� dy
++ 1
+�pk
+�
+f(y)
+�
+�
+2d−1
+�
+l=1
+∞
+�
+j=J
+�
+m∈Slj
+θQ
+ljmψljm(y)
+�
+� dy +
+�
+f(y)ν∗
+k,Q(y)dy ·
+�
+1 − pk
+�pk
+�
+≤ 1
+�pk
+������
+1
+n
+n
+�
+i=1
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(Xi)) ρk(Xi) − E
+� 2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(X)) ρk(X)
+�������
+�
+��
+�
+(A)
++ 1
+�pk
+�����
+1
+n
+n
+�
+i=1
+�
+m∈S
+bmφm(Q(Xi))ρk(Xi) − E
+� �
+m∈S
+bmφm(Q(X))ρk(X)
+������
+�
+��
+�
+(B)
++ 1
+�pk
+2d−1
+�
+l=1
++∞
+�
+j=J
+�
+m∈Slj
+fljmθQ
+ljm
+�
+��
+�
+(C)
++
+��1 − pk
+�pk
+��
+�
+��
+�
+(D)
+.
+(16)
+First for term (D),by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤
+C
+�
+log n
+n , then by pk > 0, for large enough n, we have
+��1 − pk
+�pk
+�� ≤ C
+�
+log n
+n . Moreover, for term (C), since
+|fljm| ≲ (2−dj)
+1
+d + 1
+2 , |θQ
+ljm| ≲ (2−dj)
+α
+d + 1
+2 and 2dJ ≍ n
+d
+2α+d , we can get
++∞
+�
+j=J+1
+2d−1
+�
+l=1
+�
+m∈Slj
+fljmθQ
+ljm ≲ n− α+1
+2α+d .
+33
+
+Then for term (A), by standard symmetrization, we can get
+E
+�
+sup
+f∈C1
+1 (Rd)
+Q∈ �
+Qk
+����
+1
+n
+n
+�
+i=1
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(Xi)) ρk(Xi) − E
+� 2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(X)) ρk(X)
+�����
+�
+≤ 2E
+�
+sup
+f∈C1
+1 (Rd)
+Q∈ �
+Qk
+����
+1
+n
+2d−1
+�
+l=1
+n
+�
+i=1
+εi
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(Xi)) ρk(Xi)
+����
+�
+,
+where {εi}n
+i=1 are n i.i.d. copies from Rademacher distribution, i.e. P(εi = 1) = P(εi = −1) = 0.5.
+Define function set
+F =
+�
+�
+�f : M → R : f(z) =
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm(Q(x)); |fljm| ≤ (2−dj)
+1
+d + 1
+2 ; Q ∈ �Qk
+�
+�
+� .
+First we consider the function set
+F1 =
+�
+�
+�f : [−L3, L3]d → R, f(y) =
+2d−1
+�
+l=1
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm(y), |fljm| ≤ (2−dj)
+1
+d + 1
+2
+�
+�
+� .
+If 1
+d ≤ 1
+2, then there exists a constant c such that for any f ∈ F1, it holds that
+c(2dJ)
+1
+d − 1
+2 f ∈ C
+d
+2
+1 ([−L, L]d).
+So for any ϵ > 0, we can find a set N f
+ϵ ⊆ F1 such that
+�
+log |N f
+ϵ | ≲ (2dJ)( 1
+2 − 1
+d )+
+ϵ
+and for any f ∈ F1,
+there exists �f ∈ N f
+ϵ such that
+sup
+y∈[−L,L]d|f(y) − �f(y)| ≤ ϵ.
+Then since Ωk = Q∗
+k(M ∩ Sk) is compactly supported and M ∩ Sk = G∗
+k(Ωk), by lemma 6, for any ϵ > 0,
+we can find a function set N Q
+ϵ such that
+�
+log |N Q
+ϵ | ≲ ( 1
+ϵ )
+d
+2β and for any Q ∈ �Qk, there exists �Q ∈ N Q
+ϵ
+such that
+sup
+x∈M∩Sk
+∥ �Q(x) − Q(x)∥ ≤ ϵ.
+Then for any Q ∈ �Qk and f ∈ F1, there exists �Q ∈ N Q
+ϵ , �f ∈ N f
+ϵ and a constant c such that
+���f(Q(x))ρk(x) − �f( �Q(x))ρk(x)
+���
+≤
+���f(Q(x))ρk(x) − �f(Q(x))ρk(x)
+��� +
+��� �f(Q(x))ρk(x) − �f( �Q(x))ρk(x)
+���
+≤ cϵ.
+So we can get
+log N(F, ϵ, ∥ · ∥∞) ≲
+�1
+ϵ
+� d
+β
++
+�
+2dJ( 1
+2 − 1
+d )+
+ϵ
+�2
+.
+Choose δ =
+� 1
+n
+� α+1
+2α+d ∨
+� 1
+n
+� β
+d , we can get
+1
+√n
+� 1
+δ
+��log n
+ϵ
+� d
+2β
++ (2dJ( 1
+2 − 1
+d ) log n) ∨ 1
+ϵ
+�
+dϵ
+≲ log n
+√n +
+� 1
+n
+� α+1
+2α+d +
+� 1
+n
+� β
+d .
+34
+
+By Dudley’s entropy integral bound, it holds that
+E
+sup
+f∈C1
+1 (Rd)
+Q∈ �
+Qk
+������
+1
+n
+2d−1
+�
+l=1
+n
+�
+i=1
+εi
+J
+�
+j=0
+�
+m∈Slj
+fljmψljm (Q(Xi))) ρk(Xi)
+������
+≲ log n
+√n +
+� 1
+n
+� α+1
+2α+d +
+� 1
+n
+� β
+d .
+Similarly we can get
+E
+sup
+f∈C1
+1 (Rd)
+Q∈ �
+Qk
+�����
+1
+n
+n
+�
+i=1
+εi
+�
+m∈S
+bmφm (Q(Xi)) ρk(Xi)
+�����
+≲ n− β
+d + log n
+√n .
+The statement is then followed by Talagrand concentration inequality (see for example, Theorem 3.27 of
+Wainwright (2019)) and the fact that α + 1 ≤ β.
+C.3
+Proof of Lemma 4
+We fix an arbitrary k ∈ [K] in the following analysis. Since
+1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2
+2 · ρk(Xi) ≤ n− 2β
+d −1,
+we can get
+1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ 1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2
+�
+ρk(Xi)
+≤
+�
+�
+�
+� 1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2
+2 · ρk(Xi)
+≤ n− β
+d − 1
+2 .
+Define
+F2 = {f = G ◦ Q : G ∈ Cβ
+L(Rd; RD), Q ∈ Cβ
+L(RD; Rd)}.
+Then we have �G ◦ �Q ∈ F2. Moreover, when supx∈M∩Sk ∥f1(x) − f2(x)∥2 ≤ ϵ, it holds that
+sup
+x∈M
+��∥x − f1(x)∥2ρk(x) − ∥x − f2(x)∥2ρk(x)
+��
+≤
+sup
+x∈M∩Sk
+∥f2(x) − f1(x)∥2
+≤ ϵ.
+So consider the function class �F2 = {|∥x − f(x)∥2ρk(x), f ∈ F2}, by Lemma 6, it holds that log N( �F2, ∥ ·
+∥M, ϵ) ≤ log N(F2, ∥ · ∥M, ϵ) ≲ ( 1
+ϵ )
+d
+β . By Dudley’s entropy integral bound (see for example, Wainwright
+(2019)), we can get that
+E
+�
+sup
+f∈F2
+��� 1
+n
+n
+�
+i=1
+∥Xi − f(Xi)∥2ρk(Xi) − Eµ∗�
+∥X − f(X)∥2ρk(X)
+����
+�
+≤ C n− β
+d ∨ log n
+√n .
+35
+
+Then by Talagrand concentration inequality (see for example, Wainwright (2019)), we can get that there
+exists a constant c2, such that it holds with probability 1 − n−3 that
+Eµ∗�
+∥X − �Gk ◦ �Qk(X)∥2 · ρk(X)
+�
+≤ c2
+�
+n− β
+d ∨ log n
+√n
+�
+.
+For the second statement, we first fix a small enough positive constant r > 0 that will be chosen later.
+Then for any z ∈ Ωk = Q∗(M ∩ Sk), there exists σ(z) ∈ Ωk so that z ∈ Br(σ(z)) and ν∗
+k(σ(z)) ≥ g(r) > 0.
+Let Az = {σ(z) : z ∈ Ωk} and �fk = �Gk ◦ �Qk ◦ G∗
+k, we resort to the following lemma that provides an
+upper bound on ∥G∗
+k(z) − �fk(z)∥2 for all z ∈ Az.
+Lemma 7. It holds with probability at least 1 − c n−3 that for all z ∈ Az,
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j| ≤ C ·
+�
+g(r)
+�− 2β
+d −1 ·
+�log n
+n
+� β
+d ,
+(17)
+where δn = b1 · (g(r))− 2
+d ·
+� log n
+n
+� 1
+d for a constant b1 independent of n and r.
+So by Lemma 7, we can get
+sup
+z∈Az
+∥G∗
+k(z) − �fk(z)∥2 ≤ C ·
+�
+g(r)
+�− 2β
+d −1 ·
+�log n
+n
+� β
+d
+sup
+z∈Az
+∥JG∗
+k(z) − J �
+fk(z)∥F ≤ C1 ·
+�
+g(r)
+�− 2β−2
+d
+−1 ·
+�log n
+n
+� β−1
+d .
+Also by the fact that G∗
+k and �f = �Gk ◦ �Qk ◦ G∗
+k are β-Hölder smooth with β ≥ 1 + ˜α > 1, we have
+sup
+z∈Ωk
+∥G∗
+k(z) − �fk(z)∥2 ≤ C ·
+�
+g(r)
+�− 2β
+d −1 ·
+�log n
+n
+� β
+d + C2 r
+sup
+z∈Ωk
+∥JG∗
+k(z) − J �
+fk(z)∥F ≤ C1 ·
+�
+g(r)
+�− 2β−2
+d
+−1 ·
+�log n
+n
+� β−1
+d
++ C3 r.
+(18)
+By the fact that for any z ∈ Ωk, it holds that z = Q∗
+k(G∗
+k(z)), we obtain Id = JQ∗
+k(G∗
+k(z)) JG∗
+k(z). Since
+Q∗
+k is L-Lipschitz, JQ∗
+k(G∗
+k(z)) has bounded operator norm, which implies det(JT
+G∗
+k(z) JG∗
+k(z)) ≥ c for
+some positive constant c > 0 and z ∈ Ωk. Moroever, by the fact that G∗
+k is β-Hölder smooth with β > 1,
+there exists a positive constant ϵ so that for the ϵ- enlargement of Ωk: Ωk,ϵ = {y ∈ Bϵ(z) : z ∈ Ωk}, it
+holds that for any z ∈ Ωk,ϵ, det(JT
+G∗
+k(z) JG∗
+k(z)) ≥ c
+2. Therefore, the second display in (18) and β-Hölder
+smooth of �fk implies that infz∈Ωk,ϵ det(JT
+�
+fk(z) J �
+fk(z)) ≥ c
+4 for all sufficiently small ϵ, r and sufficiently
+large n. Now let �lk = �Qk ◦ G∗
+k and by using the identity �fk = �Gk ◦ �lk,
+JT
+�
+fk(z) J �
+fk(z) =
+�
+J �
+Gk(�lk(z)) J�lk(z)
+�T �
+J �
+Gk(�lk(z)) J�lk(z)
+�
+= JT
+�lk(z) JT
+�
+Gk(�lk(z)) J �
+Gk(�lk(z)) J�lk(z),
+by taking determinant we further obtain (note that J�lk(z) is a square matrix)
+det2 �
+J�lk(z)
+�
+· det
+�
+JT
+�
+Gk(�lk(z)) J �
+Gk(�lk(z))
+�
+≥ c
+4.
+Since both �Gk and �Qk are L-Lipschitz, we can further deduce that 0 < c1 ≤ det(J�lk(z)) ≤ c2 for all
+z ∈ Ωk,ϵ.
+We claim that �lk is globally invertible over Ωk,ϵ when ϵ, r are small enough and n is large enough.
+Otherwise, suppose there exist distinct z0 and z1 in Ωk,ϵ such that �lk(z0) = �lk(z1). Since 0 < c1 ≤
+36
+
+det(J�lk(z)) ≤ c2 implies �lk to be locally invertible, meaning that there exists some constant b0 > 0
+independent of ϵ such that ∥z0 − z1∥ ≥ b0. By the definition of Ωk,ϵ and the Lipschitzness of �Gk and �lk,
+there exist ¯z0 and ¯z1 in Ωk such that (for sufficiently small ϵ)
+∥¯z0 − ¯z1∥ ≥ 1
+2b0,
+∥�lk(¯z0) − �lk(¯z1)∥2 ≤ Cϵ
+and
+∥ �fk(¯z0) − �fk(¯z1)∥ ≤ Cϵ.
+(19)
+The third display above combined with the first display in (18) implies ∥G∗
+k(¯z0) − G∗
+k(¯z1)∥2 ≤ C1(ϵ + r).
+On the other hand, from the first display above and the Lipschitzness of Q∗
+k, we have
+1
+2b0 ≤ ∥¯z0 − ¯z1∥ = ∥Q∗
+k(G∗
+k(¯z0) − Q∗
+k(G∗
+k(¯z1))∥2 ≤ C∥G∗
+k(¯z0) − G∗
+k(¯z1)∥2 ≤ CC1(ϵ + r),
+which is a contradiction when ϵ, r are chosen small enough.
+Let �l−1
+k
+: �lk(Ωk,ϵ/2) → Ωk,ϵ/2 be the inverse of �lk over Ωk,ϵ/2. By using the inverse function theorem for
+Hölder space (see for example, Appendix A of (Eldering, 2013)), we can conclude �l−1
+k
+∈ Cβ
+C0(�lk(Ωk,ϵ/2); Rd)
+for some sufficiently large constant C0. Therefore, we can write the expression of the density function of
+ν∗
+k, �
+Qk = [ �Qk]#( ρkµ∗
+pk ) as
+ν∗
+k, �
+Qk(y) = ν∗
+k(�l−1
+k (y)) ·
+�
+det
+�
+JT
+�l−1
+k (x)J�l−1
+k (x)
+�� 1
+2 · 1
+�
+y ∈ �lk(Ωk)
+�
+by applying the change of variable of y = �lk(z) with z ∼ ν∗
+k. Moreover, since ν∗
+k ∈ Cα
+L(Rd), this together
+with �l−1
+k
+∈ Cβ
+C0(�lk(Ωk,ϵ/2); Rd) implies ν∗
+�
+Q ∈ C ˜α
+C1(Rd) for some constant C1 (recall ˜α = α ∧ (β − 1)).
+C.4
+Proof of Lemma 7
+The proof follows the analysis in Tang & Yang (2022). Let hn =
+� log n
+n
+� 1
+d and Nhn ⊂ Az be a minimal
+hn-covering set of Az under the ℓ2 distance, where its cardinality satisfies |Nhn| ≤ C
+n
+log n. For any
+�z ∈ Nhn, define δn = b
+� log n
+n
+� 1
+d .
+We claim that it suffices to show that for sufficiently large b, it holds with probability at least 1 − n−3
+that for any �z ∈ Nhn,
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j| ≤ C
+�log n
+n
+� β
+d
+.
+(20)
+In fact, if this inequality holds, then we can apply a standard argument of approximation by the hn-
+covering set. Concretely, for any z ∈ Az, there exists �z ∈ Nhn such that ∥z − �z∥2 ≤ hn = ( log n
+n )
+1
+d , we
+can obtain by applying Taylor expansion to G∗
+k(z) − �fk(z) that
+∥G∗
+k(z) − �fk(z)∥2 ≤ C
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2
+�log n
+n
+� |j|
+d + C
+�log n
+n
+� β
+d
+≤ C
+�log n
+n
+� β
+d
+.
+Now let us prove inequality (20). Recall that
+1
+n
+n
+�
+i=1
+∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ C n− 2β
+d −1.
+37
+
+In particular, by restricting the sum to those Q∗(Xi) in Bδn(�z) for a fixed �z ∈ Nhn, we further obtain
+(recall that �fk = �Gk ◦ �Qk ◦ G∗
+k)
+1
+n
+n
+�
+i=1
+∥G∗
+k(Q∗
+k(Xi)) − �fk(Q∗
+k(Xi))∥2 · ρk(Xi) · 1Bδn(�z)(Q∗
+k(Xi)) ≤ C n− 2β
+d −1.
+By applying the Taylor expansion to G∗
+k(z) − �fk(z) around �z in the preceding display and using the fact
+that G∗
+k − �fk ∈ Cβ
+C0(Bd
+1; RD) with some sufficiently large constant C0, we can get the following localized
+basic inequality after some algebra calculation
+Un(�z, �fk) : =
+1
+n
+n
+�
+i=1
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+(Q∗
+k(Xi) − �z)j
+����
+2
+2
+· 1Bδn(�z)(Q∗
+k(Xi)) · ρk(Xi)
+≤ c
+�
+(δn)2β + (δn)β
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+��G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+��
+2 (δn)|j|
+�
+· 1
+n
+n
+�
+i=1
+1Bδn(�z)(Q∗
+k(Xi)) · ρk(Xi).
+(21)
+The second factor on the right hand side of (21) can be bounded by applying a simple union bound
+argument and Bernstein’s inequality for bounded function as follows. First, we can bound the expectation
+Eµ∗�
+1Bδn(�z)(Q∗
+k(X)) · ρk(X)
+�
+(i)
+= pk
+�
+Bδn(�z)
+ν∗
+k(z) dz ≤ C pk δd
+n
+≤ C bd log n
+n
+,
+(22)
+where step (i) follows by the fact that ν∗
+k = (Q∗
+k)#( µ∗·ρk
+pk ). Since the random variable 1Bδn(�z)(Q∗
+k(X))·ρk(X)
+is uniformly bounded by 1, and inequality (22) and ρk ≤ 1 implies its variance to be bounded by C1 bd log n
+n ,
+we may apply the Bernstein inequality and a simple union bound argument over all �z ∈ Nhn (with
+|Nhn| ≤ C
+n
+log n) to obtain that with probability at least 1 − n−c,
+sup
+�z∈Nhn
+����
+1
+n
+n
+�
+i=1
+1Bδn(�z)(Q∗
+k(Xi)) · ρk(Xi) − Eµ∗�
+1Bδn(�z)(Q∗
+k(X)) · ρk(X)
+����� ≤ C2b
+d
+2 · log n
+n
+,
+(23)
+which together with (22) leads to
+sup
+�z∈Nhn
+� 1
+n
+n
+�
+i=1
+1Bδn(�z)(Q∗
+k(Xi))
+�
+≤ C bd · log n
+n
+.
+(24)
+To analyze the quantity Un(�z, �fk) on the left hand side of the localized basic inequality (21), we will
+resort to the following lemma.
+Lemma 8. With probability at least 1 − n−3, the following inequality holds for any β-smooth function
+f ∈ Cβ
+L(Bd
+1; RD) and �z ∈ Nh�n,
+��Un(�z, f) − Eµ∗[Un(�z, f)]
+�� ≤ C b
+d
+2 · log n
+n
+·
+��log n
+n
+� 2β
+d +
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥f (j)(�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j|�2�
+,
+where the expectation is taken with respect to the randomness in {Xi}n
+i=1.
+38
+
+Before applying this lemma, notice that for any �z ∈ Nhn, we can bound the expectation Eµ∗[Un(�z, �fk )],
+where f has been plugged-in with �fk , by
+Eµ∗[Un(�z, �fk )]
+= Eµ∗
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+(Q∗
+k(X) − �z)j���
+2
+2 · 1Bδn(�z)(Q∗
+k(X)) · ρk(X)
+�
+(i)
+≥
+inf
+z∈Bδn(�z)ν∗
+k(z)
+�
+z∈Bδn(�z)
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+(z − �z)j���
+2
+2 dz
+(ii)
+= δd
+n
+inf
+z∈Bδn(�z)ν∗
+k(z)
+�
+Bd
+1
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! δj
+n
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+zj���
+2
+2 dz
+≥ C bd · log n
+n
+· g(r) ·
+�
+Bd
+1
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!δj
+n
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+zj���
+2
+2 dz,
+(25)
+where step (i) uses the fact that Q∗
+k(X) given X ∼ µ∗·ρk
+pk
+is distributed as ν∗
+k, step (ii) follows by applying
+the change of variable of z−�z
+δn → z, and the last step follows by the fact that ν∗
+k(�z) ≥ g(r) for �z ∈ Az and
+the smoothness of ν∗
+k. Now using the fact that for any d-variate polynomial S(y) = �
+j∈Nd
+0, |j|≤k ajyj,
+y ∈ Rd, there exists some positive constant C(d, k) only depending on (d, k) such that
+�
+Bd
+1
+S2(y) dy ≥ C(d, k)
+�
+j∈Nd
+0, |j|≤k
+a2
+j,
+we can obtain that
+�
+Bd
+1
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!δj
+n
+�
+G∗,(j)
+k
+(�z) − �f (j)
+k (�z)
+�
+zj���
+2
+2 dz ≥ c
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j|
+�2
+.
+(26)
+Finally, by combining equations (21), (22), (25), (26) and Lemma 8, we obtain that with probability at
+least 1 − cn−3, for any �z ∈ Nhn,
+bd · log n
+n
+· g(r) ·
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j|
+�2
+≤ Cb
+d
+2 · log n
+n
+·
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥ �f (j)
+k (�z) − G∗,(j)
+k
+(�z)∥2 (δn)|j|
+�2
++ Cb
+d
+2 ·
+�log n
+n
+� 2β
+d · log n
+n
++ Cbd · log n
+n
+·
+�
+(δn)2β + (δn)β
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥G∗,(j)
+k
+(�z) − �f (j)
+k (�z)∥2(δn)|j|
+�
+.
+Consequently, the claimed inequality (20) follows from the above by choosing b = b1(g(r))− 2
+d with
+sufficiently large b1 and the definition that δn = b
+� log n
+n
+� 1
+d .
+C.5
+Proof of Lemma 8
+The proof follows from the proof of Lemma 18 in Tang & Yang (2022), we include it here for completeness.
+Since f ∈ Cβ
+L(Bd
+1; RD), for any z ∈ Bd
+1 and j ∈ Nd
+0 with |j| ≤ ⌊β⌋, it holds that ∥f (j)(z)∥2 ≤
+√
+DL = C0.
+39
+
+For any fixed �z ∈ Nh�n and �δ > 0, let
+¯T (�δ) =
+�
+T = {Tj}j∈Nd
+0, |j|≤⌊β⌋ ∈ [−C0, C0]D×(
+d+⌊β⌋−1
+d
+) :
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+��Tj − G∗,(j)
+k
+(�z)
+��
+2 (δ�z)|j| ≤ �δ
+�
+.
+We also define the following supreme of an empirical process indexed by T ∈ ¯T (�δ),
+Zn(�δ) =
+sup
+T ∈ ¯T (�δ)
+����� Eµ∗
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(X) − �z)j���
+2
+2 · 1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+�
+− 1
+n
+n
+�
+i=1
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(Xi) − �z)j���
+2
+2 · 1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+������,
+and Rn(�δ) = Eµ∗�
+Zn(�δ)
+�
+. We will first prove a concentration inequality for a fixed radius �δ > 0, and then
+using the peeling technique to allow the radius to be random, which leads to the desired result.
+To apply the Talagrand concentration inequality (see, for example, Theorem 3.27 of Wainwright (2019))
+for bounding the difference |Zn(�δ) − Rn(�δ)| for a fixed �δ > 0, we notice that each additive component in
+the second empirical sum above has second moment uniformly bounded by
+Eµ∗
+�
+sup
+T ∈ ¯T (�δ)
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(X) − �z)j���
+4
+2 · 1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+��
+≤
+sup
+z∈Bδ�z (�z)
+T ∈ ¯
+T (�δ)
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(z − �z)j���
+4
+2 · Eµ∗�
+1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+�
+≤ C sup
+T ∈ ¯T (�δ)
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − G∗,(j)
+k
+(�z)∥2 (δ�z)|j|
+�4
+· bd
+2 · log n
+n
+≤ C bd
+2 �δ4 · log n
+n
+,
+where we have used inequality (22) to bound Eµ∗�
+1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+�
+. Moreover, each additive
+component can be almost surely bounded by
+sup
+z∈Bδ�z (�z)
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(X) − �z)j���
+2
+2
+≤ C
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − G∗,(j)(�z)∥2 (δ�z)|j|
+�2
+≤ C �δ2.
+Based on these two bounds, we can apply the Talagrand concentration inequality to obtain that for any
+s ≥ 0,
+P
+�
+Zn(�δ) ≥ Rn(�δ) + s2�
+≤ 2 exp
+�
+−
+c ns4
+s2 �δ2 + bd
+2 �δ4 · log n
+n
+�
+.
+(27)
+It remains to bound the expectation Rn(�δ) via the symmetrization technique and chaining. By a standard
+40
+
+symmetrization, we can get
+Rn(�δ) ≤
+2
+√n E
+�
+sup
+T ∈ ¯T (�δ)
+�����
+1
+√n
+n
+�
+i=1
+εi
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(X) − �z)j���
+2
+2 · 1Bδ�z (�z)(Q∗
+k(Xi)) · ρk(Xi)
+������
+�
+,
+where {εi}n
+i=1 are n i.i.d. copies from the Rademacher distribution, i.e. P(εi = 1) = P(εi = −1) = 0.5.
+Since given {Xi}n
+i=1, the stochastic process inside the supreme is a sub-Gaussian process with intrinsic
+metric
+d2
+n(T, �T)
+= 1
+n
+n
+�
+i=1
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(Xi) − �z)j���
+2
+2
+−
+���
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − �Tj
+�
+(Q∗
+k(Xi) − �z)j���
+2
+2
+�2
+· 1Bδ�z (�z)(Q∗
+k(Xi))) · ρk(Xi)
+≤ C �δ4 1
+n
+n
+�
+i=1
+1Bδ�z (�z)(Q∗
+k(Xi))) · ρk(Xi),
+for any T, �T ∈ ¯T (�δ), where the last step uses the definition of ¯T (�δ).
+The above combined with
+inequality (22) implies
+Eµ∗
+�
+sup
+T, �
+T ∈ ¯T (δ)
+d2
+n(T, �T)
+�
+≤ C bd
+2 �δ4 · log n
+n
+and
+dn(T, �T) ≤ C�δ
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − �Tj∥2 δ|j|
+�z .
+Lastly, let Kn(δ) =
+sup
+T, �
+T ∈ ¯T (δ)
+d2
+n(T, �T), by applying the standard chaining via Dudley’s inequality, we can
+get
+Rn(�δ) ≤ C
+1
+√n Eµ∗
+� � Kn(�δ)
+0
+�
+log
+�δ
+u du
+�
+= C
+1
+√n Eµ∗
+�
+Kn(�δ) ·
+� 1
+0
+�
+log
+�δ
+u · Kn(�δ)
+du
+�
+= C
+1
+√n Eµ∗
+�
+Kn(�δ) · 1(Kn(�δ) ≤ b
+d
+2
+2 �δ2
+�
+log n
+n
+)
+� 1
+0
+�
+log
+�δ
+u · Kn(�δ)
+du
+�
++ C
+1
+√n Eµ∗
+�
+Kn(�δ) · 1(Kn(�δ) > b
+d
+2
+2 �δ2
+�
+log n
+n
+)
+� 1
+0
+�
+log
+�δ
+u · Kn(�δ)
+du
+�
+≤ C1 b
+d
+2
+2 · log(n/�δ)
+n
+· �δ2,
+(28)
+where we have used the fact that the u-covering entropy of ¯T (�δ) relative to metric dn is at most C2 log
+�δ
+u
+for u ∈ (0, 1) where C2 depends on (d, D) (at most polynomial dependence on D). By combining this
+with inequality (27), we obtain that for all t ≥ 1,
+P
+�
+Zn(�δ) ≥ C t2 b
+d
+2
+2 · log(n/�δ)
+n
+· �δ2�
+≤ 2 exp
+�
+− c t2 log(n/�δ)
+�
+.
+(29)
+41
+
+Finally, we apply the peeling technique to extend the above high probability bound on Zn(�δ) to
+the random radius �δ = �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�� �f (j)
+k
+− G∗,(j)
+k
+(�z)
+��
+2 (δ�z)|j|. Specifically, we first set the basic level
+¯δ =
+� log n
+n
+� β
+d , and for s = 1, · · · , S with S ≤ C log 1
+¯δ , define sets
+�T0 =
+�
+T = {Tj}j∈Nd
+0,|j|≤⌊β⌋ ∈ [−C0, C0]D×(
+d+⌊β⌋−1
+d
+) :
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − G∗,(j)
+k
+(�z)∥2 (δ�z)|j| ≤ ¯δ
+�
+;
+�Ts =
+�
+T = {Tj}j∈Nd
+0,|j|≤⌊β⌋ ∈ [−C0, C0]D×(
+d+⌊β⌋−1
+d
+) : 2s−1¯δ ≤
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − G∗,(j)
+k
+(�z)∥2 (δ�z)|j| ≤ 2s¯δ
+�
+.
+By applying inequality (29) to �δ = 2s¯δ for s ∈ [S] with sufficiently large constant t > 0, as C1 ≤
+− log(2s¯δ) ≤ C2 log n, we obtain that
+P
+�
+Zn(¯δ) ≥ C b
+d
+2
+2
+log n
+n
+¯δ2
+�
++
+S
+�
+s=1
+P
+�
+Zn(2s¯δ) ≥ C b
+d
+2
+2
+log n
+n
+4s¯δ2
+�
+≤ n−(c+1).
+Note that for any T ∈ �Ts and any s ∈ {0} ∪ [S], the event Zn(2s¯δ) ≤ C b
+d
+2
+2
+log n
+n
+4s¯δ2 implies
+����� Eµ∗
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(X) − �z)j���
+2
+2 · 1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+�
+− 1
+n
+n
+�
+i=1
+����
+�
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j!
+�
+G∗,(j)
+k
+(�z) − Tj
+�
+(Q∗
+k(Xi) − �z)j���
+2
+2 · 1Bδ�z (�z)(Q∗
+k(X)) · ρk(X)
+������
+≤ c1 b
+d
+2
+2
+log n
+n
+�
+¯δ2 +
+� �
+j∈Nd
+0
+|j|≤⌊β⌋
+1
+j! ∥Tj − G∗,(j)
+k
+(�z)∥2 (δ�z)|j|
+�2�
+.
+Finally, since for any f ∈ Cβ
+L(Bd
+1; RD), Tf : = {Tf,j = f (j)}j∈Nd
+0,|j|≤⌊β⌋ must belong to some �Ts, the
+claimed result is a consequence of the two preceding displays and a simple union bound over �z ∈ Nhn
+where |Nhn| ≤ C
+n
+log n ≤ C n.
+C.6
+Proof of Lemma 5
+Firstly by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤ C
+�
+log n
+n ,
+then by pk > 0, for large enough n, we have
+��1 − pk
+�pk
+�� ≤ C
+�
+log n
+n . Thus
+sup
+Q∈Cβ
+L(RD;Rd)
+sup
+f∈Lip1(Rd)
+� 1
+pk
+�
+f(Q(x))ρk(x) dµ∗ −
+�
+f(z) �νk,Q(z) dz
+�
+=
+sup
+Q∈Cβ
+L(RD;Rd)
+sup
+f∈Lip1(Rd)
+� 1
+pk
+�
+f(Q(x))ρk(x) dµ∗ −
+1
+�pkn
+n
+�
+i=1
+f(Q(Xi))ρk(Xi)
+�
+≤ C
+�
+log n
+n
++ 1
+pk
+sup
+Q∈Cβ
+L(RD;Rd)
+sup
+f∈Lip1(Rd)
+� �
+f(Q(x))ρk(x) dµ∗ − 1
+n
+n
+�
+i=1
+f(Q(Xi))ρk(Xi)
+�
+≤C
+�
+log n
+n
++ C
+sup
+f∈Lip1(RD)
+� �
+f(x)ρk(x) dµ∗ − 1
+n
+n
+�
+i=1
+f(Xi)ρk(Xi)
+�
+,
+42
+
+where the last inequality is due to the assumption that β ≥ 1. Then consider the pseudo-metric for
+f, f ′ ∈ Lip1(RD)
+dn(f, f ′) =
+�
+�
+�
+� 1
+n
+n
+�
+i=1
+�
+f(Xi)ρk(Xi) − f ′(Xi)ρk(Xi)
+�2 ≤
+sup
+x∈Sk∩M
+|f(x)−f ′(x)| =
+sup
+x∈G∗(Q∗(Sk∩M))
+|f(x)−f ′(x)|.
+Then by Lemma 6, we have log N(Lip1(RD), ∥ · ∥L∞(Sk∩M), ϵ) ≤ C ϵ−d. Choose δ = ( 1
+n)
+1
+d , we can get
+1
+√n
+� 1
+δ
+�
+log N(Lip1(RD), ∥ · ∥L∞(Sk∩M), ϵ) dϵ ≤ C n− 1
+d ∨ log n
+√n .
+Thus similar as the analysis in the proof of Lemma 3, using Dudley’s entropy integral bound and Talagrand
+concentration inequality, we can obtain the desired result.
+D
+Proof of Technical Details
+D.1
+Proof of Lemma 1
+Consider an aritrary µ∗ ∈ P∗(d, D, α, β, L∗). Denote M = supp(µ∗), to begin with, we consider the
+following lemma.
+Lemma 9 (Lemma 17 of Tang & Yang (2022)). There exist positive constants (τ1, L1) such that for any
+x0 ∈ M, define Qx0 : RD → Rd as Qx0(x) = W T
+x0(x−x0) where Wx0 ∈ RD×d is an arbitrary orthonormal
+basis of the tangent space of M at x0, then there exists a set �Ux0 satisfying Bτ1(x0) ∩ M ⊂ �Ux0 ⊂ M
+and function Gx0 ∈ Cβ
+L1(Rd; RD) so that
+(1). Gx0(Bd
+1) = �Ux0 and for any z ∈ Bd
+1, Qx0(Gx0(z)) = z;
+(2). µ∗ ◦ Gx0|Bd
+1 ∈ Cα
+L1(Bd
+1) and for any z ∈ ∂Bd
+1, ∥Gx0(z) − x0∥ ≥ τ1.
+By Sobolev extension theorem, there exists a constant L2 so that for any x0 ∈ M, there exists Qx0 ∈
+Cβ
+L2(RD) so that Qx0| �Ux0 = Qx0| �Ux0. Since JGx0(z)T JGx0(z) has uniformly lower bounded eigenvalues
+on z ∈ Bd
+1 and Gx0 is β-smooth with β > 1 and uniformly bounded Hölder norm, there exists a small
+enough positive constant r0 so that for any x0 ∈ M,
+sup
+v∈Sd−1
+1
+={v∈Rd : ∥v∥=1}
+sup
+z,z′∈Bdr0
+∥(JGx0 (z) − JGx0 (z′))v∥
+∥JGx0 (0)v∥
+≤ 1
+3.
+(30)
+We then choose r∗ ≤ τ1/2 to be a small enough positive constant so that for any x0 ∈ M, Qx0(Br∗(x0) ∩
+M) ⊂ Bd
+r0. For an arbitrary k ∈ [K], consider x0 = ak ∈ M. Define Gk = Gx0 and Qk = Qx0. Then
+we have Sk ∩ M ⊂ Bτ1(x0) ∩ M ⊂ �Ux0, and for any x ∈ M ∩ Sk, x = Gk(Qk(x)). Moreover, let
+pk = Eµ∗[ρk(X)], since the density of µ∗ is uniformly bounded from below and rk ≥ L∗, we have pk is
+also uniformly bounded from below. Furthermore, we can write νk = (Qk)#( µ∗ρk
+pk ) as
+νk(z) =
+�
+�
+�
+µ∗(Gk(z))·ρk(Gk(z))√
+det(JGk (z)T JGk (z))
+�
+Qk(M∩Sk) µ∗(Gk(z))·ρk(Gk(z))√
+det(JGk (z)T JGk (z)) dz
+z ∈ Bd
+1;
+0
+o.w.
+Then by the α-smoothness of �ρk(·) and the uniformly lower boundness of �
+k∈[K] �ρk(·), we have νk(z)|Bd
+1 ∈
+Cα
+L(Bd
+1) for some constant L.
+On the other hand, by the second statement of Lemma 9 and the
+Lipschitzness of Gm, there exists a positive constant ϵ so that Qk(M ∩ Sk) ⊂ Bd
+1−ϵ. Combined with
+43
+
+the fact that νk(z) = 0 when z /∈ Qk(M ∩ Sk), we can obtain νk(z) ∈ Cα
+L∗(Rd). In addition, for any
+z ∈ Ωk = Qk(M ∩ Sk) and any r > 0, we will show that there exists z′ ∈ Br(z) so that νk(z′) ≥ c (rγ ∧ 1).
+Firstly if ∥Gk(z) − x0∥ ≤ r∗/2, then we have
+νk(z) ≥ c1 ρk(Gk(z)) ≥ c1
+( 3r2
+k
+4 )γ
+M(r∗)2γ .
+On the other hand, if ∥Gk(z) − Gk(0)∥ = ∥Gk(z) − x0∥ ≥ r∗/2, denote z = av with v = z/∥z∥ and
+a = ∥z∥. Then we have a ≤ r0 and
+r∗/2 ≤ ∥Gk(z) − Gk(0)∥ ≤ c2 ∥z∥ = c2 |a|.
+If r ≥ a, then z = av ∈ Br(0) and νk(0) ≥ c for some positive constant c. If r < a, choose z′ = (a − r)v,
+then z′ ∈ Br(z) and
+∥Gk(z′) − Gk(0)∥ = sup
+l∈Sd−1
+1
+�
+lT Gk(z′) − lT Gk(0)
+�
+= sup
+l∈Sd−1
+1
+�
+lT Gk(z) − lT Gk(0) + lT Gk(z′) − lT Gk(z)
+�
+(i)
+=
+sup
+l∈Sd−1
+1
+�
+lT JGk(zl)av − lT JGk(z′
+l)rv
+�
+= sup
+l∈Sd−1
+1
+�
+lT JGk(zl)(a − r)v + lT (JGk(zl) − JGk(z′
+l))rv
+�
+(ii)
+≤
+sup
+l∈Sd−1
+1
+�
+lT JGk(zl)(a − r)v + r
+2a∥Gk(z) − Gk(0)∥)
+= a − r/2
+a
+∥Gk(z) − Gk(0)∥
+≤ rk − rrk
+2r0
+,
+where (i) uses mean-value theorem and (ii) uses equation (30) and Taylor’s theorem to obtain
+r
+2a∥Gk(z) − Gk(0)∥ = r
+2
+���
+� 1
+0
+JGk(tz) dt · v
+���
+≥ r
+2∥JGk(0) · v∥ − r
+2
+���
+� 1
+0
+JGk(0) − JGk(tz) dt · v
+���
+≥ 3r
+2
+sup
+z,z′∈Bd
+r0
+∥(JGx0 (z) − JGx0 (z′))v∥ − r
+2
+� 1
+0
+∥(JGk(0) − JGk(tz))v∥ dt
+≥ r
+sup
+z,z′∈Bdr0
+∥(JGx0 (z) − JGx0 (z′))v∥
+≥ sup
+l∈Sd−1
+1
+lT (JGk(zl) − JGk(z′
+l))rv.
+So we have
+�ρk(Gk(z′)) = (r2
+k − ∥Gk(z′) − x0∥2)γ· ≥ (rr2
+k
+2r0
+)γ ≥ ( r2
+k
+2r0
+)γrγ ≥ ((L∗
+1)2
+2r0
+)γrγ.
+(31)
+Thus there exists constant c so that
+νk(z′) ≥ c rγ.
+Therefore, we have Assumption A holds for µ∗ with G∗
+k = Gk, Q∗
+k = Qk and ν∗
+k = νk with k ∈ [K]. For
+44
+
+the second statement, note that by the β-smoothness of G∗
+k = Gk, Q∗
+k = Qk and the α-smoothness of
+ν∗
+k = νk, Assumption B trivially holds for approximation family G = G1. Moreover, consider
+νk(z) =
+µ∗(Gk(z)) ·
+�
+det(JGk(z)T JGk(z))
+�
+Bd
+1 µ∗(Gk(z)) ·
+�
+det(JGk(z)T JGk(z)) dz
+,
+z ∈ Bd
+1.
+Then we have νk(z) = νk(z)·ρk(Gk(z))
+Eνk [ρk(Gk(z))] , νk(z) ∈ Cα
+L(Bd
+1) and infz∈Bd
+1νk(z) ≥ L3 > 0. So there exists an
+(α + 1)-smooth invertible function Vk : Bd
+1 → Bd
+1 (see for example, (Caffarelli, 1996; Villani, 2009)) so that
+ν0 = Vm#νk and νk = V −1
+m
+#ν0. Therefore, G2 suffices to model µ∗.
+For the family G3, let V k be an (α + 1)-smooth extension of Vk|Bd
+1−ϵ/2 to Rd. Note that Vk has (α + 1)-
+smooth inverse and Vk(Bd
+1−ϵ/2) ⊂ Bd
+1−ϵ1 for some positive constant ϵ1. We can consider V −1
+k
+as an
+α-smooth extension of V −1
+k
+|Vk(Bd
+1−ϵ/2) to Rd. Then we can define G′
+k = Gk ◦ V −1
+k
+and Q′
+k = V k ◦ Qk, by
+the fact that Qk(M ∩ Sk) ⊂ Bd
+1−ϵ, we have for any x ∈ M ∩ Sk, G′
+k(Q′
+k(x)) = x. Moreover, let
+ν′
+k(z) = (Q′
+k)#
+µ∗ρk
+pk
+=
+�
+�
+�
+ν0(z)·ρk(Gk◦V −1
+k
+(z))
+�
+Bd
+1 ν0(z)·ρk(Gk◦V −1
+k
+(z)) dz,
+z ∈ Vk(Bd
+1−ϵ/2),
+0,
+o.w.
+Using the fact that V −1
+k
+|Vk(Bd
+1−ϵ/2) is (α + 1)-smooth with bounded Hölder norm and ν′
+k(z) = 0 when
+z /∈ Vk(Bd
+1−ϵ), we have ν′
+k ∈ Cα
+L(Rd) for some constant L. In addition, recall that for any z0 ∈ Qk(M∩Sk)
+and r > 0, there exists z′
+0 ∈ Qk(M ∩ Sk) so that z0 ∈ Br(z′
+0) and �ρk(Gk(z′
+0)) ≥ c1(rγ ∧ 1). Note that by
+the Lipschitzness of Vk, there exists a constant L4 ≥ 1 so that
+∥Vk(z0) − Vk(z′
+0)∥ ≤ L4∥z0 − z′
+0∥.
+Therefore, for any z ∈ Vk(Qk(M ∩ Sk)) and r > 0, there exists z′ ∈ Vk(Qk(M ∩ Sk)) ∩ Br(z), so that
+�ρk(Gk ◦ V −1
+k
+(z′)) ≥ c1
+Lγ
+4 (rγ ∧ 1) and ν′
+k(z) ≥ c (rγ ∧ 1). Therefore, when α = β − 1. Assumption A holds
+with G∗
+k = G′
+k, Q∗
+k = Q′
+k and ν∗
+k = ν′
+k, and Assumption B holds with G = G3.
+D.2
+Proof of Lemma 2
+Let Nϵ be the minimal ϵ-covering set of M, where ϵ is a number that will be chosen later, then by
+Lemma 9 and the compactness of M, we have |Nϵ| ≤ C1 ( 1
+ϵ )d where C1 is a positive constant that only
+depends on (d, D, β, L∗). Then if ϵ ≤ τ1, by Lemma 9, we have for any x0 ∈ M
+Pµ∗(Bϵ(x0)) =
+�
+Qx0(Bϵ(x0))
+µ∗(Gx0(z))
+�
+det(JGx0 (z)T JGx0 (z)) dz ≥ C2 ϵd.
+Then, by Bernstein’s inequality and a simple union bound argument, it holds with probability at least
+1 − n−c
+1
+that for any x0 ∈ Nϵ,
+��� 1
+n1
+�
+i∈I1
+1(∥Xi − x0∥ ≤ ϵ) − Pµ∗(Bϵ(x0))
+��� ≤
+1
+3n1
+log(δ) +
+�
+2C2ϵd log(δ)
+n1
+,
+δ = 2C1nc
+1(1
+ϵ )d.
+Therefore, there exists a constant C3, C so that when n1 ≥ C, by choosing ϵ = C3 ( log n1
+n1 )
+1
+d , we have it
+holds with probability at least 1 − n−c
+1
+that for any x0 ∈ Nϵ,
+��� 1
+n1
+�
+i∈I1
+1(∥Xi − x0∥ ≤ ϵ) − Pµ∗(Bϵ(x0))
+��� ≤ C2
+2 ϵd ≤ 1
+2Pµ∗(Bϵ(x0)).
+45
+
+Therefore, for any x0 ∈ Nϵ, there exists i ∈ I1 so that ∥Xi − x0∥ ≤ ϵ. We can then obtain that for any
+x ∈ M, there exists i ∈ I1 so that ∥Xi − x0∥ ≤ 2ϵ. Proof of the first statement is then completed. For the
+second statement, when n1 is large enough, we have ϵ = C3 ( log n1
+n1 )
+1
+d ≤ r∗
+16. Let �
+Nr∗/4 denote the minimal
+r∗/4-covering set of �
+i∈I1 B2ϵ(Xi). Then | �
+Nr∗/4| is controlled by the minimal r∗/8-covering number of
+M. For any x0 ∈ | �
+Nr∗/4|, there exists an index i ∈ I1 so that B2ϵ(Xi) ∩ Br∗/4(x0) ̸= ∅. Let I2 be the set
+of such index i for x0 ∈ | �
+Nr∗/4|. Then for any x ∈ �
+i∈I1 B2ϵ(Xi), there exists i ∈ I2 so that
+∥x − Xi∥ ≤ r∗/4 + r∗/4 + 2ϵ ≤ 5r∗
+8 .
+Therefore, set M = |I2| and {ak}K
+k=1 = {Xi}i∈I2, we have
+�
+i∈I1
+B2ϵ(Xi) ⊂
+�
+k∈[K]
+Br∗(ak),
+and
+inf
+x∈M
+�
+k∈[K]
+�ρk(x) ≥ ((r∗)2 − (5r∗
+8 )2)γ > ((r∗)2
+2
+)γ.
+Proof is completed.
+46
+
diff --git a/cNAyT4oBgHgl3EQf-PrC/content/tmp_files/load_file.txt b/cNAyT4oBgHgl3EQf-PrC/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4a98b30f16d7bf4012827f17c963c2b3ba16ee52
--- /dev/null
+++ b/cNAyT4oBgHgl3EQf-PrC/content/tmp_files/load_file.txt
@@ -0,0 +1,1912 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf,len=1911
+page_content='Estimating Distributions with Low-dimensional Structures  Using Mixtures of Generative Models  Rong Tang and Yun Yang  University of Illinois Urbana Champaign  Abstract  There has been a growing interest in statistical inference from data satisfying the so-called manifold  hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a  submanifold of much lower dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In machine learning, encoder-decoder pair based generative  modelling approaches have been successful in learning complicated high-dimensional distributions  such as those over images and texts by explicitly imposing the low-dimensional manifold structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  In this work, we introduce a new approach for estimating distributions on unknown submanifolds via  mixtures of generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We show that conventional generative modeling approaches using a  single encoder-decoder pair are generally unable to capture data distributions under the manifold  hypothesis, unless the underlying manifold admits a global parametrization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' however, this issue can  be solved by using a collection of encoder-decoder pairs for learning different local patches of the  data supporting manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' A rigorous theoretical analysis is developed to demonstrate that the  proposed estimator attains the minimax-optimal rate of convergence for the implicit estimation of  data distributions with manifold structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Our experiments show that, by utilizing parameter  sharing, the proposed method can significantly improve the performance of conventional auto-encoder  based generative modelling approaches with minimal additional computational efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Keywords: Autoencoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' distribution estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' generative model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' minimax-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  1  Introduction  Modelling and estimating complicated high-dimensional distributions with low-dimensional structures  remains one of the major challenges in modern statistical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Suppose we observe n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' samples  {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' , Xn} living in an ambient Euclidean space RD according to some unknown distribution  µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We wish to estimate µ∗ based on the samples for conducting statistical inference and generating  new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' One of the most popular nonparametric methods for distribution estimation is kernel  density estimation (KDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' It has been shown that when µ∗ admits a density function relative to the  Lebesgue measure of RD, and the density function is β-smooth, then KDE can achieve the optimal rate  n−  β  2β+D for recovering the density value at any point in RD (Silverman, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Tsybakov, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' However,  the non-parametric rate n−  β  2β+D suffers from the curse of dimensionality as the ambient dimension D  appears in the rate exponent and can be enormous in machine learning applications involving images and  texts (Brock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In order to avoid this exponential blow-up of the dimension,  a common practice is to assume some additional structure in the data so that the effective dimension of  the data space is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  One such structure that has attracted much attention recently is the so-called manifold hypothesis,  which assumes the date to live on a d-dimensional submanifold M embedded in the possibly high-  dimensional ambient space RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Although submanifolds have more complicated geometry than the  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='00890v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ME]  2 Jan 2023  conventional Euclidean spaces, the manifold hypothesis is a natural assumption to make in a number of  areas of science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For example, in computer vision and medical imaging, data are usually  images represented as vectorized pixel intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Although images may contain millions of pixels, it is  usually determined by a comparatively smaller set of global characteristics such as camera projection,  lighting condition, texture, object position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Other examples of high-dimensional complex  data with low-dimensional manifold structures appear in natural language processing (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2017), protein-protein interaction detection (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Terradot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2004), and  astronomy and shape analysis (Mardia, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Jupp & Mardia, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Statistical theory and methodology for modeling manifold valued data have been developed in various  contexts (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2020, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Divol, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Tang & Yang, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Berenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Specifically, the problem of estimating a probability measure lying on an  unknown low-dimensional Riemannian submanifold has been studied in a number of recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For  example, Divol (2022) consider a kernel density type estimator based on a preliminary step of estimating  the volume measure of the submanifold using local polynomial estimation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' They prove that  the developed estimator can achieve the minimax-optimal error bound under the Wasserstein loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Tang &  Yang (2022) construct a two-step estimator: the first step estimates the data supporting submanifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and  the second step recovers the distribution on the estimated submanifold based on wavelet type estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  They also show that such an estimation is minimax-optimal with respect to certain adversarial loss  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Berenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2022) develop a Bayesian procedure based on location-scale mixtures of  Gaussians for estimating the density of data living close to an unknown submanifold with theoretical  guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' However, although these existing methods are theoretically appealing, they usually have poor  computational scalability with the ambient dimensionality and the sample size, making them costly to  implement for modeling massive and high-dimensional real data, such as images and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Auto-encoder based deep generative modeling approaches in the machine learning literature, such as  variational auto-encoder (VAE) (Kingma & Welling, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rezende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2016),  Wasserstein auto-encoder (WAE) (Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2019), InfoVAE (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2019) and inferential  Wasserstein generative adversarial networks (iWGAN) (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2022), have achieved great successes  in generating synthetic realistic-looking images and texts, and are usually very efficient to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  However, despite their empirical successes, a general theoretical framework explaining whether and how  these generative modelling approaches benefit from the low-dimensional manifold structure is lacking,  and it is also not clear whether these existing methods are theoretically optimal in the minimax sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For example, the key step in the auto-encoder is the extraction of d (d ≪ D) latent features (via an  encoder Q : RD → Rd) that can be used for accurately reconstructing the original data (via a decoder  G : Rd → RD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In other words, these auto-encoder based methods implicitly assume data X to have a  low-dimensional structure so that they can be accurately reconstructed in the sense that X ≈ G(Q(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Moreover, it is often the case that real-world data falls on a manifold that does not admit a global  parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For example, when the data space is a boundaryless manifold such as a sphere, or  disconnected (Khayatkhoei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' This lack of global parametrization makes conventional auto-  encoder methods equipped with a single encoder/decoder pair incapable of recovering the entire data  space without incurring distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Our empirical results (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 1) also suggest that conventional  auto-encoder based generative modelling approaches tend to generate off real-manifold samples with  unrealistic appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  In this article, we propose a new generative modelling approach for learning manifold-supported  distributions that is theoretically minimax-optimal, computationally efficient, and empirically promising  in generating complicated yet realistic-looking data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Unlike most existing generative modelling procedures  that rely on the strong assumption of the existence of a global parametrization of the data space, we  employ multiple encoder/decoder pairs, where each pair corresponds to the parametrization of a local  2  patch of the data supporting manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, we utilize the partition of unity technique for gluing  local probability measures estimated in the patches to form a global estimation of the probability measure  on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In addition, most existing methods simply plug in the data empirical distribution in  constructing the objective function for defining a GAN (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2014) type estimator, which  may lead to theoretical deficiency due to the failure of taking the smoothness of the target distribution  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We instead propose to plug in a smoothness-regularized version that provably improves the  estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Concretely, we show that when the target distribution is α-smooth and lies in a  β-smooth d-dimensional submanifold in RD, then the corresponding estimator �µ based on n data points  achieves a non-asymptotic error bound of order O  � log n  √n ∨ n−  α∧(β−1)+1  2(α∧(β−1))+d �  (here a ∧ b denotes min{a, b})  under the 1-Wasserstein distance, which corresponds to the minimax rate modulo a logarithmic factor  when α ≤ β − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The implied rate of convergence does not suffer from the “curse of dimensionality” and  only depends on the intrinsic dimensionality d of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Our numerical results also show that the  proposed method tends to be more accurate than conventional auto-encoder based generative modelling  approaches and classic kernel density estimators for learning target distributions with low-intrinsic  dimensional structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Notation  We summarize some necessary notations and definitions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any positive integer k, we use the  shorthand [k] := {1, 2, · · · , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use ∥ · ∥p to denote the usual vector ℓp norm, and reserve ∥ · ∥ for  the ℓ2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use Sd  1 = {x ∈ Rd+1 : ∥x∥ = 1} to denote the d-dimensional unit sphere in Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For  a probability measure µ, we use supp(µ) to denote its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any measure ν and map G, the  push-forward measure µ = G#ν is defined as the unique measure such that µ(A) = ν(G−1(A)) holds  for any measurable set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For two probability measures µ, ν, the 1-Wasserstein distance between µ and  ν is defined as W1(µ, ν) = inf  � �  ∥x − T(x)∥1 dµ(x) : T#µ = ν  �  = sup  � �  f(x)d(µ − ν) : Lip(f) ≤ 1  �  ,  where Lip(f) denotes the minimal Lipschitz constant for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' When no ambiguity arises, for an absolutely  continuous probability measure ν, we may also use ν to refer to its density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use P(Ω) to  denote the set of probability measures on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use Br(x) to denote the closed ball centered at x  with radius r under the ℓ2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use Cα  r (Ω) to denote the set of all α-Hölder smooth functions  with Hölder norm ∥ · ∥Cα(Ω) being bounded by r (see for example, Evans (2010a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Similarly, we use  Cα  r (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) =  �  f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' , fD) : Ω → RD �� ∀ j ∈ [D], fj ∈ Cα  r (Ω)  �  to denote the vector valued function  space counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  Organization  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In Section 2, we give a brief introduction to the auto-encoder  based generative modelling approaches and define smooth distributions on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Our proposed  model is introduced in Section 3, and its implementation and theoretical properties are described in  Section 4 and Section 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Simulations and a real data application are provided in Section 6  and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Auto-encoder based generative modelling approaches  Assume i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' data X(n) = {X1, X2, · · · , Xn} sampled from an unknown target distribution µ∗ over  data space X ⊂ RD are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In the literature of generative modelling, the target distribution µ∗  is implicitly specified by its sampling scheme, represented by a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Mathematically, a  3  generative model is defined as a pair (ν0, G), where ν0 is a distribution on a low-dimensional latent space  Z ⊂ Rd, called generative distribution, that is easy to sample from;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and G : Z → X is a map from Z to  X, called generative map, so that if Z ∼ ν0, then G(Z) ∼ µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The goal of generative modelling is to fit a  generative model that specifies a stochastic process whose simulated data look indistinguishable from real  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In particular, auto-encoder based generative modelling approaches introduce a family of encoders  Q that send the data X to the (low-dimensional) latent variables Z, and a family of decoders G that  reconstruct the data from the latent variables, so that they jointly minimizes the following objective:  1  n  n  �  i=1  c  �  Xi, G(Q(Xi))  �  + Penalty term,  where c(·, ·) is a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The first component n−1 �n  i=1 c(Xi, G(Q(Xi))) of the objective function  corresponds to the reconstruction cost: a common choice is the squared loss c(x, y) = ∥x − y∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' This  reconstruction cost enforces the push-forward measure of (G ◦ Q)#µ∗ to be as close to µ∗ as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  On the other hand, the second component involves a penalty term for regularizing the encoder/decoder  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For example, in VAE (Kingma & Welling, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rezende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2016), the  penalty term is chosen to be an averaged Kullback-Leibler (KL) divergence between the latent variable  distribution induced by the (probabilistic) encoder and a prior distribution ν0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' in iWGAN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=',  2022), the penalty term is chosen to be an approximation to the 1-Wasserstein distance between the  reconstructed data distribution (G ◦ Q)#µ∗ and induced distribution from the generative model G#ν0  using prior ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Other approaches choose penalty terms for directly matching the encoder-induced latent  variable distribution and a given prior in the latent space (for example, WAE, Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  infoVAE, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Sliced WAE, Kolouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2018)), which leads to the following training  objective:  1  n  n  �  i=1  c  �  Xi, G(Q(Xi))  �  + λ · D(Q#�µem, ν0),  (1)  where �µem : = n−1 �n  i=1 δXi denotes the discrete empirical distribution of the data X(n), and D is a  generic discrepancy metrics characterizing closeness between distributions over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We will  call any method whose training objective takes the form as (1) a latent distribution matched auto-encoder  (LDMAE) for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Ideally, the learned decoder �G from minimizing (1) has the property  that �µ = �G#ν0 ≈ µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Comparing with approaches directly dealing with distributions on the ambient  space (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Khayatkhoei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018), the latent distribution  matching schemes bring several computational benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  First, the choice of D for quantifying the  discrepancy between distributions in the latent space Z is more flexible since these distributions usually  admit a density function relative to the Lebesgue measure over Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' in contrast, many commonly used  discrepancies metrics such as the total variation distance, the Hellinger distance and the KL divergence  are known to be unsuitable for characterizing closeness between nearly singular measures over the ambient  space (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Second, computing a discrepancy between distributions in the  relatively low-dimensional latent space is much more efficient and does not suffer from the curse of  dimensionality, make the training process more stable and less time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In addition, the extra  flexibility of D allows one to employ those metrics that have simple and explicit computational formulas,  such as the squared maximum mean discrepancy (MMD) (Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2019) and  the sliced Wasserstein distance (Kolouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  At the end of this subsection, we describe two important limitations of LDMAE, which motivate our  proposed method to be introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Concretely, based on the aforementioned decomposition  perspective of the training objective (1), the estimation error of �µ from the target distribution µ∗ depends  on two terms: (1) the “distance” between µ∗ and ( �G ◦ �Q)#µ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) the “distance” between �Q#µ∗ and ν0,  4  where �Q denotes the learned encoder from minimizing (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Let M denote the support of the true data  generating distribution µ∗ as a d-dimensional submanifold embedded in RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To control the first distance,  one needs M to have a global parametrization, that is, we can find some continuous maps G∗ : Rd → RD  and Q∗ : RD → Rd so that for any x ∈ M, G∗(Q∗(x)) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' This global-parametrization condition does  not hold for many common manifolds, such as disconnected manifolds and boundaryless manifolds like  spheres and torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The second distance depends on how well the empirical data distribution �µem induced  empirical latent distribution �Q#�µem can approximate the population level distribution �Q#µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' However,  even though we assume that manifold M admits a global parameterization such that µ∗ = G∗  #ν0 and  Q∗ = (G∗)−1 for some G∗, Q∗ in the decoder and encoder families, the discrete empirical distribution  �µem may suffer from statistical deficiency for approximating a smooth measure (Liang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Tang &  Yang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' As a result, simply plugging-in Q#�µem in the penalty term may lead to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  Partition of unity and distributions on manifolds  Intuitively speaking, a manifold is a topological space that locally resembles the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Formally,  we have the following mathematical definition of a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' A d-dimensional manifold M is defined as a topological space satisfying:(1) There exists  an atlas on M consisting of a collection of d-dimensional charts A = {(Uλ, ϕλ)}λ∈Λ covering M, that  is, M = �  λ∈Λ Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) Each chart 1(U, ϕ) in atlas A consists of a homeomorphism ϕ : U → �U, called  coordinate map, from an open set U ⊂ M to an open set �U ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We call a manifold M a (β-smooth) submanifold embedded on RD if M ⊂ RD, and the coordinate  map ϕ and its inverse ϕ−1 in each chart are β-smooth maps when identified as functions defined on  subsets of Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Another useful notion related to the manifold is partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' A partition of unity of a manifold M is a collection of functions {ρλ}λ∈Λ satisfying  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 0 ≤ ρλ ≤ 1 for all λ ∈ Λ, and �  λ∈Λ ρλ(x) = 1 for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Each point x ∈ M has a neighborhood which intersects supp(ρλ) for only finitely many λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Using the partition of unity, one can glue constructions in the local charts to form a global construction  on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' A partition of unity can be constructed from any open cover {Uλ}λ∈Λ of the manifold  in a way where the partition {ρλ}λ∈Λ is indexed over the same set and supp(ρλ) ⊂ Uλ for any λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Such a partition of unity is said to be subordinate to the open cover {Uλ}λ∈Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For a manifold M with atlas A = {(Uλ, ϕλ)}λ∈Λ, suppose Λ is finite and we write it as Λ = [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Given a partition of unity {ρk}k∈[K] subordinate to the open cover {Uk}k∈[K], one can decompose any  distribution µ∗ on M as  µ∗ =  �  k∈[K]  ρkµ∗ =  �  k∈[K]  (ϕ−1  k )#  �  (ϕk)#(ρkµ∗)  �  ,  (2)  where the first inequality uses �  k∈[K] ρk(x) = 1 for all x ∈ M, and the second inequality uses the fact  that supp(ρk) ⊂ Uk and ϕk is a homeomorphism on Uk → ϕk(Uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 Then if we write ν∗  k = (ϕk)#  �  ρkµ∗  Eµ∗[ρk]  �  ,  µ∗ can be expressed as the following (mixture of) generative models:  µ∗ =  �  k∈[K]  Eµ∗[ρk] · (ϕ−1  k )#ν∗  k,  (3)  1Subscript λ is suppressed for the simplicity of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2If (ϕk)#(ρkµ∗) admits an α-smooth density function for α ∈ [0, β − 1] relative to the Lebesgue measure on Rd for each  k ∈ [K], then µ∗ is said to be an α-smooth distribution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  5  Decomposition (3) suggests that any distribution µ∗ lying on a d-dimensional submanifold embedded on  RD whose atlas composed of at most K-number of charts belongs to the following mixture of generative  models class:  S∗ =  �  µ =  �  k∈[K]  pk · (Gk)#νk  ��� Gk : Rd → RD, νk ∈ P(Rd), 0 ≤ pk ≤ 1,  �  k∈[K]  pk = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  This space forms our model space representing distributions on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  3  Mixture of latent distribution matched auto-encoder  From discussions in Section 2, we see that conventional auto-encoder based generative modelling approaches  may suffer from low representation power when the target distribution to be estimated lies on a general  submanifold without global parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  However, the property that any manifold-supported  distribution can be expressed in the form of a mixture of generative models (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' decomposition (3))  motivates us to employ multiple encoder/decoder pairs, and to use the partition of unity to glue them  together with proper weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Recall that we have a set of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d observations X(n) = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' , Xn} sampled from the target  distribution µ∗ lying on a d-dimensional submanifold M embedded in RD with d ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Let {Sk}k∈[K]  be a suitably chosen open cover to M = supp(µ∗) ⊂ RD, fix a partition of unity {ρk}k∈[K] subordinate  to {Sk}k∈[K],3 which can be chosen without the knowledge of M (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any generic  approximation family G consists of (G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' where G = (G1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' GK) with Gk : Rd → RD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Q = (Q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' QK) with Qk : RD → Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and v = (ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ν2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' νK) with νk ∈ P(Rd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we define the  following estimator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' which we call mixture of latent distribution matched auto-encoder (MLDMAE)  estimator:  �µ =  �  k∈[K]  �pk · ( �Gk)#�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  with  �pk = 1  n  n  �  i=1  ρk(Xi)  and  ( �G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �v) = arg min  (G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='v)∈G  K  �  k=1  � 1  n  n  �  i=1  c  �  Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Gk(Qk(Xi))  �  ρk(Xi) + λk · D  �  �νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' νk  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (4)  where recall that c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ·) is the cost function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Qk is a (smoothness-regularized) estimator to the density  of (Qk)#( µ∗·ρk  Eµ∗[ρk]) (the precise definition is available in Appendix C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and D(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ·) is a generic discrepancy  measure between distributions on the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Different from conventional LDMAE estimators,  MLDMAE can also use empirical Bayes method to select data-dependent prior distributions for local  latent variables, which adds extra flexibility in the modeling and may potentially reduce the approximation  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We show in Theorem 1 that for some carefully chosen approximation family G, cost function c,  discrepancy metric D, and smoothness-regularized estimator �νk,Qk, the resulting estimator �µ attains the  minimax rate of convergence under the 1-Wasserstein distance as W1(�µ, µ∗) ≤ C n−  α∧(β−1)+1  2(α∧(β−1))+d ∨ log n  √n  when µ∗ is an α-smooth distribution on an unknown β-smooth d-dimensional submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The objective function of MLDMAE can be decomposed into two parts: the reconstruction cost  n−1 �K  k=1  �n  i=1 c  �  Xi, Gk(Qk(Xi))  �  ρk(Xi) and the penalty �K  k=1 λk · D  �  �νk,Qk, νk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We may also allow  the latent dimension d to be different across encoder/decoder pairs over k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The reconstruction cost  aims to learn local parametrizations of the supporting manifold of µ∗ by enforcing the encoder/decoder  pair ( �Qk, �Gk) to represent some coordinate system (ϕk, ϕ−1  k ) of local patch Uk = M ∩ Sk of M in  decomposition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, it corresponds to the support recovery of µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By employing multiple  3Here we may consider any function ρk : RD → [0, 1] so that supp(ρk) ⊂ Sk ⊂ RD and �  k∈[K] ρk(x) = 1 for any  x ∈ ∪k∈[K]Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Note that {ρk|Sk∩M}k∈[K] would form a partition of unity to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  6  encoder/decoder pairs, the MLDMAE estimator avoids the restrictive global-parametrization assumption  that is implicitly assumed in conventional LDMAE estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' As a result, MLDMAE is suitable for  a wider range of problems (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 1 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' On the other hand, the penalty term aims  to enforce the reweighted local latent distribution ( �Qk)#  � µ∗·ρk  Eµ∗[ρk]  �  to match some member �νk in the  pre-specified prior family, so that �νk is close to the ν∗  k in decomposition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (a) Real data  (b) LDMAE  (c) MLDMAE  Figure 1: Comparison between LDMAE and MLDMAE when the target distribution is the uniform  distribution on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Figure (a) plots the real data, and Figures (b), (c) plot the randomly generated  samples from the MLDMAE and LDMAE estimators respectively, based on 10000 training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The  partition of unity chosen in MLDMAE is the smooth partition of unity described in Section 4 with K = 10  and γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The discrepancy metric D(·, ·) is chosen to be the MMD with Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can  see that LDMAE fails to capture the correct shape of a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The reason is that the sphere (or any  boundaryless manifold) requires at least two covering charts in its describing atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The LDMAE model  uses a single pair of encoder/decoder, and thus it returns a curve that has start/end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' On the  contrary, our estimator is able to learn general manifolds that can not be globally parametrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  In practice, instead of selecting the best data-dependent priors, we can also fix the prior as a simple  distribution ν0 such as an isotropic Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, D(·, ·) can be chosen as certain squared maximum  mean discrepancy (MMD) loss4 that can be efficiently computed in a closed-form formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The k-th  smoothness-regularized distribution �νk,Qk in (4) can be constructed by applying kernel-smoothing to its  (weighted) empirical counterpart (Qk)#�µk  n with �µk  n = (n�pk)−1 �n  i=1 ρk(Xi) δXi, leading to �νk,Qk(z) =  (n�pk)−1 �n  i=1 �k(z, Qk(Xi))ρk(Xi) for a suitable kernel �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Note that when kernel �k is the Gaussian  kernel �k(x, y) = (2πh)− d  2 exp(− ∥x−y∥2  2h  ) with bandwidth parameter h, then �νk,Qk corresponds to the  Gaussian-smoothed distribution ( �Qk)#�µk  n, where �Qk is the randomly perturbed encoder defined by  �Qk(X) = Qk(X) +  √  h · N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 Employing such a smoothness-regularized distribution can be viewed  as applying a randomized data augmentation to increase the variability of the encoded training samples,  which mitigates potential overfitting to data and improves the generalization ability of the resulting  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Introducing the encoder-decoder structure as in our estimator brings several benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Computationally,  the encoders turn the high-dimensional data into low-dimensional latent variables so that we only need to  compute a penalty term over low-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, the MLDMAE framework brings  less computational burden compared with generative modelling approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' iWGAN) that directly  deal with distributions in the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Theoretically, when d ≪ D, the data distribution µ∗ becomes  a singular measure in RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' As a consequence, with the information about the supporting manifold of µ∗,  which is explicitly induced by the encoder-decoder pairs, it is possible to utilize classical techniques of  nonparametric density estimation, such as wavelet truncation, to construct a minimax-optimal estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Specifically, the underlying true latent variable distribution (Qk)#( µ∗·ρk  Eµ∗[ρk]) defined in the mixture of  generative models (3) is, with high probability, absolutely continuous with respect to the Lebesgue  measure on Rd (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Lemma 4 in Appendix C), which enables us to develop smoothness-regularized  4For a positive-definite reproducing kernel k, the MMD loss is defined as MMD2(µ1, µ2) = EX,X′∈µ1[k(X, X′)] +  EY,Y ′∈µ2[k(Y, Y ′)] − 2EX∈µ1,Y ∈µ2[k(X, Y )]  5Here for randomized map Q, the push forward measure Q#µ is defined as the measure so that for any measureable  function f,  �  f(x) d[Q#µ] = E  � �  f(Q(x)) dµ  �  where the expectation is with respect to the randomness of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='00  0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='00estimators by borrowing techniques from Liang (2020) and Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Indeed, as suggested  by Liang (2020) and Tang & Yang (2022), the rate O  �  n− α+1  2α+d ∨ log n  √n  �  achieved by the MLDMAE estimator  when α ≤ β − 1 is minimax-optimal up to logarithmic factor relative to the 1-Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  4  Computation  In this section, we discuss some important computational aspects of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Choice of partition of unity: One issue we need to address is how to choose a reasonable partition of  unity {ρk}k∈[K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To do this, we can first run a clustering algorithm such as (mini-batch) K-means to  the data using a sufficiently large cluster number K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Based on the clustering result, one straightforward  choice of ρk is the indicator function 1(x ∈ k-th cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can also choose a smooth partition of  unity by the following: firstly we record the centroid of the k-th cluster as ak and the smallest radius rk  so that data points in the k-th cluster are included in Brk(ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we can construct an open cover  {Sk = Brk+ε(ak)◦}k∈[K] where ε is a small positive number so that {Sk}k∈[K] can cover the unknown  support of µ∗ with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Given the open cover, for each k ∈ [K], we define a local partition  function as �ρk(x) = ((rk + ε)2 − ∥x − ak∥2)γ · 1(x ∈ Sk), where γ > 1 is a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The resulting  {ρk}k∈[K] forms a partition of unity for M with ρk = �ρk/  � �K  k′=1 �ρk′�  for k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Note that �ρk tends to  give less weight to points away from the centroid ak for large γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Choice of penalty terms: We choose the smoothness-regularized distribution �νk,Qk as the Gaussian  kernel-smoothed version of (Qk)#�µk  n as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' This �νk,Qk relates to the commonly-used  Gaussian encoder in VAE, and thus we can utilize the reparametrization trick in VAE to optimize the  desired objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For the priors v, we can consider a simple distribution ν0 such as standard  Gaussian N(0, Id) as is usually done in conventional generative modelling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, to  ensure the smoothness of the learned manifold, we can consider data-driven priors described in Remark 1  in Section 5 below, so that νk and �νk, �  Qk can be ensured to have matching tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For the discrepancy metric  D(·, ·), to prevent instability in the adversarial training, we can: (1) choose D(·, ·) to be the (squared)  MMD characterized by a positive-definite kernel k, such as the inverse multiquadratics (IMQ) kernel  k(x, y) = Cim/(Cim + ∥x − y∥2  2) and the RBF kernel k(x, y) = exp(−∥x − y∥2/C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) when the intrinsic  (latent) dimension is of order O(1), we can choose D(·, ·) as the 1-Wasserstein distance computed by the  “POT” package (Flamary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2021), which returns the optimal transport map between two discrete  measures using network simplex algorithm (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Construction of decoders and encoders: The decoders G and encoders Q can be realized through  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' However, for a large K, we will have a large number of parameters to train if we see  each decoder/encoder as an independent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To address this issue, we enable parameter  sharing inside the set of decoders G and the set of encoders Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Specifically, for the set of encoders, we set  the last (output) layer to be free among the encoders {Qk}k∈[K] while other layers to be tied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover,  since the decoder-encoder structure aims to reconstruct the data, we want the decoder Gk to be close  to the inverse of the encoder Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To achieve this, for the set of decoders, we oppositely set the first  (input) layer to be free among the decoders {Gk}k∈[K] and tie other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The rationale behind our  parameter sharing scheme is that in the encoder, the first (convolutional) layer focuses on “low-level”  features extraction and other layers extract “high-level” features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, the described parameter  sharing scheme enables the networks to leverage the common “low-level” information among different  clusters of the data, hence improves the model training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Optimization of objective function: Recall that �νk,Qk = �Qk(�µk  n), where �Qk is a randomized perturbed  encoder �Qk(X) = Qk(X) +  √  h · N(0, Id) and �µk  n is the re-weighted empirical measure �µk  n = �µn·ρk  �pk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Given  a discrepancy metric D(·, ·) and cost function c(·, ·) defining the penalty and reconstruction cost, we can  8  rewrite the objective function as  K  �  k=1  �  �pk  �  c  �  x, Gk(Qk(x))  �  d�µk  n + λk · D  �  νk, ( �Qk)#�µk  n  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  To approximate the gradient of this objective function, we sample from the measure �µk  n by introducing  auxiliary random variables u ∈ Unif(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Specifically, we include data Xi into the empirical measure �µk  n  if u < ρk(Xi) and otherwise we exclude the data from �µk  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, the penalty term can be estimated  using finite samples from νk and ( �Qk)#�µm  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Algorithm 1: Algorithm for implementing MLDMAE  Input: Regularization coefficient {λk}k∈[K], partition of unity {ρk}k∈[K], discrepancy metric  D(·, ·) and cost function c(·, ·), priors {νk}k∈[K], latent (intrinsic) dimension d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Data:X(n) = {X1, X2 · · · , Xn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  repeat  Sample a mini-batch dataset D from X(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  for k ← 1 to K do  Initialize an empty dataset �  Dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  for X ∈ D do  Generate random variable u form Unif(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  if u ≤ ρk(X) then  Add X to dataset �  Dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Generate dataset Lk from prior νk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Generate dataset Ek from N(Qφ,k(X), h Id) for X uniformly picked in �  Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Update φ and θ by one step first-order method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Adam, Kingma & Ba (2014)) with  objective function:  K  �  k=1  � �pk  | �  Dk|  �  X∈ �  Dk  c  �  X, Gθ,k(Qφ,k(X))  �  + λk · D  �  1  |Lk|  �  z∈Lk  δz,  1  |Ek|  �  z∈Ek  δz  ��  ,  where δz is the Dirac measure on point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  until (φ, θ) converges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Based on the above discussion, we can now develop the algorithm as described in Algorithm 1  for implementing the MLDMAE estimation, where we use Gθ = {Gθ,1, Gθ,2, · · · , Gθ,M} and Qφ =  {Qφ,1, Qφ,2, · · · , Qφ,M} to denote the decoders and encoders parametrized by θ and φ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  5  Theoretical Analysis  In this section, we derive the finite sample error of the MLDMAE estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We first state the following  assumptions on the target distribution µ∗ and the approximation family used in defining the estimator (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Assumption A (Target distribution): The target distribution µ∗ on manifold M satisfies that: (1)  M ⊂ ∪k∈[K]Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) for any k ∈ [K], there exists G∗  k ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) and Q∗  k ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd) so that  x = G∗  k(Q∗  k(x)) holds for any x ∈ M ∩ Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (3) for any m ∈ [K], let pk = Eµ∗[ρk], then pk > 0 and  ν∗  k = (Q∗  k)#( µ∗·ρk  pk ) ∈ Cα  L(Rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' let Ωk = Q∗  k(M ∩ Sk), there exists a function gk : R+ → R+, so that for  any r > 0 and z ∈ Ωk, there exists z′ ∈ Ωk such that z ∈ Br(z′) and ν∗  k(z′) ≥ gk(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Assumption B (Approximation family): The approximation family G satisfies that (1) (G∗, Q∗, v∗) ∈  G with G∗ = (G∗  1, G∗  2, · · · , G∗  K), Q∗ = (Q∗  1, Q∗  2, · · · , Q∗  K) and v∗ = (ν∗  1, ν∗  2, · · · , ν∗  K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) for any  9  (G, Q, v) ∈ G, it holds that Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) and Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd) for any k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Example (Manifold-supported distributions): For any α-smooth distribution µ∗ on a β-smooth  d-dimensional boundaryless compact submanifold embedded in RD and with a positive density, we  can find a suitable open cover {Sk}k∈[K] and partition of unity {ρk}k∈[K] so that Assumption A holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Moreover, the (mixture of) generative model class induced by the approximation family G =  �  (G, Q, v) :  ∀k ∈ [K], Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), νk = (Vk#ν0)·ρk(Gk(z))  Eν0[ρk(Gk(Vk(z)))], Vk ∈ Cα+1  L  (Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Bd  1)  �  with ν0 being  any fixed α-smooth distribution on Bd  1 whose density value bounded away from zero (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', uniform  distribution), suffices to model the manifold-supported distributions µ∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Assumption B holds for the  approximation family G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In particular, when α + 1 = β, we can consider the compositions Gk ◦ Vk as  β-smooth encoders for preventing the estimation of priors, that is, we can use the approximation family  G =  �  (G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), νk =  ν0·ρk(Gk(z))  Eν0[ρk(Gk(z))]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Further details are  available in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The choice of the approximation family suggests that, for learning a manifold-supported  distribution, instead of taking the priors νk to be some fixed simple distribution ν0, we may rescale ν0 by  the weight ρk(Gk(·)) so that the resulting distribution has a matching tail as Qk#(ρkµ∗/Eµ∗[ρk]) after con-  vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' However, incorporating such a prior family in MLDMAE may lead to an unstable training due to  the high irregularity of functions ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To address this issue, we can consider “data-driven” priors as follows:  we first fix νk to be some simple fixed distribution ν0, such as uniform distribution or truncated normal,  then we run the MLDMAE algorithm to obtain estimators of encoder/decoder pairs {( �G[1]  k , �Q[1]  k )}k∈[K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Now we fix the priors νk to be ν0 rescaled by ρk(G[1]  k (·)), and run the MLDMAE algorithm with initializa-  tion {( �G[1]  k , �Q[1]  k )}k∈[K] to obtain estimators of encoder/decoder pairs {( �G[2]  k , �Q[2]  k )}k∈[K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The above steps  can continue by fixing the priors νk to be ν0 · ρk(G[l]  k (·)) and obtaining estimators {( �G[l+1]  k  , �Q[l+1]  k  )}k∈[K],  and stop until no improvement in validation error is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Using such a data-driven prior can largely  improve the performance of MLDMAE at the intersections of the support of different partition functions,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (a) MLDMAE: truncated normal prior (b) MLDMAE: data-driven prior (once  update)  (c)  MLDMAE:  data-driven  prior  (twice updates)  Figure 2: Performance of MLDMAE with different choices of priors when the target measure is the  uniform distribution on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Figure (a) plots the generated samples from MLDMAE estimator when  the priors are truncated normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Figures (b) and (c) plot the generated samples from MLDMAE estimator  with data-driven priors described in Remark 1 under once and twice updates respectively, where ν0 is the  truncated normal as in Figure (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can see with a simple truncated normal prior, the generated plot  tends to be non-smooth at the intersection of different partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' While once update of priors  using the strategy described in Remark 1 can lead to much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Example (Distributions with clustering structures): Another example is a distribution induced  by the mixture of generative models µ∗ = �K  k=1 pk · (G∗  k)#ν∗  k, where supports of generative models are  disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In this case, the supporting manifold of µ∗ is a disconnected manifold, and we can simply  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='00choose {Sk}k∈[K] to be any disjoint sets that can cover each support of the generative model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=',  supp((G∗  k)#ν∗  k) ⊂ Sk for k ∈ [K]), and take ρk to be the indicator function 1(x ∈ Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' With such choices,  Assumption A holds if for each k ∈ [K], ν∗  k is α-smooth with a compact support, and G∗  k is β-smooth  with a β-smooth inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Fix α ≥ 0, β ≥ 1 and a partition of unity {ρk}k∈[K] subordinate to the open cover  {Sk}k∈[K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Suppose the target distribution µ∗ satisfies Assumption A and the approximation family G  satisfies Assumption B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' If we choose the discrepancy metric D(·, ·) to be the 1-Wasserstein distance W1  and cost function c(·, ·) to be the squared ℓ2 loss, then there exists a choice of regularization coefficients  {λk}k∈[K] so that for large enough n,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' if α = 0, then by choosing the re-weighted empirical measure �νk,Qk =  1  �pkn  �n  i=1 δQ(Xi)ρk(Xi) with  �pk = 1  n  �n  i=1 ρk(Xi) as the plug-in, the resulting estimator �µ satisfies with probability at least 1−n−1  that  W1(�µ, µ∗) ≤ C n− 1  d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' if α > 0, β > 1, then there exists a smoothness-regularized empirical measure �νk,Qk as the plug-in,  so that the resulting estimator �µ satisfies with probability at least 1 − n−1 that  W1(�µ, µ∗) ≤ C n− (α∧(β−1))+1  2(α∧(β−1))+d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (6)  The smoothness-regularized empirical measure �νk,Qk adopted in the proof of Theorem 1 can either be  based on wavelet truncation or kernel density estimator of the measure Qk#( µ∗·ρk  pk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Based on the minimax  lower bound developed in Liang (2020) and Tang & Yang (2022), the convergence rate in Theorem 1 is  minimax-optimal relative to the 1-Wasserstein distance when α ≤ β −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' If Assumption A holds for K = 1,  then statement (5) can provide a theoretical guarantee to the LDMAE estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The ambient dimension  D does not appear in the exponent of the developed rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' thus Theorem 1 shows the adaptiveness of  MLDMAE to low-dimensional submanifold structures since the bound does not suffer from the “curse of  dimensionality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, the MLDMAE estimator can take advantage of the smoothness of the target  measure to further enhance the estimation accuracy by regularizing the empirical measure in the penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  6  Simulation  In this section, we present some visual results of the MLDMAE approach when apply to common manifolds:  2D-spiral and torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The precise data generating distributions are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The penalty  term D(·, ·) is chosen to be the 1-Wasserstein distance computed by the “POT” package (Flamary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=',  2021) and the cost function c(·, ·) is the squared ℓ2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' As benchmarks, we also consider (1) the classic  kernel density estimator (KDE) with Gaussian kernel that is commonly employed in statistics literature  for density estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) the LDMAE estimator with the same kind of cost function and discrepancy  metric as MLDMAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The generated samples from the generators learned by different approaches are  given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The 1-Wasserstein distance between the estimated distribution and the true distribution  are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can see that employing multiple encoders and decoders can lead to much better  performance than employing a single pair of encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In particular, we can see even though  we increase the complexity of the encoder/decoder family, LDMAE still can not capture the correct shape  of these standard manifolds in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' On the other hand, by allowing multiple encoders/decoders  in MLDMAE, the manifold structure can be correctly learned with a relatively simple encoder/decoder  structure and a smaller number of training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, MLDMAE can beat the classic KDE  in both examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  11  Training sample  KDE  MLDMAE (1-NN)  LDMAE (1-NN)  LDMAE (2-NN)  Figure 3: The figure illustrates the performance of MLDMAE, LDMAE and KDE when the target  measures lying on a spiral and torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We observe n = 1000 and n = 3000 training points for the example  of spiral (Top row) and torus (Bottom row) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The first column plots the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The  second column plots the generated samples from the classic kernel density estimator, which corresponds  to adding Gaussian noises to the original training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The third column plots the generated samples  from our proposed MLDMAE estimator (K = 10 for spiral and K = 15 for torus), the encoders and  decoders are parameterized by one-hidden layer neural networks with hidden layer size being 128, the  partition of unity is chosen to be the smooth partition of unity described in Section 4 with γ = 10, the  priors are selected as described in Remark 1 with ν0 being a truncated normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The fourth and fifth  columns plot the generated samples from the LDMAE estimator where the encoder-decoder pair are  parameterized by one-hidden layer and two-hidden layer neural networks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  KDE  MLDMAE  LDMAE (1-NN)  LDMAE (2-NN)  Spiral  W1 distance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='4666  Number of training parameters  /  4492  1027  34051  Torus  W1 distance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='6193  Number of training parameters  /  10529  1412  34565  Table 1: The table gives the 1-Wasserstein distance between the target measure and the distribution  estimators of different approaches for the spiral and torus examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We also provide numbers of training  parameters for different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  7  Real Data Application  In this section, we empirically evaluate the proposed MLDMAE approach using three real-world datasets:  MNIST handwritten digit (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 1995), Fashion-MNIST (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2017) and CelebA 64 ×  64 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For comparison, we consider LDMAE approach and variational auto-enocoder  (VAE) (Kingma & Welling, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rezende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2016), which are commonly used auto-  encoder based generative modelling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In all reported experiments, we set the reconstruction cost  to be the squared ℓ2 loss, and fix the priors νk to be a standard Gaussian N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The partition of unity  is chosen to be the indicator functions described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The encoders and decoders are modelled by  convolutional deep neural networks with parameter sharing as described in Section 4, and further details  are available in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We consider two kinds of discrepancy metric D(·, ·) in the penalty terms  of LDMAE and MLDMAE, one is the 1-Wasserstein distance computed through “POT” package, and  the other one is MMD with the inverse multiquadratics (IMQ) kernel k(x, y) = 2d/(2d + ∥x − y∥2  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' As  described in Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2019), the inverse multiquadratics kernel has a much heavier tail than the  12  0  2  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='5  Z Label  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  4  六人  3  3  2  1  0  X Label  1  1  2  2  w  3  4  44  2  0  2  4  6  2  1  0  1  2  3conventional RBF kernel k(x, y) = exp(−∥x − y∥2/C), so it can provide more meaningful gradients for  outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The numbers K of clusters for MLDMAE are selected so that the number of free parameters  is around 1  5 ∼ 1  2 of the number of sharing parameters, and it turns out K = 5 works well for all the  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Another important factor is the latent dimension, which is not explicit for the real dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Selecting a too small latent dimension would lead to large reconstruction errors and thus result in noisy  generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' On the contrary, selecting a too large latent dimension would lead to the singularity  of the encoded distribution and thus result in numerical instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use d = 4 for Fashion-MNIST,  d = 8 for MNIST handwritten digit and d = 64 for CelebA, which seems to work reasonably well, and  trainings of MLDMAE are stable and robust to initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  To quantitatively assess the MLDMAE estimator, for the dataset of MNIST handwritten digit and  Fashion-MNIST, we consider two kinds of evaluation metrics: one is the test log-likelihood (Test LL) used  in Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2014) by fitting a Gaussian Parzen window to the generated samples and reporting  the log-likelihood evaluated at the test samples, and the other one is the 1-Wasserstein distance (W1)  between the test samples and generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The generated samples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 4 and the Test  LL and W1 distance are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can see for the Fashion-MNIST dataset, MLDMAE  with W1 or MMD penalty obviously outperforms VAE and LDMAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For the MNIST handwritten digit  dataset, MLDMAE with MMD penalty outperforms the Wasserstein penalty in the W1 metric, and it  may attribute to the fact that the MNIST digit dataset has a relatively larger latent dimension d = 8, so  we need a very large batch size for accurately estimating the Wasserstein penalty, which will reduce the  number of gradient updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 5 gives the trends of the Test LL and W1 distance as the  cluster number K increases for the MNIST digit dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can see the trends of both metrics become  smooth when M ≥ 10, this is consistent with the underlying fact that the MNIST digit dataset contains  10 digits (clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Furthermore, we can see an obvious improvement in both metrics when increasing K  in the range of [1, 10], while the total number of training parameters only increases by 7% when cluster  number K increases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Table 2: MNIST handwritten digit and Fashion-MNIST dataset:  Test LL and W1 distance for different approaches, “+MMD” and  “+W1” represent the choices of the discrepancy metric D(·, ·) in  the penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Fashion-MNIST  MNIST  Test LL ↑  W1 ↓  Test LL ↑  W1 ↓  VAE  533  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='7  393  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  LDMAE+MMD  549  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  381  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='0  LDMAE+W1  562  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='6  400  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  MLDMAE+MMD  557  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='0  443  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='6  MLDMAE+W1  562  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8  456  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Table 3: CelebA dataset: FID and  KID for different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The  penalty term in LDMAE and MLD-  MAE are chosen to be MMD penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  FID ↓  KID ↓  VAE  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='063  LDMAE+MMD  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='057  LDMAE+SW1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='064  MLDMAE+MMD  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='051  MLDMAE+SW1  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='052  13  VAE  LDMAE+MMD  LDMAE+W1  MLDMAE+MMD  MLDMAE+W1  Figure 4: The generated samples from different approaches for Fashion-MNIST (Top row) and MNIST  handwritten digit dataset (Bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The first column corresponds to the VAE estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' the second  and third columns correspond to the LMDAE estimator with MMD and W1 penalty respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' the  fourth and firth columns correspond to the MLMDAE estimator with MMD and W1 penalty respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Figure 5: Negative test log-likelihood (red) and W1 distance (black) of MLDMAE with MMD penalty  and different cluster numbers K for the MNIST handwritten digit dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For the celebA dataset, since the ambient dimension D = 64 × 64 × 3 is extremely large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Instead  of using test log-likelihood or W1 distance, which are evaluated in the ambient space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we consider  two commonly-used metrics for color image data: FID (Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2017) and KID (Bińkowski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', 2018), in which the original high-dimensional data is fed into an ImageNet-pretrained inception  network to obtain 2048-dimensional inception (feature) representations, and the FID and KID are the  fréchet distance and the squared MMD between inception representations of generated samples and test  samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, as described previously, the W1 penalty is unsuitable for large intrinsic  dimensions, for avoiding the curse of dimensionality, we consider the so-called sliced Wasserstein distance:  SW1(µ, ν) := Eθ∼Unif(Sd−1  1  )  �  W1(Projθ#µ, Projθ#ν)  �  , where Projθ denotes the projection function to the  direction θ and Unif(Sd−1  1  ) denotes the uniform distribution on Sd−1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The expectation over Unif(Sd−1  1  )  can be estimated by Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The sliced-Wasserstein distance slices high-dimensional  probability densities into sets of one-dimensional marginal distributions and compare these marginal  distributions via the Wasserstein distance, it has similar qualitative properties to the Wasserstein distance,  but is much easier to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The generated samples and FID, KID are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='250We can see that the MLDMAE estimator can achieve the best performance under all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The MLDMAE with MMD penalty performs slightly better than the SW1 penalty, while it also requires  more computation time (the computation time for MLDMAE+MMD is 150s per epoch using NVIDIA  A100-SXM4-40GB GPU, while that is 120s per epoch for MLDMAE+SW1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (a) VAE  (b) LDMAE+MMD  (c) LDMAE+SW1  (d) MLDMAE+MMD  (e) MLDMAE+SW1  Figure 6: The generated samples from different approaches (first column: VAE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' second and third columns:  LDMAE with MMD and SW1 penalty respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' fourth and fifth columns: MLDMAE with MMD and  SW1 penalty respectively) for the CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  8  Conclusion  In this work, we proposed a new approach, mixture of latent distribution matched auto-encoder (MLD-  MAE), to improve the conventional auto-encoder based generative modelling approaches for learning  manifold-supported distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We showed theoretically that the proposed estimator can learn manifold-  supported distributions with a minimax-optimal convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, we conducted experiments  to show that by employing multiple encoder/decoder pairs, the estimators derived from MLDMAE can  substantially boost the target distribution estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In our theoretical analysis, we consider  the case where the penalty term is chosen to be the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We leave the theoretical  analysis for some other adversarial losses, such as the MMD distance considered in our experiments, to  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content=', Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' & Weinberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' An empirical  study on evaluation metrics of generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='07755 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  You, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Gui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' & Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Using manifold embedding for  assessing and predicting protein interactions from high-throughput experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Bioinformatics  26, 2744–2751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Ogden, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Picard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' & Srivastava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Nonparametric k-sample test on  shape spaces with applications to mitochondrial shape analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Journal of the Royal Statistical Society:  Series C (Applied Statistics) 71, 51–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' & Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Infovae: Balancing learning and inference in variational  autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Proceedings of the AAAI Conference on Artificial Intelligence 33, 5885–5892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  18  Appendix  Notations: We adopt the notations in the manuscript, and further introduce the following additional  notations for technical proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use N(F, �d, ϵ) to denote the ϵ-covering number of function space  F with respect to pseudo-metric �d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We use Br(x) to denote the closed ball centered at x with radius  r under the ℓ2 distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' in particular, we use Bd  r denote Br(0d) when no ambiguity may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We  denote Sd−1  1  = {x ∈ Rd : ∥x∥ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For a function f : Ω → Rd, we use Jf(x) to denote the  d × m Jacobian matrix of f at x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For a function f : Rd → R, we use f (a) to denote its mixed  partial derivative ∂|a|f/∂xa1  1 · · · ∂xad  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We define the α-smooth Hölder (function) class (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=', Evans  (2010b)) with radius r > 0 over Ω as Cα  r (Ω) :=  �  f : Ω → R  �� ∥f∥Cα(Ω) = �  |a|≤⌊α⌋ maxx∈Ω |f (a)(x)| +  �  |a|=⌊α⌋ maxx,y∈Ω, x̸=y  ��f (a)(x) − f (a)(y)  �� /∥x − y∥˜α−⌊α⌋ ≤ r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Similarly, we use Cα  r (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) =  �  f =  (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' , fD) : Ω → RD �� ∀ j ∈ [D], fj ∈ Cα  r (Ω)  �  to denote the vector valued function space counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For an f ∈ Cα  r (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) and a multi-index a ∈ Nd  0, we denote f (a) as the D dimensional vector whose j-th  component is the mixed partial derivative [fj](a) of fj for j ∈ [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Throughout, C, c, C0, c0, C1, c1,  C2, c2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' are generically used to denote positive constants whose values might change from one line to  another, but are independent from everything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  A  Remaining implementation details  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Simulation  The training points D in spiral are generated via the following steps: (1) generate φ0 ∼ N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) set  φ = 3πφ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (3) generate data point X though X = [ cos(φ+2)·φ  π  , 2 sin(φ+2)·φ  π  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The training points D in torus  are generated via the following steps: (1) generate φ0, φ1 ∼ N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) set φ = 2πφ0 and θ = 2πφ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (3)  generate data point X through X = [(3 + cos(θ)) cos(φ), (3 + cos(θ)) sin(φ), sin(θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The cluster number  M in MLDMAE is M = 10 for the dataset of spiral and M = 15 for the dataset of torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The partition of  unity is given by ρk = �ρk/  � �K  k′=1 �ρk′�  with �ρk = (r2  k − ∥x − ak∥2)10 · 1(x ∈ Sk), where {ak}k∈[K] are the  centers returned by the K-means algorithm, rk = sup{∥x−ak∥ : x ∈ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∀k1 ∈ [K], ∥x−ak∥ ≤ ∥x−ak1∥},  and Sk = Brk(ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  Real data application  The specification of our models trained on MNIST handwritten digit, Fashion-MNIST and CelebA are  described in Table 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' “Shared” is short for parameter sharing among encoders or among decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  All models are optimized using Adam optimization with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='001, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='9, and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The  partition of unity for all datasets is chosen as the indicator function ρk(x) = 1(x ∈ cluster m) for k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The codes for reproducing the experiments are available in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='com/rtang1997/MLDMAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  19  Operation  Kernel  Strides  Feature maps  Activation  Shared?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Decoder Gk(z) : k ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' z ∈ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Id)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='7 × 7 × 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='14 × 14 × 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='28 × 28 × 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Encoder Qk(x) : k ∈ [K]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='28 × 28 × 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='26 × 26 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5 × 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Cluster number K for MLDMAE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='256 for MLDMAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and 128 for WAE and VAE  Number of epochs  50  Leaky ReLU slope  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Regularization coefficients (λk)  100 for MMD penalty and 10 for W1 penalty  Bandwidth (h) for MLDMAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='01  Number of training samples  60k  Table 4: Encoder/decoder Network architecture and hyperparameters for the MNIST handwritten digit  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Operation  Kernel  Strides  Feature maps  Activation  Shared?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Decoder Gk(z) : k ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' z ∈ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Id)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='7 × 7 × 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='14 × 14 × 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Encoder Qk(x) : k ∈ [K]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='28 × 28 × 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='26 × 26 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='12 × 12 × 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5 × 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='LeakyReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Cluster number K for MLDMAE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='256 for MLDMAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and 128 for WAE and VAE  Number of epochs  50  Leaky ReLU slope  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Regularization coefficients (λk)  100 for MMD penalty and 10 for W1 penalty  Bandwidth (h) for MLDMAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='01  Number of training samples  60k  Table 5: Encoder/decoder architecture and hyperparameters for the Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  20  Operation  Kernel  Strides  Feature maps  Activation  Shared?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Decoder Gk(z) : k ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' z ∈ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Id)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8 × 8 × 1024  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='32 × 32 × 256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='64 × 64 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Transposed convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1 × 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='64 × 64 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Tanh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Encoder Qk(x) : k ∈ [K]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='64 × 64 × 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='32 × 32 × 128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='16 × 16 × 256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='8 × 8 × 512  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Convolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5 × 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2 × 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='4 × 4 × 1023  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='ReLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch normalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Fully connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Cluster number K for MLDMAE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Batch size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='256 for MLDMAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and 128 for WAE and VAE  Number of epochs  50  Regularization coefficients (λk)  100  Bandwidth (h) for SWAE and MLDMAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='01  Number of training samples  180k  Table 6: Encoder/decoder architecture and hyperparameters for the CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  B  Generative modelling of Distributions on submanifolds  A submanifold in the ambient space RD can be viewed as a nonlinear “subspace”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Borrow the definition  in Tang & Yang (2022), we define the family of smooth distributions on d-dimensional smooth compact  submanifolds without boundaries on RD as the set P∗ = P∗(d, D, α, β, L∗) with d ≤ D, β > 1 and  α ∈ (0, β − 1] composed of all probability measures µ ∈ P(RD) satisfying:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' µ is an α-smooth distribution on a β-smooth d-dimensional compact submanifold M embedded in  RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' The density µ relative to the volume measure of M is uniformly bounded from below by 1/L∗ on  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' M is covered by an atlas A = {(Uλ, φλ)}λ∈Λ on M such that: a) each chart (U, φ) in atlas A  satisfies ∥φ−1∥Cβ(φ(U)) ≤ L∗ and ∥µ ◦ φ−1∥Cα(φ(U)) ≤ L∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' b) for any z ∈ φ(U), the Jacobian of φ−1(z)  is full rank and all its singular values are lower bounded by 1/L∗ in absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, for any  x ∈ M, there exists a λ ∈ Λ such that Uλ and φλ(Uλ) covers B1/L∗(x)∩M and B1/L∗(φλ(x)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We have the following lemma describing the mixture of generative model classes that can model the  submanifold-supported distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  21  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Consider OK = {Sk = B◦  rk(ak)}K  m=1, for k ∈ [K], choosing ρk(x) =  �ρk(x)  �  k∈[K] �ρk(x) with  �ρk(x) = (r2  k − ∥x − ak∥2)γ · 1(x ∈ Sk) for γ ≥ α + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' There exists a constant r∗ that only depends  on (L∗, β, α, d, D) so that for any µ∗ ∈ P∗(d, D, α, β, L∗), if (1) supp(µ∗) ⊂ ∪k∈[K]Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (2) for any  k ∈ [K], rk ≤ r∗ and ak ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (3) there exists some positive constants L∗  1 so that mink∈[K] rk ≥ L∗  1 and  infx∈M  �  k∈[K] �ρk(x) ≥ L∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  there exist some universal constants (L, c) that only depend on (L∗, L∗  1, β, α, d, D, γ) so that  Assumption A holds for µ∗ with upper bound L and function gk(r) = c (rγ ∧ 1) for any k ∈ [K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' consider ν0 ∈ P(Bd  1) whose density being α-smooth and bounded below from zero, and approximation  families  (a)  G1 =  �  (G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), νk ∈ P(Bd  1) with νk ∈  Cα  L(Bd  1)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  G2 =  �  (G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), vk = (Vm#ν0)·ρk(Gk(z))  Eν0[ρk(Gk(Vk(z)))] , Vk ∈  Cα+1  L  (Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Bd  1)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  G3 =  �  (G, Q, v) : ∀k ∈ [K], Gk ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Qk ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), vk =  ν0·ρk(Gk(z))  Eν0[ρk(Gk(z))]  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  then for sufficiently large L, we have Assumption B holds for G1 or G2, that is, the approximation families  G1 and G2 are both sufficient to model distributions inside P∗(d, D, α, β, L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, if α = β − 1,  then Assumption B holds for G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The next lemma shows that OK satisfying conditions of Lemma 1 can be found based on a small portion  of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Consider any µ∗ ∈ P∗(d, D, α, β, L∗) with support M, fix r∗ being an arbitrary positive  constant and let n1 ≤ n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then for any positive constant c, there exist constants C, c1  that only depend on (d, D, β, L∗, r∗, c) so that when C ≤ n1 ≤ n, let I1 be any subset of [n] with |I1| = n1,  it holds with probability larger than 1 − n−c  1  that  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' M ⊂ �  i∈I1 Bc1( log n1  n1  )  1  d (xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' there exists a constant K that only depends on (d, D, β, L∗, r∗) and a subset {ak}K  k=1 ⊂ {Xi}i∈I1  so that  (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  i∈I1 Bc1( log n1  n1  )  1  d (xi) ⊂ �K  k=1 Br∗(ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' infx∈M  �  k∈[K] �ρk(x) > ( r∗  √  2)2γ, where �ρk(x) = ((r∗)2 − ∥x − ak∥2)γ · 1(x ∈ Br∗(ak)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  We first consider the case α > 0 and β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' To simplify the notation, we write ˜α = α ∧ (β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We  consider two kinds of smoothness-regularized empirical measure �νk,Qk, one is based on kernel density  estimator and one is based on wavelet estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Kernel density estimator: Define  �νk,Qk(y) =  1  n�pkhd  n  �  i=1  �k  �y − Qk(Xi)  h  �  ρk(Xi),  �pk = 1  n  n  �  i=1  ρk(Xi),  (7)  with h = n−1/(2�α+d) and �k : Rd → R satisfies that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�k(·) is ⌈˜α⌉ ∨ ⌈ d  2⌉ smooth in Rd and has support contained in [−1, 1]d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  Rd �k(z) dz = 1 and for any j ∈ Nd  0 with 1 ≤ |j| ≤ ⌊α⌋ + 1,  � �k(z) · zj dz = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' for any z ∈ Rd, �k(z) = �k(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Wavelet estimator: Define �νk,Qk(y) as  �νk,Qk(y) = 1  �pk  � �  m∈S  �aQk  m φm(y) +  2d−1  �  l=1  J  �  j=0  �  m∈Slj  �θQk  ljmψljm(y)  �  ,  (8)  with  �pk = 1  n  n  �  i=1  ρk(Xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  S = {m ∈ Zd | supp(φm) ∩ [−L, L]d ̸= ∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Slj = {m ∈ Zd | supp(ψljm) ∩ [−L, L]d ̸= ∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �aQk  m = 1  n  n  �  i=1  φm(Qk(Xi))ρk(Xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �θQk  ljm = 1  n  n  �  i=1  ψljm(Qk(Xi))ρk(Xi),  where 2dJ ≍ n  d  2α+d and {φm, ψljm : l = 1, · · · , 2d − 1, j ∈ N, m ∈ Zd} is the orthonormal wavelet  basis for Besov space on Rd defined as φm(y) = φ(y − m) and ψljm(y) = 2  jd  2 ψl(2jy − m), and it holds  that φ(·) and ψl(·) are compactly supported and have bounded ⌈α ∨ ( d  2 − α)⌉ order derivatives for any  1 ≤ l ≤ 2d − 1 (Bouzebda & Didi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We will show both choices of �νk,Qk can lead to the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By Assumption A of µ∗, for any k ∈ [K],  there exist G∗  k ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) and Q∗  k ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd) so that and for any x ∈ M ∩ Sk, G∗  k(Q∗  k(x)) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By  the optimality of �G, �Q and �v for the training objective, we can get that  K  �  k=1  � 1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) + λk ·  sup  f∈Lip1(Rd)  � �  f(z)�νk(z) dz −  �  f(z)�νk, �  Qk(z) dz  ��  ≤  K  �  k=1  λk ·  sup  f∈Lip1(Rd)  � �  f(z)ν∗  k(z) dz −  �  f(z)�νk,Q∗  k(z) dz  �  ,  (9)  where recall ν∗  k = (Q∗  k)#( µ∗ρk  pk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any fixed c1 and c2, define  �Qk = {Q ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd) : the density ν∗  k,Q of Q#  �µ∗ · ρk  pk  �  exists and ν∗  k,Q ∈ C ˜α  c2(Rd)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then there exists a constant c3 such that it holds with probability larger than 1 −  1  n2 that for any k ∈ [K],  sup  Q∈ �  Qk  sup  f∈Lip1(Rd)  � �  f(z)ν∗  k,Q(z) dz −  �  f(z)�νk,Q(z) dz  �  ≤ c4  �  n− ˜  α+1  2 ˜  α+d + log n  √n  �  ,  where �νk,Q can either be the kernel density estimator in (7) or wavelet estimator in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So we choose λk = λ =  �  n− ˜  α+1  2 ˜  α+d + log n  √n  �−1 · n− 2β  d −1 for any k ∈ [K], then by the second statement of  23  Lemma 3 and Q∗  k ∈ �Qk, it holds with probability larger than 1 − M n−2 that,  K  �  k=1  � 1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) + λ ·  sup  f∈Lip1(Rd)  � �  f(z)�νk(z) dz −  �  f(z)�νk, �  Qk(z) dz  ��  ≤ C λ ·  �  n− ˜  α+1  2 ˜  α+d + log n  √n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So it holds with probability larger than 1 − M n−2 that for any k ∈ [K],  1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ C n− 2β  d −1,  (10)  and  sup  f∈Lip1(Rd)  � �  f(z)�νk(z) dz −  �  f(z)�νk, �  Qk(z) dz  �  ≤ C  �  n− ˜  α+1  2 ˜  α+d + log n  √n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (11)  Then we use the following lemma for bounding the population level reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For the estimator �G and �Q, there exist positive constants N, c1, c2 and c3 such that when  n ≥ N, for any k ∈ [K],  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' it holds with probability larger than 1−c1 n−3 that E[∥X− �Gk( �Qk(X))∥2·ρk(X)] ≤ c2 n− β  d ∨ log n  √n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  if ˜α > 0, then it holds with probability larger than 1 − c1 n−3 that the density ν∗  k, �  Qk of  ( �Qk)#  � µ∗·ρk  pk  �  exists and belongs to C ˜α  c3(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then by Lemma 3 and second statement of Lemma 4, there exist constants c, c1 such that it holds  with probability larger than 1 − c n−2 that  sup  f∈Lip1(Rd)  � �  f(z)ν∗  k, �  Qk(z) dz −  �  f(z)�νk, �  Qk(z) dz  �  ≤ c1  �  n− ˜  α+1  2 ˜  α+d + log n  √n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (12)  24  So combined with equation (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (11) and (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' it holds with probability larger than 1 − 1  n that  W1(�µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' µ∗) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f∈Lip1(RD)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk · f(X) d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�µ∗ · ρk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(X) d( �Gk)#�νk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='(i)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='≤ C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='log n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f∈Lip1(RD)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(X) d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�µ∗ · ρk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(X) d( �Gk)#�νk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='≤ C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='log n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f∈Lip1(RD)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(X) d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�µ∗ · ρk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(x) d( �Gk)#�νk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='≤ C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='log n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f∈Lip1(RD)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� � �pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f(X)ρk(X)dµ∗ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� �pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f( �Gk( �Qk(X))ρk(X)dµ∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� �pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='pk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f( �Gk( �Qk(X))ρk(X)dµ∗ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�pk · f(x) d( �Gk)#�νk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='≤ C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='log n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Eµ∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='∥X − �Gk( �Qk(X))∥ · ρk(X)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f∈Lip1(RD)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='f( �Gk(z))ν∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz −  �  f( �Gk(z)) �νk(z) dz  �  ≤ C1 n− β  d ∨ log n  √n +  K  �  k=1  �  sup  f∈Lip1(RD)  � �  f( �Gk(z))ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz −  �  f( �Gk(z))�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz  �  +  sup  f∈Lip1(RD)  � �  f( �Gk(z))�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz −  �  f( �Gk(z)) �νk(z) dz  ��  (ii)  ≤ C1 n− β  d ∨ log n  √n + C2  K  �  k=1  �  sup  f∈Lip1(Rd)  � �  f(z)ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz −  �  f(z)�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz  �  +  sup  f∈Lip1(Rd)  � �  f(z)�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  Qk(z) dz −  �  f(z) �νk(z) dz  ��  ≤ C2 n− ˜  α+1  2 ˜  α+d ∨ log n  √n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (13)  where (i) uses Bernstein’s inequality to obtain that  ���pk − pk  �� ≤ C  �  log n  n  holds with probability at least  1 − n−3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and (ii) uses the fact that β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then we consider the case α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Define �νk,Q =  1  �pkn  �n  i=1 δQ(Xi)ρk(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any f ∈ Lip1(Rd), we  have  �  f(z) d�νk,Q =  1  �pkn  n  �  i=1  f(Q(Xi))ρk(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' There exists a constant c so that it holds with probability larger than 1 −  1  n2 that  sup  Q∈Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Rd)  sup  f∈Lip1(Rd)  � 1  pk  �  f(Q(x))ρk(x) dµ∗ −  �  f(z)�νk,Q(z) dz  �  ≤ c  �log n  √n + n− 1  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then by Lemma 5 and equation (9), choose λk = λ =  �  n− 1  d + log n  √n  �−1 · n− 2β  d −1, we have that  statement (10) holds and  sup  f∈Lip1(Rd)  � �  f(z)�νk(z) dz −  �  f(z)�νk, �  Qk(z) dz  �  ≤ C  �  n− 1  d + log n  √n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  25  Then by Lemma 4, we have  E[∥X − �Gk( �Qk(X))∥2ρk(X)] ≤ c2n− β  d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So combined with Lemma 5, following equation (13), we have  W1(�µ, µ∗) ≤ C  �  log n  n  + 2  K  �  k=1  Eµ∗  �  ∥X − �Gk( �Qk(X))∥ · ρk(X)  �  +  K  �  k=1  sup  f∈Lip1(RD)  � �  f( �Gk(z))ν∗  k, �  Qk(z) dz −  �  f( �Gk(z)) �νk(z) dz  �  ≤ C1 n− β  d ∨ log n  √n + C2  K  �  k=1  �  sup  f∈Lip1(Rd)  � �  f(z)ν∗  k, �  Qk(z) dz −  �  f(z)�νk, �  Qk(z) dz  �  +  sup  f∈Lip1(RD)  � �  f(z)�νk, �  Qk(z) dz −  �  f(z) �νk(z) dz  ��  ≤ C2 n− 1  d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Proof of Lemma 3: kernel density estimator  Fix an arbitrary k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since �Qk ⊆ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), it holds that for any Q ∈ �Qk, ν∗  k,Q ∈ C ˜α  L(Rd) and  supp(ν∗  k,Q) ∈ [−L, L]d, where ν∗  k,Q is the density of the push-forward measure of µ∗·ρk  pk  by map Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Recall  that  �pk · �νk,Q(y) =  1  nhd  n  �  i=1  �k(y − Q(Xi)  h  )ρk(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Since ν∗  k,Q and �νk,Q are both compactly supported, there exists a constant C so that for any Q ∈ �Qk,  sup  f∈Lip1(Rd)  � �  f(y)ν∗  k,Q(y) dy −  �  f(y)�νk,Q(y) dy  �  ≤ C  sup  f∈C1  1(Rd)  � �  f(y)ν∗  k,Q(y) dy −  �  f(y)�νk,Q(y) dy  �  ,  where C1  1(Rd) =  �  f : Rd → R | supz∈Rd �  j∈Nd  0,|j|≤1 |f (j)(z)| ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we consider f ∈ C1  1(Rd), we can  get  �  f(y)ν∗  k,Q(y) dy −  �  f(y)�νk,Q(y) dy  = 1  �pk  � �  pk · f(y)ν∗  k,Q(y) dy −  �  �pk · f(y)�νk,Q(y) dy  �  +  �  f(y)ν∗  k,Q(y) dy ·  �  1 − pk  �pk  �  ≤ 1  �pk  ���  �  f(y) · pk · ν∗  k,Q(y) dy −  �  f(y) · EX(n)  �  �pk · �νk,Q(y)  �  dy  ���  �  ��  �  (A)  + 1  �pk  ���  �  f(y) · EX(n)  �  �pk · �νk,Q(y)  �  dy −  �  f(y) · �pk · �νk,Q(y) dy  ���  �  ��  �  (B)  +  ��1 − pk  �pk  ��  �  ��  �  (C)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (14)  First for term (C),by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤  C  �  log n  n , then by pk > 0, for large enough n, we have  ��1 − pk  �pk  �� ≤ C  �  log n  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For bounding the term (A),  we use a similar strategy as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3 of Divol (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Recall ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q = Q#[ µ∗·ρk  pk ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we can  26  write  �  f(y) · EX(n)  �  �pk · �νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)  �  dy =  �  f(y) · 1  hd · Eµ∗��k(y − Q(X)  h  ) · ρk(X)  �  dy  =  � �  f(y) · 1  hd · �k(y − z  h  ) · pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(z) dz dy  Denote υ(·) = pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we can obtain  ���  �  f(y) · pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y) dy −  �  f(y) · EX(n)  �  �pk · �νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)  �  dy  ���  =  ���  �  f(y) · υ(y) dy −  � �  f(y) · 1  hd · �k(y − z  h  ) · υ(z) dz dy  ���  =  ���  � �  f(y) · 1  hd · �k(y − z  h  ) · (υ(z) − υ(y)) dz dy  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  When ⌊˜α⌋ is even, denote s = ⌊˜α⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' when ⌊˜α⌋ is odd, denote s = ⌊˜α⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then using Taylor’s theorem,  we can decompose  υ(z) − υ(y) =  �  j∈Nd  0  1≤|j|<s  υ(j)(y)  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (z − y)j +  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1υ(j)(y + t(z − y)) dt · (z − y)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Using the fact that for any j ∈ Nd  0 with 1 ≤ |j| ≤ ⌊˜α⌋,  �  Rd �k(z) · zj dz = 0 and �k(·) = �k(−·), we can obtain  � �  f(y) · 1  hd · �k(y − z  h  ) ·  �  j∈Nd  0  |j|<s  υ(j)(y)  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (z − y)j dz dy  =  � �  f(y) · �k(z) ·  �  j∈Nd  0  |j|<s  υ(j)(y)  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  zj · h|j| dz dy = 0,  and  � �  f(y) · 1  hd · �k(y − z  h  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1υ(j)(y) · (z − y)j dt dz dy  =  � �  f(y) · �k(z) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1υ(j)(y) · zj · hs dt dz dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, we have  � �  f(y) · 1  hd · �k(y − z  h  ) · (υ(z) − υ(y)) dz dy  =  � �  f(y) · 1  hd · �k(y − z  h  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(y + t(z − y)) − υ(j)(y)  �  (z − y)j dt dz dy  (i)  =  � �  f(y) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y)  � �x − y  t  �j dt dx dy,  27  where (i) uses the change of variable x = y + t(z − y) and �k(·) = �k(−·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then by switching the variable x  and y, using the facts that s is a even number and k is an even function, we have  � �  f(y) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y)  � �x − y  t  �j dt dx dy  =  � �  f(x) ·  1  tdhd · �k(y − x  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(y) − υ(j)(x)  � �y − x  t  �j dt dx dy  = −  � �  f(x) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y)  � �x − y  t  �j dt dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, when ⌊˜α⌋ is even, we can obtain  ���  � �  f(y) · 1  hd · �k(y − z  h  ) · (υ(z) − υ(y)) dz dy  ���  =  ���1  2  � �  (f(y) − f(x)) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y)  � �x − y  t  �j dt dx dy  ���  =  ���1  2  � �  (f(y) − f(y + wth)) · �k(w) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(y + wth) − υ(j)(y)  � �  wh  �j dt dw dy  ���  (ii)  ≲ h1+˜α ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  [−L−h,L+h]d  �  [−1,1]d ∥w∥1+˜α · �k(w) ·  � 1  0  (1 − t)s−1 dt dw dy  ≲ n− 1+ ˜  α  2 ˜  α+d ,  where (ii) uses υ(·) = pk · ν∗  k,Q(·) has support contained in [−L, L]d and �k(·) has support contained in  [−1, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' On the other hand, when ⌊˜α⌋ is odd, recall s = ⌊˜α⌋ − 1 is even, we have  ���  � �  f(y) · 1  hd · �k(y − z  h  ) · (υ(z) − υ(y)) dz dy  ���  =  ���1  2  � �  (f(x) − f(y)) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y)  � �x − y  t  �j dt dx dy  ���  ≤  ����  1  2  � �  (f(x) − f(y)) ·  1  tdhd · �k(x − y  th  ) �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1�  υ(j)(x) − υ(j)(y) −  �  q∈Nd  0  |q|=1  υ(j+q)(y) · (x − y)q� �x − y  t  �j dt dx dy  ����  +  ����  1  2  � �  (f(x) − f(y)) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) · (x − y)q ·  �x − y  t  �j dt dx dy  ����  ≤ C h1+˜α  +  ����  1  2  � �  (f(x) − f(y)) ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) · (x − y)q ·  �x − y  t  �j dt dx dy  ����  �  ��  �  (D)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  28  Then for the term (D), using Taylor’s theorem and f ∈ C1  1(Rd), we can write  f(x) − f(y) =  �  j∈Nd  0  |j|=1  � 1  0  f (j)(y + t(x − y)) dt · (x − y)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (15)  So using the fact that for any j ∈ Nd  0 with 1 ≤ |j| ≤ ⌊˜α⌋ + 1,  �  Rd �k(z) · zj dz = 0, we can obtain  � � �  l∈Nd  0  |l|=1  f (l)(y) · (x − y)l ·  1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) · (x − y)q ·  �x − y  t  �j dt dx dy  =  �  l∈Nd  0  |l|=1  �  j∈Nd  0  |j|=s  �  q∈Nd  0  |q|=1  � 1  0  1  tdhd · s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ·  � �  f (l)(y) · (x − y)l · �k(x − y  th  )(1 − t)s−1υ(j+q)(y) · (x − y)q ·  �x − y  t  �j dx dy dt  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore,  (D) =  ����  1  2  � � �  l∈Nd  0  |l|=1  � 1  0  �  f (l)(y + ω(x − y)) − f (l)(y)  �  dω · (x − y)l 1  tdhd · �k(x − y  th  ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  � 1  0  (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) · (x − y)q ·  �x − y  t  �j dt dx dy  ����  (iii)  =  ����  1  2  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(θ) − f (l)(y)  � �θ − y  ω  �l 1  ωdtdhd · �k(θ − y  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) ·  �θ − y  ω  �q ·  �θ − y  ωt  �j dω dt dθ dy  ����,  where (iii) uses the change of variable y + ω(x − y) = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By switching the variable θ and y, we have  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(θ) − f (l)(y)  � �θ − y  ω  �l 1  ωdtdhd · �k(θ − y  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) ·  �θ − y  ω  �q ·  �θ − y  ωt  �j dω dt dθ dy  =  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(y) − f (l)(θ)  � �y − θ  ω  �l 1  ωdtdhd · �k(y − θ  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(θ) ·  �y − θ  ω  �q ·  �y − θ  ωt  �j dω dt dθ dy  = −  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(θ) − f (l)(y)  � �θ − y  ω  �l 1  ωdtdhd · �k(θ − y  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(θ) ·  �θ − y  ω  �q ·  �θ − y  ωt  �j dω dt dθ dy,  29  which leads to  ����  1  2  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(θ) − f (l)(y)  � �θ − y  ω  �l 1  ωdtdhd · �k(θ − y  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  υ(j+q)(y) ·  �θ − y  ω  �q ·  �θ − y  ωt  �j dω dt dθ dy  ����,  =  ����  1  4  � � � 1  0  � 1  0  �  l∈Nd  0  |l|=1  �  f (l)(θ) − f (l)(y)  � �θ − y  ω  �l 1  ωdtdhd · �k(θ − y  ωth ) ·  �  j∈Nd  0  |j|=s  s  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (1 − t)s−1 �  q∈Nd  0  |q|=1  �  υ(j+q)(y) − υ(j+q)(θ)  � �θ − y  ω  �q ·  �θ − y  ωt  �j dω dt dθ dy  ����  ≲ h1+˜α,  where the last inequality uses the fact that s = ⌊˜α⌋ − 1 and υ is ˜α is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can then obtain that  (A) ≲ n− ˜  α+1  2 ˜  α+d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  It remains to bound term (B), which is  (B) =  ���  �  f(y) · EX(n)  �  �pk · �νk,Q(y)  �  dy −  �  f(y) · �pk · �νk,Q(y) dy  ���  =  ����  1  n  n  �  i=1  �  f(y) · 1  hd · �k  �y − Q(Xi)  h  �  ρk(Xi) dy − Eµ∗  � �  f(y) · 1  hd · �k  �y − Q(X)  h  �  ρk(X) dy  �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  By standard symmetrization, we can get  E  �  sup  f∈C1  1(Rd)  Q∈ �  Qk  ����  1  n  n  �  i=1  �  f(y) · 1  hd · �k  �y − Q(Xi)  h  �  ρk(Xi) dy − E  � �  f(y) · 1  hd · k  �y − Q(X)  h  �  ρk(X) dy  �����  �  ≤ 2E  �  sup  f∈C1  1(Rd)  Q∈ �  Qk  ����  1  n  n  �  i=1  εi  �  f(y) · 1  hd · �k  �y − Q(Xi)  h  �  ρk(Xi) dy  ����  �  ,  where {εi}n  i=1 are n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' copies from Rademacher distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' P(εi = 1) = P(εi = −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Define function set  F =  �  f : M → R | f(x) =  �  g(y) · 1  hd · �k  �y − Q(x)  h  �  ρk(x) dy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' g ∈ C1  1(Rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Q ∈ �Qk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Since there exists a constant L so that supp(Q) = [−L, L]d, we can first consider the function set  F1 =  �  f : [−L, L]d → R | f(z) =  �  g(y) · 1  hd · �k  �y − z  h  �  dy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' g ∈ C1  1(Rd)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For any f ∈ F1 and j ∈ Nd  0 with 1 ≤ |j| ≤ ⌊ d  2⌋, we have  f (j)(z) =  �  g(y)  1  hd+|j| �k(j)(z − y  h  ) dy  =  1  h|j|  �  g(z − ht)�k(j)(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  30  Since |j| ≥ 1, there exists j1 ∈ Nd  0 with |j1| = 1 so that every element in j − j1 is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we  have  |f (j)(z)| =  ���  1  h|j|−1  �  g(j1)(z − ht)�k(j−j1)(t) dt  ��� ≲  1  h|j|−1 ,  where the last inequality uses g ∈ C1  1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' when d ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' for any j ∈ Nd  0 with |j| = ⌊ d  2⌋ and  z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' z2 ∈ Rd it holds that  |f (j)(z1) − f (j)(z2)| =  ��  1  h⌊ d  2 ⌋−1  �  g(j1)(z1 − ht)�k(j−j1)(t) dt −  1  h⌊ d  2 ⌋−1  �  g(j1)(z2 − ht)�k(j−j1)(t) dt  ��  =  ���  1  h⌊ d  2 ⌋−1  �  1  hd g(j1)(y)  ��k(j−j1)(z1 − y  h  ) − �k(j−j1)(z2 − y  h  )  �  dy  ���  If ∥z1 − z2∥ ≤ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' then  |f (j)(z1) − f (j)(z2)| =  ���  1  h⌊ d  2 ⌋−1  �  1  hd g(j1)(y)  ��k(j−j1)(z1 − y  h  ) − �k(j−j1)(z2 − y  h  )  �  dy  ���  ≤ C  1  h⌊ d  2 ⌋ ∥z1 − z2∥ ≤ C  1  h  d  2 −1 ∥z1 − z2∥  d  2 −⌊ d  2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  If ∥z1 − z2∥ ≥ h, then  |f (j)(z1) − f (j)(z2)| ≤ |f (j)(z1)| + |f (j)(z2)| ≤ C  1  h⌊ d  2 ⌋−1 ≤ C  1  h  d  2 −1 ∥z1 − z2∥  d  2 −⌊ d  2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So we can get when d ≥ 2, there exists a constant C such that for any f ∈ F1, it holds that  C h  d  2 −1f ∈ C  d  2  1 ([−L, L]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Similarly, when d = 1, we have for any z1, z2 ∈ Rd it holds that  |f(z1) − f(z2)| =  ��  �  (g(z1 − ht) − g(z2 − ht)) · �k  �  t  �  dy  �� ≲ ∥z1 − z2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, we can conclude that  F1 ∈ C  d  2  L1 h−( d  2 −1)+ ([−L, L]d),  for a constant L1, where (a)+ = max(a, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' So for any ϵ > 0, we can find a set N f  ϵ ⊆ F1 such that  �  log |N f  ϵ | ≲ h−( d  2 −1)+  ϵ  and for any f ∈ F1, there exists �f ∈ N f  ϵ such that  sup  y∈[−L,L]d|f(y) − �f(y)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then, to derive an ϵ-covering number for F, we introduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' (Lemma 12 of Tang & Yang (2022)) Let XG =  �  x ∈ RD : x = G(z), z ∈ Bd  1  �  be a d-  dimensional submanifold induced by a Lipschitz continuous map G : Rd → RD, then it holds for any  �γ > 0 that  log N  �  C�γ  1 (RD), ∥ · ∥L∞(XG), ϵ  �  ≤ C ϵ− d  �γ ,  ∀ϵ > 0,  where N(F, �d, ϵ) denotes the ϵ-covering number of function space F with respect to pseudo-metric �d, and  ∥f∥L∞(XG) = sup  x∈XG  ��f(x)  �� denotes the functional supreme norm constrained on set XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then since Ωk = Q∗  k(M ∩ Sk) is compactly supported and M ∩ Sk = G∗  k(Ωk), for any ϵ > 0, we can  31  find a function set N Q  ϵ such that  �  log |N Q  ϵ | ≲ ( 1  ϵ )  d  2β and for any Q ∈ �Qk, there exists �Q ∈ N Q  ϵ such that  sup  x∈M∩Sk  ∥ �Q(x) − Q(x)∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then for any Q ∈ �Qk and f ∈ F1, there exists �Q ∈ N Q  ϵ , �f ∈ N f  ϵ and a constant c such that  ���f(Q(x))ρk(x) − �f( �Q(x))ρk(x)  ���  ≤  ���f(Q(x))ρk(x) − �f(Q(x))ρk(x)  ��� +  ��� �f(Q(x))ρk(x) − �f( �Q(x))ρk(x)  ���  ≤ cϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So we can get  log N(F, ϵ, ∥ · ∥∞) ≲  �1  ϵ  � d  β  +  �  h−( d  2 −1)+  ϵ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Choose δ =  � 1  n  � ˜  α+1  2 ˜  α+d ∨  � 1  n  � β  d , we can get  1  √n  � 1  δ  ��1  ϵ  � d  2β  + h1− d  2 ∨ 1  ϵ  �  dϵ  ≲ log n  √n +  � 1  n  � ˜  α+1  2 ˜  α+d +  � 1  n  � β  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  By Dudley’s entropy integral bound (see for example, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='22 of Wainwright (2019)), it holds that  E  �  sup  f∈C1  1(Rd)  Q∈ �  Qk  ����  1  n  n  �  i=1  εi  �  f(y) · 1  hd · �k  �y − Q(Xi)  h  �  ρk(Xi) dy  ����  �  ≲ log n  √n +  � 1  n  � ˜  α+1  2 ˜  α+d +  � 1  n  � β  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The statement is then followed by Talagrand concentration inequality (see for example, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='27 of  Wainwright (2019)) and the fact that ˜α + 1 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  Proof of Lemma 3: Wavelet estimator  Fix an arbitrary k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since �Qk ⊆ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd), it holds that for any Q ∈ �Qk, supp(ν∗  k,Q) ⊆ [−L, L]d,  where ν∗  k,Q is the density of the push-forward measure of µ∗·ρk  pk  by map Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' by ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q ∈ Cα  L(Rd)  with support contained in [−L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' L]d and 0 < pk = Eµ∗[ρk] ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we can write pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y) as  pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y) =  �  m∈S  aQ  mφm(y) +  2d−1  �  l=1  +∞  �  j=0  �  m∈Slj  θQ  ljmψljm(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  where {φm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ψljm : l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 2d − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' j ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' m ∈ Zd} is the orthonormal wavelet basis for Besov space on  Rd defined as φm(y) = φ(y − m) and ψljm(y) = 2  jd  2 ψl(2jy − m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and it holds that φ(·) and ψl(·) for any  1 ≤ l ≤ 2d − 1 are compactly supported and have bounded β order derivatives (Bouzebda & Didi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then there exists a constant C such that |θQ  ljm| ≤ C(2−dj)  α  d + 1  2 and aQ  m ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Recall that  �pk · �νk,Q(y) =  �  m∈S  �aQ  mφm(y) +  2d−1  �  l=1  J  �  j=0  �  m∈Slj  �θQ  ljmψljm(y),  32  with  �aQ  m = 1  n  n  �  i=1  φm(Q(Xi))ρk(Xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �θQ  ljm = 1  n  n  �  i=1  ψljm(Q(Xi))ρk(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We have  E[�aQ  m] = pk ·  �  φm(y)ν∗  k,Q(y)dy = aQ  m,  E[�θQ  ljm] = pk ·  �  ψljm(y)ν∗  k,Q(y)dy = θQ  ljm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Moreover, by the fact that φ(·) and ψl(·) are compactly supported, we can get that there exists a constant  C such that for any 1 ≤ l ≤ 2d − 1 and j ∈ N, it holds that |Slj| ≤ C2dj and |S| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since ν∗  k,Q and  �νk,Q are both compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' There exists a constant C so that for any Q ∈ �Qk,  sup  f∈Lip1(Rd)  � �  f(y)ν∗  k,Q(y) dy −  �  f(y)�νk,Q(y) dy  �  ≤ C  sup  f∈C1  1(Rd)  � �  f(y)ν∗  k,Q(y) dy −  �  f(y)�νk,Q(y) dy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then we consider f ∈ C1  1(Rd), similarly, we can rewrite  f(y) =  �  m∈Zd  bmφm(y) +  2d−1  �  l=1  +∞  �  j=0  �  m∈Zd  fljmψljm(y)  where |fljm| ≤ C1(2−dj)  1  d + 1  2 and |bm| ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' So we can get  �  f(y)ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy −  �  f(y)�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy  = 1  �pk  � �  f(y)pk · ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy −  �  �pk · f(y)�νk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy  �  +  �  f(y)ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy ·  �  1 − pk  �pk  �  = 1  �pk  �  f(y)  �  ��  m∈S  (�aQ  m − E�aQ  m)φm(y) +  2d−1  �  l=1  J  �  j=0  �  m∈Slj  (�θQ  ljm − E�θQ  ljm)ψljm(y)  �  � dy  + 1  �pk  �  f(y)  �  �  2d−1  �  l=1  ∞  �  j=J  �  m∈Slj  θQ  ljmψljm(y)  �  � dy +  �  f(y)ν∗  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Q(y)dy ·  �  1 − pk  �pk  �  ≤ 1  �pk  ������  1  n  n  �  i=1  2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm (Q(Xi)) ρk(Xi) − E  � 2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm (Q(X)) ρk(X)  �������  �  ��  �  (A)  + 1  �pk  �����  1  n  n  �  i=1  �  m∈S  bmφm(Q(Xi))ρk(Xi) − E  � �  m∈S  bmφm(Q(X))ρk(X)  ������  �  ��  �  (B)  + 1  �pk  2d−1  �  l=1  +∞  �  j=J  �  m∈Slj  fljmθQ  ljm  �  ��  �  (C)  +  ��1 − pk  �pk  ��  �  ��  �  (D)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (16)  First for term (D),by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤  C  �  log n  n , then by pk > 0, for large enough n, we have  ��1 − pk  �pk  �� ≤ C  �  log n  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, for term (C), since  |fljm| ≲ (2−dj)  1  d + 1  2 , |θQ  ljm| ≲ (2−dj)  α  d + 1  2 and 2dJ ≍ n  d  2α+d , we can get  +∞  �  j=J+1  2d−1  �  l=1  �  m∈Slj  fljmθQ  ljm ≲ n− α+1  2α+d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  33  Then for term (A), by standard symmetrization, we can get  E  �  sup  f∈C1  1 (Rd)  Q∈ �  Qk  ����  1  n  n  �  i=1  2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm (Q(Xi)) ρk(Xi) − E  � 2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm (Q(X)) ρk(X)  �����  �  ≤ 2E  �  sup  f∈C1  1 (Rd)  Q∈ �  Qk  ����  1  n  2d−1  �  l=1  n  �  i=1  εi  J  �  j=0  �  m∈Slj  fljmψljm (Q(Xi)) ρk(Xi)  ����  �  ,  where {εi}n  i=1 are n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' copies from Rademacher distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' P(εi = 1) = P(εi = −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Define function set  F =  �  �  �f : M → R : f(z) =  2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm(Q(x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' |fljm| ≤ (2−dj)  1  d + 1  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Q ∈ �Qk  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  First we consider the function set  F1 =  �  �  �f : [−L3, L3]d → R, f(y) =  2d−1  �  l=1  J  �  j=0  �  m∈Slj  fljmψljm(y), |fljm| ≤ (2−dj)  1  d + 1  2  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  If 1  d ≤ 1  2, then there exists a constant c such that for any f ∈ F1, it holds that  c(2dJ)  1  d − 1  2 f ∈ C  d  2  1 ([−L, L]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So for any ϵ > 0, we can find a set N f  ϵ ⊆ F1 such that  �  log |N f  ϵ | ≲ (2dJ)( 1  2 − 1  d )+  ϵ  and for any f ∈ F1,  there exists �f ∈ N f  ϵ such that  sup  y∈[−L,L]d|f(y) − �f(y)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then since Ωk = Q∗  k(M ∩ Sk) is compactly supported and M ∩ Sk = G∗  k(Ωk), by lemma 6, for any ϵ > 0,  we can find a function set N Q  ϵ such that  �  log |N Q  ϵ | ≲ ( 1  ϵ )  d  2β and for any Q ∈ �Qk, there exists �Q ∈ N Q  ϵ  such that  sup  x∈M∩Sk  ∥ �Q(x) − Q(x)∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then for any Q ∈ �Qk and f ∈ F1, there exists �Q ∈ N Q  ϵ , �f ∈ N f  ϵ and a constant c such that  ���f(Q(x))ρk(x) − �f( �Q(x))ρk(x)  ���  ≤  ���f(Q(x))ρk(x) − �f(Q(x))ρk(x)  ��� +  ��� �f(Q(x))ρk(x) − �f( �Q(x))ρk(x)  ���  ≤ cϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So we can get  log N(F, ϵ, ∥ · ∥∞) ≲  �1  ϵ  � d  β  +  �  2dJ( 1  2 − 1  d )+  ϵ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Choose δ =  � 1  n  � α+1  2α+d ∨  � 1  n  � β  d , we can get  1  √n  � 1  δ  ��log n  ϵ  � d  2β  + (2dJ( 1  2 − 1  d ) log n) ∨ 1  ϵ  �  dϵ  ≲ log n  √n +  � 1  n  � α+1  2α+d +  � 1  n  � β  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  34  By Dudley’s entropy integral bound, it holds that  E  sup  f∈C1  1 (Rd)  Q∈ �  Qk  ������  1  n  2d−1  �  l=1  n  �  i=1  εi  J  �  j=0  �  m∈Slj  fljmψljm (Q(Xi))) ρk(Xi)  ������  ≲ log n  √n +  � 1  n  � α+1  2α+d +  � 1  n  � β  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Similarly we can get  E  sup  f∈C1  1 (Rd)  Q∈ �  Qk  �����  1  n  n  �  i=1  εi  �  m∈S  bmφm (Q(Xi)) ρk(Xi)  �����  ≲ n− β  d + log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The statement is then followed by Talagrand concentration inequality (see for example, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='27 of  Wainwright (2019)) and the fact that α + 1 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='3  Proof of Lemma 4  We fix an arbitrary k ∈ [K] in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since  1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2  2 · ρk(Xi) ≤ n− 2β  d −1,  we can get  1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ 1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2  �  ρk(Xi)  ≤  �  �  �  � 1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2  2 · ρk(Xi)  ≤ n− β  d − 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Define  F2 = {f = G ◦ Q : G ∈ Cβ  L(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Q ∈ Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then we have �G ◦ �Q ∈ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, when supx∈M∩Sk ∥f1(x) − f2(x)∥2 ≤ ϵ, it holds that  sup  x∈M  ��∥x − f1(x)∥2ρk(x) − ∥x − f2(x)∥2ρk(x)  ��  ≤  sup  x∈M∩Sk  ∥f2(x) − f1(x)∥2  ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So consider the function class �F2 = {|∥x − f(x)∥2ρk(x), f ∈ F2}, by Lemma 6, it holds that log N( �F2, ∥ ·  ∥M, ϵ) ≤ log N(F2, ∥ · ∥M, ϵ) ≲ ( 1  ϵ )  d  β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By Dudley’s entropy integral bound (see for example, Wainwright  (2019)), we can get that  E  �  sup  f∈F2  ��� 1  n  n  �  i=1  ∥Xi − f(Xi)∥2ρk(Xi) − Eµ∗�  ∥X − f(X)∥2ρk(X)  ����  �  ≤ C n− β  d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  35  Then by Talagrand concentration inequality (see for example, Wainwright (2019)), we can get that there  exists a constant c2, such that it holds with probability 1 − n−3 that  Eµ∗�  ∥X − �Gk ◦ �Qk(X)∥2 · ρk(X)  �  ≤ c2  �  n− β  d ∨ log n  √n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For the second statement, we first fix a small enough positive constant r > 0 that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then for any z ∈ Ωk = Q∗(M ∩ Sk), there exists σ(z) ∈ Ωk so that z ∈ Br(σ(z)) and ν∗  k(σ(z)) ≥ g(r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Let Az = {σ(z) : z ∈ Ωk} and �fk = �Gk ◦ �Qk ◦ G∗  k, we resort to the following lemma that provides an  upper bound on ∥G∗  k(z) − �fk(z)∥2 for all z ∈ Az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' It holds with probability at least 1 − c n−3 that for all z ∈ Az,  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2 (δn)|j| ≤ C ·  �  g(r)  �− 2β  d −1 ·  �log n  n  � β  d ,  (17)  where δn = b1 · (g(r))− 2  d ·  � log n  n  � 1  d for a constant b1 independent of n and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So by Lemma 7, we can get  sup  z∈Az  ∥G∗  k(z) − �fk(z)∥2 ≤ C ·  �  g(r)  �− 2β  d −1 ·  �log n  n  � β  d  sup  z∈Az  ∥JG∗  k(z) − J �  fk(z)∥F ≤ C1 ·  �  g(r)  �− 2β−2  d  −1 ·  �log n  n  � β−1  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Also by the fact that G∗  k and �f = �Gk ◦ �Qk ◦ G∗  k are β-Hölder smooth with β ≥ 1 + ˜α > 1, we have  sup  z∈Ωk  ∥G∗  k(z) − �fk(z)∥2 ≤ C ·  �  g(r)  �− 2β  d −1 ·  �log n  n  � β  d + C2 r  sup  z∈Ωk  ∥JG∗  k(z) − J �  fk(z)∥F ≤ C1 ·  �  g(r)  �− 2β−2  d  −1 ·  �log n  n  � β−1  d  + C3 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (18)  By the fact that for any z ∈ Ωk, it holds that z = Q∗  k(G∗  k(z)), we obtain Id = JQ∗  k(G∗  k(z)) JG∗  k(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since  Q∗  k is L-Lipschitz, JQ∗  k(G∗  k(z)) has bounded operator norm, which implies det(JT  G∗  k(z) JG∗  k(z)) ≥ c for  some positive constant c > 0 and z ∈ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moroever, by the fact that G∗  k is β-Hölder smooth with β > 1,  there exists a positive constant ϵ so that for the ϵ- enlargement of Ωk: Ωk,ϵ = {y ∈ Bϵ(z) : z ∈ Ωk}, it  holds that for any z ∈ Ωk,ϵ, det(JT  G∗  k(z) JG∗  k(z)) ≥ c  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, the second display in (18) and β-Hölder  smooth of �fk implies that infz∈Ωk,ϵ det(JT  �  fk(z) J �  fk(z)) ≥ c  4 for all sufficiently small ϵ, r and sufficiently  large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Now let �lk = �Qk ◦ G∗  k and by using the identity �fk = �Gk ◦ �lk,  JT  �  fk(z) J �  fk(z) =  �  J �  Gk(�lk(z)) J�lk(z)  �T �  J �  Gk(�lk(z)) J�lk(z)  �  = JT  �lk(z) JT  �  Gk(�lk(z)) J �  Gk(�lk(z)) J�lk(z),  by taking determinant we further obtain (note that J�lk(z) is a square matrix)  det2 �  J�lk(z)  �  det  �  JT  �  Gk(�lk(z)) J �  Gk(�lk(z))  �  ≥ c  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Since both �Gk and �Qk are L-Lipschitz, we can further deduce that 0 < c1 ≤ det(J�lk(z)) ≤ c2 for all  z ∈ Ωk,ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We claim that �lk is globally invertible over Ωk,ϵ when ϵ, r are small enough and n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Otherwise, suppose there exist distinct z0 and z1 in Ωk,ϵ such that �lk(z0) = �lk(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since 0 < c1 ≤  36  det(J�lk(z)) ≤ c2 implies �lk to be locally invertible, meaning that there exists some constant b0 > 0  independent of ϵ such that ∥z0 − z1∥ ≥ b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By the definition of Ωk,ϵ and the Lipschitzness of �Gk and �lk,  there exist ¯z0 and ¯z1 in Ωk such that (for sufficiently small ϵ)  ∥¯z0 − ¯z1∥ ≥ 1  2b0,  ∥�lk(¯z0) − �lk(¯z1)∥2 ≤ Cϵ  and  ∥ �fk(¯z0) − �fk(¯z1)∥ ≤ Cϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (19)  The third display above combined with the first display in (18) implies ∥G∗  k(¯z0) − G∗  k(¯z1)∥2 ≤ C1(ϵ + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  On the other hand, from the first display above and the Lipschitzness of Q∗  k, we have  1  2b0 ≤ ∥¯z0 − ¯z1∥ = ∥Q∗  k(G∗  k(¯z0) − Q∗  k(G∗  k(¯z1))∥2 ≤ C∥G∗  k(¯z0) − G∗  k(¯z1)∥2 ≤ CC1(ϵ + r),  which is a contradiction when ϵ, r are chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Let �l−1  k  : �lk(Ωk,ϵ/2) → Ωk,ϵ/2 be the inverse of �lk over Ωk,ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By using the inverse function theorem for  Hölder space (see for example, Appendix A of (Eldering, 2013)), we can conclude �l−1  k  ∈ Cβ  C0(�lk(Ωk,ϵ/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd)  for some sufficiently large constant C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, we can write the expression of the density function of  ν∗  k, �  Qk = [ �Qk]#( ρkµ∗  pk ) as  ν∗  k, �  Qk(y) = ν∗  k(�l−1  k (y)) ·  �  det  �  JT  �l−1  k (x)J�l−1  k (x)  �� 1  2 · 1  �  y ∈ �lk(Ωk)  �  by applying the change of variable of y = �lk(z) with z ∼ ν∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, since ν∗  k ∈ Cα  L(Rd), this together  with �l−1  k  ∈ Cβ  C0(�lk(Ωk,ϵ/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Rd) implies ν∗  �  Q ∈ C ˜α  C1(Rd) for some constant C1 (recall ˜α = α ∧ (β − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='4  Proof of Lemma 7  The proof follows the analysis in Tang & Yang (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Let hn =  � log n  n  � 1  d and Nhn ⊂ Az be a minimal  hn-covering set of Az under the ℓ2 distance, where its cardinality satisfies |Nhn| ≤ C  n  log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any  �z ∈ Nhn, define δn = b  � log n  n  � 1  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We claim that it suffices to show that for sufficiently large b, it holds with probability at least 1 − n−3  that for any �z ∈ Nhn,  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2 (δn)|j| ≤ C  �log n  n  � β  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (20)  In fact, if this inequality holds, then we can apply a standard argument of approximation by the hn-  covering set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Concretely, for any z ∈ Az, there exists �z ∈ Nhn such that ∥z − �z∥2 ≤ hn = ( log n  n )  1  d , we  can obtain by applying Taylor expansion to G∗  k(z) − �fk(z) that  ∥G∗  k(z) − �fk(z)∥2 ≤ C  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2  �log n  n  � |j|  d + C  �log n  n  � β  d  ≤ C  �log n  n  � β  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Now let us prove inequality (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Recall that  1  n  n  �  i=1  ∥Xi − �Gk( �Qk(Xi))∥2ρk(Xi) ≤ C n− 2β  d −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  37  In particular, by restricting the sum to those Q∗(Xi) in Bδn(�z) for a fixed �z ∈ Nhn, we further obtain  (recall that �fk = �Gk ◦ �Qk ◦ G∗  k)  1  n  n  �  i=1  ∥G∗  k(Q∗  k(Xi)) − �fk(Q∗  k(Xi))∥2 · ρk(Xi) · 1Bδn(�z)(Q∗  k(Xi)) ≤ C n− 2β  d −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  By applying the Taylor expansion to G∗  k(z) − �fk(z) around �z in the preceding display and using the fact  that G∗  k − �fk ∈ Cβ  C0(Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) with some sufficiently large constant C0, we can get the following localized  basic inequality after some algebra calculation  Un(�z, �fk) : =  1  n  n  �  i=1  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  (Q∗  k(Xi) − �z)j  ����  2  2  1Bδn(�z)(Q∗  k(Xi)) · ρk(Xi)  ≤ c  �  (δn)2β + (δn)β  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  ��G∗,(j)  k  (�z) − �f (j)  k (�z)  ��  2 (δn)|j|  �  1  n  n  �  i=1  1Bδn(�z)(Q∗  k(Xi)) · ρk(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (21)  The second factor on the right hand side of (21) can be bounded by applying a simple union bound  argument and Bernstein’s inequality for bounded function as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' First, we can bound the expectation  Eµ∗�  1Bδn(�z)(Q∗  k(X)) · ρk(X)  �  (i)  = pk  �  Bδn(�z)  ν∗  k(z) dz ≤ C pk δd  n  ≤ C bd log n  n  ,  (22)  where step (i) follows by the fact that ν∗  k = (Q∗  k)#( µ∗·ρk  pk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since the random variable 1Bδn(�z)(Q∗  k(X))·ρk(X)  is uniformly bounded by 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' and inequality (22) and ρk ≤ 1 implies its variance to be bounded by C1 bd log n  n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  we may apply the Bernstein inequality and a simple union bound argument over all �z ∈ Nhn (with  |Nhn| ≤ C  n  log n) to obtain that with probability at least 1 − n−c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  sup  �z∈Nhn  ����  1  n  n  �  i=1  1Bδn(�z)(Q∗  k(Xi)) · ρk(Xi) − Eµ∗�  1Bδn(�z)(Q∗  k(X)) · ρk(X)  ����� ≤ C2b  d  2 · log n  n  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (23)  which together with (22) leads to  sup  �z∈Nhn  � 1  n  n  �  i=1  1Bδn(�z)(Q∗  k(Xi))  �  ≤ C bd · log n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (24)  To analyze the quantity Un(�z, �fk) on the left hand side of the localized basic inequality (21), we will  resort to the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' With probability at least 1 − n−3, the following inequality holds for any β-smooth function  f ∈ Cβ  L(Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) and �z ∈ Nh�n,  ��Un(�z, f) − Eµ∗[Un(�z, f)]  �� ≤ C b  d  2 · log n  n ��log n  n  � 2β  d +  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥f (j)(�z) − G∗,(j)  k  (�z)∥2 (δn)|j|�2�  ,  where the expectation is taken with respect to the randomness in {Xi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  38  Before applying this lemma, notice that for any �z ∈ Nhn, we can bound the expectation Eµ∗[Un(�z, �fk )],  where f has been plugged-in with �fk , by  Eµ∗[Un(�z, �fk )]  = Eµ∗  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  (Q∗  k(X) − �z)j���  2  2 · 1Bδn(�z)(Q∗  k(X)) · ρk(X)  �  (i)  ≥  inf  z∈Bδn(�z)ν∗  k(z)  �  z∈Bδn(�z)  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  (z − �z)j���  2  2 dz  (ii)  = δd  n  inf  z∈Bδn(�z)ν∗  k(z)  �  Bd  1  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' δj  n  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  zj���  2  2 dz  ≥ C bd · log n  n  g(r) ·  �  Bd  1  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='δj  n  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  zj���  2  2 dz,  (25)  where step (i) uses the fact that Q∗  k(X) given X ∼ µ∗·ρk  pk  is distributed as ν∗  k, step (ii) follows by applying  the change of variable of z−�z  δn → z, and the last step follows by the fact that ν∗  k(�z) ≥ g(r) for �z ∈ Az and  the smoothness of ν∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Now using the fact that for any d-variate polynomial S(y) = �  j∈Nd  0, |j|≤k ajyj,  y ∈ Rd, there exists some positive constant C(d, k) only depending on (d, k) such that  �  Bd  1  S2(y) dy ≥ C(d, k)  �  j∈Nd  0, |j|≤k  a2  j,  we can obtain that  �  Bd  1  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='δj  n  �  G∗,(j)  k  (�z) − �f (j)  k (�z)  �  zj���  2  2 dz ≥ c  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2 (δn)|j|  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (26)  Finally, by combining equations (21), (22), (25), (26) and Lemma 8, we obtain that with probability at  least 1 − cn−3, for any �z ∈ Nhn,  bd · log n  n  g(r) ·  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2 (δn)|j|  �2  ≤ Cb  d  2 · log n  n � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥ �f (j)  k (�z) − G∗,(j)  k  (�z)∥2 (δn)|j|  �2  + Cb  d  2 ·  �log n  n  � 2β  d · log n  n  + Cbd · log n  n �  (δn)2β + (δn)β  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥G∗,(j)  k  (�z) − �f (j)  k (�z)∥2(δn)|j|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Consequently, the claimed inequality (20) follows from the above by choosing b = b1(g(r))− 2  d with  sufficiently large b1 and the definition that δn = b  � log n  n  � 1  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5  Proof of Lemma 8  The proof follows from the proof of Lemma 18 in Tang & Yang (2022), we include it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Since f ∈ Cβ  L(Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), for any z ∈ Bd  1 and j ∈ Nd  0 with |j| ≤ ⌊β⌋, it holds that ∥f (j)(z)∥2 ≤  √  DL = C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  39  For any fixed �z ∈ Nh�n and �δ > 0, let  ¯T (�δ) =  �  T = {Tj}j∈Nd  0, |j|≤⌊β⌋ ∈ [−C0, C0]D×(  d+⌊β⌋−1  d  ) :  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  ��Tj − G∗,(j)  k  (�z)  ��  2 (δ�z)|j| ≤ �δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  We also define the following supreme of an empirical process indexed by T ∈ ¯T (�δ),  Zn(�δ) =  sup  T ∈ ¯T (�δ)  ����� Eµ∗  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(X) − �z)j���  2  2 · 1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  �  − 1  n  n  �  i=1  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(Xi) − �z)j���  2  2 · 1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  ������,  and Rn(�δ) = Eµ∗�  Zn(�δ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We will first prove a concentration inequality for a fixed radius �δ > 0, and then  using the peeling technique to allow the radius to be random, which leads to the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  To apply the Talagrand concentration inequality (see, for example, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='27 of Wainwright (2019))  for bounding the difference |Zn(�δ) − Rn(�δ)| for a fixed �δ > 0, we notice that each additive component in  the second empirical sum above has second moment uniformly bounded by  Eµ∗  �  sup  T ∈ ¯T (�δ)  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(X) − �z)j���  4  2 · 1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  ��  ≤  sup  z∈Bδ�z (�z)  T ∈ ¯  T (�δ)  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (z − �z)j���  4  2 · Eµ∗�  1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  �  ≤ C sup  T ∈ ¯T (�δ)  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − G∗,(j)  k  (�z)∥2 (δ�z)|j|  �4  bd  2 · log n  n  ≤ C bd  2 �δ4 · log n  n  ,  where we have used inequality (22) to bound Eµ∗�  1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, each additive  component can be almost surely bounded by  sup  z∈Bδ�z (�z)  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(X) − �z)j���  2  2  ≤ C  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − G∗,(j)(�z)∥2 (δ�z)|j|  �2  ≤ C �δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Based on these two bounds, we can apply the Talagrand concentration inequality to obtain that for any  s ≥ 0,  P  �  Zn(�δ) ≥ Rn(�δ) + s2�  ≤ 2 exp  �  −  c ns4  s2 �δ2 + bd  2 �δ4 · log n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (27)  It remains to bound the expectation Rn(�δ) via the symmetrization technique and chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By a standard  40  symmetrization, we can get  Rn(�δ) ≤  2  √n E  �  sup  T ∈ ¯T (�δ)  �����  1  √n  n  �  i=1  εi  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(X) − �z)j���  2  2 · 1Bδ�z (�z)(Q∗  k(Xi)) · ρk(Xi)  ������  �  ,  where {εi}n  i=1 are n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' copies from the Rademacher distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' P(εi = 1) = P(εi = −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Since given {Xi}n  i=1, the stochastic process inside the supreme is a sub-Gaussian process with intrinsic  metric  d2  n(T, �T)  = 1  n  n  �  i=1  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(Xi) − �z)j���  2  2  −  ���  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − �Tj  �  (Q∗  k(Xi) − �z)j���  2  2  �2  1Bδ�z (�z)(Q∗  k(Xi))) · ρk(Xi)  ≤ C �δ4 1  n  n  �  i=1  1Bδ�z (�z)(Q∗  k(Xi))) · ρk(Xi),  for any T, �T ∈ ¯T (�δ), where the last step uses the definition of ¯T (�δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  The above combined with  inequality (22) implies  Eµ∗  �  sup  T, �  T ∈ ¯T (δ)  d2  n(T, �T)  �  ≤ C bd  2 �δ4 · log n  n  and  dn(T, �T) ≤ C�δ  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − �Tj∥2 δ|j|  �z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' let Kn(δ) =  sup  T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �  T ∈ ¯T (δ)  d2  n(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' �T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' by applying the standard chaining via Dudley’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' we can  get  Rn(�δ) ≤ C  1  √n Eµ∗  � � Kn(�δ)  0  �  log  �δ  u du  �  = C  1  √n Eµ∗  �  Kn(�δ) ·  � 1  0  �  log  �δ  u · Kn(�δ)  du  �  = C  1  √n Eµ∗  �  Kn(�δ) · 1(Kn(�δ) ≤ b  d  2  2 �δ2  �  log n  n  )  � 1  0  �  log  �δ  u · Kn(�δ)  du  �  + C  1  √n Eµ∗  �  Kn(�δ) · 1(Kn(�δ) > b  d  2  2 �δ2  �  log n  n  )  � 1  0  �  log  �δ  u · Kn(�δ)  du  �  ≤ C1 b  d  2  2 · log(n/�δ)  n  �δ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (28)  where we have used the fact that the u-covering entropy of ¯T (�δ) relative to metric dn is at most C2 log  �δ  u  for u ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' 1) where C2 depends on (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' D) (at most polynomial dependence on D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' By combining this  with inequality (27), we obtain that for all t ≥ 1,  P  �  Zn(�δ) ≥ C t2 b  d  2  2 · log(n/�δ)  n  �δ2�  ≤ 2 exp  �  − c t2 log(n/�δ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (29)  41  Finally, we apply the peeling technique to extend the above high probability bound on Zn(�δ) to  the random radius �δ = �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �� �f (j)  k  − G∗,(j)  k  (�z)  ��  2 (δ�z)|j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Specifically, we first set the basic level  ¯δ =  � log n  n  � β  d , and for s = 1, · · · , S with S ≤ C log 1  ¯δ , define sets  �T0 =  �  T = {Tj}j∈Nd  0,|j|≤⌊β⌋ ∈ [−C0, C0]D×(  d+⌊β⌋−1  d  ) :  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − G∗,(j)  k  (�z)∥2 (δ�z)|j| ≤ ¯δ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �Ts =  �  T = {Tj}j∈Nd  0,|j|≤⌊β⌋ ∈ [−C0, C0]D×(  d+⌊β⌋−1  d  ) : 2s−1¯δ ≤  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − G∗,(j)  k  (�z)∥2 (δ�z)|j| ≤ 2s¯δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  By applying inequality (29) to �δ = 2s¯δ for s ∈ [S] with sufficiently large constant t > 0, as C1 ≤  − log(2s¯δ) ≤ C2 log n, we obtain that  P  �  Zn(¯δ) ≥ C b  d  2  2  log n  n  ¯δ2  �  +  S  �  s=1  P  �  Zn(2s¯δ) ≥ C b  d  2  2  log n  n  4s¯δ2  �  ≤ n−(c+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Note that for any T ∈ �Ts and any s ∈ {0} ∪ [S], the event Zn(2s¯δ) ≤ C b  d  2  2  log n  n  4s¯δ2 implies  ����� Eµ∗  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(X) − �z)j���  2  2 · 1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  �  − 1  n  n  �  i=1  ����  �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  �  G∗,(j)  k  (�z) − Tj  �  (Q∗  k(Xi) − �z)j���  2  2 · 1Bδ�z (�z)(Q∗  k(X)) · ρk(X)  ������  ≤ c1 b  d  2  2  log n  n  �  ¯δ2 +  � �  j∈Nd  0  |j|≤⌊β⌋  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' ∥Tj − G∗,(j)  k  (�z)∥2 (δ�z)|j|  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Finally, since for any f ∈ Cβ  L(Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD), Tf : = {Tf,j = f (j)}j∈Nd  0,|j|≤⌊β⌋ must belong to some �Ts, the  claimed result is a consequence of the two preceding displays and a simple union bound over �z ∈ Nhn  where |Nhn| ≤ C  n  log n ≤ C n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='6  Proof of Lemma 5  Firstly by Bernstein’s inequality, it holds with probability at least 1 − n−3 that |pk − �pk| ≤ C  �  log n  n ,  then by pk > 0, for large enough n, we have  ��1 − pk  �pk  �� ≤ C  �  log n  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Thus  sup  Q∈Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Rd)  sup  f∈Lip1(Rd)  � 1  pk  �  f(Q(x))ρk(x) dµ∗ −  �  f(z) �νk,Q(z) dz  �  =  sup  Q∈Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Rd)  sup  f∈Lip1(Rd)  � 1  pk  �  f(Q(x))ρk(x) dµ∗ −  1  �pkn  n  �  i=1  f(Q(Xi))ρk(Xi)  �  ≤ C  �  log n  n  + 1  pk  sup  Q∈Cβ  L(RD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='Rd)  sup  f∈Lip1(Rd)  � �  f(Q(x))ρk(x) dµ∗ − 1  n  n  �  i=1  f(Q(Xi))ρk(Xi)  �  ≤C  �  log n  n  + C  sup  f∈Lip1(RD)  � �  f(x)ρk(x) dµ∗ − 1  n  n  �  i=1  f(Xi)ρk(Xi)  �  ,  42  where the last inequality is due to the assumption that β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then consider the pseudo-metric for  f, f ′ ∈ Lip1(RD)  dn(f, f ′) =  �  �  �  � 1  n  n  �  i=1  �  f(Xi)ρk(Xi) − f ′(Xi)ρk(Xi)  �2 ≤  sup  x∈Sk∩M  |f(x)−f ′(x)| =  sup  x∈G∗(Q∗(Sk∩M))  |f(x)−f ′(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then by Lemma 6, we have log N(Lip1(RD), ∥ · ∥L∞(Sk∩M), ϵ) ≤ C ϵ−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Choose δ = ( 1  n)  1  d , we can get  1  √n  � 1  δ  �  log N(Lip1(RD), ∥ · ∥L∞(Sk∩M), ϵ) dϵ ≤ C n− 1  d ∨ log n  √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Thus similar as the analysis in the proof of Lemma 3, using Dudley’s entropy integral bound and Talagrand  concentration inequality, we can obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  D  Proof of Technical Details  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='1  Proof of Lemma 1  Consider an aritrary µ∗ ∈ P∗(d, D, α, β, L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Denote M = supp(µ∗), to begin with, we consider the  following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Lemma 9 (Lemma 17 of Tang & Yang (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' There exist positive constants (τ1, L1) such that for any  x0 ∈ M, define Qx0 : RD → Rd as Qx0(x) = W T  x0(x−x0) where Wx0 ∈ RD×d is an arbitrary orthonormal  basis of the tangent space of M at x0, then there exists a set �Ux0 satisfying Bτ1(x0) ∩ M ⊂ �Ux0 ⊂ M  and function Gx0 ∈ Cβ  L1(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' RD) so that  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Gx0(Bd  1) = �Ux0 and for any z ∈ Bd  1, Qx0(Gx0(z)) = z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' µ∗ ◦ Gx0|Bd  1 ∈ Cα  L1(Bd  1) and for any z ∈ ∂Bd  1, ∥Gx0(z) − x0∥ ≥ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  By Sobolev extension theorem, there exists a constant L2 so that for any x0 ∈ M, there exists Qx0 ∈  Cβ  L2(RD) so that Qx0| �Ux0 = Qx0| �Ux0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Since JGx0(z)T JGx0(z) has uniformly lower bounded eigenvalues  on z ∈ Bd  1 and Gx0 is β-smooth with β > 1 and uniformly bounded Hölder norm, there exists a small  enough positive constant r0 so that for any x0 ∈ M,  sup  v∈Sd−1  1  ={v∈Rd : ∥v∥=1}  sup  z,z′∈Bdr0  ∥(JGx0 (z) − JGx0 (z′))v∥  ∥JGx0 (0)v∥  ≤ 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (30)  We then choose r∗ ≤ τ1/2 to be a small enough positive constant so that for any x0 ∈ M, Qx0(Br∗(x0) ∩  M) ⊂ Bd  r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For an arbitrary k ∈ [K], consider x0 = ak ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Define Gk = Gx0 and Qk = Qx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then  we have Sk ∩ M ⊂ Bτ1(x0) ∩ M ⊂ �Ux0, and for any x ∈ M ∩ Sk, x = Gk(Qk(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, let  pk = Eµ∗[ρk(X)], since the density of µ∗ is uniformly bounded from below and rk ≥ L∗, we have pk is  also uniformly bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Furthermore, we can write νk = (Qk)#( µ∗ρk  pk ) as  νk(z) =  �  �  �  µ∗(Gk(z))·ρk(Gk(z))√  det(JGk (z)T JGk (z))  �  Qk(M∩Sk) µ∗(Gk(z))·ρk(Gk(z))√  det(JGk (z)T JGk (z)) dz  z ∈ Bd  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  0  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then by the α-smoothness of �ρk(·) and the uniformly lower boundness of �  k∈[K] �ρk(·), we have νk(z)|Bd  1 ∈  Cα  L(Bd  1) for some constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  On the other hand, by the second statement of Lemma 9 and the  Lipschitzness of Gm, there exists a positive constant ϵ so that Qk(M ∩ Sk) ⊂ Bd  1−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Combined with  43  the fact that νk(z) = 0 when z /∈ Qk(M ∩ Sk), we can obtain νk(z) ∈ Cα  L∗(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In addition, for any  z ∈ Ωk = Qk(M ∩ Sk) and any r > 0, we will show that there exists z′ ∈ Br(z) so that νk(z′) ≥ c (rγ ∧ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Firstly if ∥Gk(z) − x0∥ ≤ r∗/2, then we have  νk(z) ≥ c1 ρk(Gk(z)) ≥ c1  ( 3r2  k  4 )γ  M(r∗)2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  On the other hand, if ∥Gk(z) − Gk(0)∥ = ∥Gk(z) − x0∥ ≥ r∗/2, denote z = av with v = z/∥z∥ and  a = ∥z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we have a ≤ r0 and  r∗/2 ≤ ∥Gk(z) − Gk(0)∥ ≤ c2 ∥z∥ = c2 |a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  If r ≥ a, then z = av ∈ Br(0) and νk(0) ≥ c for some positive constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' If r < a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' choose z′ = (a − r)v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  then z′ ∈ Br(z) and  ∥Gk(z′) − Gk(0)∥ = sup  l∈Sd−1  1  �  lT Gk(z′) − lT Gk(0)  �  = sup  l∈Sd−1  1  �  lT Gk(z) − lT Gk(0) + lT Gk(z′) − lT Gk(z)  �  (i)  =  sup  l∈Sd−1  1  �  lT JGk(zl)av − lT JGk(z′  l)rv  �  = sup  l∈Sd−1  1  �  lT JGk(zl)(a − r)v + lT (JGk(zl) − JGk(z′  l))rv  �  (ii)  ≤  sup  l∈Sd−1  1  �  lT JGk(zl)(a − r)v + r  2a∥Gk(z) − Gk(0)∥)  = a − r/2  a  ∥Gk(z) − Gk(0)∥  ≤ rk − rrk  2r0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  where (i) uses mean-value theorem and (ii) uses equation (30) and Taylor’s theorem to obtain  r  2a∥Gk(z) − Gk(0)∥ = r  2  ���  � 1  0  JGk(tz) dt · v  ���  ≥ r  2∥JGk(0) · v∥ − r  2  ���  � 1  0  JGk(0) − JGk(tz) dt · v  ���  ≥ 3r  2  sup  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='z′∈Bd  r0  ∥(JGx0 (z) − JGx0 (z′))v∥ − r  2  � 1  0  ∥(JGk(0) − JGk(tz))v∥ dt  ≥ r  sup  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='z′∈Bdr0  ∥(JGx0 (z) − JGx0 (z′))v∥  ≥ sup  l∈Sd−1  1  lT (JGk(zl) − JGk(z′  l))rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  So we have  �ρk(Gk(z′)) = (r2  k − ∥Gk(z′) − x0∥2)γ· ≥ (rr2  k  2r0  )γ ≥ ( r2  k  2r0  )γrγ ≥ ((L∗  1)2  2r0  )γrγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  (31)  Thus there exists constant c so that  νk(z′) ≥ c rγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, we have Assumption A holds for µ∗ with G∗  k = Gk, Q∗  k = Qk and ν∗  k = νk with k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For  44  the second statement, note that by the β-smoothness of G∗  k = Gk, Q∗  k = Qk and the α-smoothness of  ν∗  k = νk, Assumption B trivially holds for approximation family G = G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, consider  νk(z) =  µ∗(Gk(z)) ·  �  det(JGk(z)T JGk(z))  �  Bd  1 µ∗(Gk(z)) ·  �  det(JGk(z)T JGk(z)) dz  ,  z ∈ Bd  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then we have νk(z) = νk(z)·ρk(Gk(z))  Eνk [ρk(Gk(z))] , νk(z) ∈ Cα  L(Bd  1) and infz∈Bd  1νk(z) ≥ L3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' So there exists an  (α + 1)-smooth invertible function Vk : Bd  1 → Bd  1 (see for example, (Caffarelli, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Villani, 2009)) so that  ν0 = Vm#νk and νk = V −1  m  #ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, G2 suffices to model µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  For the family G3, let V k be an (α + 1)-smooth extension of Vk|Bd  1−ϵ/2 to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Note that Vk has (α + 1)-  smooth inverse and Vk(Bd  1−ϵ/2) ⊂ Bd  1−ϵ1 for some positive constant ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can consider V −1  k  as an  α-smooth extension of V −1  k  |Vk(Bd  1−ϵ/2) to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then we can define G′  k = Gk ◦ V −1  k  and Q′  k = V k ◦ Qk, by  the fact that Qk(M ∩ Sk) ⊂ Bd  1−ϵ, we have for any x ∈ M ∩ Sk, G′  k(Q′  k(x)) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Moreover, let  ν′  k(z) = (Q′  k)#  µ∗ρk  pk  =  �  �  �  ν0(z)·ρk(Gk◦V −1  k  (z))  �  Bd  1 ν0(z)·ρk(Gk◦V −1  k  (z)) dz,  z ∈ Vk(Bd  1−ϵ/2),  0,  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Using the fact that V −1  k  |Vk(Bd  1−ϵ/2) is (α + 1)-smooth with bounded Hölder norm and ν′  k(z) = 0 when  z /∈ Vk(Bd  1−ϵ), we have ν′  k ∈ Cα  L(Rd) for some constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' In addition, recall that for any z0 ∈ Qk(M∩Sk)  and r > 0, there exists z′  0 ∈ Qk(M ∩ Sk) so that z0 ∈ Br(z′  0) and �ρk(Gk(z′  0)) ≥ c1(rγ ∧ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Note that by  the Lipschitzness of Vk, there exists a constant L4 ≥ 1 so that  ∥Vk(z0) − Vk(z′  0)∥ ≤ L4∥z0 − z′  0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, for any z ∈ Vk(Qk(M ∩ Sk)) and r > 0, there exists z′ ∈ Vk(Qk(M ∩ Sk)) ∩ Br(z), so that  �ρk(Gk ◦ V −1  k  (z′)) ≥ c1  Lγ  4 (rγ ∧ 1) and ν′  k(z) ≥ c (rγ ∧ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Therefore, when α = β − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Assumption A holds  with G∗  k = G′  k, Q∗  k = Q′  k and ν∗  k = ν′  k, and Assumption B holds with G = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='2  Proof of Lemma 2  Let Nϵ be the minimal ϵ-covering set of M, where ϵ is a number that will be chosen later, then by  Lemma 9 and the compactness of M, we have |Nϵ| ≤ C1 ( 1  ϵ )d where C1 is a positive constant that only  depends on (d, D, β, L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then if ϵ ≤ τ1, by Lemma 9, we have for any x0 ∈ M  Pµ∗(Bϵ(x0)) =  �  Qx0(Bϵ(x0))  µ∗(Gx0(z))  �  det(JGx0 (z)T JGx0 (z)) dz ≥ C2 ϵd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Then, by Bernstein’s inequality and a simple union bound argument, it holds with probability at least  1 − n−c  1  that for any x0 ∈ Nϵ,  ��� 1  n1  �  i∈I1  1(∥Xi − x0∥ ≤ ϵ) − Pµ∗(Bϵ(x0))  ��� ≤  1  3n1  log(δ) +  �  2C2ϵd log(δ)  n1  ,  δ = 2C1nc  1(1  ϵ )d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, there exists a constant C3, C so that when n1 ≥ C, by choosing ϵ = C3 ( log n1  n1 )  1  d , we have it  holds with probability at least 1 − n−c  1  that for any x0 ∈ Nϵ,  ��� 1  n1  �  i∈I1  1(∥Xi − x0∥ ≤ ϵ) − Pµ∗(Bϵ(x0))  ��� ≤ C2  2 ϵd ≤ 1  2Pµ∗(Bϵ(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  45  Therefore, for any x0 ∈ Nϵ, there exists i ∈ I1 so that ∥Xi − x0∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' We can then obtain that for any  x ∈ M, there exists i ∈ I1 so that ∥Xi − x0∥ ≤ 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Proof of the first statement is then completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For the  second statement, when n1 is large enough, we have ϵ = C3 ( log n1  n1 )  1  d ≤ r∗  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Let �  Nr∗/4 denote the minimal  r∗/4-covering set of �  i∈I1 B2ϵ(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then | �  Nr∗/4| is controlled by the minimal r∗/8-covering number of  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' For any x0 ∈ | �  Nr∗/4|, there exists an index i ∈ I1 so that B2ϵ(Xi) ∩ Br∗/4(x0) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Let I2 be the set  of such index i for x0 ∈ | �  Nr∗/4|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content=' Then for any x ∈ �  i∈I1 B2ϵ(Xi), there exists i ∈ I2 so that  ∥x − Xi∥ ≤ r∗/4 + r∗/4 + 2ϵ ≤ 5r∗  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Therefore, set M = |I2| and {ak}K  k=1 = {Xi}i∈I2, we have  �  i∈I1  B2ϵ(Xi) ⊂  �  k∈[K]  Br∗(ak),  and  inf  x∈M  �  k∈[K]  �ρk(x) ≥ ((r∗)2 − (5r∗  8 )2)γ > ((r∗)2  2  )γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  Proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
+page_content='  46' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAyT4oBgHgl3EQf-PrC/content/2301.00890v1.pdf'}
diff --git a/cNAzT4oBgHgl3EQfZvyS/content/tmp_files/2301.01357v1.pdf.txt b/cNAzT4oBgHgl3EQfZvyS/content/tmp_files/2301.01357v1.pdf.txt
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+arXiv:2301.01357v1  [math.RA]  3 Jan 2023
+A NOETHERIAN CRITERION FOR SEQUENCES OF MODULES
+WEE LIANG GAN AND KHOA TA
+Abstract. Patzt defined linear categories associated to certain families of
+diagram algebras. We prove that finitely generated modules over these linear
+categories are noetherian.
+Gan and Li proved a noetherian criterion for infinite EI categories in [1]. In this
+note, we adapt their proof to a slightly more general setting and apply it to the
+linear categories studied by Patzt in [2].
+We work over a field k. Denote by N the set of non-negative integers. Let A be
+a sequence {Ai}i∈N where Ai is a k-algebra for each i ∈ N.
+Definition 1. An A-module M is a sequence of pairs {(Mi, φM
+i )}i∈N where, for
+each i ∈ N,
+• Mi is an Ai-module;
+• φM
+i
+: Mi → Mi+1 is a k-linear map.
+Definition 2. Let M and N be A-modules. A morphism f : M → N of A-modules
+is a sequence {fi}i∈N where, for each i ∈ N,
+• fi : Mi → Ni is a homomorphism of Ai-modules;
+• φN
+i ◦ fi = fi+1 ◦ φM
+i .
+Definition 3. Let M be an A-module.
+(i) An A-module N is an A-submodule of M if each Ni is a subset of Mi and
+the sequence of inclusion maps {Ni ֒→ Mi}i∈N is a morphism of A-modules
+(called the inclusion morphism).
+(ii) Let E ⊆ �
+i∈N Mi. The A-submodule of M generated by E is the smallest
+A-submodule N of M such that E ⊆ �
+i∈N Ni.
+Definition 4.
+(i) An A-module is finitely generated if it can be generated by a
+finite set.
+(ii) An A-module is noetherian if each of its A-submodules is finitely generated.
+Notation 5.
+(i) Denote by Mod(k) the category of k-modules.
+(ii) Let M be an A-module. Denote by Sub(M) the category whose objects are
+the A-submodules of M and whose morphisms are the inclusion morphisms.
+For each i ∈ N, denote by F M
+i
+the functor
+F M
+i
+: Sub(M) → Mod(k),
+N �→ HomAi(Mi, Ni).
+(An inclusion morphism N ′ ֒→ N between A-submodules of M induces an
+inclusion map F M
+i (N ′) ֒→ F M
+i (N).)
+Lemma 6. Let M be an A-module, let N be an A-submodule of M, and let N ′ be
+an A-submodule of N. Let i ∈ N and assume that:
+• Mi is a semisimple Ai-module;
+1
+
+2
+WEE LIANG GAN AND KHOA TA
+• N ′
+i ⫋ Ni.
+Then F M
+i (N ′) ⫋ F M
+i (N).
+Proof. Since N ′
+i ⊆ Ni ⊆ Mi, and Mi is a semisimple Ai-module, there exist Ai-
+submodules Q and Q′ of Mi such that
+Mi = Ni ⊕ Q
+and
+Ni = N ′
+i ⊕ Q′.
+Since N ′
+i ̸= Ni, we have Q′ ̸= 0. Now Mi = N ′
+i ⊕ Q′ ⊕ Q. The projection map from
+Mi onto Q′ along N ′
+i ⊕ Q belongs to F M
+i (N) but not F M
+i (N ′).
+□
+Our noetherian criterion is:
+Theorem 7. Let M be an A-module such that each Mi is finite dimensional. Let
+d ∈ N. Assume that for each integer i ⩾ d,
+(7.1) Mi is a semisimple Ai-module;
+(7.2) there exists a morphism of functors νi : F M
+i
+→ F M
+i+1 such that
+νi(M) : F M
+i (M) → F M
+i+1(M)
+is a bijection.
+Then M is noetherian.
+Proof. We follow the proof of [1, Proposition 4.10].
+Assume on the contrary that there exists an A-submodule N of M which is not
+finitely generated. For each ℓ ∈ N, let N (ℓ) be the A-submodule of N generated by
+�ℓ
+i=0 Ni. Observe that if i ∈ {0, . . ., ℓ}, then N (ℓ)
+i
+= Ni.
+Since each Ni is finite dimensional, the A-module N (ℓ) is finitely generated; but
+N is not finitely generated, so N (ℓ) ⫋ N. It follows that there exists an integer
+dℓ > ℓ such that N (ℓ)
+dℓ ⫋ Ndℓ. We have N (ℓ)
+dℓ ⫋ N (dℓ)
+dℓ
+.
+Define a sequence of integers {ℓi}i∈N recursively by ℓ0 = d and
+ℓi+1 = dℓi
+for each i ∈ N.
+We have d = ℓ0 < ℓ1 < ℓ2 < · · · . For each i ∈ N,
+N (ℓi)
+ℓi+1 ⫋ N (ℓi+1)
+ℓi+1
+.
+Hence, by Lemma 6 and (7.1),
+F M
+ℓi+1(N (ℓi)) ⫋ F M
+ℓi+1(N (ℓi+1)).
+In particular,
+(8)
+dim F M
+ℓi+1(N (ℓi)) < dim F M
+ℓi+1(N (ℓi+1)).
+By (7.2), we also have, for each i ∈ N, a commuting diagram
+F M
+d (N (ℓi))
+νd(N (ℓi))
+�
+� �
+�
+F M
+d+1(N (ℓi))
+νd+1(N (ℓi)) �
+� �
+�
+F M
+d+2(N (ℓi))
+νd+2(N (ℓi)) �
+� �
+�
+· · ·
+F M
+d (M)
+νd(M)
+� F M
+d+1(M)
+νd+1(M)
+� F M
+d+2(M)
+νd+2(M)
+� · · ·
+In the above diagram, the maps in the second row are bijective, hence the maps in
+the first row are injective. We deduce that, for each i ∈ N,
+(9)
+dim F M
+ℓi (N (ℓi)) ⩽ dim F M
+ℓi+1(N (ℓi))
+
+A NOETHERIAN CRITERION FOR SEQUENCES OF MODULES
+3
+and
+(10)
+dim F M
+ℓi (N (ℓi)) ⩽ dim F M
+d (M).
+From (8) and (9), we obtain, for each i ∈ N,
+dim F M
+ℓi (N (ℓi)) < dim F M
+ℓi+1(N (ℓi+1)).
+Hence,
+dim F M
+ℓ0 (N (ℓ0)) < dim F M
+ℓ1 (N (ℓ1)) < dim F M
+ℓ2 (N (ℓ2)) < · · · ,
+a contradiction to (10).
+□
+Example 11. Let k = C. Let {Ai}i∈N be one of the three sequences of C-algebras
+{T Li}i∈N, {Bri}i∈N, or {Pi}i∈N associated to a parameter δ ∈ C (see [2] for defini-
+tions); assume that δ is chosen so that each Ai is a semisimple C-algebra. Let A be
+the sequence {Ai}i∈N. Let CA be the linear category defined in [2, Theorem 2.1].
+We claim that every finitely generated CA-module is noetherian. By a standard
+argument1, it suffices to prove that for each m ∈ N, the CA-module M(m) defined
+in [2, Theorem 3.15] is noetherian.
+Let m ∈ N. Define an A-module M as follows: for each i ∈ N, let Mi = M(m)i
+and let φM
+i
+: Mi → Mi+1 be the map induced by the element 1 ⊗ 1 ∈ Ai+1 ⊗A1 C
+(see [2, Lemma 2.8]). Then each Mi is a finite dimensional semisimple Ai-module.
+To apply Theorem 7, it remains to verify that there exists d ∈ N such that for each
+i ⩾ d, condition (7.2) holds.
+For each integer i ⩾ m, define a functor F ′
+i by
+F ′
+i : Sub(M) → Mod(C),
+N �→ C ⊗Ai−m Ni.
+Claim 11.1. For each integer i ⩾ m, the functors F M
+i
+and F ′
+i are isomorphic.
+Proof. We have:
+F M
+i (N) = HomAi(Ai ⊗Ai−m C, Ni)
+∼= HomAi−m(C, Ni).
+Since Ni is a semisimple Ai−m-module, it is a direct sum of isotypic components.
+Let N triv
+i
+be the isotypic component of Ni spanned by the submodules isomorphic
+to the trivial Ai−m-module C. Then we have:
+F M
+i (N) ∼= N triv
+i
+∼= HomC(HomC(N triv
+i
+, C), C)
+∼= HomC(HomAi−m(Ni, C), C)
+∼= HomC(HomC(C ⊗Ai−m Ni, C), C)
+∼= C ⊗Ai−m Ni
+= F ′
+i(N).
+□
+For each integer i ⩾ m and R-submodule N of M, the map φN
+i
+: Ni → Ni+1
+induces a map
+ν′
+i(N) : C ⊗Ai−m Ni → C ⊗Ai+1−m Ni+1.
+1See, for example, [1, §4.2]
+
+4
+WEE LIANG GAN AND KHOA TA
+This defines a functor ν′
+i : F ′
+i → F ′
+i+1. By [2, Theorem 3.15], if i ⩾ 3m, the map
+ν′
+i(M) : F ′
+i(M) → F ′
+i+1(M)
+is a bijection. Hence, by Claim 11.1, if i ⩾ 3m, condition (7.2) holds.
+We conclude by Theorem 7 that the A-module M (and hence the CA-module
+M(m)) is noetherian.
+Remark 12. Similarly to Example 11, we can deduce [1, Theorem 3.7] from The-
+orem 7.
+References
+[1] Wee Liang Gan and Liping Li, Noetherian property of infinite EI categories, New York J.
+Math. 21 (2015), 369–382.
+[2] Peter
+Patzt,
+Representation
+stability
+for
+diagram
+algebras,
+available
+at
+https://arxiv.org/abs/2009.06346.
+Department of Mathematics, University of California, Riverside, CA 92521, USA
+Email address: wlgan@ucr.edu
+Department of Mathematics, University of California, Riverside, CA 92521, USA
+Email address: kta003@ucr.edu
+
diff --git a/cNAzT4oBgHgl3EQfZvyS/content/tmp_files/load_file.txt b/cNAzT4oBgHgl3EQfZvyS/content/tmp_files/load_file.txt
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@@ -0,0 +1,120 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf,len=119
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='01357v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='RA]  3 Jan 2023  A NOETHERIAN CRITERION FOR SEQUENCES OF MODULES  WEE LIANG GAN AND KHOA TA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Patzt defined linear categories associated to certain families of  diagram algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' We prove that finitely generated modules over these linear  categories are noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Gan and Li proved a noetherian criterion for infinite EI categories in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' In this  note, we adapt their proof to a slightly more general setting and apply it to the  linear categories studied by Patzt in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  We work over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Denote by N the set of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let A be  a sequence {Ai}i∈N where Ai is a k-algebra for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' An A-module M is a sequence of pairs {(Mi, φM  i )}i∈N where, for  each i ∈ N,  Mi is an Ai-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  φM  i  : Mi → Mi+1 is a k-linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let M and N be A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' A morphism f : M → N of A-modules  is a sequence {fi}i∈N where, for each i ∈ N,  fi : Mi → Ni is a homomorphism of Ai-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  φN  i ◦ fi = fi+1 ◦ φM  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let M be an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (i) An A-module N is an A-submodule of M if each Ni is a subset of Mi and  the sequence of inclusion maps {Ni ֒→ Mi}i∈N is a morphism of A-modules  (called the inclusion morphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (ii) Let E ⊆ �  i∈N Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' The A-submodule of M generated by E is the smallest  A-submodule N of M such that E ⊆ �  i∈N Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (i) An A-module is finitely generated if it can be generated by a  finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (ii) An A-module is noetherian if each of its A-submodules is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (i) Denote by Mod(k) the category of k-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (ii) Let M be an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Denote by Sub(M) the category whose objects are  the A-submodules of M and whose morphisms are the inclusion morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  For each i ∈ N, denote by F M  i  the functor  F M  i  : Sub(M) → Mod(k),  N �→ HomAi(Mi, Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (An inclusion morphism N ′ ֒→ N between A-submodules of M induces an  inclusion map F M  i (N ′) ֒→ F M  i (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=')  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let M be an A-module, let N be an A-submodule of M, and let N ′ be  an A-submodule of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let i ∈ N and assume that:  Mi is a semisimple Ai-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  1  2  WEE LIANG GAN AND KHOA TA  N ′  i ⫋ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Then F M  i (N ′) ⫋ F M  i (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Since N ′  i ⊆ Ni ⊆ Mi, and Mi is a semisimple Ai-module, there exist Ai-  submodules Q and Q′ of Mi such that  Mi = Ni ⊕ Q  and  Ni = N ′  i ⊕ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Since N ′  i ̸= Ni, we have Q′ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Now Mi = N ′  i ⊕ Q′ ⊕ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' The projection map from  Mi onto Q′ along N ′  i ⊕ Q belongs to F M  i (N) but not F M  i (N ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  □  Our noetherian criterion is:  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let M be an A-module such that each Mi is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let  d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Assume that for each integer i ⩾ d,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='1) Mi is a semisimple Ai-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='2) there exists a morphism of functors νi : F M  i  → F M  i+1 such that  νi(M) : F M  i (M) → F M  i+1(M)  is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Then M is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' We follow the proof of [1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Assume on the contrary that there exists an A-submodule N of M which is not  finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' For each ℓ ∈ N, let N (ℓ) be the A-submodule of N generated by  �ℓ  i=0 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Observe that if i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=', ℓ}, then N (ℓ)  i  = Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Since each Ni is finite dimensional, the A-module N (ℓ) is finitely generated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' but  N is not finitely generated, so N (ℓ) ⫋ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' It follows that there exists an integer  dℓ > ℓ such that N (ℓ)  dℓ ⫋ Ndℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' We have N (ℓ)  dℓ ⫋ N (dℓ)  dℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Define a sequence of integers {ℓi}i∈N recursively by ℓ0 = d and  ℓi+1 = dℓi  for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  We have d = ℓ0 < ℓ1 < ℓ2 < · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' For each i ∈ N,  N (ℓi)  ℓi+1 ⫋ N (ℓi+1)  ℓi+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Hence, by Lemma 6 and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='1),  F M  ℓi+1(N (ℓi)) ⫋ F M  ℓi+1(N (ℓi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  In particular,  (8)  dim F M  ℓi+1(N (ℓi)) < dim F M  ℓi+1(N (ℓi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='2), we also have, for each i ∈ N, a commuting diagram  F M  d (N (ℓi))  νd(N (ℓi))  �  � �  �  F M  d+1(N (ℓi))  νd+1(N (ℓi)) �  � �  �  F M  d+2(N (ℓi))  νd+2(N (ℓi)) �  � �  �  · ·  F M  d (M)  νd(M)  � F M  d+1(M)  νd+1(M)  � F M  d+2(M)  νd+2(M)  � · · ·  In the above diagram, the maps in the second row are bijective, hence the maps in  the first row are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' We deduce that, for each i ∈ N,  (9)  dim F M  ℓi (N (ℓi)) ⩽ dim F M  ℓi+1(N (ℓi))  A NOETHERIAN CRITERION FOR SEQUENCES OF MODULES  3  and  (10)  dim F M  ℓi (N (ℓi)) ⩽ dim F M  d (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  From (8) and (9), we obtain, for each i ∈ N,  dim F M  ℓi (N (ℓi)) < dim F M  ℓi+1(N (ℓi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Hence,  dim F M  ℓ0 (N (ℓ0)) < dim F M  ℓ1 (N (ℓ1)) < dim F M  ℓ2 (N (ℓ2)) < · · · ,  a contradiction to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  □  Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let {Ai}i∈N be one of the three sequences of C-algebras  {T Li}i∈N, {Bri}i∈N, or {Pi}i∈N associated to a parameter δ ∈ C (see [2] for defini-  tions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' assume that δ is chosen so that each Ai is a semisimple C-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let A be  the sequence {Ai}i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Let CA be the linear category defined in [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  We claim that every finitely generated CA-module is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' By a standard  argument1, it suffices to prove that for each m ∈ N, the CA-module M(m) defined  in [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='15] is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Let m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Define an A-module M as follows: for each i ∈ N, let Mi = M(m)i  and let φM  i  : Mi → Mi+1 be the map induced by the element 1 ⊗ 1 ∈ Ai+1 ⊗A1 C  (see [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Then each Mi is a finite dimensional semisimple Ai-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  To apply Theorem 7, it remains to verify that there exists d ∈ N such that for each  i ⩾ d, condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  For each integer i ⩾ m, define a functor F ′  i by  F ′  i : Sub(M) → Mod(C),  N �→ C ⊗Ai−m Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' For each integer i ⩾ m, the functors F M  i  and F ′  i are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' We have:  F M  i (N) = HomAi(Ai ⊗Ai−m C, Ni)  ∼= HomAi−m(C, Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Since Ni is a semisimple Ai−m-module, it is a direct sum of isotypic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Let N triv  i  be the isotypic component of Ni spanned by the submodules isomorphic  to the trivial Ai−m-module C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Then we have:  F M  i (N) ∼= N triv  i  ∼= HomC(HomC(N triv  i  , C), C)  ∼= HomC(HomAi−m(Ni, C), C)  ∼= HomC(HomC(C ⊗Ai−m Ni, C), C)  ∼= C ⊗Ai−m Ni  = F ′  i(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  □  For each integer i ⩾ m and R-submodule N of M, the map φN  i  : Ni → Ni+1  induces a map  ν′  i(N) : C ⊗Ai−m Ni → C ⊗Ai+1−m Ni+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  1See, for example, [1, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='2]  4  WEE LIANG GAN AND KHOA TA  This defines a functor ν′  i : F ′  i → F ′  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' By [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='15], if i ⩾ 3m, the map  ν′  i(M) : F ′  i(M) → F ′  i+1(M)  is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Hence, by Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='1, if i ⩾ 3m, condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  We conclude by Theorem 7 that the A-module M (and hence the CA-module  M(m)) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' Similarly to Example 11, we can deduce [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='7] from The-  orem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  References  [1] Wee Liang Gan and Liping Li, Noetherian property of infinite EI categories, New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content=' 21 (2015), 369–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  [2] Peter  Patzt,  Representation  stability  for  diagram  algebras,  available  at  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='org/abs/2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='06346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='  Department of Mathematics, University of California, Riverside, CA 92521, USA  Email address: wlgan@ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='edu  Department of Mathematics, University of California, Riverside, CA 92521, USA  Email address: kta003@ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfZvyS/content/2301.01357v1.pdf'}
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+Multi armed bandits and quantum channel oracles
+Simon Buchholz, Jonas M. Kübler, Bernhard Schölkopf
+23rd January 2023
+Abstract
+Multi armed bandits are one of the theoretical pillars of reinforcement learning.
+Recently,
+the investigation of quantum algorithms for multi armed bandit problems was started, and it was
+found that a quadratic speed-up is possible when the arms and the randomness of the rewards
+of the arms can be queried in superposition. Here we introduce further bandit models where we
+only have limited access to the randomness of the rewards, but we can still query the arms in
+superposition. We show that this impedes any speed-up of quantum algorithms.
+1
+Introduction
+Quantum computing is a model of computation that is based on quantum properties of matter. By
+using superposition and entanglement, it offers potentially large speed-ups when compared to classical
+algorithms. For some problems exponential speed-ups haven been shown under the assumption of
+widely believed hardness results for classical computing, the two most prominent examples being
+Shor’s algorithm for factoring integers [32] and the HHL algorithm for sampling from the solution of
+sparse linear equations [17]. On the other hand, it was shown that for many problems only polynomial
+speed-ups are possible, in particular Grover’s algorithm [16] for the unstructured search problem only
+offers a quadratic speed-up and no greater improvement is possible.
+Recently, quantum machine learning has emerged as one potential area of application for quantum
+computers [7]. It was suggested to use quantum computers for linear algebra subroutines but also
+complete implementations of well-known classical algorithms for supervised learning were designed,
+e.g., quantum support vector machines [29], quantum principal component analysis [23], and recom-
+mender systems [20]. There has also been some work on unsupervised learning and reinforcement
+learning [2, 19, 12]. For a recent review, we refer to [11].
+Reinforcement learning is a machine learning paradigm that is concerned with learning actions that
+maximize a reward. Arguable, the simplest model capturing the essentials of reinforcement learning
+is the multi-armed bandit problem. There one tries to identify the arm with the highest mean reward
+from several alternatives. Classical algorithms for the multi armed bandit problem have been studied
+for a long time and almost optimal results were derived [8].
+The problem of best arm identification using quantum computers was considered recently in [34].
+They show that a quadratic speed-up compared to the classical algorithm is possible and optimal in
+their setting. Their implementation of the bandit arms is, however, not directly comparable to the
+classical setting because the bandit arms are implemented through an oracle that acts deterministically
+and the randomness originates from the difficulty to discern quantum states.
+In this work, we will discuss when this multi armed bandit model is applicable and we will discuss
+further models of multi armed bandits that are more suitable in different settings. Those models will
+involve additional randomness, i.e., the oracle implementing the bandit will act as a quantum channel
+instead of a unitary map. This provides a link to the channel discrimination problem, however, there
+Max Planck Institute for Intelligent Systems, Tübingen
+{sbuchholz, jmkuebler, bs}@tue.mpg.de
+1
+arXiv:2301.08544v1  [quant-ph]  20 Jan 2023
+
+the focus has been on very different channels that are either generic or more related to the transmission
+of information [26].
+Here we show that the additional randomness prevents any polynomial speed-up compared to
+classical algorithms even though the bandits can be queried in superposition. It was shown in [30]
+that no speed-up for unstructured search is possible if the oracle has a fixed probability of error. Our
+results can be seen as an extension to a substantially more general setting. Thus, this work highlights
+that quantum speed-ups can be impeded by (small) amounts of classical randomness present in the
+algorithm, thus also underlining again that having parts of the computation routine a that are not
+error corrected pose challenges. From a technical side, we connect classical methods from quantum
+information theory with coupling techniques from probability theory.
+The rest of this paper is structured as follows. In the next section, we introduce different settings of
+quantum bandits and give an overview of their query complexities. Then we give a quantum inspired
+proof of the classical result in Section 3. Afterward, in Section 4 we consider quantum channel oracles
+and in particular the setting of the faulty Grover oracle. In Section 5 we then consider non-adaptive
+algorithms for the bandit problem. Finally, in Section 6 we prove our main result Theorem 6. The
+proof combines the techniques used in the three sections before.
+2
+Main results and setting
+We now discuss the setting for our main results. In this section, we review the relevant background
+on (classical) bandits and our main results for quantum bandits.
+2.1
+Classical bandits
+We now discuss the well-known classical multi-armed bandit problem, give the key results and the
+underlying heuristics, to give the necessary background to readers not familiar with the setting. In
+the multi-armed bandit problem, an agent can choose an arm i ∈ {1, . . . , N} in every turn and
+receives a probabilistic reward νi depending on the chosen arm. Typically, the goal is to maximize
+the reward or, equivalently, minimize the regret which denotes the difference to the reward obtained
+when always choosing the most favorable arm. This is one of the simplest models for reinforcement
+learning featuring the exploration-exploitation tradeoff. A slightly simpler problem is the best-arm
+identification problem in the fixed confidence setting. Here the goal is to identify the arm with the
+highest expected reward with a given probability 1 − δ for some δ > 0 with the fewest number of
+rounds possible. We consider mean reward vectors p = (p1, . . . , pN) ∈ RN, indicating that arm i has
+average reward pi. We usually assume that the rewards are ordered, i.e., p1 ≥ p2 ≥ . . . ≥ pN and that
+the rewards are Bernoulli Ber(pi) distributed. In other words, when pulling arm i we receive reward 1
+with probability pi and reward 0 with probability 1 − pi. We will use the shorthand ∆i = pi − p1 for
+the difference in mean rewards between the best arm and arm i. We define the important quantity
+H(p) =
+�
+i>1
+1
+(p1 − pi)2 =
+�
+i>1
+∆−2
+i .
+(1)
+It can be shown that H(p) governs the optimal time to identify the best arm (up to logarithmic terms)
+in a fixed confidence setting. Moreover, this is optimal when pi ∈ [η, 1 − η] for all i and some η > 0.
+To make this concrete, we state this as a well-known theorem.
+Theorem 1 (Thm. 2 in [13], Thm. 5 in [24]). Consider an algorithm that identifies the best arm of a
+multi armed bandit with probability at least 1 − δ for Bernoulli distributed rewards with reward vector
+p ∈ [0, 1]N such that pi ∈ [η, 1 − η]. Then this algorithm requires O(H(p)) rounds in the worst case.
+On the other hand, there exists such an algorithm requiring ˜O(H(p)) steps.
+Let us emphasize that identification of the best arm is not simpler if we know the rewards up to
+a permutation, in fact, the following result holds.
+2
+
+Theorem 2 (Thm. 4 in [5]). Let p ∈ [η, 1−η]N be a reward vector. Then any algorithm that identifies
+the best arm for Bernoulli distributed rewards with means p′ where p′ is any permutation of p requires
+at least O(H(p)) rounds and such an algorithm exists.
+Let us add several remarks to these results. There is a long list of results improving upon the two
+results above by deriving also bounds on the logarithmic correction and analyzing various algorithms,
+see e.g., [14, 18, 10]. The assumption that the variables are Bernoulli distributed is for convenience,
+many generalizations are possible. One advantage is that Bernoulli distributions readily generalize to
+the quantum setting.
+Let us explain for the readers not so familiar with the literature on bandit problems the intuition
+underlying the results mentioned above.
+A Ber(p) variable has variance p(1 − p) which is lower
+bounded by η(1 − η) for all arms by our assumptions. Suppose we pulled arm i for ti times. Then the
+empirical mean ˆpi will be approximately distributed according to a Gaussian variable (by the central
+limit theorem) with mean pi and standard deviation
+�
+tipi(1 − pi)/ti = O(t
+− 1
+2
+i
+), i.e., in distribution
+ˆpi ≈ pi +
+�
+pi(1 − pi)
+√ti
+N
+(2)
+where N denotes a standard normal variable. If we want to conclude that pi < p1 with high probability
+we need P( ˆpi > p1) ≪ 1. This is the case iff
+t
+− 1
+2
+i
+≲ p1 − pi = ∆i.
+(3)
+We conclude that ti ≈ (p1 − pi)−2 = ∆−2
+i
+pulls on lever i are necessary to exclude the possibility that
+arm i has the highest reward with high probability. Applying this reasoning to all arms j suggests
+the lower bound H(p) for the query complexity.
+Let us emphasize that this argument crucially uses that all rewards are away from 1 and 0.
+Otherwise, the variance term pi(1 − pi) cannot necessarily assumed to be a fixed constant in the
+analysis. To illustrate this, we consider the particular case that p1 = p > 0 and pi = 0 for i > 1. Then
+H(p) = N/p2 but only O(N/p) pulls are required to identify the best arm. To see this, note that it
+takes with high probability O(p−1) pulls to get a single success on and arm with Ber(p) distributed
+rewards. This is the setting considered in [30] and we will come back to it in Section 4.
+2.2
+Quantum bandits
+The investigation of quantum bandits was started recently [9, 34] and so far focused on the best arm
+identification problem. In this section, we want to review the results and definitions in the literature
+and in particular highlight that different assumptions on the available oracles are reasonable in different
+settings. We assume that there are n arms with Bernoulli distributed rewards with mean Ber(pi). To
+query the arms we consider a Hilbert-space HA for the arms with dimension |Ha| = N. Moreover,
+we assume that the internal randomness of the reward is captured through an additional Hilbert-
+space HP and the output indicating the reward for a certain arm and state of internal randomness is
+collected in a two-dimensional Hilbert space HR. We then consider rewards that are given through an
+oracle O acting on a state |i⟩ |ω⟩ |c⟩ inHA ⊗ HP ⊗ HR, i.e., arm i, internal randomness ω an element
+from a fixed basis of HP , and initial reward state c by
+O |i⟩ |ω⟩ |c⟩ = |i⟩ |ω⟩ |c + ri(ω)⟩
+(4)
+where ri(ω) ∈ {0, 1} denotes the reward for arm i with internal randomness ω and addition is modulo
+2 (a similar definition was recently considered in [33]). Moreover, we assume that when averaging over
+this basis of HP we obtain
+|HP |−1 �
+ω
+ri(ω) = pi,
+(5)
+3
+
+i.e., the average reward of arm i is pi. As before, we collect the mean rewards pi in a vector p ∈ [0, 1]N
+with pi = pi. Then it is sometimes convenient to clarify the mean rewards of the oracle by writing
+Op. Note that this does not determine the oracle as the dependence on ω is not fixed.
+We now consider different settings that are characterized by the degree of control that we have
+over the space HP determining the internal randomness of the bandits. We consider the following
+three settings.
+1.
+We have full control over the space HP , i.e., we can apply arbitrary unitary operators to the
+system HA ⊗ HP ⊗ HR and a potential work space.
+2.
+We can prepare any pure basis state ω ∈ HP but have no further access to HP except through
+the oracle O.
+3.
+The state of HP will always be a pure state ω which is unobserved and selected uniformly at
+random from the fixed basis and resampled after each invocation of O.
+We now give a different (almost equivalent) characterisation of the second and third setting described
+above which eliminates the Hilbert space HP modelling the internal randomness. For this it is con-
+venient to define for X ∈ {0, 1}N the oracle OX acting on HA ⊗ HR by
+OX |i⟩ |c⟩ → |i⟩
+��c + Xi�
+.
+(6)
+Let us assume that Xt is a sequence of i.i.d. random variables where Xt
+i is Ber(pi) distributed. Then
+we can phrase the second scenario above almost equivalently as having access to the sequence of oracles
+OXt |i⟩ |c⟩ = |i⟩
+��c + Xi
+t
+�
+,
+for t = 1, 2, . . ..
+(7)
+The third setting corresponds to having access to an oracle channel that acts by
+E(ρ) =
+�
+X∈{0,1}n
+P(X) OXρ(OX)†
+where P(X) =
+�
+i
+pXi
+i (1 − pi)1−Xi.
+(8)
+The minor difference between the description above is that there we considered a finite dimensional
+space HP for the internal randomness, the settings become equivalent as the dimension of HP grows.
+Note that the second and the third setting are equivalent if we are allowed to use each of the oracles
+OXt only once. Multiple invocations can be useful to uncompute parts of the computation, avoiding
+that the state of the system decoheres. As we will discuss next the appropriate model depends on the
+concrete situation at hand. They cannot directly be compared to the setting of classical bandits.
+2.3
+Models
+We now motivate the three settings described above in a bit more detail and discuss our main results.
+Let us emphasize that quantum bandits can only be useful when the reward is given by the observable
+of a quantum system or the evaluation of a computation on a quantum device as the acquisition of
+the data is commonly seen as the expensive part. It does not seem plausible that we collect classical
+data store them in, e.g., a QRAM to query them in superposition to identify the best arm as this
+will always be more expensive than classically evaluating the mean of the collected rewards. Thus, we
+will motivate all three settings from a quantum perspective. Note that the classical setting roughly
+corresponds to the case where only queries of the form |i⟩ |ω⟩ without superposition are allowed, i.e.,
+we can query neither a single arm for a single reward. Let us also mention that minimization of regret
+(at least when defined in the standard way) only makes sense in the third setting where we have no
+access to the sampling randomness. Otherwise, we can just resample a fortunate realization of the
+reward to incur no regret.
+4
+
+Oracle
+Lower bound
+Upper bound
+Classical
+O(H(p))
+˜O(H(p))
+ERM (eq. (4))
+O(
+�
+H(p)) (Thm. 4)
+˜O(
+�
+H(p)) (Thm. 1 in [34])
+Reusable (eq. (7))
+?
+˜O(
+��
+i ∆−4
+i ) (Thm. 5)
+One-time (eq. (8))
+O(H(p)) (Thm. 6)
+˜O(H(p))
+Table 1: Overview of query complexity bounds. The upper bound for the one-time oracle and the
+reusable oracle follow from the classical result. We conjecture (see Conjecture 1) that the upper bound
+for the reusable oracle are optimal.
+2.3.1
+Empirical risk minimisation
+As already explained in [34], one setting where such an oracle could arise is empirical risk minimisation.
+To make this concrete, assume we have a dataset (xj, yj) ∈ X ×{1, . . . , K} and a finite set of candidate
+functions fi. We now want to find the index i0 such that
+i0 = arg min
+i
+�
+j
+1fi(xj)̸=yj = arg max
+i
+�
+j
+1fi(xj)=yj,
+(9)
+i.e., for simplicity we consider 0-1 loss in a classification problem or equivalently, we maximize the
+accuracy over the functions fi. If we can access xj and evaluate fi this provides us with an oracle
+acting by
+|i⟩ |ω⟩ |0⟩ → |i⟩ |ω⟩ |ri(ω)⟩ ,
+(10)
+where the reward for ω = (x, y) is given by ri(ω) = 1y=fi(x). Now the problem of best arm identi-
+fication with respect to this oracle is equivalent to empirical risk minimization. Moreover, this oracle
+is exactly of the form introduced in (4). When considering this setting it is a reasonable assumption
+that the dataset can be accessed in arbitrary superposition, i.e., is stored in our computing device,
+and we can also evaluate functions fi and thus losses in superposition. The problem of empirical risk
+minimization and the relation to bandit problems was considered before in [34]. where they consider
+an oracle that acts as
+|i⟩ |0⟩ → |i⟩ (√pi |1⟩ +
+�
+1 − pi |0⟩).
+(11)
+The relation to our setting is that when applying our oracle to a uniform superposition over the ω
+register we obtain
+�
+ω
+|i⟩ |ω⟩ |0⟩ → √pi |i⟩ |v⟩ |1⟩ +
+�
+1 − pi |i⟩ |u⟩ |0⟩
+(12)
+where u and v are suitable junk states which can be neglected as argued in [34] (at least when restricting
+attention to query complexity). Note that this superposition eliminates the statistical randomness of
+the bandits. Then the following result holds.
+Theorem 3 (Theorem 1 in [34]). There is a quantum algorithm that identifies the best arm of a
+quantum oracle as in (11) for any reward vector p ∈ [η, 1 − η]n with probability 1 − δ with query
+complexity
+T = ˜O(
+�
+H(p)).
+(13)
+Moreover, this bound is optimal up to logarithmic terms.
+5
+
+Thus we obtain a quadratic speed-up compared to the classical setting. They prove the lower
+bound only for the oracle (11). For completeness we prove the same lower bound when the more
+general oracle (4) is available, i.e., the ability to query arbitrary superpositions of the data points does
+not allow any speedups compared to always considering the uniform superposition.
+Theorem 4 (Informal version). Any algorithm that identifies the best arm for any reward vector
+p ∈ [η, 1 − η]n with confidence 1 − δ given access to an oracle as in (4) requires at least O(
+�
+H(p))
+calls to the oracle.
+The proof and a formal statement of this result are in Appendix D. Note that while it is intuitively
+clear that it is optimal to query over the uniform mixture of ω as in (12) a rigorous proof requires
+a careful tracking of the classical randomness of the oracle and its interaction with the quantum
+algorithm.
+2.3.2
+Reusable oracles
+We now consider oracles as in (7), i.e., we can query arms in superposition but we can only retrieve
+the reward for one chosen realization of the randomness. A similar type of oracle was considered in
+[28]. They show that hedging algorithms can be implemented using these oracles which have runtime
+√
+N in the arms, thus offering a quadratic speedup compared to the classical algorithm. In this paper
+we evaluate the best arm identification problem. We only give partial results for this case.
+A motivation to consider this is the setting of quantum sensing. Quantum sensing refers to the
+general use of quantum phenomena to measure physical quantities. In our case the Hilbert space HP
+is the state space of a quantum system whose properties we want to probe through the oracle O.
+We assume that we can prepare states in HP according to some distribution (which for simplicity we
+assumed to be the uniform distribution over some basis). However, HP is not part of our computing
+device so we cannot query, e.g., in superposition. We have the following result.
+Theorem 5. For confidence δ ∈ (0, 1) there is a quantum algorithm that outputs the best arm with
+probability 1 − δ using
+T ≤ ˜O
+�
+�
+��
+i>1
+∆−4
+i
+�
+�
+(14)
+queries to an oracle as in (7) where ∼ indicates terms that are polynomially in ln(N/(δ∆2)).
+A sketch of the proof of this result is in Appendix F. It relies on a small modification of the
+algorithm in [34]. We conjecture that this bound is also optimal.
+Conjecture 1. Any quantum algorithm that identifies the best arm for any reward vector p ∈ [η, 1−η]n
+for some η > 0 with probability 1 − δ for some δ < 1
+2 requires at least
+T ≥ c
+��
+i>1
+∆−4
+i
+(15)
+calls to an oracle as in (7).
+The main difficulty to prove a lower bound is that the applied oracles can be reused so that the
+fidelity between the quantum states obtained for different mean rewards p and p′ is not necessarily
+monotonically decreasing. This makes it hard to extend our other proofs that rely on the loss of
+fidelity in a single step to this setting. In general, standard techniques to obtain lower bounds and
+the techniques used in this work do not appear to be sufficient to address this problem.
+6
+
+2.3.3
+One time oracles
+Finally we consider the oracle defined in (8). The motivation to consider this type of oracle is similar
+as described in Section 2.3.2. The difference is that we have no control over the investigated quantum
+system with state space HP . Examples of such a setting are that the state on the subsystem HP
+follows some Hamiltonian evolution with a fixed Hamiltonian or it is distributed according to some
+Gibbs measure and we assume that the system equilibrates after each oracle invocation.
+Formally this setting is similar to [30], where they essentially consider the case pi = 0 for all i ̸= i0
+and pi0 > 0. This can be viewed as Grover search with a faulty oracle with failure probability 1 − pi0.
+As we focus on this case we will review their results in Section 4. Here we just give the our extension
+of their result: For oracles as in (8) no speedup is possible for the best arm identification.
+Theorem 6. Any algorithm that identifies the best arm for any reward vector p ∈ [η, 1 − η]N for
+some η > 0 with probability 1 − δ for some δ < 1/2 based on calls to an oracle as in (8) requires at
+least
+T ≥ c(δ, η)H(p)
+(16)
+calls to the oracle.
+Again, this result is not a consequence of the generality of reward vectors p allowed but even when
+the set of mean rewards is known no better result is possible. The precise statement can be found in
+Theorem 10 below which is slightly stronger than the result above. Note that the assumption that
+the rewards are Bernoulli distributed might be unrealistic in this setting. However, our main result
+shows that even if the rewards are Bernoulli distributed no improvements over classical algorithms is
+possible in terms of query complexity.
+2.4
+Comparison of settings
+Let us further discuss these results to give a better intuition of the result. We remark that access to
+the oracle (4) is strictly more powerful than access to oracles as in (7) which in turn is more powerful
+than access to the channel oracle (8). This is reflected in the following chain of inequalities for the
+query complexities
+��
+i>1
+∆−2
+i
+≤
+��
+i>1
+∆−4
+i
+≤
+�
+i>1
+∆−2
+i .
+(17)
+On the other hand we have the following reverse bound for the last two terms
+��
+i>1
+∆−4
+i
+≥
+1
+√
+N
+�
+i>1
+∆−2
+i .
+(18)
+So the speed-up is bounded by
+√
+N but can be less depending on p. To illustrate this further we
+consider as a prototypic example a reward vector p with p1 > p2 = . . . = pn with p1 − p2 = ε Then
+the query complexities are
+Term ≈
+�
+N
+ε2 =
+√
+N
+ε
+< Treusable ≈
+√
+N
+ε2
+< Tclassical ≈ Tone−time ≈ N
+ε2 .
+(19)
+For this setting the two difficulties can be well separated: The reward of the arms needs to be estimated
+(statistical complexity) and the correct arm needs to be searched. Classically the statistical complexity
+is ε−2 but it can be reduced to ε−1 in the empirical risk minimization setting (similar to quantum
+metrology [15]).
+Not having access to a superposition of samples ω prevents this speed-up.
+The
+7
+
+complexity of the search of the best arm is
+√
+N in a quantum setting compared to the complexity of
+N in the classical setting. Note that it is possible to obtain this speed-up using the reusable oracles but
+not when having only access to the one time oracle. Note that while some assumptions, e.g., Bernoulli
+distributed rewards might be very simplistic. However, even under these favorable assumptions for
+the quantum setting we show that quantum algorithms offer no improvement in query complexity
+when HP is not part of the computing device.
+3
+Complexity bounds for classical bandits with a quantum per-
+spective
+Before addressing the case of quantum bandits we revisit the classical bandit problem and give a
+different proof for the required number of rounds in the fixed confidence setting. This section serves
+two purposes. It shows that the fidelity of probability distributions is a useful distance measure to
+analyze classical bandit problems which offers the additional advantage that it readily generalizes to
+quantum states. Moreover, this section is a preparation that introduces some notation for the more
+involved proof in the quantum setting. In fact the proof for the quantum result is essentially based on
+a combination of the proof given in this section with the results from the following section on random
+oracles.
+Let us fix some notation. It is convenient to slightly deviate from the notation used before and
+consider a slightly more restricted version of the bandit problem, for which we show a lower bound.
+We consider probability vectors p = (p0, . . . , pn) ∈ [0, 1]N+1 with p0 > p1 > p2 ≥ . . . ≥ pn. We denote
+∆i = p0 − pi. Note that this is sightly different from the earlier definition. We consider the vectors
+pi ∈ [0, 1]N given by pi
+j = pj for i ̸= j and pi
+i = p0 and p0 given by p0
+i = pi. Note that for every i ≥ 1
+the vectors p0 and pi differ only in the entry i. Moreover, we assume that p0 − p1 = p1 − p2 = ∆1.
+Note that then for j ≥ 1
+H(pj) =
+�
+i̸=j
+(p0 − pi)−2 ≤
+�
+i≥1
+∆−2
+i
+= (p0 − p1)−2 +
+�
+i>1
+(p0 − pi)−2
+≤ 2
+�
+i>1
+(p1 − pi)2 = 2H(p0)
+(20)
+where we used p0 − p1 = p1 − p2 in the last step. Similarly, we obtain
+�
+i≥1
+∆−2
+i
+≥ H(pj) =
+�
+i̸=j
+(p0 − pi)−2 =
+�
+i̸=j
+(p1 − pi + ∆1)−2
+≥
+�
+i>1
+(p1 − pi + ∆1)−2 ≥ 1
+4
+�
+i>1
+(p1 − pi)2 = 1
+4H(p0).
+(21)
+Here we used in the third step that p1−p1+∆1 ≤ p1−pj+∆1 for any j ≥ 1 and p1−pi+∆1 ≤ 2(p1−pi)
+in the following inequality. This shows that H(pj) is up to constants given by �
+i ∆−2
+i
+for all j. Then
+the following result implies Theorem 1.
+Theorem 7. Let δ < 1/2.
+Assume that pj are as above with pi ∈ [η, 1 − η] for some η > 0.
+Any classical algorithm that identifies the best arm when it is known that the reward vector is in
+{p0, . . . , pn} with probability at least 1 − δ requires at least
+T ≥ cH(p1) = c
+n
+�
+j=2
+∆−2
+j
+(22)
+rounds where c = c(δ, η) > 0.
+8
+
+Since this result is well known, the proof serves merely pedagogical purposes to illustrate our
+approach to the quantum setting. Therefore, we do not give the most concise presentation but instead
+highlight the main difference to the standard proofs of this result.
+Proof. Suppose we are given an algorithm A (we do not need to consider randomized algorithms as
+we focus on the worst case). In each step t it picks an arm at depending on all earlier outcomes and
+receives a (binary) reward rt ∈ {0, 1}. We introduce the variables xt = (at, rt) ∈ [N]×{0, 1} encoding
+the path of the algorithm. We denote by zt = (x1, . . . , xt) the entire history of the exploration. Note
+that at only depends on the outcomes of the previous rounds and therefore is a deterministic function
+of zt−1, i.e., at = at(zt−1). When the rewards follow the distribution pj this induces a distribution on
+xt and zt and we denote the corresponding random variables by Zj
+t and Xj
+t and the distribution by
+Pj. The main idea of the proof is to bound the fidelity of random variables Zj
+t and Z0
+t for each t from
+below. On the other hand, we can upper bound the fidelity because the algorithm can identify the best
+arm for the reward distributions pj and p0 and those arms are different for j > 1. Together, those
+two bounds will imply the claim. The proof will rely on the fidelity
+√
+F of two discrete probability
+distributions px and qx which is defined by
+√
+F(p, q) =
+�
+x
+√pxqx.
+(23)
+We refer to Appendix A for a brief summary of distance measures, here we only need the definition
+and the bound by the total variation distance (defined by the first equality)
+dTV(p, q) = 1
+2
+�
+x
+|px − qx| ≤
+�
+1 − F(p, q).
+(24)
+We now discuss the simple upper bound on the fidelity after the final round T coming from the
+assumption that the algorithm succeeds with high probability. Let Mj be the disjoint sets of outcomes
+zT such that arm j is selected by the algorithm. By assumption Pj(Mj) > 1 − δ and P0(M1) > 1 − δ
+and therefore P0(Mj) < δ. This implies for j > 1 (see Appendix A for a brief summary of distance
+measures)
+1 − 2δ < Pj(Mj) − P0(Mj) ≤ dTV(Zj
+T , Z0
+T ) ≤
+�
+1 − F(Zj
+T , Z0
+T ).
+(25)
+We conclude that
+√
+F(Zj
+T , Z0
+T ) ≤ 2
+�
+δ(1 − δ).
+(26)
+We now bound the fidelity from below. The first bound will not be sufficient to conclude but it is
+nevertheless instructive to understand the difficulties of the quantum setting and the relation to earlier
+proofs. We will later refine the following estimates. We bound
+√
+F(Zj
+t , Z0
+t ) =
+�
+zt
+�
+Pj(zt)P0(zt)
+=
+1
+�
+rt=0
+�
+zt−1
+�
+Pj(rt|at(zt−1))Pj(zt−1)P0(rt|at(zt−1)P0(zt−1))
+=
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)
+1
+�
+rt=0
+�
+Pj(rt|at(zt−1))P0(rt|at(zt−1)).
+(27)
+We now consider two cases at(zt−1) = j and at(zt−1) ̸= j. In the latter case
+Pj(rt|at(zt−1)) = P0(rt|at(zt−1)) = pj
+(28)
+9
+
+and thus for at(zt−1) ̸= j
+1
+�
+rt=0
+�
+Pj(rt|at(zt−1))P0(rt|at(zt−1)) =
+1
+�
+rt=0
+P0(rt|at(zt−1)) = 1.
+(29)
+For at(zt−1) = j we use the simple bound (162) from Lemma 4 in Appendix B. bounding for p, q ∈
+[c, 1 − c]
+√
+F(Ber(p), Ber(q)) ≥ 1 − |p − q|2
+4c(1 − c).
+(30)
+This implies
+1
+�
+rt=0
+�
+Pj(rt|at = j)P0(rt|at = j) ≥ 1 − |pj − p0|2
+4η(1 − η).
+(31)
+We can now use the last two displays to continue to estimate (27)
+√
+F(Zj
+t , Z0
+t ) ≥
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1) −
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)1at(zt−1)=j
+∆2
+j
+4η(1 − η)
+=
+√
+F(Zj
+t−1, Z0
+t−1) −
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)1at(zt−1)=j
+∆2
+j
+4η(1 − η).
+(32)
+Using this iteratively we obtain (using
+√
+F(Zj
+0, Z0
+0) = 1) the bound
+2
+�
+δ(1 − δ) ≥
+√
+F(Zj
+0, Z0
+0) ≥ 1 −
+T
+�
+t=1
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)1at(zt−1)=j
+∆2
+j
+4η(1 − η).
+(33)
+Now the standard way to proceed from here is to show that with high probability Pj(zt−1) and P0(zt−1)
+are similar using tail bounds for random variables (note that we already control the fidelity). Suppose
+that up to small errors we could replace Pj by P0. Then we could conclude from (33) that
+∆2
+j
+4η(1 − η)
+T
+�
+t=1
+�
+zt−1
+P0(zt−1)1at(zt−1)=j ≥ 1 − 2
+�
+δ(1 − δ) > 0.
+(34)
+Dividing by ∆2
+j and summing over j this would imply that there is a constant c > 0 such that
+�
+j
+∆−2
+j
+≤ c
+T
+�
+t=1
+�
+zt−1
+P0(zt−1)
+�
+j
+1at(zt−1)=j = cT.
+(35)
+This approach cannot be extended to the quantum setting. The reason is that the tail bounds rely
+on the fact that when we use a total of O(H) queries then we cannot query all arms more often than
+∆−2
+j . On the other hand, in the quantum setting we query in superposition so that we cannot simply
+count the number of pulls on an arm. We now show how the tail bounds can be avoided in a way that
+can similarly be generalized to the quantum setting. Let us denote by nj(zt) = |{s : as(zt) = j} the
+number of times we queried the j-th arm. We introduce the decay factor
+dj(zt) =
+�
+1 −
+∆2
+j
+4η(1 − η)
+�nj(zt)
+.
+(36)
+10
+
+Introducing this decay factor artificially will allow us to derive stronger bounds. Note that
+dj(zt) = dj(zt−1)
+�
+1 −
+∆2
+j
+4η(1 − η)1at(zt−1)=j
+�
+.
+(37)
+Now we can bound using the same reasoning as in (27) and (32) and the display above
+�
+zt
+�
+Pj(zt)P0(zt)dj(zt) =
+�
+zt
+�
+Pj(zt)P0(zt)dj(zt−1)
+�
+1 −
+∆2
+j
+4η(1−η)1at(zt−1)=j
+�
+≥
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1)
+�
+1 −
+∆2
+j
+4η(1−η)1at(zt−1)=j
+�
+−
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1)
+�
+1 −
+∆2
+j
+4η(1−η)1at(zt−1)=j
+�
+1at(zt−1)=j
+∆2
+j
+4η(1−η)
+≥
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1) −
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1)
+∆2
+j
+2η(1−η)1at(zt−1)=j
+≥
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1) −
+∆2
+j
+2η(1−η)
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1))1at(zt−1)=j
+(38)
+Using (26) and a telescopic series we conclude that
+2
+�
+δ(1 − δ) ≥
+√
+F(Zj
+T , Z0
+T ) ≥
+�
+zT
+�
+Pj(zT )P0(zT )dj(zT )
+≥ 1 −
+∆2
+j
+2η(1−η)
+T
+�
+t=1
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1))1at(zt−1)=j.
+(39)
+Equivalently, this can be rewritten as
+(1 − 2
+�
+δ(1 − δ))2η(1 − η)
+∆2
+j
+≤
+T
+�
+t=1
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1)21at(zt−1)=j
+(40)
+It remains to bound the right hand side of this inequality. We write for t < T and zT |t = (x1, . . . , xt)
+where zT = (x1, . . . , xt, . . . , xT ), i.e., zT |t denotes the restriction of the history zT to the first t steps.
+Then we have by definition of P
+�
+zT
+Pj(zT )f(zT |t) =
+�
+zt
+Pj(zt)f(zt).
+(41)
+Moreover, we note that for all zT we can bound
+T
+�
+t=1
+dj(zT |(t−1))21
+�
+at(zT |(t−1)) = j
+�
+=
+nj(zT )−1
+�
+n=0
+�
+1 −
+∆2
+j
+4η(1 − η)
+�2n
+≤
+∞
+�
+n=0
+�
+1 −
+∆2
+j
+4η(1 − η)
+�n
+≤ 4η(1 − η)
+∆2
+j
+.
+(42)
+11
+
+We can bound using the Cauchy-Schwarz estimate, (41), and
+T
+�
+t=1
+�
+zt−1
+�
+Pj(zt−1)P0(zt−1)dj(zt−1))12
+at(zt−1)=j
+≤
+�
+�
+T
+�
+t=1
+�
+zt−1
+P0(zt−1)1at(zt−1)=j
+�
+�
+1
+2 �
+�
+T
+�
+t=1
+�
+zt−1
+Pj(zt−1)dj(zt−1)21at(zt−1)=j
+�
+�
+1
+2
+≤
+� T
+�
+t=1
+P0(At = j)
+� 1
+2 ��
+zT
+Pj(zT )
+T
+�
+t=1
+dj(zT |(t−1))21at(zT |(t−1))=j
+� 1
+2
+≤
+� T
+�
+t=1
+P0(At = j)
+� 1
+2 ��
+zT
+Pj(zT )4η(1 − η)
+∆2
+j
+� 1
+2
+≤
+� T
+�
+t=1
+P0(At = j)
+� 1
+2 2
+�
+η(1 − η)
+∆j
+.
+(43)
+Plugging this in (40), dividing by 2
+�
+η(1 − η) and summing over j > 1 gives
+(1 − 2
+�
+δ(1 − δ))
+�
+η(1 − η)
+N
+�
+j=2
+∆−2
+j
+≤
+N
+�
+j=2
+∆−1
+j
+� T
+�
+t=1
+P0(At = j)
+� 1
+2
+≤
+�
+�
+N
+�
+j=2
+∆−2
+j
+�
+�
+1
+2 �
+�
+n
+�
+j=2
+T
+�
+t=1
+P0(At = j)
+�
+�
+1
+2
+=
+�
+�
+n
+�
+j=2
+∆−2
+j
+�
+�
+1
+2 �
+�
+T
+�
+t=1
+n
+�
+j=1
+P0(At = j)
+�
+�
+1
+2
+=
+�
+H(p1)
+√
+T.
+(44)
+Squaring this relation ends the proof.
+4
+Quantum channel oracles
+In this section we consider the query complexity for the identification of certain oracles that act as a
+quantum channel. This is a simplified setting of the more general bandit problem. Most of the work
+related to oracle query complexity has focused on oracles that act as a unitary map. There is a large
+body of research work on quantum channels also with a focus on quantum channel discrimination and
+general lower and upper bounds were derived [1, 27, 35]. However, the application of these general
+bounds is mostly targeted towards rather simple channels, in particular channels that implement error
+mechanisms present in quantum devices. Those general results are not directly applicable here and
+we will derive bounds targeted at our specific setting. The work closest to ours is [30] where a noisy
+oracle for quantum search was investigated, i.e., instead of the usual oracle we have access to the
+quantum channel
+Fi(ρ) = (1 − p)ρ + pOiρO†
+i
+(45)
+where Oi is the usual oracle that performs a phase flip on the (unknown) state i. Let us compare
+this to the setting we introduced before. A careful look reveals that this oracle agrees with (8) when
+considering the mean reward vector p given by pj = pδij.
+12
+
+For this oracle it is shown in [30] that N/p queries are required to identify i. We give a slightly
+more general proof of this result which will serve as a basis for the more general bandit setting we
+consider later on. The intuition of the proof is that the progress we make is directly related to the
+decoherence of the state as measured by its purity. As the purity is lower bounded this gives us tight
+control of the progress. To cover general oracles we assume that the oracle acts on a Hilbert given by
+H = HA ⊗ HR where HA = ⟨|i⟩ , 1 ≤ i ≤ N⟩ and HR is the space where the output is written which
+will typically be a single qubit space. We assume that we are given oracles Oi acting by
+Oi |i⟩ ⊗ |w⟩ = |i⟩ ⊗ |Uw⟩ ,
+Oi |j⟩ ⊗ |w⟩ = |j⟩ ⊗ |w⟩
+for j ̸= i
+(46)
+where U denotes a unitary map. This covers the case of the phase flip and the bit flip oracle. As before
+we consider that we have oracle access to one of the quantum channels Fi(ρ) = (1 − p)ρ + pOiρO†
+i
+which we seek to identify. For convenience we define F0 = id. Then we have the following slightly
+extended version of Theorem 1 in [30].
+Theorem 8. Any algorithm that can decide whether F = Fi for some i > 0 or F = F0 with probability
+1 − δ requires at least
+T ≥ (1 − p)(1 − 4δ(1 − δ))2
+p
+n
+(47)
+calls to the channel.
+Remark 1.
+1.
+We emphasize again that the original proof still applies but we think that our proof
+clarifies the main ideas and prepares for the more general results.
+2.
+Note that the bound becomes vacuous for p → 1, where we recover the setting of Grover’s
+algorithm where the well known lower bound scales as
+√
+N. So for p =
+√
+N the bound (47)
+agrees with the Grover lower bound so it is possible that a small error probability depending
+on the number of arms still allows the same query complexity as in the noise-free case. Similar
+questions were investigated in [31].
+Proof. Let us denote by ΦU the quantum channel acting by the unitary U, i.e., ΦU(ρ) = UρU †. As in
+the first proof we consider an algorithm acting by ΦUT ◦ F ◦ ΦUT −1 ◦ . . . ◦ ΦU1 ◦ F ◦ ΦU0(ρ0) on some
+initial state ρ0 = |Ω⟩ ⟨Ω| for some pure state Ω. We define
+˜ρi
+t = ΦUt(ρi
+t),
+ρi
+t = Fi(˜ρi
+t−1),
+ρi
+0 = ρ0.
+(48)
+Note that for i = 0 the state remains pure during the entire algorithm and we denote it by ψt and ˜ψt.
+We now define
+Ri
+t = Tr
+�
+ρi
+t
+�2,
+(49)
+F i
+t = F(ρi
+t, ρ0
+t),
+(50)
+i.e., the purity of the state and the fidelity (defined by F(σ, ρ) =
+�
+Tr �√ρσ√ρ
+�2) of the state with
+respect to the state corresponding to the trivial oracle. For a brief overview of properties of the fidelity
+we refer to Appendix A. We show that the changes of the two quantities are directly related, i.e., for
+every increase in distance (loss in fidelity) we have to pay with a loss in purity, i.e., decoherence.
+Fidelity and purity are invariant under unitary maps and therefore Ri
+t = Tr
+�
+˜ρi
+t
+�2 and F i
+t = F(˜ρi
+t, ˜ρ0
+t).
+13
+
+We control, using that Oi is unitary,
+Ri
+t−1 − Ri
+t = Tr
+�
+˜ρi
+t−1
+�2 − Tr
+�
+ρi
+t
+�2
+= Tr
+�
+˜ρi
+t−1
+�2 − Tr
+�
+pOi˜ρi
+t−1O†
+i + (1 − p)˜ρi
+t−1
+�2
+= Tr
+�
+˜ρi
+t−1
+�2 − (p2 + (1 − p)2) Tr
+�
+˜ρi
+t−1
+�2 − 2p(1 − p) Tr
+�
+Oi˜ρi
+t−1O†
+i ˜ρt−1
+�
+= 2p(1 − p)
+�
+Tr
+�
+˜ρi
+t−1
+�2 − Tr
+�
+Oi˜ρi
+t−1O†
+i ˜ρt−1
+��
+= p(1 − p) Tr
+�
+˜ρi
+t−1 − Oi˜ρi
+t−1O†
+i
+�2
+(51)
+Similarly we estimate the change in fidelity using that ρt
+0 is a pure state and ˜ψt−1 = ψt (because
+E0 = id)
+F i
+t−1 − F i
+t = F(˜ρi
+t−1, ˜ρ0
+t−1) − F(ρi
+t, ρ0
+t)
+= ⟨ ˜ψt−1, ˜ρi
+t−1 ˜ψt−1⟩ − ⟨ψt, ρi
+tψt⟩
+= ⟨ψt, (˜ρi
+t−1 − (1 − p)˜ρi
+t−1 − pOi˜ρi
+t−1O†
+i )ψt⟩
+= p⟨ψt, (˜ρi
+t−1 − Oi˜ρi
+t−1O†
+i )ψt⟩.
+(52)
+We now relate the change of Rt and Ft Suppose that there is an orthogonal projection P and a unitary
+O such that (Id − P)O = Id − P, i.e., O acts trivially on the complement of the image of P. Then
+Lemma 3 in Appendix B establishes the bound
+���
+ϕ
+���
+OσO† − σ
+�
+ϕ
+��� ≤ 2∥Pϕ∥∥ϕ∥
+�
+tr
+�
+OσO† − σ
+�2� 1
+2 .
+(53)
+We define projections Pi = |i⟩ ⟨i| ⊗ Id Note that by definition of Oi we have (Id − Pi)Oi = Id − Pi.
+Applying Lemma 3, i.e., (53) (with P = Pi, σ = ˜ρi
+t−1, ϕ = ψt) we can continue to estimate (52) as
+follows
+F i
+t−1 − F i
+t ≤ 2p∥Piψt∥
+�
+Tr
+�
+˜ρi
+t−1 − Oi˜ρi
+t−1O†
+i
+�2� 1
+2
+≤ 2∥Piψt∥
+�
+p
+1 − p
+�
+Ri
+t−1 − Ri
+t
+(54)
+Note that the initial values of F and R are F i
+0 = Ri
+0 = 1 and Ri
+t ≥ 0. Thus we can conclude
+�
+i
+(F i
+0 − F i
+T ) =
+�
+i,t
+(F i
+t−1 − F i
+t )
+≤
+�
+i,t
+2∥Piψt∥
+�
+p
+1 − p
+�
+Ri
+t−1 − Ri
+t
+≤ 2
+�
+p
+1 − p
+�
+��
+i,t
+∥Piψt∥2
+�
+�
+1
+2 �
+��
+i,t
+Ri
+t−1 − Ri
+t
+�
+�
+1
+2
+≤ 2
+�
+p
+1 − p
+��
+t
+∥ψt∥2
+� 1
+2 ��
+i
+Ri
+0 − Ri
+T
+� 1
+2
+≤ 2
+�
+p
+1 − p
+√
+T
+√
+N.
+(55)
+14
+
+Finally we use our assumption that the algorithm is able to decide whether the oracle is trivial E = E0
+or not with probability 1 − δ for some δ < 1/2. From this we can conclude that the output of the
+algorithm for oracles F0 and Fj must be sufficiently different. Formally success of the algorithm
+implies (see (149) and (156) in Appendix A) that for each i
+1 − 2δ ≤ T(ρi
+T , ρ0
+T ) ≤
+�
+1 − F(ρi
+T , ρ0
+T )
+(56)
+where T(ρ, σ) = 1
+2∥ρ − σ∥tr denotes the trace distance. This implies the bound
+F i
+T ≤ 4δ(1 − δ).
+(57)
+We conclude that
+2
+�
+p
+1 − p
+√
+T
+√
+N ≥ N(1 − 4δ(1 − δ)) ⇒ T ≥ N(1 − p)(1 − 4δ(1 − δ))2
+p
+.
+(58)
+A natural question suggested by this result and also our main result is whether speed-ups can be
+obtained for non-unitary oracles. This question was also posed in [30]. We now show that this is not
+true in this generality, the simplest example is a faulty oracle that indicates its own failure, i.e. we
+consider an oracle Oi acting by
+Oi |i⟩ |0⟩ = − |i⟩ |1⟩ ,
+Oi |j⟩ |0⟩ = |j⟩ |1⟩
+for j ̸= i
+(59)
+and channels
+Fi(ρ) = pOiρO†
+i + (1 − p)ρ.
+(60)
+Then we can obtain the same speed-up as with the usual oracle except that we need to correct for the
+number of times the oracle is not working. This is not in contradiction to the previous result as this
+oracle is not of the form defined in (46). In particular, the action of the oracle in (59) is not trivial
+on |j⟩ ⊗ |c⟩.
+Theorem 9. The channel i can be identified with probability at least 1/4 using ⌊π/(2θp)⌋ queries to
+the oracle where θ = 2 arcsin
+�√
+N
+−1�
+≈ 2
+√
+N
+−1.
+Remark 2. Note that up to constant factors we need
+√
+N/p queries of which typically
+√
+N queries
+work, this is the same scaling as the usual Grover algorithm. As usual this bound can also be obtained
+if p is unknown by iteratively increasing the number of iterations in the algorithm described below.
+The proof can be found in Appendix E. While this result is simple and not very surprising it
+underlines that it will be difficult to obtain general results showing that no quantum speedup with
+quantum channel oracles.
+5
+Complexity bounds for non-adaptive strategies for quantum
+bandits
+In this section we discuss the complexity of non-adaptive strategies to the quantum bandit problem.
+While those results also follow from the more general treatment below we think that it is worth to
+include this section as a preparation for the rather involved arguments below. We introduce some
+additional notation. As before we consider for 0 ≤ p ≤ 1 oracles Fp
+i acting by
+Fp
+i (ρ) = (1 − p)ρ + pOiρO†
+i
+(61)
+15
+
+where Oi is as in (46). Note that the channels Fpi
+i
+and Fpj
+j
+commute for i ̸= j because Oi and Oj
+commute for i ̸= j. We consider probability vectors p ∈ [0, 1]N and define further
+Ep = Fp1
+1
+◦ . . . ◦ FpN
+n
+.
+(62)
+Note that for the special case that Oi denotes the bit flip on arm |i⟩ the channel Ep agrees with the
+definition in (8) where the vector p indicates the mean rewards, i.e.,
+Ep(ρ) =
+�
+x∈{0,1}N
+P(x)OxρO†
+x
+where
+Ox =
+�
+i:xi=1
+Oi
+P(x) =
+�
+pxi
+i (1 − pi)1−xi.
+(63)
+We now state a lemma that controls the fidelity between applications of the oracle. This result provides
+a sharp bound that might be of independent interest.
+Lemma 1. Assume that Oi is self-adjoint and unitary and acts as in (46). For density matrices ρ, σ
+and p, q ∈ [η, 1 − η] the bound
+√
+F(Fp
+i (ρ), Fq
+i (σ)) ≥
+√
+F(ρ, σ) − (p − q)2
+η(1 − η)
+�
+tr(Piρ) tr(Piσ)
+(64)
+where Pi = |i⟩ ⟨i| ⊗ Id denotes as before the projection on state |i⟩.
+The proof of this lemma can be found in Appendix C. Let us for completeness state one direct
+consequence.
+Corollary 1. Let p, p′ ∈ [η, 1 − η]N. Then
+√
+F(Ep(ρ), Ep′(σ)) ≥
+√
+F(ρ, σ) −
+�
+i
+(pi − p′
+i)2
+2η(1 − η)
+�
+tr(Piρ) tr(Piσ).
+(65)
+Proof. We note that [Pi, Oj] = 0 for i ̸= j and as Oj is unitary we get
+tr
+�
+PiFp
+j (ρ)
+�
+= tr
+�
+Pi((1 − p)ρ + pOjρO†
+j)
+�
+= tr
+�
+Pi((1 − p)ρ + pρO†
+jOj)
+�
+= tr(Piρ).
+(66)
+Then Lemma 1 can be applied inductively to the relation (62) to obtain the claim.
+From this result we conclude that any non-adaptive algorithm requires the same amount of oracle
+queries as the best classical algorithm. Note that Ep corresponds to the oracle in (8)
+Corollary 2. Assume that pj are as introduced at the beginning of Section 3 with pi ∈ [η, 1 − η] for
+some η > 0. Fix a density matrix ρ. We are given access to m copies of the state E(ρ) and it is
+known that E is as in (8) where the mean reward vector is in {p0, . . . , pN}. If the best arm of E can
+be identified with probability at least 1 − δ for some δ < 1
+2 then
+m ≥ η(1 − η)(1 − 2
+�
+δ(1 − δ))
+16
+H(p) = O(H(p)).
+(67)
+Proof. We write Ei = Epi.
+We obtain using Corollary 1 (which can be applied since the bit flip
+operation is self adjoint) for m copies
+√
+F(Ei(ρ)
+� m, E0(ρ)
+� m) =
+√
+F(Ei(ρ), E0(ρ))m
+≥
+�
+1 − (p0 − pi)2
+c(1 − c) tr(Pkρ)
+�m
+≥ 1 − m
+4∆2
+k
+η(1 − η) tr(Pkρ)
+(68)
+16
+
+where we used the Bernoulli inequality in the last step. From
+1 − 2δ ≤ T(Ei(ρ)
+� m, E0(ρ)
+� m) ≤
+�
+1 − F(Ei(ρ)
+� m, E0(ρ)
+� m)
+(69)
+we conclude that
+2
+�
+δ(1 − δ) ≥
+√
+F(Ei(ρ)
+� m, E0(ρ)
+� m) ≥ 1 − m
+4∆2
+i
+η(1 − η) tr(Piρ).
+(70)
+Equivalently
+tr(Piρ) ≥ η(1 − η)(1 − 2
+�
+δ(1 − δ))
+4m∆2
+i
+(71)
+Using tr(�
+i Piρ) = 1 and summing over i ≥ 2 we conclude
+1 ≥ η(1 − η)(1 − 2
+�
+δ(1 − δ))
+4m
+N
+�
+i≥2
+∆−2
+i .
+(72)
+Using �N
+i≥2 ∆−2
+i
+≥ 1
+4
+�N
+i=2(p1 − pi)−2 ends the proof.
+We can similarly derive a suboptimal bound for adaptive algorithms. Here we lose a
+√
+N factor.
+In the proof of Theorem 6 we will show that the bound in fact holds without that sqrtN factor. The
+reason that the arguments used in the proof of Theorem 8 do not extend is that it uses in an essential
+way that one of the density matrices is pure.
+Then we show that the states for the other oracle
+decoherence with respect to this pure reference state. In the setting here both density matrices are
+highly mixed so it is more subtle to formalize the their decoherence. Note that this result is already
+suboptimal in the case of Theorem 8.
+Corollary 3. Assume that pj are as introduced at the beginning of Section 3 with pi ∈ [η, 1 − η] for
+some η > 0. Any quantum algorithm that identifies the best arm when it is known that the reward
+vector is in {p0, . . . , pN} with probability at least 1 − δ for some δ < 1
+2 requires at least
+T ≥ η(1 − η)(1 − 2
+�
+δ(1 − δ))
+16
+√
+N
+H(p) = O(H(p)/
+√
+N)
+(73)
+calls to the oracle Ei(ρ) = Epi(ρ).
+Proof. The proof is close to the proof of Corollary 2. We assume we are given any algorithm (Ei ⊗Id)◦
+EUT ◦ . . . ◦ (Ei ⊗ Id) ◦ EU1 where Ui are arbitrary unitary maps. We denote the state using the oracle i
+before the t-th invocation of the oracle by ρi
+t. Using the invariance of the fidelity under unitary maps
+and Lemma 1 we bound
+√
+F(ρi
+T , ρ0
+T ) ≥ 1 −
+�
+t
+4∆2
+i
+η(1 − η)
+�
+tr
+�
+Piρi
+t
+�
+tr(Piρ0
+t).
+(74)
+As in equation (70) we conclude
+2
+�
+δ(1 − δ) ≥ 1 −
+�
+t
+4∆2
+i
+η(1 − η)
+�
+tr
+�
+Piρi
+t
+�
+tr(Piρ0
+t).
+(75)
+17
+
+Thus we get using �
+i tr(Piρt
+0) = 1 and tr
+�
+Piρi
+t
+�
+≤ 1
+η(1 − η)(1 − 2
+�
+δ(1 − δ))
+4
+�
+i≥2
+∆−2
+i
+≤
+�
+t,i
+�
+tr
+�
+Piρi
+t
+�
+tr(Piρ0
+t)
+≤
+�
+��
+t,i
+tr
+�
+Piρi
+t
+�
+�
+�
+1
+2 �
+��
+t,i
+tr
+�
+Piρ0
+t
+�
+�
+�
+1
+2
+≤
+√
+NT
+√
+T.
+(76)
+This implies that
+T ≥ η(1 − η)(1 − 2
+�
+δ(1 − δ))
+4
+√
+N
+�
+i≥2
+∆−2
+i .
+(77)
+As in (20) and (21) this ends the proof.
+6
+Complexity bounds for quantum bandits
+In this section we finally prove our main result Theorem 6.
+The proof essentially combines the
+ingredients of the three preceding sections. We use the general reasoning from the proof for the faulty
+Grover oracle in Theorem 8 and combine it with decompositions as in the proof of the classical bandit
+result in Theorem 7 and optimal fidelity estimates as in Lemma 1.
+As in the proof of Theorem 8 we want to exploit the decoherence of the density matrices ρk
+t and
+ρ0
+t. However, in contrast to the setting of Theorem 8 also the reference state ρ0
+t is not pure, so it
+is a-priori not clear how the decoherence can be measured in this setting. We did not succeed to
+find a suitable replacement of the purity used in Theorem 8 that can be defined intrinsically in terms
+of ρk
+t and ρ0
+t. Instead, our proof relies on a coupling argument, a technique that is standard in the
+theory of stochastic processes but not so much in quantum information theory. For an overview of this
+technique, we refer to [22]. To implement the coupling we need an improvement of Lemma 1 which is
+given in the following (slightly technical) lemma.
+Lemma 2. Consider a density matrix ρ and a pure state ψ. Consider a unitary and self-adjoint map
+U and the channel Ep
+U defined by Ep
+U(ρ) = pUρU † + (1 − p)ρ. Let 0 < η < 1/2 and p, q ∈ [η, 1 − η].
+We define
+¯ρ0 = (1 − p)ρ,
+ρ0 = ρ,
+¯ρ1 = pUρU †,
+ρ1 = UρU †
+(78)
+and
+¯ψ0 =
+�
+1 − q cos(α) |ψ⟩ + √q sin(α) |Uψ⟩ ,
+¯ψ1 = √q cos(α) |Uψ⟩ −
+�
+1 − q sin(α) |ψ⟩
+(79)
+ψ0 = ¯ψ0/∥ ¯ψ0∥,
+ψ1 = ¯ψ1/∥ ¯ψ1∥
+(80)
+where
+cos(α) = √pq +
+�
+(1 − p)(1 − q),
+sin(α) =
+�
+(1 − p)q −
+�
+(1 − q)p.
+(81)
+We define q′ = ∥ ¯ψ1∥2. Then
+Eq
+U(|ψ⟩ ⟨ψ|) = q′ |ψ1⟩ ⟨ψ1| + (1 − q′) |ψ0⟩ ⟨ψ0| .
+(82)
+Let S =
+√
+F(ρ, |ψ⟩ ⟨ψ|) =
+�
+⟨ψ| ρ |ψ⟩ denote the initial fidelity. Then the following bound holds
+√
+F
+�
+Ep
+U(ρ), Eq
+U(|ψ⟩ ⟨ψ|)
+�
+≥
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, |ψ0⟩ ⟨ψ0|) +
+�
+pq′√
+F(ρ1, |ψ1⟩ ⟨ψ1|)
+≥ S − (p − q)2|(ψ, (UρU † − ρ)ψ)|
+2ηS
+− (p − q)2|Re(ψ, Uρψ) − (ψ, ρψ)|2
+8η2S3
+.
+(83)
+18
+
+Remark 3. Let us give some explanation regarding this lemma.
+1.
+Note that for p = q we recover the result that quantum channels can only increase the fidelity
+for our specific channel.
+2.
+The problem that this lemma solves is that we need a bound on the loss in fidelity that has,
+firstly, the optimal quadratic rate in (p − q), secondly, the bound needs to have a form that
+allows to exploit the specific structure of the oracles Oi which act nontrivially only on a small
+subspace, and, thirdly, we later want to use that the density matrices ρ decohere with respect to
+the state ψ. The last point will become clearer in the proof of Theorem 10 below but note that
+the expression (ψ, (UρU †−ρ)ψ) appeared already in the proof of Theorem 8 which indicates that
+similar arguments can be applied. We remark that Lemma 1 above already satisfied the first two
+requirements but the lack of the third requirement allowed us to only show the suboptimal bound
+in Corollary 3. It is also quite straightforward to satisfy the second and the third requirement
+with the suboptimal rate |p − q|. But this also gives only a suboptimal bound.
+3.
+The second error term does not require all desiderata outlined above but it is of higher order
+(note the extra square) which is sufficient to control it.
+4.
+To clarify the origin of the expressions for sin(α) and cos(α) we remark that if we choose β, γ
+such that √p = sin(β), √q = sin(γ), then α = γ − β. This also explains the simplifications in
+the formula below, in particular (195) and (198) below are just the trigonometric identities for
+angle sums.
+The proof of this lemma can be found in Appendix C. In fact we need a slight improvement of this
+result stated in the following corollary.
+Corollary 4. Assume the same setting as in Lemma 1, with the following changes. We in addition
+consider another self adjoint traceless operator σ and we define
+¯ρ0 = (1 − p)(ρ + pσ)
+ρ0 = ρ + pσ
+¯ρ1 = pUρU † + p(1 − p)σ
+ρ1 = UρU † + (1 − p)σ.
+(84)
+We assume that ρ0 and ρ1 are density matrices, i.e., non-negative. Then the following bound holds
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, |ψ0⟩ ⟨ψ0|) +
+�
+pq′√
+F(ρ1, |ψ1⟩ ⟨ψ1|) − S ≥
+− (p − q)2|(ψ, (UρU † − ρ)ψ)|
+2ηS
+− (p − q)2|Re(ψ, Uρψ) − (ψ, ρψ)|2
+8η2S3
+− p|⟨ ¯ψ0, σ ¯ψ0⟩| + (1 − p)|⟨ ¯ψ1, σ ¯ψ1⟩|
+S
+.
+(85)
+We sketch the proof in Appendix C. Now we can state the formal version of Theorem 6.
+Theorem 10. Let δ < 1/2. Assume that pj are as introduced at the beginning of Section 3 with
+pi ∈ [η, 1 − η] for some η > 0. Any quantum algorithm that identifies the best arm when it is known
+that the reward vector is in {p0, . . . , pn} with probability at least 1 − δ requires at least
+T ≥ cH(p1) = c
+n
+�
+i=2
+∆−2
+i
+(86)
+calls to the oracle Ek where c = c(δ, η) where an explicit expression under the condition ∆2
+N/η < 1/2
+is given by
+c(δ, η) =
+�
+η(1 − 2
+�
+δ(1 − δ))
+20
+�2
+.
+(87)
+19
+
+Remark 4. We only do the proof under the condition that ∆2
+N/η ≤ 1/2 (the behavior for small ∆i is
+the main interest anyway). If this does not hold some definitions below need to be slightly adjusted
+(starting with (94) and (95)) but the final result will be the same except that the constant has a
+poorer dependence on η.
+Proof. We use the same notation as before in the proofs of Corollary 2, and Corollary 3. Recall in
+particular the definition of Fp
+i and Ei in (61) and (62) and the definition of pj (in particular pj
+i = pi
+for i ̸= j and pi
+i = p0). To notate the rewards in step t we consider xt ∈ {0, 1}N and we collect
+those rewards in the vector zt = (x1, . . . , xt). As before we consider the measure Pi on sequences zT
+which has the property that Pi((xt)j = 1) = pi
+j and Pi((xt)j = 0) = 1 − pi
+j and those variables are
+independent. We assume the algorithm is given by an circuit acting by
+Ti = (Ei ⊗ Id) ◦ EUT ◦ . . . ◦ (Ei ⊗ Id) ◦ EU1
+(88)
+with one of the unknown oracle Ek to be identified followed by a POVM-measurement. As before we
+denote the state of the system after t invocations of the oracle Ei by ρi
+t and by ˜ρi
+t the state directly
+before the t + 1-th invocation of the oracle. Let us for now fix i ∈ {1, . . . , n}. We try to lower bound
+the fidelity
+√
+F(ρi
+t, ρ0
+t). This will be based on decompositions of the density matrices, where we split
+the state in two after each invocation of any of the oracles Fpj
+j
+depending on the realization of the
+randomness. For ρi
+t we consider a decomposition given by
+ρi
+t =
+�
+zt
+Pi(zt)ρ(zt).
+(89)
+For ρ0
+t we consider a decomposition into pure states depending on i, i.e., we construct a distribution
+Qi on sequences zT and states ψ(zt) such that
+ρ0
+t =
+�
+zt
+Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)| .
+(90)
+To avoid a too complex notation we dropped the i dependence of ρ(zt) and ψ(zt).
+This time the decomposition for ρ0 involves the complexity while the decomposition for the oracle Ei
+will be relatively straightforward. The high-level idea for the decomposition is based on the observation
+that the optimal loss in fidelity when applying Ei and E0 is of the order ∆2
+i as shown in Lemma 2
+above. We essentially use the decomposition constructed there.
+We decompose ρi
+t roughly as
+ρt(zt) ≈ |ϕ(zt)⟩ ⟨ϕ(zt)| ,
+ϕ(zt = (x1, . . . , xt)) = (Oxt ⊗ Id)Ut . . . (Ox1 ⊗ Id)U1,
+(91)
+i.e., we just decompose it a according to the realizations of the rewards. However, we in addition
+need to ensure that the density matrices ρt(zt) decohere with respect to ψ(zt). For the definition we
+introduce the notation ˆxt ∈ {0, 1}N for the vector xt with the i-th entry set to 0, i.e., (ˆxt)j = (xt)j
+for j ̸= i and (xt)i = 0. We define (recall that ρ0 denotes the initial state of the algorithm)
+ρ(z0) = ρ0,
+(92)
+˜ρ(zt) = EUt+1(ρ(zt)),
+(93)
+˜ρ0(zt) =
+�
+1 − p0∆2
+i
+η
+�
+˜ρ(zt) + p0∆2
+i
+η
+Oi˜ρ(zt)O†
+i ,
+(94)
+˜ρ1(zt) =
+�
+1 − (1 − p0)∆2
+i
+η
+�
+Oi˜ρ(zt)O†
+i + (1 − p0)
+η
+∆2
+i ˜ρ(zt),
+(95)
+ρ(zt+1) = EOˆxt+1 (˜ρ(xt+1)i(zt)).
+(96)
+20
+
+The reason to slightly perturb ˜ρ0(zt) and ˜ρ1 from their natural definitions ˜ρ(zt) and Oi˜ρ(zt)O†
+i is that
+our definition ensures up to a constant the same loss in fidelity of order ∆2
+i but in addition ensures
+decoherence of ˜ρ so that we can argue as in the proof of Theorem 8. The factor η leads to slightly
+simpler terms but is not strictly necessary.
+We now claim that this decomposition satisfies (89) and
+˜ρi
+t =
+�
+zt
+P(zt)˜ρ(zt).
+(97)
+We argue by induction. We have
+˜ρi
+t = EUt+1(ρi
+t) = EUt+1
+��
+zt
+Pi(zt)ρ(zt)
+�
+=
+�
+zt
+Pi(zt)EUt+1(ρ(zt)) =
+�
+zt
+Pi(zt)˜ρ(zt).
+(98)
+Next we note that
+p0˜ρ1(zt) + (1 − p0)˜ρ0(zt) = p0 Oi˜ρ(zt)O†
+i + (1 − p0) ˜ρ(zt) = Fp0
+i (˜ρ(zt)).
+(99)
+This implies together with the definition of Pi that
+�
+xt+1
+Pi(xt+1)ρ((zt, xt+1)) =
+�
+xt+1
+Pi(xt+1)EOˆxt+1 (˜ρ(xt+1)i(zt)) =
+�
+xt+1
+Pi(xt+1)EOˆxt+1 ◦ Fp0
+i (˜ρ(zt))
+=
+�
+xt+1
+Pi(xt+1)EOxt+1 (˜ρ(zt)) = Ei(˜ρ(zt)).
+(100)
+Here we used that Pi((xt+1)i = 1) = pi
+i = p0. Using the induction hypothesis we conclude that
+�
+zt+1
+Pi(zt+1)ρ(zt+1) =
+�
+zt
+Pi(zt)
+�
+xt+1
+P(xt+1)ρ((zt, xt+1)) =
+�
+zt
+Pi(zt)Ei(˜ρ(zt)) = Ei(˜ρi
+t) = ρi
+t+1.
+(101)
+For the oracle E0 we consider a decomposition depending on i. We consider a probability distri-
+bution Qi on sequences zt = (x1, . . . , xt) as before. It will factorize according to
+Qi(zt+1) = Qi(zt)Qi(xt+1|zt) = Qi(zt)Qi((xt+1)i|zt)
+�
+j̸=i
+p(xt+1)j
+j
+(1 − pj)1−(xt+1)j.
+(102)
+We define
+˜ψ(zt) = Ut
+�
+ψ(zt−1)
+�
+.
+(103)
+We now define, essentially as in Lemma 2,
+¯ψ1(zt) =
+�
+1 − pi cos(α) ˜ψ(zt) + √pi sin(α)Oi ˜ψ(zt)
+(104)
+˜ψ1(zt) = ¯ψ1(zt)/∥ ¯ψ1(zt)∥
+(105)
+¯ψ0(zt) = −
+�
+1 − pi sin(α) ˜ψ(zt) + √pi cos(α)Oi ˜ψ(zt)
+(106)
+˜ψ0(zt) = ¯ψ0(zt)/∥ ¯ψ0(zt)∥
+(107)
+where α is as in (81) with p and q replaced by pi and p0. Finally we set
+ψ(zt+1) = Oˆxt+1 ˜ψ(xt+1)i(zt).
+(108)
+21
+
+We define
+Qi�
+(xt+1)i = 1 | zt
+�
+= ∥ ¯ψ1(zt)∥2
+Qi�
+(xt+1)i = 0 | zt
+�
+= ∥ ¯ψ0(zt)∥2 = 1 − ∥ ¯ψ1(zt)∥2.
+(109)
+Moreover, we assume that the (xt)j for i ̸= j are independent and distributed according to Ber(p0
+j) =
+Ber(pj). Together, those properties define a unique distribution Qi. We claim that
+ρ0
+t =
+�
+zt
+Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)| ,
+(110)
+˜ρ0
+t =
+�
+zt
+Qi(zt)
+��� ˜ψ(zt)
+� �
+˜ψ(zt)
+��� .
+(111)
+To show this we argue by induction. The first step is simple
+�
+zt
+Qi(zt)
+��� ˜ψ(zt)
+� �
+˜ψ(zt)
+��� =
+�
+zt
+Qi(zt) |Ut+1ψ(zt)⟩ ⟨Uψ(zt)|
+= Ut+1
+��
+zt
+Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)|
+�
+U †
+t+1 = EUt+1(ρ0
+t) = ˜ρ0
+t.
+(112)
+By (82) in Lemma 2 the following relation holds in view of
+Fpi
+i ( ˜ψ(zt)) = Qi�
+(xt+1)i = 1 | zt
+� ��� ˜ψ1(zt)
+� �
+˜ψ1(zt)
+��� + Qi�
+(xt+1)0 = 1 | zt
+� ��� ˜ψ0(zt)
+� �
+˜ψ0(zt)
+��� .
+(113)
+Using this relation we conclude that
+�
+zt+1
+Qi(zt+1) |ψ0(zt+1)⟩ ⟨ψ0(zt+1)| =
+�
+zt+1
+Qi(zt+1)
+���Oˆxt+1 ˜ψ(xt+1)i(zt)
+� �
+Oˆxt+1 ˜ψ(xt+1)i(zt)
+���
+= Fp1
+1 ◦ . . . ◦ Fpi−1
+i−1 ◦ Fpi+1
+i+1 ◦ . . . ◦ FpN
+N
+��
+zt
+Qi(zt)
+1
+�
+s=0
+Qi((xt+1)i = s|zt)
+��� ˜ψs(zt)
+� �
+˜ψs(zt)
+���
+�
+= Fp1
+1 ◦ . . . ◦ Fpi−1
+i−1 ◦ Fpi+1
+i+1 ◦ . . . ◦ FpN
+N ◦ Fpi
+i
+��
+zt
+Qi(zt)
+��� ˜ψ(zt)
+� �
+˜ψ(zt)
+���
+�
+= E0(˜ρ0
+t) = ρ0
+t+1.
+(114)
+We now start to estimate the fidelity of ρk
+t and ρ0
+t. Let us remark that invariance of the fidelity
+under unitary maps implies
+√
+F
+�
+ρ((zt, xt+1)), |ψ((zt, xt+1))⟩
+�
+=
+√
+F
+�
+˜ρ(xt+1)i(zt),
+��ψ(xt+1)i(zt)
+� �
+,
+(115)
+√
+F
+�
+ρ(zt), |ψ(zt)⟩
+�
+=
+√
+F
+�
+˜ρ(zt),
+��� ˜ψ(zt)
+� �
+.
+(116)
+The loss in fidelity occurs when passing from ˜ρ(zt) and ˜ψ(zt) to ˜ρ0/1(zt) and ˜ψ0/1(zt). We will bound
+this loss using Corollary 4 above. The additional flexibility of the σ term in this corollary allows
+to apply this to our setting where ˜ρ0/1 are defined as in (94) and (95). Specifically, we apply this
+Corollary with ρ = ˜ρ(zt), ψ = ˜ψ(zt), p = p0, q = pi, U = Oi and σ = ∆2
+i (Oi˜ρ(zt)O†
+i − ˜ρ(zt)). Then
+we note that the definition of ˜ψ0/1 agrees with the definition of ψ0/1 in Lemma 1 and ˜ρ0/1(zt) agrees
+22
+
+with ρ0/1 and q′ = Qi((xt+1)i = 1|zt), p = Pi(xi = 1). We write S =
+√
+F(˜σ(zt), ˜ψ(zt)) and assume
+S ≥ 1
+2. Thus we conclude from Corollary 4 that
+�
+(1 − p0)Qi((xt+1)i = 0|zt)
+√
+F(˜ρ0(zt), ˜ψ0(zt)) +
+�
+p0Qi((xt+1)i = 1|zt)
+√
+F(˜ρ1(zt), ˜ψ1(zt))
+≥ S − ∆2
+i
+�|( ˜ψ(zt), (Oi˜ρ(zt)O†
+i − ˜ρ(zt)) ˜ψ(zt))|
+η
++
+���Re
+�
+( ˜ψ(zt), (Oi˜ρ(zt) − ˜ρ(zt)) ˜ψ(zt))
+����
+2
+η2
++ 2(1 − p)
+η
+���( ¯ψ1(zt),
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�
+¯ψ1(zt))
+��� + 2p
+η
+���( ¯ψ0(zt),
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�
+¯ψ0(zt))
+���
+�
+(117)
+We control the right hand side of this expression by exploiting the specific structure of the oracle Oi.
+As in the proof of Theorem 8 we use that Pi = |i⟩ ⟨i| ⊗ Id satisfies (1 − Pi)Oi = (1 − Pi) and apply
+Lemma 3. We get
+|( ˜ψ(zt), (Oi˜ρ(zt)O†
+i − ˜ρ(zt)) ˜ψ(zt))| ≤ 2∥Pi ˜ψ(zt)∥
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+.
+(118)
+For the second term we use that Oi − Id = (Id − Pi + Pi)(Oi − Id) = Pi(Oi − Id) which implies after
+an application of Cauchy-Schwarz
+���Re
+�
+( ˜ψ(zt), (Oi˜ρ(zt) − ˜ρ(zt)) ˜ψ(zt))
+����
+2
+≤ ∥Pi ˜ψ(zt)∥2 · ∥(Oi − Id)˜ρ(zt) ˜ψ(zt)∥2 ≤ 4∥Pi ˜ψ(zt)∥2.
+(119)
+For the third term we use again Lemma 3, the definition (104), and [Oi, Pi] = 0 and bound
+���( ¯ψ1(zt),
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�
+¯ψ1(zt))
+��� ≤ ∥Pi ¯ψ1(zt)∥ · ∥ ¯ψ1(zt)∥ ·
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+≤
+��
+1 − pi cos(α)∥Pi ˜ψ(zt)∥ + √pi sin(α)∥PiOi ˜ψ(zt)∥
+�
+∥ ¯ψ1(zt)∥ ·
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+≤ 2∥Pi ˜ψ(zt)∥
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+.
+(120)
+The same reasoning implies
+���( ¯ψ0(zt),
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�
+¯ψ0(zt))
+��� ≤ 2∥Pi ˜ψ(zt)∥
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+.
+(121)
+Plugging (118), (119), (120), and (121) in (117) we obtain
+�
+(1 − p0)Qi((xt+1)i = 0|zt)
+√
+F(˜ρ0(zt), ˜ψ0(zt)) +
+�
+p0Qi((xt+1)i = 1|zt)
+√
+F(˜ρ1(zt), ˜ψ1(zt))
+≥ S − 5∆2
+i
+η ∥Pi ˜ψ(zt)∥
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+− 4∆2
+i
+η2 ∥Pi ˜ψ(zt)∥2.
+(122)
+This is the key relation that allows us to control the fidelity between ρ0
+t and ρi
+t. We now sum this
+bound over all possible values of xt+1 to move from t + 1 to step t.
+We denote ˆxt+1 the vector
+xt+1 with entry i removed. Note that under Pi and Qi this vector is independent of zt, (xt+1)i and
+Pi(ˆxt+1) = Qi(ˆxt+1). We also introduce the notation ˆxc
+t+1 for the vector that has entry i equal to c.
+23
+
+Then we get using (115) and (116) for any zt and c ∈ {0, 1}
+�
+ˆxt+1
+�
+Pi(ˆxt+1Qi(ˆxt+1)
+√
+F
+�
+˜ρ((zt, ˆxc
+t+1)), ˜ψ((zt, ˆxc
+t+1))
+�
+=
+�
+ˆxt+1
+�
+Pi(ˆxt+1Pi(ˆxt+1)
+√
+F
+�
+˜ρc(zt), ˜ψc(zt)
+�
+=
+√
+F
+�
+˜ρc(zt), ˜ψc(zt)
+�
+(123)
+Summing this over c and using (122) we get for all zt such that
+√
+F(˜ρ(zt),
+˜
+ψ(zt)) ≥ 1/2 the bound
+�
+xt+1
+�
+Pi(xt+1)Qi(xt+1|zt)
+√
+F
+�
+˜ρ((zt, xt+1)), ˜ψ((zt, xt+1))
+�
+1
+�
+c=0
+�
+Pi((xt+1)i = c)Qi((xt+1)i = c|zt)
+�
+ˆxt+1
+�
+Pi(ˆxt+1Qi(ˆxt+1)
+√
+F
+�
+˜ρ((zt, ˆxc
+t+1)), ˜ψ((zt, ˆxc
+t+1))
+�
+≥
+√
+F(˜ρ(zt),
+˜
+ψ(zt)) − 5∆2
+i
+η ∥Pi ˜ψ(zt)∥
+�
+tr
+�
+Oi˜ρ(zt)O†
+i − ˜ρ(zt)
+�2� 1
+2
+− 4∆2
+i
+η2 ∥Pi ˜ψ(zt)∥2.
+(124)
+Equipped with this relation we now move on to control the total loss in fidelity.
+Let us first
+explain how we deal with the trace term in the loss of fidelity which will be very similar to the proof
+of Theorem 8. Thus we define (again not indicating the i dependence)
+R(zt) = tr
+�
+˜ρ(zt)2�
+.
+(125)
+Invariance of the purity under unitary operations implies that
+R(zt+1) = tr
+�
+˜ρ(xt+1)i(zt)2�
+.
+(126)
+Calculations as in (51) give us for (xt+1)i = 0
+R(zt) − R(zt+1) =
+�
+1 − p0∆2
+i
+η
+� p0∆2
+i
+η
+· Tr
+�
+˜ρ(zt) − Oi˜ρ(zt)O†
+i
+�2
+.
+(127)
+A similar identity for (xt)i = 1 together with the assumption ∆2
+i /η ≤ 1/2 and min(p0, 1 − p0) ≥ η
+imply
+Tr(˜ρ(zt) − Oi˜ρ(zt))O†
+i )2 ≤
+1
+2∆2
+i
+(R(zt) − R(zt+1)).
+(128)
+Note that the right hand side only depends on zt so we conclude
+Tr(˜ρ(zt) − Oi˜ρ(zt))O†
+i )2 ≤
+1
+2∆2
+i
+(R(zt) − max
+xt+1 R((zt, xt+1)).
+(129)
+The general strategy is now to use joint concavity of the fidelity and then inductive application
+of the estimate (122) above to lower bound the fidelity. There are two technical difficulties: When
+directly trying to bound fidelity loss we do not get the optimal scaling and we need to apply an
+approach as in the proof of Theorem 7. The second difficulty is that the change in fidelity in Lemma 2
+involves the inverse of the initial fidelity and so we derived (124) only for S ≥ 1/2. The high level
+argument that this is sufficient is that by lower bounding the fidelity we also show that for most
+sequences zt the fidelity
+√
+F(ρ(zt), ψ(zt)) is lower bounded by 1/2 (say). Put differently, the fidelity
+has already decreased substantially when we can no longer apply this bound.
+Technically, we address both difficulties by introducing additional sequences d(zt), s(zt), and h(zt).
+24
+
+We define d(z0) = 1 and then recursively for zt = (zt−1, xt)
+d(zt) = d(zt−1)
+�
+1 − ∆2
+i ∥Pi ˜ψ(zt−1)∥2
+η2
+�
+= d(zt−1) − d(zt−1)∆2
+i ∥Pi ˜ψ(zt−1)∥2
+η2
+.
+(130)
+Note that the η factor was introduced for convenience, it could be dropped at the price of a slightly
+worse η dependence. We define s(zt) = 0 for zt = (x1, . . . , xt) if there is t′ ≤ t such that zt′ =
+(x1, . . . , xt′) satisfies
+√
+F(ρ(zt), ψ(zt)) =
+√
+F(˜ρ(zt), ˜ψ(zt)) < 1/2 and s(zt) = 1 otherwise. Moreover,
+we define h(zt) = 1 if s(zt) = 0 but s(zt′) = 1 for all zt′ as above, put differently for zt = (zt−1, xt) the
+relation h(zt) = s(zt−1) − s(zt) holds, i.e., s(zt) keeps track when the fidelity
+√
+F(ρ(zt), ψ(zt)) falls
+below 1/2 for the first time.
+We can now bound (loosely speaking) the change in fidelity
+√
+F(ρk
+T −1, ρ0
+T −1)−
+√
+F(ρk
+T , ρ0
+T ) (actually
+we only bound the difference on the lower bounds). This will be achieved by combining the estimates
+above and the introduction of various error terms. We again use the notation zT |t = (x1, . . . , xt) for
+t ≤ T and zT = (x1, . . . , xT ). We first note that the definitions of s, h, and d imply
+√
+F(˜ρk
+T , ˜ρ0
+T ) ≥
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT )
+=
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT )(s(zT |T −1) − h(zT ))
+=
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT |T −1) − E1
+T .
+= −E1
+T +
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT |T −1)s(zT |T −1)
+− ∆2
+i
+η2
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))∥Pi ˜ψ(zT |T −1)∥2d(zT |T −1)s(zT |T −1)
+= −E1
+T − E2
+T +
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT |T −1)s(zT |T −1)
+(131)
+Where the error terms E1
+T and E2
+T are defined by those equations. Now we bound this expression
+using (124) and (129). Here we use that either the factor s(zT |T −1) vanishes and the inequality below
+is trivially true or the bound
+√
+F(˜ρ(zT |T −1), ˜ψ(zT |T −1)) ≥ 1/2 holds so that (124) can be applied
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψT (zT ))d(zT |T −1)s(zT |T −1)
+≥
+�
+zT −1
+�
+Pi(zT −1)Qi(zT −1)d(zT −1)s(zT −1)
+�
+√
+F(˜ρ(zT −1), ˜ψ(zT −1))
+− 5∆i
+√
+2η ∥Pi ˜ψ(zT −1)∥
+�
+R(zT −1) − max
+xt R((zT −1, xt))
+� 1
+2
+− 4∆2
+i
+η2 ∥Pi ˜ψ(zT −1)∥2
+�
+≥
+�
+zT −1
+�
+Pi(zT −1)Qi(zT −1)
+√
+F(˜ρ(zT −1), ˜ψ(zT −1))d(zT −1)s(zT −1) − E3
+T − E4
+T
+(132)
+where we again define the error terms E3
+T −1 and E4
+T −1 implicitly through these equations. Applying
+this inductively together with
+√
+F(˜ρi
+0, ˜ρ0
+0) = 1 we get
+√
+F(˜ρi
+T , ˜ρ0
+T ) ≥
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT ) ≥ 1 −
+T
+�
+t=1
+E1
+t + E2
+t + E3
+t + E4
+t (133)
+25
+
+We now bound the error terms. We start with E2
+t which we bound by (using
+√
+F ≤ 1, s(zt) ≤ 1)
+E2
+t ≤ ∆2
+i
+η2
+�
+zt−1
+�
+Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)
+�
+xt
+�
+Pi(xt|zt−1)Qi(xt|zt−1)
+≤ ∆2
+i
+η2
+�
+zt−1
+�
+Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)
+(134)
+Clearly the error term E4
+t satisfies a similar bound and we obtain using Cauchy Schwarz and d(zt) ≤ 1
+T
+�
+t=1
+E2
+t + E4
+t ≤ 5∆2
+i
+η2
+T
+�
+t=1
+�
+zt−1
+�
+Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)
+≤ 5∆i
+η
+�
+�
+T
+�
+t=1
+�
+zt−1
+Qi(zt−1)∥Pi ˜ψ(zt−1)∥2
+�
+�
+1
+2 �
+�
+T
+�
+t=1
+�
+zt−1
+Pi(zt−1)∆2
+i
+η2 ∥Pi ˜ψ(zt−1)∥2d(zt−1)
+�
+�
+1
+2
+≤ 5∆i
+η
+� T
+�
+t=1
+Qi(zt−1) tr
+�
+Pi
+��� ˜ψ(zt−1)
+� �
+˜ψ(zt−1)
+��� Pi
+�� 1
+2
+�
+�
+T
+�
+t=1
+�
+zt−1
+Pi(zt−1)
+�
+d(zt−1) −
+�
+xt
+Pi(xt)d((zt−1, xt))
+��
+�
+1
+2
+≤ 5∆i
+η
+� T
+�
+t=1
+tr
+�
+Pi˜ρ0
+t
+�
+� 1
+2 �
+�
+T
+�
+t=1
+�
+zt−1
+Pi(zt−1)d(zt−1) −
+T
+�
+t=1
+�
+zt
+Pi(zt)d(zt)
+�
+�
+1
+2
+≤ 5∆i
+η
+� T
+�
+t=1
+tr
+�
+Pi˜ρ0
+t
+�
+� 1
+2 �
+d(z0) −
+�
+zT
+Pi(zT )d(zT )
+� 1
+2
+≤ 5∆i
+η
+� T
+�
+t=1
+tr
+�
+Pi˜ρ0
+t
+�
+� 1
+2
+(135)
+where we used (111) and the telescopic sum together with d(z0) = 1 and 0 ≤ d(zT ) ≤ 1 in the last
+step. Next we address the error term E3
+t . We bound, using again Cauchy Schwarz,
+T
+�
+t=1
+E3
+t = 5∆i
+√
+2η
+T −1
+�
+t=0
+�
+zt
+�
+Pi(zt)Qi(zt)d(zt)s(zt)∥Pi ˜ψ(zt)∥
+�
+R(zt) − max
+xt+1 R((zt, xt+1))
+� 1
+2
+≤ 5∆i
+√
+2η
+�T −1
+�
+t=0
+�
+zt
+Qi(zt)∥Pi ˜ψ(zt)∥2
+� 1
+2 � T
+�
+t=1
+�
+zt
+Pi(zt)R(zt) − max
+xt+1 R((zt, xt+1))
+� 1
+2
+≤ 5∆i
+√
+2η
+�T −1
+�
+t=0
+tr
+�
+Pi˜ρ0
+t
+�
+� 1
+2 �T −1
+�
+t=0
+�
+zt
+Pi(zt)R(zt) −
+T
+�
+t=1
+Pi(zt)R(zt)
+� 1
+2
+≤ 5∆i
+√
+2η
+�T −1
+�
+t=0
+tr
+�
+Pi˜ρ0
+t
+�
+� 1
+2
+.
+(136)
+Here we again used the telescopic sum and the fact that 0 ≤ R(zt) ≤ 1.
+26
+
+It remains to bound the last remaining error term E1
+t . The key ingredient to bound E1
+t is the
+observation that if h(zt) = 1 we have
+√
+F(˜ρ(zt), ˜ψ(zt)) < 1/2 which implies
+√
+F(˜ρ(zt), ˜ψ(zt)) < 1 −
+√
+F(˜ρ(zt), ˜ψ(zt)).
+(137)
+From here we conclude using the definition of E1
+t
+T
+�
+t+1
+E1
+t =
+T
+�
+t=1
+�
+zt
+�
+Pi(zt)Qi(zt)
+√
+F(˜ρ(zt), ˜ψ(zt))d(zt)h(zt)
+≤
+T
+�
+t=1
+�
+zt
+�
+Pi(zt)Qi(zt)
+�
+1 −
+√
+F(˜ρ(zt), ˜ψ(zt))
+�
+d(zt)h(zt)
+≤
+T
+�
+t=1
+�
+zt
+�
+Pi(zt)Qi(zt)h(zt) −
+T
+�
+t=1
+E1
+t .
+(138)
+Starting from (133) and plugging in the display above we find
+�
+zT
+�
+Pi(zT )Qi(zT )s(zT ) ≥
+�
+zT
+�
+Pi(zT )Qi(zT )
+√
+F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT )
+≥ 1 −
+T
+�
+t=1
+E1
+t + E2
+t + E3
+t + E4
+t
+≥ 1 −
+T
+�
+t=1
+E2
+t + E3
+t + E4
+t +
+T
+�
+t=1
+E1
+t −
+T
+�
+t=1
+�
+zt
+�
+Pi(zt)Qi(zt)h(zt)
+(139)
+Now it is easy to see that
+T
+�
+t=1
+�
+zt
+Pi(zt)h(zt) +
+�
+zT
+Pi(zT )s(zT ) = 1
+(140)
+because this is the probability that the fidelity drops below 1/2 under the measure Pi plus the prob-
+ability of the complement and the same is true for Qi. Thus, we can identify those expressions with
+probability vectors and since the (classical) fidelity of any two probability distributions is bounded by
+1 we conclude that
+�
+zT
+�
+Pi(zT )Qi(zT )s(zT ) +
+T
+�
+t=1
+�
+zt
+�
+Pi(zt)Qi(zt)h(zt) ≤ 1.
+(141)
+Using this relation in (139) we find
+1 ≥ 1 −
+T
+�
+t=1
+E2
+t + E3
+t + E4
+t +
+T
+�
+t=1
+E1
+t
+(142)
+from which
+T
+�
+t=1
+E1
+t ≤
+T
+�
+t=1
+E2
+t + E3
+t + E4
+t
+(143)
+follows.
+27
+
+Plugging first (143) and then (135) and (136) in (133) we obtain
+√
+F(ρk
+T , ρ0
+T ) ≥ 1 −
+T
+�
+t=1
+E1
+t + E2
+t + E3
+t + E4
+t ≥ 1 − 2
+T
+�
+t=1
+E2
+t + E3
+t + E4
+t
+≥ 1 − 20η−1∆i
+� T
+�
+t=1
+Tr Pk ˜ρ0
+t
+� 1
+2
+.
+(144)
+The rest of the proof is identical to the reasoning in the proofs before. From 2
+�
+δ(1 − δ) ≥
+√
+F
+�
+ρk
+T , ρ0
+T
+�
+we infer
+η(1 − 2
+�
+δ(1 − δ))
+20
+�
+i≥2
+∆−2
+i
+≤
+�
+i≥2
+∆−1
+i
+� T
+�
+t=1
+Tr Piρ0
+t
+� 1
+2
+≤
+�
+��
+i≥2
+∆−2
+i
+�
+�
+1
+2 �
+��
+i≥2
+T
+�
+t=1
+Tr Piρ0
+t
+�
+�
+1
+2
+≤
+�
+��
+i≥2
+∆−2
+i
+�
+�
+1
+2 √
+T.
+(145)
+This finally ends the proof.
+7
+Discussion
+In this work we investigated quantum algorithms for multi armed bandit problems. It was shown
+earlier in [34] that quantum algorithms for best arm identification with fixed confidence can have a
+quadratic speed-up compared to their classical counterparts. This result is based on the assumption
+that the arms and the randomness of the rewards of the arms can be both queried in superposition.
+These assumptions are reasonable in the setting of empirical risk minimization where we can evaluate
+loss values in superposition. However, there are many settings, e.g., motivated by quantum sensing
+where it might not be possible to query the internal randomness of the bandits in superposition.
+Instead every pull of the lever returns a single random reward. We then show that in this setting no
+speed-up compared to classical algorithms is possible.
+This highlights that classical randomness pose a major challenge for quantum algorithms. In our
+case the randomness of the rewards even prevent Grover type speed-ups of the search problem that
+one would naively expect to arise from the search part of the multi-armed bandit problem. Note
+that such a speed-up is possible in the intermediate regime that we considered. When we can select
+the state of the internal randomness of the oracle (but not query it in superposition) the statistical
+complexity of the problem remains the same but we can search through the arms faster providing
+some speed-up.
+There are many open questions related to this work and we will briefly mention two. Firstly,
+classical randomness appears frequently in different settings.
+This has been studied a lot in the
+context of noise channels but not so much in other contexts, e.g., machine learning.
+Secondly, our proofs proceed by directly controlling the fidelity between the quantum states when
+invoking different oracles. From a methodological side, it would be interesting to see to what degree the
+well known strategies to lower bound the query complexity like the polynomial [6] or the adversarial
+method [3] extend to non-unitary oracles.
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+A
+A brief review of distance measures for quantum states
+For the convenience of the reader we give a brief review of distance measures for quantum states.
+Textbooks on quantum computation, e.g., [25] discuss this thoroughly. For a review on fidelities we
+refer to [21]. We consider the trace distance which is defined by
+T(ρ, σ) = 1
+2∥ρ − σ∥tr
+(146)
+where the norm indicates the trace norm defined by ∥A∥tr = tr
+�√
+A†A
+�
+. It has the property that for
+any POVM {Ei} the outcome probabilities
+pi = tr(Eiρ),
+qi = tr(Eiσ)
+(147)
+the total variation distance between the probability vectors pi and qi satisfy
+1
+2
+�
+i
+|pi − qi| ≤ T(ρ, σ).
+(148)
+The Helmstrom measurement gives the optimal discrimination probability of two states and has success
+probability
+psuccess = 1
+2 + 1
+2T(ρ, σ).
+(149)
+For many applications the fidelity is a more useful distance measure to obtain optimal bounds. It is
+defined by
+√
+F(ρ, σ) = tr
+��
+ρ
+1
+2 σρ
+1
+2
+�
+= ∥ρ
+1
+2 σ
+1
+2 ∥tr.
+(150)
+Some authors instead call the square of this expression the fidelity and to clarify our convention we
+added the square root. As suggested by the notation we set F =
+√
+F
+2. We collect some properties of
+the fidelity that we will use frequently.
+1.
+For a density matrix ρ and a pure state ψ the fidelity is given by
+√
+F(|ψ⟩ ⟨ψ| , ρ) =
+�
+⟨ψ, ρψ⟩.
+(151)
+31
+
+2.
+For any density matrices ρ, σ and a quantum channel E the following bound holds
+√
+F(ρ, σ) ≤
+√
+F(E(ρ), E(σ)).
+(152)
+3.
+If the quantum channel E acts by a unitary matrix, i.e., E(ρ) = UρU † then
+√
+F(ρ, σ) =
+√
+F(E(ρ), E(σ)).
+(153)
+4.
+The fidelity is strongly concave
+√
+F(
+�
+i
+piρi,
+�
+i
+qiσi) ≥
+�
+i
+√piqi
+√
+F(ρi, σi).
+(154)
+This directly implies concavity
+√
+F(
+�
+i
+piρi,
+�
+i
+piσi) ≥
+�
+i
+pi
+√
+F(ρi, σi).
+(155)
+5.
+Fidelity and trace distance are related by
+1 −
+√
+F(ρ, σ) ≤ T(ρ, σ) ≤
+�
+1 − F(ρ, σ).
+(156)
+Those properties can be proved using Uhlmann’s Theorem which states that
+√
+F(ρ, σ) = max
+ϕ,ψ ⟨ϕ, ψ⟩
+(157)
+where the maximum is over all purifications ψ and ϕ of ρ and σ, respectively. We use this result to
+bound the fidelity change of certain quantum operations (see Lemma 1 and 2).
+B
+Auxiliary lemmas
+Here we include simple mostly algebraic lemmas that are used in the proof of Theorem 8 and in the
+proofs in Appendix C. The first lemma is a simple Cauchy-Schwarz estimate that in addition exploits
+invariant subspaces of an operator O. It is used in the proof of Theorem 8.
+Lemma 3. Let O be a unitary operator and P a self-adjoint orthogonal projections such that (1 −
+P)O = 1 − P, i.e., O acts trivially on the orthogonal complement of the projection P. Then, for any
+vector |ϕ⟩ and density matrix σ the bound
+���
+ϕ
+���
+OσO† − σ
+�
+ϕ
+��� ≤ 2∥Pϕ∥∥ϕ∥
+�
+tr
+�
+OσO† − σ
+�2� 1
+2
+(158)
+holds.
+Proof. We define Q = Id − P. Then we have QO = Q. By assumption we can decompose
+σ − OσO† = (P + Q)(σ − OσO†)(P + Q)
+= (σ − OσO†)P + P(σ − OσO†)Q
+(159)
+where we used QOσO†Q = QσQ. Using Cauchy-Schwarz for the Hilbert-Schmidt scalar product we
+can bound for M = M †
+|⟨ϕ1, Mϕ2⟩| = |Tr(|ϕ2⟩ ⟨ϕ1| M)| ≤
+�
+Tr(|ϕ2⟩ ⟨ϕ1|ϕ1⟩ ⟨ϕ2|) Tr M 2� 1
+2 = ∥ϕ1∥ · ∥ϕ2∥
+�
+Tr M 2� 1
+2 .
+(160)
+32
+
+Using (159) and (160) we can continue to estimate
+���
+ϕ
+���
+OσO† − σ
+�
+ϕ
+��� ≤ (∥Pϕ∥ ∥ϕ∥ + ∥Pϕ∥ ∥Qϕ∥)
+�
+tr
+�
+OσO† − σ
+�2� 1
+2 ≤ 2∥Pϕ∥
+�
+tr
+�
+OσO† − σ
+�2� 1
+2 .
+(161)
+This ends the proof.
+The next lemma states two simple algebraic bounds.
+Lemma 4. For p, q ∈ [c, 1 − c] the following bounds hold
+�
+(1 − p)(1 − q) + √pq ≥ 1 − |p − q|2
+4c(1 − c)
+(162)
+���
+�
+(1 − p)q −
+�
+(1 − q)p
+��� ≤
+|p − q|
+2
+�
+c(1 − c)
+.
+(163)
+Proof. We first consider the second inequality. We note that |√x−√y| ≤ |x−y|/(√x+√y) and thus
+���
+�
+(1 − p)q −
+�
+(1 − q)p
+��� ≤
+|p − q|
+�
+p(1 − q) +
+�
+q(1 − p)
+≤
+|p − q|
+2 4�
+p(1 − q)q(1 − p)
+≤
+|p − q|
+2
+�
+c(1 − c)
+(164)
+where we used the arithmetic geometric mean inequality in the middle step. To prove the first bound
+we note that
+��
+(1 − p)(1 − q) + √pq
+�2
++
+��
+(1 − p)q −
+�
+(1 − q)p
+�2
+= 1
+(165)
+This implies
+�
+(1 − p)(1 − q) + √pq =
+�
+1 −
+��
+(1 − p)q −
+�
+(1 − q)p
+�2
+≥ 1 −
+��
+(1 − p)q −
+�
+(1 − q)p
+�2
+≥ 1 − |p − q|2
+4c(1 − c).
+(166)
+The following simple lemma provides a lower bound on the square root that is used in the proof
+of Lemma 2 below.
+Lemma 5. Let s, t be real numbers such that s + t ≥ −1. Then the bound
+√
+1 + s + t ≥ 1 − |s| + t
+2 − t2
+2
+(167)
+holds.
+Proof. First we note that elementary manipulations show that for all t ∈ R the bound
+�
+max(1 + t, 0) ≥ 1 + t
+2 − t2
+2
+(168)
+holds. First we consider s > 0. In this case, we can conclude
+√
+1 + s + t ≥
+�
+max(1 + t, 0) ≥ 1 + t
+2 − t2
+2 ≥ 1 + t
+2 − t2
+2 − |s|.
+(169)
+33
+
+Note that x − y = (√x − √y)(√x + √y) > (√x − √y) if x > 1 and x > y. This implies for s < 0 and
+t ≥ 0 that
+√
+1 + t + s ≥
+√
+1 + t − |s| ≥ 1 + t
+2 − t2
+2 − |s|.
+(170)
+Finally, we consider the case t, s < 0 where we get from (168)
+√
+1 + t + s ≥ 1 + t + s
+2
+− (t + s)2
+2
+= 1 + t
+2 − t2
+2 + s
+�1 − t − s
+2
+�
+≥ 1 + t
+2 − t2
+2 − |s|
+(171)
+using −t − s ≤ 1 in the last step.
+Finally, we state a simple fact on the relation of partial trace and operators.
+Lemma 6. Let ρ be an operator on the system Q ⊗ R. Let O be a linear operator on Q. Then
+trR((O × Id)ρ) = O trR(ρ).
+(172)
+Proof. By linearity it is sufficient to consider ρ = S ⊗ T. But then
+trR((O × Id)ρ) = trR(OS × T) = tr(T)OS = O trR(ρ).
+(173)
+C
+Fidelity loss of oracle calls
+In this section we bound the loss in fidelity when applying the oracles Fp
+i and Fq
+i to density matrices
+ρ and σ. Recall that Fp
+i (ρ) = (1 − p)ρ + pOiρO†
+i .
+Fidelity bound for invariant operators
+Here we discuss a lemma that is useful to bound the loss
+in fidelity
+√
+F(ρ, σ) −
+√
+F(OρO†, σ) when it is known that Oψ = ψ for many states ψ (the eigenvalue
+1 has large multiplicity). Note that in terms of the oracles Fp
+i this corresponds to the case p = 1 and
+q = 0.
+Lemma 7. Let O be a unitary operator and P a hermitian projection such that
+P(O − Id) = (O − Id),
+(174)
+i.e., 1 − P projects on a subspace of the eigenspace of eigenvalue 1 of O and [P, O] = 0. Let ρ, σ be
+two density matrices. Then the bound
+√
+F(ρ, σ) −
+√
+F(ρ, OσO†) ≤ 2
+�
+tr(Pρ) tr(Pσ)
+(175)
+holds.
+Proof. Call the system on which ρ, σ act Q Let R be a copy of Q. Let ϕ and ψ be purifications of ρ
+and σ on the system QR such that
+√
+F(ρ, σ) = ⟨ϕ, ψ⟩. Then (O ⊗ Id)ψ is a purification of σ and we
+get
+√
+F(ρ, OσO†) ≥ ⟨ϕ, (O ⊗ Id)ψ⟩ = ⟨ϕ, ψ⟩ − ⟨ϕ, ((O − Id) ⊗ Id)ψ⟩
+=
+√
+F(ρ, σ) − ⟨ϕ, (P(O − Id) ⊗ Id)ψ⟩
+(176)
+34
+
+We now bound using [P, O] = 0 and Lemma 6
+⟨ϕ, (P(O − Id) ⊗ Id)ψ⟩ ≤ ⟨(P ⊗ Id)ϕ, (P ⊗ Id)((O − Id) ⊗ Id)ψ⟩
+≤ ∥(P ⊗ Id)ϕ∥ · ∥((O − Id) ⊗ Id)(P ⊗ Id)ψ∥
+≤ 2
+�
+tr trR((P ⊗ Id) |ϕ⟩ ⟨ϕ| (P ⊗ Id)†) tr trR((P ⊗ Id) |ψ⟩ ⟨ψ| (P ⊗ Id)†)
+� 1
+2
+≤ 2
+�
+tr(Pρ) tr(Pσ).
+(177)
+Proof of Lemma 1
+Proof. Call the system on which ρ, σ act Q and we denote ¯ρ = Fp
+i (ρ) and ¯σ = Fq
+i (σ) Let R be a copy
+of Q. Let ϕ and ψ be purifications of ρ and σ on the system QR such that
+√
+F(ρ, σ) = ⟨ϕ, ψ⟩. We
+use the shorthand ¯Oi = Oi ⊗ IdR in the following. Let S be a system consisting of a single qubit. We
+consider the following state on the system QRS
+ω =
+�
+1 − p |ϕ, 0⟩ + √p
+�� ¯Oiϕ, 1
+�
+.
+(178)
+It is easy to check that ω is a purification of ¯ρ
+trQR |ω⟩ ⟨ω| = trQ
+�
+(1 − p) |ϕ⟩ ⟨ϕ| + p
+�� ¯Oiϕ
+� � ¯Oiϕ
+���
+= (1 − p)ρ + pOiρO†
+i = ¯ρ.
+(179)
+To obtain a purification of ¯σ we define for an angle α the state
+ζ =
+�
+1 − q cos(α) |ψ, 0⟩ + √q sin(α)
+�� ¯Oiψ, 0
+�
+−
+�
+1 − q sin(α) |ψ, 1⟩ + √q cos(α)
+�� ¯Oiψ, 1
+�
+.
+(180)
+It is easy to check that ∥ζ∥ = 1. We now check that this is a purification of ¯σ. Note that the cross
+terms |ψ⟩
+� ¯Oiψ
+�� and
+�� ¯Oiψ
+�
+⟨ψ| cancel and thus
+trS |ζ⟩ ⟨ζ| = (1 − q) cos2(α) |ψ⟩ ⟨ψ| + q sin2(ω)
+�� ¯Oiψ
+� � ¯Oiψ
+��
++ (1 − q) sin2(α) |ψ⟩ ⟨ψ| + q cos2(α)
+�� ¯Oiψ
+� � ¯Oiψ
+�� .
+= (1 − q) |ψ⟩ ⟨ψ| + q
+�� ¯Oiψ
+� � ¯Oiψ
+�� .
+(181)
+We calculate
+⟨ω|ζ⟩ =
+�
+(1 − p)(1 − q) cos(α) ⟨ϕ|ψ⟩ +
+�
+(1 − p)q sin(α)
+�
+ϕ
+�� ¯Oiψ
+�
+−
+�
+p(1 − q) sin(α)
+� ¯Oiϕ
+��ψ
+�
++ √pq cos(α)
+� ¯Oiϕ
+�� ¯Oiψ
+�
+.
+(182)
+Using that Oi is self-adjoint and unitary we obtain
+⟨η|ζ⟩ =
+��
+(1 − p)(1 − q) + √pq
+�
+cos(α) ⟨ϕ|ψ⟩ +
+��
+(1 − p)q −
+�
+p(1 − q)
+�
+sin(α)
+�
+ϕ
+�� ¯Oiψ
+�
+.
+(183)
+Now we set
+sin(α) =
+�
+(1 − p)q −
+�
+p(1 − q)
+(184)
+cos(α) =
+�
+(1 − p)(1 − q) + √pq
+(185)
+and obtain
+⟨ω|ζ⟩ = cos2(α) ⟨ϕ|ψ⟩ + sin(α)2 � ¯Oiϕ
+��ψ
+�
+= ⟨ϕ|ψ⟩ + sin(α)2 � ¯Oiϕ − ϕ
+��ψ
+�
+.
+(186)
+35
+
+We conclude that
+√
+F(¯ρ, ¯σ) ≥ | ⟨η|ζ⟩ | ≥
+√
+F(ρ, σ) − | sin2(α)
+� ¯Oiϕ − ϕ
+��ψ
+�
+|.
+(187)
+As above we write ¯Pi = Pi ⊗ Id and P−i = Id − Pi and we get
+¯Oϕ − ϕ = ( ¯Pi + ¯P−i)( ¯Oiϕ − ϕ) = ¯Pi( ¯Oiϕ − ϕ) + ¯P−i( ¯Oiϕ − ϕ) = ¯Pi( ¯Oiϕ − ϕ).
+(188)
+Then we control using Lemma 6
+��� ¯Oiϕ − ϕ
+��ψ
+���2 =
+���
+( ¯Oiϕ − ϕ)
+�� ¯Piψ
+���2 ≤ ∥ ¯Pi( ¯Oiϕ − ϕ)∥2 ∥ ¯Piψ∥2
+= tr ¯Pi
+��� ¯Oiϕ − ϕ
+� � ¯Oiϕ − ϕ
+�� ¯P †
+i
+�
+tr
+��� ¯Piψ
+� � ¯Piψ
+���
+= tr trR
+�
+( ¯Oi − Id) ¯Pi |ϕ⟩ ⟨ϕ| ¯P †
+i ( ¯O†
+i − Id)
+�
+tr trR
+�
+¯Pi |ψ⟩ ⟨ψ| ¯P †
+i
+�
+≤ 4 tr (Piρ) tr (Piσ) .
+(189)
+Using the bound (163) from Lemma 4 implies
+| sin(α)| =
+�
+(1 − p)q −
+�
+p(1 − q) ≤
+|p − q|
+2
+�
+η(1 − η)
+.
+(190)
+From (187), (189), and (190) we conclude that
+√
+F(¯ρ, ¯σ) ≥ | ⟨η|ζ⟩ | ≥
+√
+F(ρ, σ) − (p − q)2
+�
+tr (Piρ) tr (Piσ)
+2η(1 − η)
+.
+(191)
+Proof of Lemma 2
+Proof. First, simple algebra shows ∥ ¯ψ0∥2 = 1 − q′ and
+Ep
+U(ρ) = ¯ρ0 + ¯ρ1 = (1 − p)ρ0 + pρ1
+(192)
+Eq
+U(|ψ⟩ ⟨ψ|) =
+�� ¯ψ0
+� � ¯ψ0
+�� +
+�� ¯ψ1
+� � ¯ψ1
+�� = (1 − q′) |ψ0⟩ ⟨ψ0| + q′ |ψ1⟩ ⟨ψ1| ,
+(193)
+in particular (82) holds. Strong concavity of the fidelity implies the first bound of (83). We now
+address the second estimate. We can express, using that U = U † is self adjoint
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, ψ0) =
+�
+(1 − p)(1 − q′) ⟨ψ0| ρ |ψ0⟩ =
+�
+(1 − p)
+� ¯ψ0
+�� ρ
+�� ¯ψ0
+�
+=
+�
+(1 − p)
+�
+(1 − q) cos2(α) ⟨ψ| ρ |ψ⟩ + q sin2(α) ⟨ψ| UρU † |ψ⟩
++ 2
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| ρU |ψ⟩)
+� 1
+2
+=
+�
+(1 − p)
+� �
+(1 − q) cos2(α) + q sin2(α) + 2
+�
+q(1 − q) cos(α) sin(α)
+�
+S2
++ q sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩
++ 2
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+� 1
+2
+(194)
+Now we calculate using the definition of α
+�
+(1 − p)
+��
+1 − q cos(α) + √q sin(α)
+�
+=
+�
+(1 − p)(1 − q)(
+�
+(1 − p)(1 − q) + √pq) +
+�
+(1 − p)q(
+�
+(1 − p)q −
+�
+(1 − q)p)
+= (1 − p)(1 − q) + (1 − p)q = 1 − p.
+(195)
+36
+
+Then we obtain
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, ψ0)
+= (1 − p)S
+�
+1 + q sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ + 2
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+(1 − p)S2
+� 1
+2 .
+(196)
+Similarly the second term can be expressed as
+�
+pq′√
+F(ρ1, ψ1) = √p
+� �
+q cos2(α) + (1 − q) sin2(α) − 2
+�
+q(1 − q) cos(α) sin(α)
+�
+S2
++ (1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ − 2
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+� 1
+2 .
+(197)
+We can calculate
+√p
+�√q cos(α) −
+�
+1 − q sin(α)
+�
+= √p
+�√q(
+�
+(1 − p)(1 − q) + √pq) −
+�
+1 − q(
+�
+(1 − p)q −
+�
+(1 − q)p
+�
+= pq + (1 − q)p = p.
+(198)
+we get
+�
+pq′√
+F(ρ1, ψ1) =
+= pS
+�
+1 + (1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ − 2
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+pS2
+� 1
+2 .
+(199)
+We continue to estimate the square root terms. Note that by first order Taylor expansion the mixed
+last terms of the two expressions (196) and (199) cancel and only the first term and higher order
+corrections remains. To make this rigorous we use Lemma 5 in Appendix B which states that for
+s + t ≥ −1 the bound
+√
+1 + s + t ≥ 1 + t
+2 − t2
+2 − |s|
+(200)
+holds. We apply this to (196) and (199) where s corresponds to the term involving ⟨ψ| (UρU † − ρ) |ψ⟩
+and t corresponds to the term involving Re(⟨ψ| (ρU − ρ) |ψ⟩)2. Then we get from (196)
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, ψ0)
+≥ (1 − p)S
+�
+1 + −q sin2(α)| ⟨ψ| (UρU † − ρ) |ψ⟩ | +
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+(1 − p)S2
+− q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)2
+2(1 − p)2S4
+�
+.
+(201)
+From (199) we get similarly
+�
+pq′√
+F(ρ1, ψ1) =
+≥ pS
+�
+1 + −(1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ −
+�
+q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+pS2
+− q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)
+2p2S4
+�
+.
+(202)
+37
+
+We notice that the linear terms in Re(⟨ψ| (ρU − ρ) |ψ⟩) cancel and we get
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, ψ0) +
+�
+pq′√
+F(ρ1, ψ1)
+≥ S − sin2(α)| ⟨ψ| (UρU † − ρ) |ψ⟩ |
+S
+−
+�1
+p +
+1
+1 − p
+� q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ). |ψ⟩)2
+2S3
+.
+(203)
+Finally we can estimate using the assumption p, q ∈ [η, 1 − η] and Lemma 4
+�
+(1 − p)(1 − q′)
+√
+F(ρ0, ψ0) +
+�
+pq′√
+F(ρ1, ψ1)
+≥ S − |p − q|2 · | ⟨ψ| (UρU † − ρ) |ψ⟩ |
+4η(1 − η)S
+−
+�1
+η +
+1
+1 − η
+� |p − q|2Re(⟨ψ| (ρU − ρ). |ψ⟩)2
+32η(1 − η)S3
+.
+(204)
+This ends the proof.
+We now sketch the proof of Corollary 4. Unfortunately we cannot directly derive the result but
+we need to slightly modify the proof of Lemma 1.
+Proof of Corollary 4. The proof proceeds exactly in as the proof of Lemma 1 with the following minor
+modifications. We have to insert an additional term p⟨ ¯ψ0, σ ¯ψ0⟩ in (194) and a term −(1 − p)⟨ ¯ψ1, σ ¯ψ1⟩
+in (197). We carry those terms and when we estimate the square-root terms we add them to the
+s part in the bound (200), thus we end up with an additional term −p|⟨ ¯ψ0, σ ¯ψ0⟩|/S in (201) and
+−(1 − p)|⟨ ¯ψ1, σ ¯ψ1⟩|/S in (202) and thus their sum appears in (204). This ends the proof.
+D
+Proof of Theorem 4
+Here we prove the lower bound in Theorem 4. Let us first give a precise statement of the result. We
+consider the same probability vectors pi as introduced at the beginning of Section 3. First, we note that
+the result does not hold for fixed oracles Oi with reward vector pi because the algorithm could exploit
+the specific structure of the oracles. There could be, e.g., a state ω0 such that O(pi) |j⟩ |ω0⟩ |0⟩ =
+|j⟩ |ω0⟩ |δij⟩. Then the problem reduces to the unstructured search problem when ω0 is known. Thus,
+the result only holds when we assume that Oi is a random oracle with the fixed reward vector pi,
+emulating the situation where we have no additional information about the oracles. We consider for
+a reward vector r ∈ {0, 1}|HA|·|HP | the oracle Or acting as in (4), i.e.,
+Or |i⟩ |ω⟩ |c⟩ = |i⟩ |ω⟩ |c + ri(ω)⟩ .
+(205)
+We consider random reward distribution r(i) where r(i) is the uniform distribution over all reward
+vectors with mean reward vector pi, i.e., |HP |−1 �
+ω r(i)j(ω) = (pi)j. Then a more precise version of
+Theorem 4 reads as follows.
+Theorem 11. Let δ < 1/2. Assume that pj are as before with pi ∈ [η, 1 − η] for some η > 0. Any
+algorithm that identifies the best arm with probability at least 1 − δ given an oracle Or(i) where r(i) is
+distributed as above requires at least
+T ≥ 1
+2η(1 − 2
+�
+δ(1 − δ))
+� n
+�
+i=2
+∆−2
+i
+� 1
+2
+≥ c(δ, η)
+�
+H(p1)
+(206)
+calls to the oracle.
+This is still true even when it is known that the vector of mean rewards is
+{p0, . . . , pn}.
+38
+
+Proof. The proof is close to the proof of Theorem 8. We assume we are given any algorithm acting
+by (EO ⊗ Id) ◦ EUT ◦ . . . ◦ (EO ⊗ Id) ◦ EU1 where Ut are arbitrary unitary maps where O denotes the
+given oracle. Assume that the initial state is a fixed density matrix ρ. We denote the state using the
+oracle Or(i) before the t + 1-th invocation of the oracle by ρr(i)
+t
+, i.e., ρr(i)
+0
+= EU1(ρ). We introduce the
+notation Ei when we average over the reward distribution r(i) and we write
+ρi
+t = Eiρr(i)
+t
+.
+(207)
+By assumption we can identify i given ρi
+T with probability at least 1 − δ. This implies as in (26), (57)
+for i > 1
+2
+�
+δ(1 − δ) ≥
+√
+F(ρi
+T , ρ0
+T ).
+(208)
+One main ingredient of the proof is to define a suitable coupling of the random variables r(0) and
+r(i). Note that its reward vectors are p0 and pi such that (pi)j = pj = (p0)j for j ̸= i and (pi)i = p0,
+(p0)i = pi. We now consider a coupling where r(0)j = r(i)j for i ̸= j and r(i)i ≥ r(0)i and the
+distribution of r(i)i ∈ {0, 1}|HP | is uniform over all rewards under this constraint for a fixed r(0).
+Note that this entails that the distribution of r(0) for a fixed r(i) is also uniform over all rewards
+with the right mean reward satisfying r(0) ≤ r(i). It is straightforward to see that such a coupling
+exists (a possible explicit construction is to draw i.i.d. random numbers ui(ω) and set r(0)i(ω) = 1 iff
+ui(ω) is in the pi-th quantile of the numbers ui(·) and similarly for r(i)). We denote this coupling by
+pi(r(i), r(0)). Observe that for this coupling satisfies, for all ω,
+P(r(i)j(ω) = 1|r(0)j(ω) = 0) = P(r(i)j = 1|r(0)j = 0) =
+�
+0
+for j ̸= i
+p0−pi
+1−pi =
+∆i
+1−pi
+for j = i.
+(209)
+Similarly, we get for all ω
+P(r(0)j(ω) = 0|r(i)j(ω) = 1) = P(r(0)j = 0|r(i)j = 1) =
+�
+0
+for j ̸= i
+p0−pi
+p0
+= ∆i
+p0
+for j = i.
+(210)
+We can bound using the concavity of the fidelity
+√
+F(ρi
+t, ρ0
+t) ≥
+�
+r(i),r(0)
+pi(r(i), r(0))
+√
+F(ρr(i)
+t
+, ρr(0)
+t
+).
+(211)
+Now we lower bound the fidelity terms on the right hand side based on our construction of the coupling.
+By construction we have r(i) ≥ r(0) which implies that r(i) − r(0) ∈ {0, 1}|HA|·|HP |. Note that
+Or(i)Or(0) |j, ω, c⟩ = |j, ω, c + (r(i))j(ω) + (r(0))j(ω)⟩ = |j, ω, c + (r(i))j(ω) − (r(0))j(ω)⟩
+= Or(i)−r(0) |j, ω, c⟩ .
+(212)
+Let us introduce the self adjoint projection P r(i),r(0) given by
+P r(i),r(0) |j, ω, c⟩ = 1(r(i))j(ω)̸=(r(0))j(ω) |j, ω, c⟩ .
+(213)
+Note that then
+(Id − P r(i),r(0))Or(i)−r(0) |j, ω, c⟩ = (Id − P r(i),r(0)) |j, ω, c⟩ .
+(214)
+We get, using the invariance of the fidelity under unitary transformations, (Or(i))2 = Id, and (212)
+√
+F(ρr(i)
+t+1, ρr(0)
+t+1 ) =
+√
+F(EOr(i)(ρr(i)
+t
+), EOr(0)(ρr(0)
+t
+)) =
+√
+F(ρr(i)
+t
+, EOr(i) ◦ EOr(0)(ρr(0)
+t
+))
+=
+√
+F(ρr(i)
+t
+, EOr(i)−r(0)(ρr(0)
+t
+)).
+(215)
+39
+
+Next we apply Lemma 7 to Or(i)−r(0) and P r(i),r(0). The assumptions of the lemma are satisfied
+because (214) and all P r(i),r(0) and Or commute. Lemma 7 together with the last display imply
+√
+F(ρr(i)
+t+1, ρr(0)
+t+1 ) ≥
+√
+F(ρr(i)
+t
+, ρr(0)
+t
+) − 2
+�
+tr
+�
+P r(i),r(0)ρr(i)
+t
+�
+tr
+�
+P r(i),r(0)ρr(0)
+t
+�
+(216)
+We obtain
+√
+F(ρi
+T , ρ0
+T ) ≥ 1 − 2
+�
+r(i),r(0)
+�
+t
+pi(r(i), r(0))
+�
+tr
+�
+P r(i),r(0)ρr(i)
+t
+�
+tr
+�
+P r(i),r(0)ρr(0)
+t
+�
+.
+(217)
+Using (208) followed by Cauchy-Schwarz we get
+1
+2 −
+�
+δ(1 − δ) ≤
+�
+r(i),r(0)
+�
+t
+pi(r(i), r(0))
+�
+tr
+�
+P r(i),r(0)ρr(i)
+t
+�
+tr
+�
+P r(i),r(0)ρr(0)
+t
+�
+≤
+�
+�
+�
+t,r(i),r(0)
+pi(r(i), r(0)) tr
+�
+P r(i),r(0)ρr(i)
+t
+�
+�
+�
+1
+2 �
+�
+�
+t,r(i),r(0)
+pi(r(i), r(0)) tr
+�
+P r(i),r(0)ρr(0)
+t
+�
+�
+�
+1
+2
+.
+(218)
+We observe that by construction of the coupling and (209) we have
+�
+r(i)
+pi(r(i), r(0))P r(i),r(0) |j, ω, c⟩ =
+�
+r(i)
+pi(r(i), r(0))1(r(i))j(ω)̸=(r(0))j(ω) |j, ω, c⟩
+= p(r(0))P((r(i))j(ω) = 1|r(0)) |j, ω, c⟩
+= p(r(0))1i=j1r(0)i(ω)=0
+∆i
+1 − pi
+|j, ω, c⟩ .
+(219)
+And similarly, using (210)
+�
+r(0)
+pi(r(i), r(0))P r(i),r(0) |j, ω, c⟩ = p(r(i))1i=j1r(i)i(ω)=1
+∆i
+p0
+|j, ω, c⟩ .
+(220)
+As before, we define Pi to be the projection on arm i, i.e., Pi |j, ω, c⟩ = δij |j, ω, c⟩. We conclude that
+�
+r(i)
+pi(r(i), r(0)) tr
+�
+P r(i),r(0)ρr(0)
+t
+�
+≤ p(r(0))∆i
+η tr
+�
+Piρr(0)
+t
+�
+,
+(221)
+�
+r(0)
+pi(r(i), r(0)) tr
+�
+P r(i),r(0)ρr(i)
+t
+�
+≤ p(r(i))∆i
+η tr
+�
+Piρr(i)
+t
+�
+.
+(222)
+Combining this with (218) and (207) we obtain
+1
+2 −
+�
+δ(1 − δ) ≤
+�
+�∆i
+η
+�
+t,r(i)
+p(r(i)) tr
+�
+Piρr(i)
+t
+�
+�
+�
+1
+2 �
+�∆i
+η
+�
+t,r(0)
+p(r(0)) tr
+�
+Piρr(0)
+t
+�
+)
+�
+�
+1
+2
+= ∆i
+η
+��
+t
+tr
+�
+Piρi
+t
+�
+� 1
+2 ��
+t
+tr
+�
+Piρ0
+t
+�
+� 1
+2
+≤ ∆i
+√
+T
+η
+��
+t
+tr
+�
+Piρ0
+t
+�
+� 1
+2
+.
+(223)
+We square this relation divide by ∆2
+i and sum over i > 1 and get
+�1
+2 −
+�
+δ(1 − δ)
+�2 �
+i>1
+∆−2
+i
+≤ T
+η2
+�
+i
+�
+t
+tr
+�
+Piρ0
+t
+�
+= T
+η2
+�
+t
+tr
+�
+ρ0
+t
+�
+= T 2
+η2 .
+(224)
+40
+
+This implies
+T ≥ 1
+2η(1 − 2
+�
+δ(1 − δ))
+��
+i>1
+∆−2
+i .
+(225)
+E
+Proof of Theorem 9
+Proof. We assume that we work on a Hilbert space H = HI ⊗ HS ⊗ HA where HI is the input space,
+HS is a single qubit state space and HA consists of T ancilla qubits where T = O(
+√
+N/p) will be
+defined below in (234). We assume that the initial state of HI is |s⟩ =
+√
+N
+−1 � |i⟩ and all remaining
+qubits are in state |0⟩. The algorithm consists of applying in turn the three operators F, a controlled
+Grover-diffusion
+Uω = id ⊗ |0⟩ ⟨0| ⊗ id + (2 |s⟩ ⟨s| − id) ⊗ |1⟩ ⟨1| ⊗ id,
+(226)
+and a swap operation St that swaps the state qubit with the t-th ancilla qubit. We can rewrite this
+concisely as
+(EST ◦ EUω ◦ F) ◦ . . . ◦ (ES1 ◦ EUω ◦ F)(ρ).
+(227)
+We denote by ˜Oi the standard oracle on HI and denote by G = (2 |s⟩ ⟨s|−id) ˜Oi the map from Grover’s
+algorithm acting on HI. We claim that the final state of the algorithm is
+ρT =
+�
+x∈{0,1}T
+p|x|1(1 − p)T −|x|1
+���G|x|1s, 0, x, 0A−T � �
+G|x|1s, 0, x, 0A−T ��� .
+(228)
+The proof of this is by induction, indeed, we note
+(EST ◦ EUω ◦ E)(
+���G|x|1s, 0, x, 0A−T � �
+G|x|1s, 0, x, 0A−T ���)
+= (EST ◦ EUω)
+�
+p
+��� ˜OG|x|1s, 1, x, 0A−T � �
+˜OG|x|1s, 1, x, 0A−T ���
++(1 − p)
+���G|x|1s, 0, x, 0A−T � �
+G|x|1s, 0, x, 0A−T ���
+�
+= EST
+�
+p
+���GG|x|1s, 1, x, 0A−T � �
+GG|x|1s, 1, x, 0A−T ���
++(1 − p)
+���G|x|1s, 0, x, 0A−T � �
+G|x|1s, 0, x, 0A−T ���
+�
+= p
+���GG|x|1s, 0, x, 1, 0A−T −1� �
+GG|x|1s, 0, x, 1, 0A−T −1���
++ (1 − p)
+���G|x|1s, 0, x, 0A−T � �
+G|x|1s, 0, x, 0A−T ���
+(229)
+which implies (228). The reduced density matrix on HI is
+ρI
+T =
+T
+�
+k=0
+pk(1 − p)T −k ��Gks
+� �
+Gks
+�� .
+(230)
+The classical analysis of the Grover algorithm proves
+Gk |s⟩ = cos
+�2k + 1
+2
+θ
+�
+|s′⟩ + sin
+�2k + 1
+2
+θ
+�
+|i⟩
+(231)
+41
+
+where s′ =
+√
+N − 1
+− 1
+2 �
+j̸=i |i⟩ and
+θ = 2 arccos
+��
+(N − 1)/N
+�
+= 2 arcsin
+�√
+N
+−1�
+.
+(232)
+We remark that θ/2 ≈
+√
+N
+−1 as N → ∞. Note that for
+k ∈ I = (π/(4θ) − 1/2, 3π/(4θ) − 1/2)
+(233)
+we have sin
+� 2k+1
+2
+θ
+�2 ≥ 1/2. Let
+T = ⌊π/(2θp)⌋.
+(234)
+It remains to be shown that with probability at least 1/2 a Bin(T, p) distributed variable is contained
+in the interval I.
+Using that the variance of X ∼ Bin(T, p) is Tp(1 − p) we can bound
+P(|X − pT| > pT/8) ≤ 64E(|X − pT|2
+p2T 2
+≤ 64
+pT ≤ 256θ
+π
+≤ 1
+2
+(235)
+for N sufficiently large. Moreover, |X − pT| < pT/8 implies
+X ∈
+�3pT
+8 , 5pT
+8
+�
+⊂
+�3π
+8θ − 1, 5π
+8θ
+�
+⊂
+� π
+4θ − 1/2, 3π
+4θ − 1/2
+�
+(236)
+for N sufficiently large. This ends the proof.
+F
+Analysis of reusable oracles
+Here we sketch a proof of Theorem 5. The proof essentially relies on the algorithm given in [34].
+The only building block that needs to be changed is the gapped amplitude estimation (Corollary 2 in
+[34]). Let us explain this algorithm along with the replacement based on oracles as in (7). Gapped
+amplitude estimation assumes we have access to an oracle Op and its adjoint acting via
+Op |0⟩ = √p |1⟩ +
+�
+1 − p |0⟩ = |coinp⟩ .
+(237)
+Then the following result holds.
+Lemma 8 (Corollary 2 in [34]). For ε > 0, l ∈ [0, 1] and δ > 0 there is a unitary procedure with
+O(ε−1 ln(δ)) queries to Op that prepares the state
+|coinp⟩ (α0 |0⟩ |ψ1⟩ + α1 |1⟩ |ψ2⟩
+(238)
+with α0, α1 ∈ [0, 1] and α1 ≤ δ if p ≤ l − 2ε and α0 ≤ δ if p ≥ l − ε.
+We replace this by classical estimation of the mean. Using tail bounds of random variables we
+obtain the following simple lemma.
+Lemma 9. Let Sk be the sum of k independent random variables with distribution Ber(p). For ε > 0
+the bounds
+P(Sk > k(p + ε)) ≤ e−2ε2k,
+P(Sk < k(p − ε)) ≤ e−2ε2k
+(239)
+hold.
+42
+
+Proof. This is just Hoeffding’s inequality.
+We obtain the following corollary.
+Corollary 5. Given access to oracles as in (7) we can construct for any ε, δ > 0 and l ∈ [0, 1] an
+algorithm A that maps with probability at least 1 − δ for all 1 ≤ i ≤ n
+A |i⟩ |0⟩ = |i⟩ |c⟩
+(240)
+where c = 1 if pi < l − 2ε and c = 0 if pi > l − ε and A requires ε−2 oracle calls.
+Proof. Set k = ⌈2ε−2 ln(n/δ)⌉. Then we consider the sequence
+|i⟩ |0⟩ |0⟩ → |i⟩
+������
+k
+�
+j=0
+Xt
+i
+�
+|0⟩ → |i⟩
+������
+k
+�
+j=0
+Xt
+i
+� ���1�k
+j=0 Xt
+i <k(l−3ε/2)
+�
+→ |i⟩ |0⟩
+���1�k
+j=0 Xt
+i <k(l−3ε/2)
+�
+.
+(241)
+Where we in the first step sum the rewards using the oracles OXt, then set the flag register conditional
+on the sum and then uncompute the work register. This requires 2k oracle calls. Using Lemma 9 we
+conclude that for pi > l − ε
+P
+�
+�
+k
+�
+j=0
+Xt
+i < l − 3ε/2
+�
+� ≤ e−2k(ε/2)2 ≤ e− ln(n/δ) = δ
+n
+(242)
+and a similar statement holds for pi < l − 2ε. The union bound implies the statement.
+Now we consider Theorem 5. Giving a full proof would require a large amount of notation that is
+not worth the effort. Thus, we give a very brief sketch of the argument and leave the details to the
+reader. Their proof relies on variable time algorithms [4] which we also do not introduce here. For
+readers not familiar with them, the following proof shall merely serve as a heuristic.
+Sketch of the proof of Theorem 5. We apply the algorithm constructed in [34] but replace the gapped
+amplitude estimation by the algorithm constructed in Corollary 5. The main strategy used in their
+proof is to construct a variable time algorithm based on the gapped amplitude estimate that flags all
+arms whose reward is smaller than a given threshold. This allows to construct algorithms to count
+the number of flagged arms and rotate on the subspace of flagged arms using variable time amplitude
+amplification [4]. Those sub-routines can be used to first estimate p1 and p2 and then identify the
+best arm.
+Their variable time algorithm based on the gapped amplitude estimate has query complexity on
+arm i is at most ∆−1
+i
+log(1/a) with a polynomial in ∆1n and the l2 averaged run-time thus amounts
+to
+t2
+av ≤ C 1
+n
+n
+�
+i=2
+∆−2
+i
+ln2(1/a).
+(243)
+When relying on the algorithm from Corollary 5 we obtain the query complexity ∆−2
+i
+log(n/δ′) for
+arm i where δ′ denotes the bound on the failure probability for the algorithm and the l2 averaged
+run-time then amounts to
+t2
+av ≤ C 1
+n
+n
+�
+i=2
+∆−4
+i
+ln2(nδ′).
+(244)
+43
+
+Then the variable time amplitude amplification gives an algorithm with success probability more than
+1/2 and query complexity tav/√psucc ln(tmax) where psucc denotes the success probability and tmax
+the maximal complexity of the initial variable algorithm (there is another term tmax ln(tmax) which is
+smaller in our case).
+Since the initial success probability is n−1 we obtain the query complexity bound
+T ≤ tav
+√n ln(tm) ≤ C
+� n
+�
+i=2
+∆−4
+i
+� 1
+2
+ln(n/δ′) ln(tmax).
+(245)
+We need to apply a total of O(ln
+�
+∆−1
+1
+�
+such amplified variable time algorithms (essentially we perform
+binary search to find lL < lR such that p2 < lL − ∆2/4, lR + ∆2/4 < p1, and lL + ∆2/4 < lR). To
+ensure that all constructed oracles as in Corollary 5 succeed with probability more than 1 − δ it is
+thus sufficient to pick δ′ = cδ/ ln
+�
+∆−1
+1
+�
+. This together with tmax = ˜O(∆−2
+1 ) ends the sketch.
+44
+
diff --git a/d9FAT4oBgHgl3EQfZR1S/content/tmp_files/load_file.txt b/d9FAT4oBgHgl3EQfZR1S/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2b04572b6d4864fef53f066bb876cf9b83feedab
--- /dev/null
+++ b/d9FAT4oBgHgl3EQfZR1S/content/tmp_files/load_file.txt
@@ -0,0 +1,1496 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf,len=1495
+page_content='Multi armed bandits and quantum channel oracles  Simon Buchholz, Jonas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Kübler, Bernhard Schölkopf  23rd January 2023  Abstract  Multi armed bandits are one of the theoretical pillars of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Recently,  the investigation of quantum algorithms for multi armed bandit problems was started, and it was  found that a quadratic speed-up is possible when the arms and the randomness of the rewards  of the arms can be queried in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Here we introduce further bandit models where we  only have limited access to the randomness of the rewards, but we can still query the arms in  superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We show that this impedes any speed-up of quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  1  Introduction  Quantum computing is a model of computation that is based on quantum properties of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' By  using superposition and entanglement, it offers potentially large speed-ups when compared to classical  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For some problems exponential speed-ups haven been shown under the assumption of  widely believed hardness results for classical computing, the two most prominent examples being  Shor’s algorithm for factoring integers [32] and the HHL algorithm for sampling from the solution of  sparse linear equations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' On the other hand, it was shown that for many problems only polynomial  speed-ups are possible, in particular Grover’s algorithm [16] for the unstructured search problem only  offers a quadratic speed-up and no greater improvement is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Recently, quantum machine learning has emerged as one potential area of application for quantum  computers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It was suggested to use quantum computers for linear algebra subroutines but also  complete implementations of well-known classical algorithms for supervised learning were designed,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', quantum support vector machines [29], quantum principal component analysis [23], and recom-  mender systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There has also been some work on unsupervised learning and reinforcement  learning [2, 19, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For a recent review, we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Reinforcement learning is a machine learning paradigm that is concerned with learning actions that  maximize a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Arguable, the simplest model capturing the essentials of reinforcement learning  is the multi-armed bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There one tries to identify the arm with the highest mean reward  from several alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Classical algorithms for the multi armed bandit problem have been studied  for a long time and almost optimal results were derived [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The problem of best arm identification using quantum computers was considered recently in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  They show that a quadratic speed-up compared to the classical algorithm is possible and optimal in  their setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Their implementation of the bandit arms is, however, not directly comparable to the  classical setting because the bandit arms are implemented through an oracle that acts deterministically  and the randomness originates from the difficulty to discern quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  In this work, we will discuss when this multi armed bandit model is applicable and we will discuss  further models of multi armed bandits that are more suitable in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Those models will  involve additional randomness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', the oracle implementing the bandit will act as a quantum channel  instead of a unitary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This provides a link to the channel discrimination problem, however, there  Max Planck Institute for Intelligent Systems, Tübingen  {sbuchholz, jmkuebler, bs}@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='de  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='08544v1  [quant-ph]  20 Jan 2023  the focus has been on very different channels that are either generic or more related to the transmission  of information [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Here we show that the additional randomness prevents any polynomial speed-up compared to  classical algorithms even though the bandits can be queried in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It was shown in [30]  that no speed-up for unstructured search is possible if the oracle has a fixed probability of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Our  results can be seen as an extension to a substantially more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus, this work highlights  that quantum speed-ups can be impeded by (small) amounts of classical randomness present in the  algorithm, thus also underlining again that having parts of the computation routine a that are not  error corrected pose challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' From a technical side, we connect classical methods from quantum  information theory with coupling techniques from probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In the next section, we introduce different settings of  quantum bandits and give an overview of their query complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we give a quantum inspired  proof of the classical result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Afterward, in Section 4 we consider quantum channel oracles  and in particular the setting of the faulty Grover oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In Section 5 we then consider non-adaptive  algorithms for the bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Finally, in Section 6 we prove our main result Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The  proof combines the techniques used in the three sections before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2  Main results and setting  We now discuss the setting for our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In this section, we review the relevant background  on (classical) bandits and our main results for quantum bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1  Classical bandits  We now discuss the well-known classical multi-armed bandit problem, give the key results and the  underlying heuristics, to give the necessary background to readers not familiar with the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In  the multi-armed bandit problem, an agent can choose an arm i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , N} in every turn and  receives a probabilistic reward νi depending on the chosen arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Typically, the goal is to maximize  the reward or, equivalently, minimize the regret which denotes the difference to the reward obtained  when always choosing the most favorable arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is one of the simplest models for reinforcement  learning featuring the exploration-exploitation tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' A slightly simpler problem is the best-arm  identification problem in the fixed confidence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Here the goal is to identify the arm with the  highest expected reward with a given probability 1 − δ for some δ > 0 with the fewest number of  rounds possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider mean reward vectors p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pN) ∈ RN, indicating that arm i has  average reward pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We usually assume that the rewards are ordered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', p1 ≥ p2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ≥ pN and that  the rewards are Bernoulli Ber(pi) distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In other words, when pulling arm i we receive reward 1  with probability pi and reward 0 with probability 1 − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We will use the shorthand ∆i = pi − p1 for  the difference in mean rewards between the best arm and arm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We define the important quantity  H(p) =  �  i>1  1  (p1 − pi)2 =  �  i>1  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (1)  It can be shown that H(p) governs the optimal time to identify the best arm (up to logarithmic terms)  in a fixed confidence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, this is optimal when pi ∈ [η, 1 − η] for all i and some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  To make this concrete, we state this as a well-known theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 1 (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 2 in [13], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 5 in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Consider an algorithm that identifies the best arm of a  multi armed bandit with probability at least 1 − δ for Bernoulli distributed rewards with reward vector  p ∈ [0, 1]N such that pi ∈ [η, 1 − η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then this algorithm requires O(H(p)) rounds in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  On the other hand, there exists such an algorithm requiring ˜O(H(p)) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us emphasize that identification of the best arm is not simpler if we know the rewards up to  a permutation, in fact, the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2  Theorem 2 (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 4 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let p ∈ [η, 1−η]N be a reward vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then any algorithm that identifies  the best arm for Bernoulli distributed rewards with means p′ where p′ is any permutation of p requires  at least O(H(p)) rounds and such an algorithm exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us add several remarks to these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There is a long list of results improving upon the two  results above by deriving also bounds on the logarithmic correction and analyzing various algorithms,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', [14, 18, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The assumption that the variables are Bernoulli distributed is for convenience,  many generalizations are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' One advantage is that Bernoulli distributions readily generalize to  the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us explain for the readers not so familiar with the literature on bandit problems the intuition  underlying the results mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  A Ber(p) variable has variance p(1 − p) which is lower  bounded by η(1 − η) for all arms by our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Suppose we pulled arm i for ti times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the  empirical mean ˆpi will be approximately distributed according to a Gaussian variable (by the central  limit theorem) with mean pi and standard deviation  �  tipi(1 − pi)/ti = O(t  − 1  2  i  ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', in distribution  ˆpi ≈ pi +  �  pi(1 − pi)  √ti  N  (2)  where N denotes a standard normal variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' If we want to conclude that pi < p1 with high probability  we need P( ˆpi > p1) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is the case iff  t  − 1  2  i  ≲ p1 − pi = ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (3)  We conclude that ti ≈ (p1 − pi)−2 = ∆−2  i  pulls on lever i are necessary to exclude the possibility that  arm i has the highest reward with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Applying this reasoning to all arms j suggests  the lower bound H(p) for the query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us emphasize that this argument crucially uses that all rewards are away from 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Otherwise, the variance term pi(1 − pi) cannot necessarily assumed to be a fixed constant in the  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To illustrate this, we consider the particular case that p1 = p > 0 and pi = 0 for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  H(p) = N/p2 but only O(N/p) pulls are required to identify the best arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To see this, note that it  takes with high probability O(p−1) pulls to get a single success on and arm with Ber(p) distributed  rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is the setting considered in [30] and we will come back to it in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  Quantum bandits  The investigation of quantum bandits was started recently [9, 34] and so far focused on the best arm  identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In this section, we want to review the results and definitions in the literature  and in particular highlight that different assumptions on the available oracles are reasonable in different  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume that there are n arms with Bernoulli distributed rewards with mean Ber(pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To  query the arms we consider a Hilbert-space HA for the arms with dimension |Ha| = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover,  we assume that the internal randomness of the reward is captured through an additional Hilbert-  space HP and the output indicating the reward for a certain arm and state of internal randomness is  collected in a two-dimensional Hilbert space HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We then consider rewards that are given through an  oracle O acting on a state |i⟩ |ω⟩ |c⟩ inHA ⊗ HP ⊗ HR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', arm i, internal randomness ω an element  from a fixed basis of HP , and initial reward state c by  O |i⟩ |ω⟩ |c⟩ = |i⟩ |ω⟩ |c + ri(ω)⟩  (4)  where ri(ω) ∈ {0, 1} denotes the reward for arm i with internal randomness ω and addition is modulo  2 (a similar definition was recently considered in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, we assume that when averaging over  this basis of HP we obtain  |HP |−1 �  ω  ri(ω) = pi,  (5)  3  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', the average reward of arm i is pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As before, we collect the mean rewards pi in a vector p ∈ [0, 1]N  with pi = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then it is sometimes convenient to clarify the mean rewards of the oracle by writing  Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that this does not determine the oracle as the dependence on ω is not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We now consider different settings that are characterized by the degree of control that we have  over the space HP determining the internal randomness of the bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider the following  three settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We have full control over the space HP , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', we can apply arbitrary unitary operators to the  system HA ⊗ HP ⊗ HR and a potential work space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We can prepare any pure basis state ω ∈ HP but have no further access to HP except through  the oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The state of HP will always be a pure state ω which is unobserved and selected uniformly at  random from the fixed basis and resampled after each invocation of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We now give a different (almost equivalent) characterisation of the second and third setting described  above which eliminates the Hilbert space HP modelling the internal randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For this it is con-  venient to define for X ∈ {0, 1}N the oracle OX acting on HA ⊗ HR by  OX |i⟩ |c⟩ → |i⟩  ��c + Xi�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (6)  Let us assume that Xt is a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' random variables where Xt  i is Ber(pi) distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  we can phrase the second scenario above almost equivalently as having access to the sequence of oracles  OXt |i⟩ |c⟩ = |i⟩  ��c + Xi  t  �  ,  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='.  (7)  The third setting corresponds to having access to an oracle channel that acts by  E(ρ) =  �  X∈{0,1}n  P(X) OXρ(OX)†  where P(X) =  �  i  pXi  i (1 − pi)1−Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (8)  The minor difference between the description above is that there we considered a finite dimensional  space HP for the internal randomness, the settings become equivalent as the dimension of HP grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Note that the second and the third setting are equivalent if we are allowed to use each of the oracles  OXt only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Multiple invocations can be useful to uncompute parts of the computation, avoiding  that the state of the system decoheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As we will discuss next the appropriate model depends on the  concrete situation at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' They cannot directly be compared to the setting of classical bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3  Models  We now motivate the three settings described above in a bit more detail and discuss our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us emphasize that quantum bandits can only be useful when the reward is given by the observable  of a quantum system or the evaluation of a computation on a quantum device as the acquisition of  the data is commonly seen as the expensive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It does not seem plausible that we collect classical  data store them in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', a QRAM to query them in superposition to identify the best arm as this  will always be more expensive than classically evaluating the mean of the collected rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus, we  will motivate all three settings from a quantum perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that the classical setting roughly  corresponds to the case where only queries of the form |i⟩ |ω⟩ without superposition are allowed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=',  we can query neither a single arm for a single reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us also mention that minimization of regret  (at least when defined in the standard way) only makes sense in the third setting where we have no  access to the sampling randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Otherwise, we can just resample a fortunate realization of the  reward to incur no regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  4  Oracle  Lower bound  Upper bound  Classical  O(H(p))  ˜O(H(p))  ERM (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (4))  O(  �  H(p)) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 4)  ˜O(  �  H(p)) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1 in [34])  Reusable (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (7))  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  ˜O(  ��  i ∆−4  i ) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 5)  One-time (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (8))  O(H(p)) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 6)  ˜O(H(p))  Table 1: Overview of query complexity bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The upper bound for the one-time oracle and the  reusable oracle follow from the classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We conjecture (see Conjecture 1) that the upper bound  for the reusable oracle are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1  Empirical risk minimisation  As already explained in [34], one setting where such an oracle could arise is empirical risk minimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  To make this concrete, assume we have a dataset (xj, yj) ∈ X ×{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , K} and a finite set of candidate  functions fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now want to find the index i0 such that  i0 = arg min  i  �  j  1fi(xj)̸=yj = arg max  i  �  j  1fi(xj)=yj,  (9)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', for simplicity we consider 0-1 loss in a classification problem or equivalently, we maximize the  accuracy over the functions fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' If we can access xj and evaluate fi this provides us with an oracle  acting by  |i⟩ |ω⟩ |0⟩ → |i⟩ |ω⟩ |ri(ω)⟩ ,  (10)  where the reward for ω = (x, y) is given by ri(ω) = 1y=fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Now the problem of best arm identi-  fication with respect to this oracle is equivalent to empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, this oracle  is exactly of the form introduced in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' When considering this setting it is a reasonable assumption  that the dataset can be accessed in arbitrary superposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', is stored in our computing device,  and we can also evaluate functions fi and thus losses in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The problem of empirical risk  minimization and the relation to bandit problems was considered before in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' where they consider  an oracle that acts as  |i⟩ |0⟩ → |i⟩ (√pi |1⟩ +  �  1 − pi |0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (11)  The relation to our setting is that when applying our oracle to a uniform superposition over the ω  register we obtain  �  ω  |i⟩ |ω⟩ |0⟩ → √pi |i⟩ |v⟩ |1⟩ +  �  1 − pi |i⟩ |u⟩ |0⟩  (12)  where u and v are suitable junk states which can be neglected as argued in [34] (at least when restricting  attention to query complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that this superposition eliminates the statistical randomness of  the bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 3 (Theorem 1 in [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There is a quantum algorithm that identifies the best arm of a  quantum oracle as in (11) for any reward vector p ∈ [η, 1 − η]n with probability 1 − δ with query  complexity  T = ˜O(  �  H(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (13)  Moreover, this bound is optimal up to logarithmic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  5  Thus we obtain a quadratic speed-up compared to the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' They prove the lower  bound only for the oracle (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For completeness we prove the same lower bound when the more  general oracle (4) is available, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', the ability to query arbitrary superpositions of the data points does  not allow any speedups compared to always considering the uniform superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 4 (Informal version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any algorithm that identifies the best arm for any reward vector  p ∈ [η, 1 − η]n with confidence 1 − δ given access to an oracle as in (4) requires at least O(  �  H(p))  calls to the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The proof and a formal statement of this result are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that while it is intuitively  clear that it is optimal to query over the uniform mixture of ω as in (12) a rigorous proof requires  a careful tracking of the classical randomness of the oracle and its interaction with the quantum  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  Reusable oracles  We now consider oracles as in (7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', we can query arms in superposition but we can only retrieve  the reward for one chosen realization of the randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' A similar type of oracle was considered in  [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' They show that hedging algorithms can be implemented using these oracles which have runtime  √  N in the arms, thus offering a quadratic speedup compared to the classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In this paper  we evaluate the best arm identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We only give partial results for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  A motivation to consider this is the setting of quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Quantum sensing refers to the  general use of quantum phenomena to measure physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In our case the Hilbert space HP  is the state space of a quantum system whose properties we want to probe through the oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We assume that we can prepare states in HP according to some distribution (which for simplicity we  assumed to be the uniform distribution over some basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, HP is not part of our computing  device so we cannot query, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For confidence δ ∈ (0, 1) there is a quantum algorithm that outputs the best arm with  probability 1 − δ using  T ≤ ˜O  �  �  ��  i>1  ∆−4  i  �  �  (14)  queries to an oracle as in (7) where ∼ indicates terms that are polynomially in ln(N/(δ∆2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  A sketch of the proof of this result is in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It relies on a small modification of the  algorithm in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We conjecture that this bound is also optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any quantum algorithm that identifies the best arm for any reward vector p ∈ [η, 1−η]n  for some η > 0 with probability 1 − δ for some δ < 1  2 requires at least  T ≥ c  ��  i>1  ∆−4  i  (15)  calls to an oracle as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The main difficulty to prove a lower bound is that the applied oracles can be reused so that the  fidelity between the quantum states obtained for different mean rewards p and p′ is not necessarily  monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This makes it hard to extend our other proofs that rely on the loss of  fidelity in a single step to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In general, standard techniques to obtain lower bounds and  the techniques used in this work do not appear to be sufficient to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3  One time oracles  Finally we consider the oracle defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The motivation to consider this type of oracle is similar  as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The difference is that we have no control over the investigated quantum  system with state space HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Examples of such a setting are that the state on the subsystem HP  follows some Hamiltonian evolution with a fixed Hamiltonian or it is distributed according to some  Gibbs measure and we assume that the system equilibrates after each oracle invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Formally this setting is similar to [30], where they essentially consider the case pi = 0 for all i ̸= i0  and pi0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This can be viewed as Grover search with a faulty oracle with failure probability 1 − pi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  As we focus on this case we will review their results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Here we just give the our extension  of their result: For oracles as in (8) no speedup is possible for the best arm identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any algorithm that identifies the best arm for any reward vector p ∈ [η, 1 − η]N for  some η > 0 with probability 1 − δ for some δ < 1/2 based on calls to an oracle as in (8) requires at  least  T ≥ c(δ, η)H(p)  (16)  calls to the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Again, this result is not a consequence of the generality of reward vectors p allowed but even when  the set of mean rewards is known no better result is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The precise statement can be found in  Theorem 10 below which is slightly stronger than the result above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that the assumption that  the rewards are Bernoulli distributed might be unrealistic in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, our main result  shows that even if the rewards are Bernoulli distributed no improvements over classical algorithms is  possible in terms of query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='4  Comparison of settings  Let us further discuss these results to give a better intuition of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We remark that access to  the oracle (4) is strictly more powerful than access to oracles as in (7) which in turn is more powerful  than access to the channel oracle (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is reflected in the following chain of inequalities for the  query complexities  ��  i>1  ∆−2  i  ≤  ��  i>1  ∆−4  i  ≤  �  i>1  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (17)  On the other hand we have the following reverse bound for the last two terms  ��  i>1  ∆−4  i  ≥  1  √  N  �  i>1  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (18)  So the speed-up is bounded by  √  N but can be less depending on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To illustrate this further we  consider as a prototypic example a reward vector p with p1 > p2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' = pn with p1 − p2 = ε Then  the query complexities are  Term ≈  �  N  ε2 =  √  N  ε  < Treusable ≈  √  N  ε2  < Tclassical ≈ Tone−time ≈ N  ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (19)  For this setting the two difficulties can be well separated: The reward of the arms needs to be estimated  (statistical complexity) and the correct arm needs to be searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Classically the statistical complexity  is ε−2 but it can be reduced to ε−1 in the empirical risk minimization setting (similar to quantum  metrology [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Not having access to a superposition of samples ω prevents this speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The  7  complexity of the search of the best arm is  √  N in a quantum setting compared to the complexity of  N in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that it is possible to obtain this speed-up using the reusable oracles but  not when having only access to the one time oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that while some assumptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', Bernoulli  distributed rewards might be very simplistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, even under these favorable assumptions for  the quantum setting we show that quantum algorithms offer no improvement in query complexity  when HP is not part of the computing device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  3  Complexity bounds for classical bandits with a quantum per-  spective  Before addressing the case of quantum bandits we revisit the classical bandit problem and give a  different proof for the required number of rounds in the fixed confidence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This section serves  two purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It shows that the fidelity of probability distributions is a useful distance measure to  analyze classical bandit problems which offers the additional advantage that it readily generalizes to  quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, this section is a preparation that introduces some notation for the more  involved proof in the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In fact the proof for the quantum result is essentially based on  a combination of the proof given in this section with the results from the following section on random  oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It is convenient to slightly deviate from the notation used before and  consider a slightly more restricted version of the bandit problem, for which we show a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We consider probability vectors p = (p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pn) ∈ [0, 1]N+1 with p0 > p1 > p2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ≥ pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We denote  ∆i = p0 − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that this is sightly different from the earlier definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider the vectors  pi ∈ [0, 1]N given by pi  j = pj for i ̸= j and pi  i = p0 and p0 given by p0  i = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that for every i ≥ 1  the vectors p0 and pi differ only in the entry i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, we assume that p0 − p1 = p1 − p2 = ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Note that then for j ≥ 1  H(pj) =  �  i̸=j  (p0 − pi)−2 ≤  �  i≥1  ∆−2  i  = (p0 − p1)−2 +  �  i>1  (p0 − pi)−2  ≤ 2  �  i>1  (p1 − pi)2 = 2H(p0)  (20)  where we used p0 − p1 = p1 − p2 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Similarly, we obtain  �  i≥1  ∆−2  i  ≥ H(pj) =  �  i̸=j  (p0 − pi)−2 =  �  i̸=j  (p1 − pi + ∆1)−2  ≥  �  i>1  (p1 − pi + ∆1)−2 ≥ 1  4  �  i>1  (p1 − pi)2 = 1  4H(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (21)  Here we used in the third step that p1−p1+∆1 ≤ p1−pj+∆1 for any j ≥ 1 and p1−pi+∆1 ≤ 2(p1−pi)  in the following inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This shows that H(pj) is up to constants given by �  i ∆−2  i  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  the following result implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Assume that pj are as above with pi ∈ [η, 1 − η] for some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Any classical algorithm that identifies the best arm when it is known that the reward vector is in  {p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pn} with probability at least 1 − δ requires at least  T ≥ cH(p1) = c  n  �  j=2  ∆−2  j  (22)  rounds where c = c(δ, η) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  8  Since this result is well known, the proof serves merely pedagogical purposes to illustrate our  approach to the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Therefore, we do not give the most concise presentation but instead  highlight the main difference to the standard proofs of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Suppose we are given an algorithm A (we do not need to consider randomized algorithms as  we focus on the worst case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In each step t it picks an arm at depending on all earlier outcomes and  receives a (binary) reward rt ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We introduce the variables xt = (at, rt) ∈ [N]×{0, 1} encoding  the path of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We denote by zt = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt) the entire history of the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note  that at only depends on the outcomes of the previous rounds and therefore is a deterministic function  of zt−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', at = at(zt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' When the rewards follow the distribution pj this induces a distribution on  xt and zt and we denote the corresponding random variables by Zj  t and Xj  t and the distribution by  Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The main idea of the proof is to bound the fidelity of random variables Zj  t and Z0  t for each t from  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' On the other hand, we can upper bound the fidelity because the algorithm can identify the best  arm for the reward distributions pj and p0 and those arms are different for j > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Together, those  two bounds will imply the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The proof will rely on the fidelity  √  F of two discrete probability  distributions px and qx which is defined by  √  F(p, q) =  �  x  √pxqx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (23)  We refer to Appendix A for a brief summary of distance measures, here we only need the definition  and the bound by the total variation distance (defined by the first equality)  dTV(p, q) = 1  2  �  x  |px − qx| ≤  �  1 − F(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (24)  We now discuss the simple upper bound on the fidelity after the final round T coming from the  assumption that the algorithm succeeds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let Mj be the disjoint sets of outcomes  zT such that arm j is selected by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' By assumption Pj(Mj) > 1 − δ and P0(M1) > 1 − δ  and therefore P0(Mj) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This implies for j > 1 (see Appendix A for a brief summary of distance  measures)  1 − 2δ < Pj(Mj) − P0(Mj) ≤ dTV(Zj  T , Z0  T ) ≤  �  1 − F(Zj  T , Z0  T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (25)  We conclude that  √  F(Zj  T , Z0  T ) ≤ 2  �  δ(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (26)  We now bound the fidelity from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The first bound will not be sufficient to conclude but it is  nevertheless instructive to understand the difficulties of the quantum setting and the relation to earlier  proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We will later refine the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We bound  √  F(Zj  t , Z0  t ) =  �  zt  �  Pj(zt)P0(zt)  =  1  �  rt=0  �  zt−1  �  Pj(rt|at(zt−1))Pj(zt−1)P0(rt|at(zt−1)P0(zt−1))  =  �  zt−1  �  Pj(zt−1)P0(zt−1)  1  �  rt=0  �  Pj(rt|at(zt−1))P0(rt|at(zt−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (27)  We now consider two cases at(zt−1) = j and at(zt−1) ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In the latter case  Pj(rt|at(zt−1)) = P0(rt|at(zt−1)) = pj  (28)  9  and thus for at(zt−1) ̸= j  1  �  rt=0  �  Pj(rt|at(zt−1))P0(rt|at(zt−1)) =  1  �  rt=0  P0(rt|at(zt−1)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (29)  For at(zt−1) = j we use the simple bound (162) from Lemma 4 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' bounding for p, q ∈  [c, 1 − c]  √  F(Ber(p), Ber(q)) ≥ 1 − |p − q|2  4c(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (30)  This implies  1  �  rt=0  �  Pj(rt|at = j)P0(rt|at = j) ≥ 1 − |pj − p0|2  4η(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (31)  We can now use the last two displays to continue to estimate (27)  √  F(Zj  t , Z0  t ) ≥  �  zt−1  �  Pj(zt−1)P0(zt−1) −  �  zt−1  �  Pj(zt−1)P0(zt−1)1at(zt−1)=j  ∆2  j  4η(1 − η)  =  √  F(Zj  t−1, Z0  t−1) −  �  zt−1  �  Pj(zt−1)P0(zt−1)1at(zt−1)=j  ∆2  j  4η(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (32)  Using this iteratively we obtain (using  √  F(Zj  0, Z0  0) = 1) the bound  2  �  δ(1 − δ) ≥  √  F(Zj  0, Z0  0) ≥ 1 −  T  �  t=1  �  zt−1  �  Pj(zt−1)P0(zt−1)1at(zt−1)=j  ∆2  j  4η(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (33)  Now the standard way to proceed from here is to show that with high probability Pj(zt−1) and P0(zt−1)  are similar using tail bounds for random variables (note that we already control the fidelity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Suppose  that up to small errors we could replace Pj by P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we could conclude from (33) that  ∆2  j  4η(1 − η)  T  �  t=1  �  zt−1  P0(zt−1)1at(zt−1)=j ≥ 1 − 2  �  δ(1 − δ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (34)  Dividing by ∆2  j and summing over j this would imply that there is a constant c > 0 such that  �  j  ∆−2  j  ≤ c  T  �  t=1  �  zt−1  P0(zt−1)  �  j  1at(zt−1)=j = cT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (35)  This approach cannot be extended to the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The reason is that the tail bounds rely  on the fact that when we use a total of O(H) queries then we cannot query all arms more often than  ∆−2  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' On the other hand, in the quantum setting we query in superposition so that we cannot simply  count the number of pulls on an arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now show how the tail bounds can be avoided in a way that  can similarly be generalized to the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us denote by nj(zt) = |{s : as(zt) = j} the  number of times we queried the j-th arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We introduce the decay factor  dj(zt) =  �  1 −  ∆2  j  4η(1 − η)  �nj(zt)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (36)  10  Introducing this decay factor artificially will allow us to derive stronger bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that  dj(zt) = dj(zt−1)  �  1 −  ∆2  j  4η(1 − η)1at(zt−1)=j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(37)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Now we can bound using the same reasoning as in (27) and (32) and the display above  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt)P0(zt)dj(zt) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt)P0(zt)dj(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='4η(1−η)1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='4η(1−η)1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='4η(1−η)1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='4η(1−η)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2η(1−η)1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2η(1−η)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pj(zt−1)P0(zt−1)dj(zt−1))1at(zt−1)=j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(38)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Using (26) and a telescopic series we conclude that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='δ(1 − δ) ≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='F(Zj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Z0  T ) ≥  �  zT  �  Pj(zT )P0(zT )dj(zT )  ≥ 1 −  ∆2  j  2η(1−η)  T  �  t=1  �  zt−1  �  Pj(zt−1)P0(zt−1)dj(zt−1))1at(zt−1)=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (39)  Equivalently, this can be rewritten as  (1 − 2  �  δ(1 − δ))2η(1 − η)  ∆2  j  ≤  T  �  t=1  �  zt−1  �  Pj(zt−1)P0(zt−1)dj(zt−1)21at(zt−1)=j  (40)  It remains to bound the right hand side of this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We write for t < T and zT |t = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt)  where zT = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xT ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', zT |t denotes the restriction of the history zT to the first t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Then we have by definition of P  �  zT  Pj(zT )f(zT |t) =  �  zt  Pj(zt)f(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (41)  Moreover, we note that for all zT we can bound  T  �  t=1  dj(zT |(t−1))21  �  at(zT |(t−1)) = j  �  =  nj(zT )−1  �  n=0  �  1 −  ∆2  j  4η(1 − η)  �2n  ≤  ∞  �  n=0  �  1 −  ∆2  j  4η(1 − η)  �n  ≤ 4η(1 − η)  ∆2  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (42)  11  We can bound using the Cauchy-Schwarz estimate, (41), and  T  �  t=1  �  zt−1  �  Pj(zt−1)P0(zt−1)dj(zt−1))12  at(zt−1)=j  ≤  �  �  T  �  t=1  �  zt−1  P0(zt−1)1at(zt−1)=j  �  �  1  2 �  �  T  �  t=1  �  zt−1  Pj(zt−1)dj(zt−1)21at(zt−1)=j  �  �  1  2  ≤  � T  �  t=1  P0(At = j)  � 1  2 ��  zT  Pj(zT )  T  �  t=1  dj(zT |(t−1))21at(zT |(t−1))=j  � 1  2  ≤  � T  �  t=1  P0(At = j)  � 1  2 ��  zT  Pj(zT )4η(1 − η)  ∆2  j  � 1  2  ≤  � T  �  t=1  P0(At = j)  � 1  2 2  �  η(1 − η)  ∆j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (43)  Plugging this in (40), dividing by 2  �  η(1 − η) and summing over j > 1 gives  (1 − 2  �  δ(1 − δ))  �  η(1 − η)  N  �  j=2  ∆−2  j  ≤  N  �  j=2  ∆−1  j  � T  �  t=1  P0(At = j)  � 1  2  ≤  �  �  N  �  j=2  ∆−2  j  �  �  1  2 �  �  n  �  j=2  T  �  t=1  P0(At = j)  �  �  1  2  =  �  �  n  �  j=2  ∆−2  j  �  �  1  2 �  �  T  �  t=1  n  �  j=1  P0(At = j)  �  �  1  2  =  �  H(p1)  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (44)  Squaring this relation ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  4  Quantum channel oracles  In this section we consider the query complexity for the identification of certain oracles that act as a  quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is a simplified setting of the more general bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Most of the work  related to oracle query complexity has focused on oracles that act as a unitary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There is a large  body of research work on quantum channels also with a focus on quantum channel discrimination and  general lower and upper bounds were derived [1, 27, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, the application of these general  bounds is mostly targeted towards rather simple channels, in particular channels that implement error  mechanisms present in quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Those general results are not directly applicable here and  we will derive bounds targeted at our specific setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The work closest to ours is [30] where a noisy  oracle for quantum search was investigated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', instead of the usual oracle we have access to the  quantum channel  Fi(ρ) = (1 − p)ρ + pOiρO†  i  (45)  where Oi is the usual oracle that performs a phase flip on the (unknown) state i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us compare  this to the setting we introduced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' A careful look reveals that this oracle agrees with (8) when  considering the mean reward vector p given by pj = pδij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  12  For this oracle it is shown in [30] that N/p queries are required to identify i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We give a slightly  more general proof of this result which will serve as a basis for the more general bandit setting we  consider later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The intuition of the proof is that the progress we make is directly related to the  decoherence of the state as measured by its purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As the purity is lower bounded this gives us tight  control of the progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To cover general oracles we assume that the oracle acts on a Hilbert given by  H = HA ⊗ HR where HA = ⟨|i⟩ , 1 ≤ i ≤ N⟩ and HR is the space where the output is written which  will typically be a single qubit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume that we are given oracles Oi acting by  Oi |i⟩ ⊗ |w⟩ = |i⟩ ⊗ |Uw⟩ ,  Oi |j⟩ ⊗ |w⟩ = |j⟩ ⊗ |w⟩  for j ̸= i  (46)  where U denotes a unitary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This covers the case of the phase flip and the bit flip oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As before  we consider that we have oracle access to one of the quantum channels Fi(ρ) = (1 − p)ρ + pOiρO†  i  which we seek to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For convenience we define F0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we have the following slightly  extended version of Theorem 1 in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any algorithm that can decide whether F = Fi for some i > 0 or F = F0 with probability  1 − δ requires at least  T ≥ (1 − p)(1 − 4δ(1 − δ))2  p  n  (47)  calls to the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We emphasize again that the original proof still applies but we think that our proof  clarifies the main ideas and prepares for the more general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Note that the bound becomes vacuous for p → 1, where we recover the setting of Grover’s  algorithm where the well known lower bound scales as  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' So for p =  √  N the bound (47)  agrees with the Grover lower bound so it is possible that a small error probability depending  on the number of arms still allows the same query complexity as in the noise-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Similar  questions were investigated in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us denote by ΦU the quantum channel acting by the unitary U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', ΦU(ρ) = UρU †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As in  the first proof we consider an algorithm acting by ΦUT ◦ F ◦ ΦUT −1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ ΦU1 ◦ F ◦ ΦU0(ρ0) on some  initial state ρ0 = |Ω⟩ ⟨Ω| for some pure state Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We define  ˜ρi  t = ΦUt(ρi  t),  ρi  t = Fi(˜ρi  t−1),  ρi  0 = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (48)  Note that for i = 0 the state remains pure during the entire algorithm and we denote it by ψt and ˜ψt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We now define  Ri  t = Tr  �  ρi  t  �2,  (49)  F i  t = F(ρi  t, ρ0  t),  (50)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', the purity of the state and the fidelity (defined by F(σ, ρ) =  �  Tr �√ρσ√ρ  �2) of the state with  respect to the state corresponding to the trivial oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For a brief overview of properties of the fidelity  we refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We show that the changes of the two quantities are directly related, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', for  every increase in distance (loss in fidelity) we have to pay with a loss in purity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Fidelity and purity are invariant under unitary maps and therefore Ri  t = Tr  �  ˜ρi  t  �2 and F i  t = F(˜ρi  t, ˜ρ0  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  13  We control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' using that Oi is unitary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Ri  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1 − Ri  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t = Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2 − Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='= Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2 − Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='pOi˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1O†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i + (1 − p)˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='= Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2 − (p2 + (1 − p)2) Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2 − 2p(1 − p) Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Oi˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1O†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i ˜ρt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='= 2p(1 − p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2 − Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Oi˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1O†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i ˜ρt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='= p(1 − p) Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1 − Oi˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1O†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(51)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Similarly we estimate the change in fidelity using that ρt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='0 is a pure state and ˜ψt−1 = ψt (because  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='E0 = id)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='F i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1 − F i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t = F(˜ρi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ρ0  t−1) − F(ρi  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ρ0  t)  = ⟨ ˜ψt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ρi  t−1 ˜ψt−1⟩ − ⟨ψt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ρi  tψt⟩  = ⟨ψt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (˜ρi  t−1 − (1 − p)˜ρi  t−1 − pOi˜ρi  t−1O†  i )ψt⟩  = p⟨ψt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (˜ρi  t−1 − Oi˜ρi  t−1O†  i )ψt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (52)  We now relate the change of Rt and Ft Suppose that there is an orthogonal projection P and a unitary  O such that (Id − P)O = Id − P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', O acts trivially on the complement of the image of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  Lemma 3 in Appendix B establishes the bound  ���  ϕ  ���  OσO† − σ  �  ϕ  ��� ≤ 2∥Pϕ∥∥ϕ∥  �  tr  �  OσO† − σ  �2� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (53)  We define projections Pi = |i⟩ ⟨i| ⊗ Id Note that by definition of Oi we have (Id − Pi)Oi = Id − Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Applying Lemma 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', (53) (with P = Pi, σ = ˜ρi  t−1, ϕ = ψt) we can continue to estimate (52) as  follows  F i  t−1 − F i  t ≤ 2p∥Piψt∥  �  Tr  �  ˜ρi  t−1 − Oi˜ρi  t−1O†  i  �2� 1  2  ≤ 2∥Piψt∥  �  p  1 − p  �  Ri  t−1 − Ri  t  (54)  Note that the initial values of F and R are F i  0 = Ri  0 = 1 and Ri  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus we can conclude  �  i  (F i  0 − F i  T ) =  �  i,t  (F i  t−1 − F i  t )  ≤  �  i,t  2∥Piψt∥  �  p  1 − p  �  Ri  t−1 − Ri  t  ≤ 2  �  p  1 − p  �  ��  i,t  ∥Piψt∥2  �  �  1  2 �  ��  i,t  Ri  t−1 − Ri  t  �  �  1  2  ≤ 2  �  p  1 − p  ��  t  ∥ψt∥2  � 1  2 ��  i  Ri  0 − Ri  T  � 1  2  ≤ 2  �  p  1 − p  √  T  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (55)  14  Finally we use our assumption that the algorithm is able to decide whether the oracle is trivial E = E0  or not with probability 1 − δ for some δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' From this we can conclude that the output of the  algorithm for oracles F0 and Fj must be sufficiently different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Formally success of the algorithm  implies (see (149) and (156) in Appendix A) that for each i  1 − 2δ ≤ T(ρi  T , ρ0  T ) ≤  �  1 − F(ρi  T , ρ0  T )  (56)  where T(ρ, σ) = 1  2∥ρ − σ∥tr denotes the trace distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This implies the bound  F i  T ≤ 4δ(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (57)  We conclude that  2  �  p  1 − p  √  T  √  N ≥ N(1 − 4δ(1 − δ)) ⇒ T ≥ N(1 − p)(1 − 4δ(1 − δ))2  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (58)  A natural question suggested by this result and also our main result is whether speed-ups can be  obtained for non-unitary oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This question was also posed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now show that this is not  true in this generality, the simplest example is a faulty oracle that indicates its own failure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' we  consider an oracle Oi acting by  Oi |i⟩ |0⟩ = − |i⟩ |1⟩ ,  Oi |j⟩ |0⟩ = |j⟩ |1⟩  for j ̸= i  (59)  and channels  Fi(ρ) = pOiρO†  i + (1 − p)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (60)  Then we can obtain the same speed-up as with the usual oracle except that we need to correct for the  number of times the oracle is not working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is not in contradiction to the previous result as this  oracle is not of the form defined in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In particular, the action of the oracle in (59) is not trivial  on |j⟩ ⊗ |c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The channel i can be identified with probability at least 1/4 using ⌊π/(2θp)⌋ queries to  the oracle where θ = 2 arcsin  �√  N  −1�  ≈ 2  √  N  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that up to constant factors we need  √  N/p queries of which typically  √  N queries  work, this is the same scaling as the usual Grover algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As usual this bound can also be obtained  if p is unknown by iteratively increasing the number of iterations in the algorithm described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The proof can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' While this result is simple and not very surprising it  underlines that it will be difficult to obtain general results showing that no quantum speedup with  quantum channel oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  5  Complexity bounds for non-adaptive strategies for quantum  bandits  In this section we discuss the complexity of non-adaptive strategies to the quantum bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  While those results also follow from the more general treatment below we think that it is worth to  include this section as a preparation for the rather involved arguments below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We introduce some  additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As before we consider for 0 ≤ p ≤ 1 oracles Fp  i acting by  Fp  i (ρ) = (1 − p)ρ + pOiρO†  i  (61)  15  where Oi is as in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that the channels Fpi  i  and Fpj  j  commute for i ̸= j because Oi and Oj  commute for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider probability vectors p ∈ [0, 1]N and define further  Ep = Fp1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ FpN  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (62)  Note that for the special case that Oi denotes the bit flip on arm |i⟩ the channel Ep agrees with the  definition in (8) where the vector p indicates the mean rewards, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=',  Ep(ρ) =  �  x∈{0,1}N  P(x)OxρO†  x  where  Ox =  �  i:xi=1  Oi  P(x) =  �  pxi  i (1 − pi)1−xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (63)  We now state a lemma that controls the fidelity between applications of the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This result provides  a sharp bound that might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that Oi is self-adjoint and unitary and acts as in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For density matrices ρ, σ  and p, q ∈ [η, 1 − η] the bound  √  F(Fp  i (ρ), Fq  i (σ)) ≥  √  F(ρ, σ) − (p − q)2  η(1 − η)  �  tr(Piρ) tr(Piσ)  (64)  where Pi = |i⟩ ⟨i| ⊗ Id denotes as before the projection on state |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The proof of this lemma can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us for completeness state one direct  consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let p, p′ ∈ [η, 1 − η]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  √  F(Ep(ρ), Ep′(σ)) ≥  √  F(ρ, σ) −  �  i  (pi − p′  i)2  2η(1 − η)  �  tr(Piρ) tr(Piσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (65)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We note that [Pi, Oj] = 0 for i ̸= j and as Oj is unitary we get  tr  �  PiFp  j (ρ)  �  = tr  �  Pi((1 − p)ρ + pOjρO†  j)  �  = tr  �  Pi((1 − p)ρ + pρO†  jOj)  �  = tr(Piρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (66)  Then Lemma 1 can be applied inductively to the relation (62) to obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  From this result we conclude that any non-adaptive algorithm requires the same amount of oracle  queries as the best classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that Ep corresponds to the oracle in (8)  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that pj are as introduced at the beginning of Section 3 with pi ∈ [η, 1 − η] for  some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Fix a density matrix ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We are given access to m copies of the state E(ρ) and it is  known that E is as in (8) where the mean reward vector is in {p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' If the best arm of E can  be identified with probability at least 1 − δ for some δ < 1  2 then  m ≥ η(1 − η)(1 − 2  �  δ(1 − δ))  16  H(p) = O(H(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (67)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We write Ei = Epi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We obtain using Corollary 1 (which can be applied since the bit flip  operation is self adjoint) for m copies  √  F(Ei(ρ)  � m, E0(ρ)  � m) =  √  F(Ei(ρ), E0(ρ))m  ≥  �  1 − (p0 − pi)2  c(1 − c) tr(Pkρ)  �m  ≥ 1 − m  4∆2  k  η(1 − η) tr(Pkρ)  (68)  16  where we used the Bernoulli inequality in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' From  1 − 2δ ≤ T(Ei(ρ)  � m, E0(ρ)  � m) ≤  �  1 − F(Ei(ρ)  � m, E0(ρ)  � m)  (69)  we conclude that  2  �  δ(1 − δ) ≥  √  F(Ei(ρ)  � m, E0(ρ)  � m) ≥ 1 − m  4∆2  i  η(1 − η) tr(Piρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (70)  Equivalently  tr(Piρ) ≥ η(1 − η)(1 − 2  �  δ(1 − δ))  4m∆2  i  (71)  Using tr(�  i Piρ) = 1 and summing over i ≥ 2 we conclude  1 ≥ η(1 − η)(1 − 2  �  δ(1 − δ))  4m  N  �  i≥2  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (72)  Using �N  i≥2 ∆−2  i  ≥ 1  4  �N  i=2(p1 − pi)−2 ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We can similarly derive a suboptimal bound for adaptive algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Here we lose a  √  N factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  In the proof of Theorem 6 we will show that the bound in fact holds without that sqrtN factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The  reason that the arguments used in the proof of Theorem 8 do not extend is that it uses in an essential  way that one of the density matrices is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Then we show that the states for the other oracle  decoherence with respect to this pure reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In the setting here both density matrices are  highly mixed so it is more subtle to formalize the their decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that this result is already  suboptimal in the case of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that pj are as introduced at the beginning of Section 3 with pi ∈ [η, 1 − η] for  some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any quantum algorithm that identifies the best arm when it is known that the reward  vector is in {p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pN} with probability at least 1 − δ for some δ < 1  2 requires at least  T ≥ η(1 − η)(1 − 2  �  δ(1 − δ))  16  √  N  H(p) = O(H(p)/  √  N)  (73)  calls to the oracle Ei(ρ) = Epi(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The proof is close to the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume we are given any algorithm (Ei ⊗Id)◦  EUT ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ (Ei ⊗ Id) ◦ EU1 where Ui are arbitrary unitary maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We denote the state using the oracle i  before the t-th invocation of the oracle by ρi  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Using the invariance of the fidelity under unitary maps  and Lemma 1 we bound  √  F(ρi  T , ρ0  T ) ≥ 1 −  �  t  4∆2  i  η(1 − η)  �  tr  �  Piρi  t  �  tr(Piρ0  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (74)  As in equation (70) we conclude  2  �  δ(1 − δ) ≥ 1 −  �  t  4∆2  i  η(1 − η)  �  tr  �  Piρi  t  �  tr(Piρ0  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (75)  17  Thus we get using �  i tr(Piρt  0) = 1 and tr  �  Piρi  t  �  ≤ 1  η(1 − η)(1 − 2  �  δ(1 − δ))  4  �  i≥2  ∆−2  i  ≤  �  t,i  �  tr  �  Piρi  t  �  tr(Piρ0  t)  ≤  �  ��  t,i  tr  �  Piρi  t  �  �  �  1  2 �  ��  t,i  tr  �  Piρ0  t  �  �  �  1  2  ≤  √  NT  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (76)  This implies that  T ≥ η(1 − η)(1 − 2  �  δ(1 − δ))  4  √  N  �  i≥2  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (77)  As in (20) and (21) this ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  6  Complexity bounds for quantum bandits  In this section we finally prove our main result Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The proof essentially combines the  ingredients of the three preceding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We use the general reasoning from the proof for the faulty  Grover oracle in Theorem 8 and combine it with decompositions as in the proof of the classical bandit  result in Theorem 7 and optimal fidelity estimates as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  As in the proof of Theorem 8 we want to exploit the decoherence of the density matrices ρk  t and  ρ0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, in contrast to the setting of Theorem 8 also the reference state ρ0  t is not pure, so it  is a-priori not clear how the decoherence can be measured in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We did not succeed to  find a suitable replacement of the purity used in Theorem 8 that can be defined intrinsically in terms  of ρk  t and ρ0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Instead, our proof relies on a coupling argument, a technique that is standard in the  theory of stochastic processes but not so much in quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For an overview of this  technique, we refer to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To implement the coupling we need an improvement of Lemma 1 which is  given in the following (slightly technical) lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Consider a density matrix ρ and a pure state ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Consider a unitary and self-adjoint map  U and the channel Ep  U defined by Ep  U(ρ) = pUρU † + (1 − p)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let 0 < η < 1/2 and p, q ∈ [η, 1 − η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We define  ¯ρ0 = (1 − p)ρ,  ρ0 = ρ,  ¯ρ1 = pUρU †,  ρ1 = UρU †  (78)  and  ¯ψ0 =  �  1 − q cos(α) |ψ⟩ + √q sin(α) |Uψ⟩ ,  ¯ψ1 = √q cos(α) |Uψ⟩ −  �  1 − q sin(α) |ψ⟩  (79)  ψ0 = ¯ψ0/∥ ¯ψ0∥,  ψ1 = ¯ψ1/∥ ¯ψ1∥  (80)  where  cos(α) = √pq +  �  (1 − p)(1 − q),  sin(α) =  �  (1 − p)q −  �  (1 − q)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (81)  We define q′ = ∥ ¯ψ1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  Eq  U(|ψ⟩ ⟨ψ|) = q′ |ψ1⟩ ⟨ψ1| + (1 − q′) |ψ0⟩ ⟨ψ0| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (82)  Let S =  √  F(ρ, |ψ⟩ ⟨ψ|) =  �  ⟨ψ| ρ |ψ⟩ denote the initial fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the following bound holds  √  F  �  Ep  U(ρ), Eq  U(|ψ⟩ ⟨ψ|)  �  ≥  �  (1 − p)(1 − q′)  √  F(ρ0, |ψ0⟩ ⟨ψ0|) +  �  pq′√  F(ρ1, |ψ1⟩ ⟨ψ1|)  ≥ S − (p − q)2|(ψ, (UρU † − ρ)ψ)|  2ηS  − (p − q)2|Re(ψ, Uρψ) − (ψ, ρψ)|2  8η2S3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (83)  18  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us give some explanation regarding this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Note that for p = q we recover the result that quantum channels can only increase the fidelity  for our specific channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The problem that this lemma solves is that we need a bound on the loss in fidelity that has,  firstly, the optimal quadratic rate in (p − q), secondly, the bound needs to have a form that  allows to exploit the specific structure of the oracles Oi which act nontrivially only on a small  subspace, and, thirdly, we later want to use that the density matrices ρ decohere with respect to  the state ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The last point will become clearer in the proof of Theorem 10 below but note that  the expression (ψ, (UρU †−ρ)ψ) appeared already in the proof of Theorem 8 which indicates that  similar arguments can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We remark that Lemma 1 above already satisfied the first two  requirements but the lack of the third requirement allowed us to only show the suboptimal bound  in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It is also quite straightforward to satisfy the second and the third requirement  with the suboptimal rate |p − q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' But this also gives only a suboptimal bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The second error term does not require all desiderata outlined above but it is of higher order  (note the extra square) which is sufficient to control it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  To clarify the origin of the expressions for sin(α) and cos(α) we remark that if we choose β, γ  such that √p = sin(β), √q = sin(γ), then α = γ − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This also explains the simplifications in  the formula below, in particular (195) and (198) below are just the trigonometric identities for  angle sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The proof of this lemma can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In fact we need a slight improvement of this  result stated in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume the same setting as in Lemma 1, with the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We in addition  consider another self adjoint traceless operator σ and we define  ¯ρ0 = (1 − p)(ρ + pσ)  ρ0 = ρ + pσ  ¯ρ1 = pUρU † + p(1 − p)σ  ρ1 = UρU † + (1 − p)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (84)  We assume that ρ0 and ρ1 are density matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the following bound holds  �  (1 − p)(1 − q′)  √  F(ρ0, |ψ0⟩ ⟨ψ0|) +  �  pq′√  F(ρ1, |ψ1⟩ ⟨ψ1|) − S ≥  − (p − q)2|(ψ, (UρU † − ρ)ψ)|  2ηS  − (p − q)2|Re(ψ, Uρψ) − (ψ, ρψ)|2  8η2S3  − p|⟨ ¯ψ0, σ ¯ψ0⟩| + (1 − p)|⟨ ¯ψ1, σ ¯ψ1⟩|  S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (85)  We sketch the proof in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Now we can state the formal version of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that pj are as introduced at the beginning of Section 3 with  pi ∈ [η, 1 − η] for some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any quantum algorithm that identifies the best arm when it is known  that the reward vector is in {p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pn} with probability at least 1 − δ requires at least  T ≥ cH(p1) = c  n  �  i=2  ∆−2  i  (86)  calls to the oracle Ek where c = c(δ, η) where an explicit expression under the condition ∆2  N/η < 1/2  is given by  c(δ, η) =  �  η(1 − 2  �  δ(1 − δ))  20  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (87)  19  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We only do the proof under the condition that ∆2  N/η ≤ 1/2 (the behavior for small ∆i is  the main interest anyway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' If this does not hold some definitions below need to be slightly adjusted  (starting with (94) and (95)) but the final result will be the same except that the constant has a  poorer dependence on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We use the same notation as before in the proofs of Corollary 2, and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Recall in  particular the definition of Fp  i and Ei in (61) and (62) and the definition of pj (in particular pj  i = pi  for i ̸= j and pi  i = p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To notate the rewards in step t we consider xt ∈ {0, 1}N and we collect  those rewards in the vector zt = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As before we consider the measure Pi on sequences zT  which has the property that Pi((xt)j = 1) = pi  j and Pi((xt)j = 0) = 1 − pi  j and those variables are  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume the algorithm is given by an circuit acting by  Ti = (Ei ⊗ Id) ◦ EUT ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ (Ei ⊗ Id) ◦ EU1  (88)  with one of the unknown oracle Ek to be identified followed by a POVM-measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As before we  denote the state of the system after t invocations of the oracle Ei by ρi  t and by ˜ρi  t the state directly  before the t + 1-th invocation of the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us for now fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We try to lower bound  the fidelity  √  F(ρi  t, ρ0  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This will be based on decompositions of the density matrices, where we split  the state in two after each invocation of any of the oracles Fpj  j  depending on the realization of the  randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For ρi  t we consider a decomposition given by  ρi  t =  �  zt  Pi(zt)ρ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (89)  For ρ0  t we consider a decomposition into pure states depending on i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', we construct a distribution  Qi on sequences zT and states ψ(zt) such that  ρ0  t =  �  zt  Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (90)  To avoid a too complex notation we dropped the i dependence of ρ(zt) and ψ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  This time the decomposition for ρ0 involves the complexity while the decomposition for the oracle Ei  will be relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The high-level idea for the decomposition is based on the observation  that the optimal loss in fidelity when applying Ei and E0 is of the order ∆2  i as shown in Lemma 2  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We essentially use the decomposition constructed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We decompose ρi  t roughly as  ρt(zt) ≈ |ϕ(zt)⟩ ⟨ϕ(zt)| ,  ϕ(zt = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt)) = (Oxt ⊗ Id)Ut .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (Ox1 ⊗ Id)U1,  (91)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', we just decompose it a according to the realizations of the rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, we in addition  need to ensure that the density matrices ρt(zt) decohere with respect to ψ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For the definition we  introduce the notation ˆxt ∈ {0, 1}N for the vector xt with the i-th entry set to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', (ˆxt)j = (xt)j  for j ̸= i and (xt)i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We define (recall that ρ0 denotes the initial state of the algorithm)  ρ(z0) = ρ0,  (92)  ˜ρ(zt) = EUt+1(ρ(zt)),  (93)  ˜ρ0(zt) =  �  1 − p0∆2  i  η  �  ˜ρ(zt) + p0∆2  i  η  Oi˜ρ(zt)O†  i ,  (94)  ˜ρ1(zt) =  �  1 − (1 − p0)∆2  i  η  �  Oi˜ρ(zt)O†  i + (1 − p0)  η  ∆2  i ˜ρ(zt),  (95)  ρ(zt+1) = EOˆxt+1 (˜ρ(xt+1)i(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (96)  20  The reason to slightly perturb ˜ρ0(zt) and ˜ρ1 from their natural definitions ˜ρ(zt) and Oi˜ρ(zt)O†  i is that  our definition ensures up to a constant the same loss in fidelity of order ∆2  i but in addition ensures  decoherence of ˜ρ so that we can argue as in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The factor η leads to slightly  simpler terms but is not strictly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We now claim that this decomposition satisfies (89) and  ˜ρi  t =  �  zt  P(zt)˜ρ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (97)  We argue by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We have  ˜ρi  t = EUt+1(ρi  t) = EUt+1  ��  zt  Pi(zt)ρ(zt)  �  =  �  zt  Pi(zt)EUt+1(ρ(zt)) =  �  zt  Pi(zt)˜ρ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (98)  Next we note that  p0˜ρ1(zt) + (1 − p0)˜ρ0(zt) = p0 Oi˜ρ(zt)O†  i + (1 − p0) ˜ρ(zt) = Fp0  i (˜ρ(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (99)  This implies together with the definition of Pi that  �  xt+1  Pi(xt+1)ρ((zt, xt+1)) =  �  xt+1  Pi(xt+1)EOˆxt+1 (˜ρ(xt+1)i(zt)) =  �  xt+1  Pi(xt+1)EOˆxt+1 ◦ Fp0  i (˜ρ(zt))  =  �  xt+1  Pi(xt+1)EOxt+1 (˜ρ(zt)) = Ei(˜ρ(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (100)  Here we used that Pi((xt+1)i = 1) = pi  i = p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Using the induction hypothesis we conclude that  �  zt+1  Pi(zt+1)ρ(zt+1) =  �  zt  Pi(zt)  �  xt+1  P(xt+1)ρ((zt, xt+1)) =  �  zt  Pi(zt)Ei(˜ρ(zt)) = Ei(˜ρi  t) = ρi  t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (101)  For the oracle E0 we consider a decomposition depending on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider a probability distri-  bution Qi on sequences zt = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It will factorize according to  Qi(zt+1) = Qi(zt)Qi(xt+1|zt) = Qi(zt)Qi((xt+1)i|zt)  �  j̸=i  p(xt+1)j  j  (1 − pj)1−(xt+1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (102)  We define  ˜ψ(zt) = Ut  �  ψ(zt−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (103)  We now define, essentially as in Lemma 2,  ¯ψ1(zt) =  �  1 − pi cos(α) ˜ψ(zt) + √pi sin(α)Oi ˜ψ(zt)  (104)  ˜ψ1(zt) = ¯ψ1(zt)/∥ ¯ψ1(zt)∥  (105)  ¯ψ0(zt) = −  �  1 − pi sin(α) ˜ψ(zt) + √pi cos(α)Oi ˜ψ(zt)  (106)  ˜ψ0(zt) = ¯ψ0(zt)/∥ ¯ψ0(zt)∥  (107)  where α is as in (81) with p and q replaced by pi and p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Finally we set  ψ(zt+1) = Oˆxt+1 ˜ψ(xt+1)i(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (108)  21  We define  Qi�  (xt+1)i = 1 | zt  �  = ∥ ¯ψ1(zt)∥2  Qi�  (xt+1)i = 0 | zt  �  = ∥ ¯ψ0(zt)∥2 = 1 − ∥ ¯ψ1(zt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (109)  Moreover, we assume that the (xt)j for i ̸= j are independent and distributed according to Ber(p0  j) =  Ber(pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Together, those properties define a unique distribution Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We claim that  ρ0  t =  �  zt  Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)| ,  (110)  ˜ρ0  t =  �  zt  Qi(zt)  ��� ˜ψ(zt)  � �  ˜ψ(zt)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (111)  To show this we argue by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The first step is simple  �  zt  Qi(zt)  ��� ˜ψ(zt)  � �  ˜ψ(zt)  ��� =  �  zt  Qi(zt) |Ut+1ψ(zt)⟩ ⟨Uψ(zt)|  = Ut+1  ��  zt  Qi(zt) |ψ(zt)⟩ ⟨ψ(zt)|  �  U †  t+1 = EUt+1(ρ0  t) = ˜ρ0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (112)  By (82) in Lemma 2 the following relation holds in view of  Fpi  i ( ˜ψ(zt)) = Qi�  (xt+1)i = 1 | zt  � ��� ˜ψ1(zt)  � �  ˜ψ1(zt)  ��� + Qi�  (xt+1)0 = 1 | zt  � ��� ˜ψ0(zt)  � �  ˜ψ0(zt)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (113)  Using this relation we conclude that  �  zt+1  Qi(zt+1) |ψ0(zt+1)⟩ ⟨ψ0(zt+1)| =  �  zt+1  Qi(zt+1)  ���Oˆxt+1 ˜ψ(xt+1)i(zt)  � �  Oˆxt+1 ˜ψ(xt+1)i(zt)  ���  = Fp1  1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ Fpi−1  i−1 ◦ Fpi+1  i+1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ FpN  N  ��  zt  Qi(zt)  1  �  s=0  Qi((xt+1)i = s|zt)  ��� ˜ψs(zt)  � �  ˜ψs(zt)  ���  �  = Fp1  1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ Fpi−1  i−1 ◦ Fpi+1  i+1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ FpN  N ◦ Fpi  i  ��  zt  Qi(zt)  ��� ˜ψ(zt)  � �  ˜ψ(zt)  ���  �  = E0(˜ρ0  t) = ρ0  t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (114)  We now start to estimate the fidelity of ρk  t and ρ0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us remark that invariance of the fidelity  under unitary maps implies  √  F  �  ρ((zt, xt+1)), |ψ((zt, xt+1))⟩  �  =  √  F  �  ˜ρ(xt+1)i(zt),  ��ψ(xt+1)i(zt)  � �  ,  (115)  √  F  �  ρ(zt), |ψ(zt)⟩  �  =  √  F  �  ˜ρ(zt),  ��� ˜ψ(zt)  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (116)  The loss in fidelity occurs when passing from ˜ρ(zt) and ˜ψ(zt) to ˜ρ0/1(zt) and ˜ψ0/1(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We will bound  this loss using Corollary 4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The additional flexibility of the σ term in this corollary allows  to apply this to our setting where ˜ρ0/1 are defined as in (94) and (95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Specifically, we apply this  Corollary with ρ = ˜ρ(zt), ψ = ˜ψ(zt), p = p0, q = pi, U = Oi and σ = ∆2  i (Oi˜ρ(zt)O†  i − ˜ρ(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  we note that the definition of ˜ψ0/1 agrees with the definition of ψ0/1 in Lemma 1 and ˜ρ0/1(zt) agrees  22  with ρ0/1 and q′ = Qi((xt+1)i = 1|zt), p = Pi(xi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We write S =  √  F(˜σ(zt), ˜ψ(zt)) and assume  S ≥ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus we conclude from Corollary 4 that  �  (1 − p0)Qi((xt+1)i = 0|zt)  √  F(˜ρ0(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ0(zt)) +  �  p0Qi((xt+1)i = 1|zt)  √  F(˜ρ1(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ1(zt))  ≥ S − ∆2  i  �|( ˜ψ(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (Oi˜ρ(zt)O†  i − ˜ρ(zt)) ˜ψ(zt))|  η  +  ���Re  �  ( ˜ψ(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' (Oi˜ρ(zt) − ˜ρ(zt)) ˜ψ(zt))  ����  2  η2  + 2(1 − p)  η  ���( ¯ψ1(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �  ¯ψ1(zt))  ��� + 2p  η  ���( ¯ψ0(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �  ¯ψ0(zt))  ���  �  (117)  We control the right hand side of this expression by exploiting the specific structure of the oracle Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  As in the proof of Theorem 8 we use that Pi = |i⟩ ⟨i| ⊗ Id satisfies (1 − Pi)Oi = (1 − Pi) and apply  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We get  |( ˜ψ(zt), (Oi˜ρ(zt)O†  i − ˜ρ(zt)) ˜ψ(zt))| ≤ 2∥Pi ˜ψ(zt)∥  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (118)  For the second term we use that Oi − Id = (Id − Pi + Pi)(Oi − Id) = Pi(Oi − Id) which implies after  an application of Cauchy-Schwarz  ���Re  �  ( ˜ψ(zt), (Oi˜ρ(zt) − ˜ρ(zt)) ˜ψ(zt))  ����  2  ≤ ∥Pi ˜ψ(zt)∥2 · ∥(Oi − Id)˜ρ(zt) ˜ψ(zt)∥2 ≤ 4∥Pi ˜ψ(zt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (119)  For the third term we use again Lemma 3, the definition (104), and [Oi, Pi] = 0 and bound  ���( ¯ψ1(zt),  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �  ¯ψ1(zt))  ��� ≤ ∥Pi ¯ψ1(zt)∥ · ∥ ¯ψ1(zt)∥ ·  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  ≤  ��  1 − pi cos(α)∥Pi ˜ψ(zt)∥ + √pi sin(α)∥PiOi ˜ψ(zt)∥  �  ∥ ¯ψ1(zt)∥ ·  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  ≤ 2∥Pi ˜ψ(zt)∥  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (120)  The same reasoning implies  ���( ¯ψ0(zt),  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �  ¯ψ0(zt))  ��� ≤ 2∥Pi ˜ψ(zt)∥  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (121)  Plugging (118), (119), (120), and (121) in (117) we obtain  �  (1 − p0)Qi((xt+1)i = 0|zt)  √  F(˜ρ0(zt), ˜ψ0(zt)) +  �  p0Qi((xt+1)i = 1|zt)  √  F(˜ρ1(zt), ˜ψ1(zt))  ≥ S − 5∆2  i  η ∥Pi ˜ψ(zt)∥  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  − 4∆2  i  η2 ∥Pi ˜ψ(zt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (122)  This is the key relation that allows us to control the fidelity between ρ0  t and ρi  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now sum this  bound over all possible values of xt+1 to move from t + 1 to step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We denote ˆxt+1 the vector  xt+1 with entry i removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that under Pi and Qi this vector is independent of zt, (xt+1)i and  Pi(ˆxt+1) = Qi(ˆxt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We also introduce the notation ˆxc  t+1 for the vector that has entry i equal to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  23  Then we get using (115) and (116) for any zt and c ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1}  �  ˆxt+1  �  Pi(ˆxt+1Qi(ˆxt+1)  √  F  �  ˜ρ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˆxc  t+1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˆxc  t+1))  �  =  �  ˆxt+1  �  Pi(ˆxt+1Pi(ˆxt+1)  √  F  �  ˜ρc(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψc(zt)  �  =  √  F  �  ˜ρc(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψc(zt)  �  (123)  Summing this over c and using (122) we get for all zt such that  √  F(˜ρ(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  ˜  ψ(zt)) ≥ 1/2 the bound  �  xt+1  �  Pi(xt+1)Qi(xt+1|zt)  √  F  �  ˜ρ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' xt+1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' xt+1))  �  1  �  c=0  �  Pi((xt+1)i = c)Qi((xt+1)i = c|zt)  �  ˆxt+1  �  Pi(ˆxt+1Qi(ˆxt+1)  √  F  �  ˜ρ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˆxc  t+1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ((zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˆxc  t+1))  �  ≥  √  F(˜ρ(zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  ˜  ψ(zt)) − 5∆2  i  η ∥Pi ˜ψ(zt)∥  �  tr  �  Oi˜ρ(zt)O†  i − ˜ρ(zt)  �2� 1  2  − 4∆2  i  η2 ∥Pi ˜ψ(zt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (124)  Equipped with this relation we now move on to control the total loss in fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Let us first  explain how we deal with the trace term in the loss of fidelity which will be very similar to the proof  of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus we define (again not indicating the i dependence)  R(zt) = tr  �  ˜ρ(zt)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (125)  Invariance of the purity under unitary operations implies that  R(zt+1) = tr  �  ˜ρ(xt+1)i(zt)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (126)  Calculations as in (51) give us for (xt+1)i = 0  R(zt) − R(zt+1) =  �  1 − p0∆2  i  η  � p0∆2  i  η  Tr  �  ˜ρ(zt) − Oi˜ρ(zt)O†  i  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (127)  A similar identity for (xt)i = 1 together with the assumption ∆2  i /η ≤ 1/2 and min(p0, 1 − p0) ≥ η  imply  Tr(˜ρ(zt) − Oi˜ρ(zt))O†  i )2 ≤  1  2∆2  i  (R(zt) − R(zt+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (128)  Note that the right hand side only depends on zt so we conclude  Tr(˜ρ(zt) − Oi˜ρ(zt))O†  i )2 ≤  1  2∆2  i  (R(zt) − max  xt+1 R((zt, xt+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (129)  The general strategy is now to use joint concavity of the fidelity and then inductive application  of the estimate (122) above to lower bound the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There are two technical difficulties: When  directly trying to bound fidelity loss we do not get the optimal scaling and we need to apply an  approach as in the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The second difficulty is that the change in fidelity in Lemma 2  involves the inverse of the initial fidelity and so we derived (124) only for S ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The high level  argument that this is sufficient is that by lower bounding the fidelity we also show that for most  sequences zt the fidelity  √  F(ρ(zt), ψ(zt)) is lower bounded by 1/2 (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Put differently, the fidelity  has already decreased substantially when we can no longer apply this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Technically, we address both difficulties by introducing additional sequences d(zt), s(zt), and h(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  24  We define d(z0) = 1 and then recursively for zt = (zt−1, xt)  d(zt) = d(zt−1)  �  1 − ∆2  i ∥Pi ˜ψ(zt−1)∥2  η2  �  = d(zt−1) − d(zt−1)∆2  i ∥Pi ˜ψ(zt−1)∥2  η2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (130)  Note that the η factor was introduced for convenience, it could be dropped at the price of a slightly  worse η dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We define s(zt) = 0 for zt = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt) if there is t′ ≤ t such that zt′ =  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt′) satisfies  √  F(ρ(zt), ψ(zt)) =  √  F(˜ρ(zt), ˜ψ(zt)) < 1/2 and s(zt) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover,  we define h(zt) = 1 if s(zt) = 0 but s(zt′) = 1 for all zt′ as above, put differently for zt = (zt−1, xt) the  relation h(zt) = s(zt−1) − s(zt) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', s(zt) keeps track when the fidelity  √  F(ρ(zt), ψ(zt)) falls  below 1/2 for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We can now bound (loosely speaking) the change in fidelity  √  F(ρk  T −1, ρ0  T −1)−  √  F(ρk  T , ρ0  T ) (actually  we only bound the difference on the lower bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This will be achieved by combining the estimates  above and the introduction of various error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We again use the notation zT |t = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xt) for  t ≤ T and zT = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , xT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We first note that the definitions of s, h, and d imply  √  F(˜ρk  T , ˜ρ0  T ) ≥  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT )  =  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT )(s(zT |T −1) − h(zT ))  =  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT |T −1) − E1  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  = −E1  T +  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT |T −1)s(zT |T −1)  − ∆2  i  η2  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))∥Pi ˜ψ(zT |T −1)∥2d(zT |T −1)s(zT |T −1)  = −E1  T − E2  T +  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT |T −1)s(zT |T −1)  (131)  Where the error terms E1  T and E2  T are defined by those equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Now we bound this expression  using (124) and (129).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Here we use that either the factor s(zT |T −1) vanishes and the inequality below  is trivially true or the bound  √  F(˜ρ(zT |T −1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ(zT |T −1)) ≥ 1/2 holds so that (124) can be applied  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψT (zT ))d(zT |T −1)s(zT |T −1)  ≥  �  zT −1  �  Pi(zT −1)Qi(zT −1)d(zT −1)s(zT −1)  �  √  F(˜ρ(zT −1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ(zT −1))  − 5∆i  √  2η ∥Pi ˜ψ(zT −1)∥  �  R(zT −1) − max  xt R((zT −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' xt))  � 1  2  − 4∆2  i  η2 ∥Pi ˜ψ(zT −1)∥2  �  ≥  �  zT −1  �  Pi(zT −1)Qi(zT −1)  √  F(˜ρ(zT −1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ(zT −1))d(zT −1)s(zT −1) − E3  T − E4  T  (132)  where we again define the error terms E3  T −1 and E4  T −1 implicitly through these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Applying  this inductively together with  √  F(˜ρi  0, ˜ρ0  0) = 1 we get  √  F(˜ρi  T , ˜ρ0  T ) ≥  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ), ˜ψ(zT ))d(zT )s(zT ) ≥ 1 −  T  �  t=1  E1  t + E2  t + E3  t + E4  t (133)  25  We now bound the error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We start with E2  t which we bound by (using  √  F ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' s(zt) ≤ 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='E2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t ≤ ∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='xt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(xt|zt−1)Qi(xt|zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≤ ∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(134)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Clearly the error term E4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t satisfies a similar bound and we obtain using Cauchy Schwarz and d(zt) ≤ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='E2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t + E4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t ≤ 5∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(zt−1)Qi(zt−1)∥Pi ˜ψ(zt−1)∥2d(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≤ 5∆i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Qi(zt−1)∥Pi ˜ψ(zt−1)∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(zt−1)∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η2 ∥Pi ˜ψ(zt−1)∥2d(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='≤ 5∆i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='η  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Qi(zt−1) tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='��� ˜ψ(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='˜ψ(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='��� Pi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�� 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='t=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='zt−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(zt−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='d(zt−1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='xt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Pi(xt)d((zt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' xt))  ��  �  1  2  ≤ 5∆i  η  � T  �  t=1  tr  �  Pi˜ρ0  t  �  � 1  2 �  �  T  �  t=1  �  zt−1  Pi(zt−1)d(zt−1) −  T  �  t=1  �  zt  Pi(zt)d(zt)  �  �  1  2  ≤ 5∆i  η  � T  �  t=1  tr  �  Pi˜ρ0  t  �  � 1  2 �  d(z0) −  �  zT  Pi(zT )d(zT )  � 1  2  ≤ 5∆i  η  � T  �  t=1  tr  �  Pi˜ρ0  t  �  � 1  2  (135)  where we used (111) and the telescopic sum together with d(z0) = 1 and 0 ≤ d(zT ) ≤ 1 in the last  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Next we address the error term E3  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We bound, using again Cauchy Schwarz,  T  �  t=1  E3  t = 5∆i  √  2η  T −1  �  t=0  �  zt  �  Pi(zt)Qi(zt)d(zt)s(zt)∥Pi ˜ψ(zt)∥  �  R(zt) − max  xt+1 R((zt, xt+1))  � 1  2  ≤ 5∆i  √  2η  �T −1  �  t=0  �  zt  Qi(zt)∥Pi ˜ψ(zt)∥2  � 1  2 � T  �  t=1  �  zt  Pi(zt)R(zt) − max  xt+1 R((zt, xt+1))  � 1  2  ≤ 5∆i  √  2η  �T −1  �  t=0  tr  �  Pi˜ρ0  t  �  � 1  2 �T −1  �  t=0  �  zt  Pi(zt)R(zt) −  T  �  t=1  Pi(zt)R(zt)  � 1  2  ≤ 5∆i  √  2η  �T −1  �  t=0  tr  �  Pi˜ρ0  t  �  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (136)  Here we again used the telescopic sum and the fact that 0 ≤ R(zt) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  26  It remains to bound the last remaining error term E1  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The key ingredient to bound E1  t is the  observation that if h(zt) = 1 we have  √  F(˜ρ(zt), ˜ψ(zt)) < 1/2 which implies  √  F(˜ρ(zt), ˜ψ(zt)) < 1 −  √  F(˜ρ(zt), ˜ψ(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (137)  From here we conclude using the definition of E1  t  T  �  t+1  E1  t =  T  �  t=1  �  zt  �  Pi(zt)Qi(zt)  √  F(˜ρ(zt), ˜ψ(zt))d(zt)h(zt)  ≤  T  �  t=1  �  zt  �  Pi(zt)Qi(zt)  �  1 −  √  F(˜ρ(zt), ˜ψ(zt))  �  d(zt)h(zt)  ≤  T  �  t=1  �  zt  �  Pi(zt)Qi(zt)h(zt) −  T  �  t=1  E1  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (138)  Starting from (133) and plugging in the display above we find  �  zT  �  Pi(zT )Qi(zT )s(zT ) ≥  �  zT  �  Pi(zT )Qi(zT )  √  F(˜ρ(zT ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ˜ψ(zT ))d(zT )s(zT )  ≥ 1 −  T  �  t=1  E1  t + E2  t + E3  t + E4  t  ≥ 1 −  T  �  t=1  E2  t + E3  t + E4  t +  T  �  t=1  E1  t −  T  �  t=1  �  zt  �  Pi(zt)Qi(zt)h(zt)  (139)  Now it is easy to see that  T  �  t=1  �  zt  Pi(zt)h(zt) +  �  zT  Pi(zT )s(zT ) = 1  (140)  because this is the probability that the fidelity drops below 1/2 under the measure Pi plus the prob-  ability of the complement and the same is true for Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus, we can identify those expressions with  probability vectors and since the (classical) fidelity of any two probability distributions is bounded by  1 we conclude that  �  zT  �  Pi(zT )Qi(zT )s(zT ) +  T  �  t=1  �  zt  �  Pi(zt)Qi(zt)h(zt) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (141)  Using this relation in (139) we find  1 ≥ 1 −  T  �  t=1  E2  t + E3  t + E4  t +  T  �  t=1  E1  t  (142)  from which  T  �  t=1  E1  t ≤  T  �  t=1  E2  t + E3  t + E4  t  (143)  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  27  Plugging first (143) and then (135) and (136) in (133) we obtain  √  F(ρk  T , ρ0  T ) ≥ 1 −  T  �  t=1  E1  t + E2  t + E3  t + E4  t ≥ 1 − 2  T  �  t=1  E2  t + E3  t + E4  t  ≥ 1 − 20η−1∆i  � T  �  t=1  Tr Pk ˜ρ0  t  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (144)  The rest of the proof is identical to the reasoning in the proofs before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' From 2  �  δ(1 − δ) ≥  √  F  �  ρk  T , ρ0  T  �  we infer  η(1 − 2  �  δ(1 − δ))  20  �  i≥2  ∆−2  i  ≤  �  i≥2  ∆−1  i  � T  �  t=1  Tr Piρ0  t  � 1  2  ≤  �  ��  i≥2  ∆−2  i  �  �  1  2 �  ��  i≥2  T  �  t=1  Tr Piρ0  t  �  �  1  2  ≤  �  ��  i≥2  ∆−2  i  �  �  1  2 √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (145)  This finally ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  7  Discussion  In this work we investigated quantum algorithms for multi armed bandit problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It was shown  earlier in [34] that quantum algorithms for best arm identification with fixed confidence can have a  quadratic speed-up compared to their classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This result is based on the assumption  that the arms and the randomness of the rewards of the arms can be both queried in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  These assumptions are reasonable in the setting of empirical risk minimization where we can evaluate  loss values in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' However, there are many settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', motivated by quantum sensing  where it might not be possible to query the internal randomness of the bandits in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Instead every pull of the lever returns a single random reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We then show that in this setting no  speed-up compared to classical algorithms is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  This highlights that classical randomness pose a major challenge for quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In our  case the randomness of the rewards even prevent Grover type speed-ups of the search problem that  one would naively expect to arise from the search part of the multi-armed bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note  that such a speed-up is possible in the intermediate regime that we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' When we can select  the state of the internal randomness of the oracle (but not query it in superposition) the statistical  complexity of the problem remains the same but we can search through the arms faster providing  some speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  There are many open questions related to this work and we will briefly mention two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Firstly,  classical randomness appears frequently in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  This has been studied a lot in the  context of noise channels but not so much in other contexts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Secondly, our proofs proceed by directly controlling the fidelity between the quantum states when  invoking different oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' From a methodological side, it would be interesting to see to what degree the  well known strategies to lower bound the query complexity like the polynomial [6] or the adversarial  method [3] extend to non-unitary oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
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+page_content='  A  A brief review of distance measures for quantum states  For the convenience of the reader we give a brief review of distance measures for quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Textbooks on quantum computation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', [25] discuss this thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For a review on fidelities we  refer to [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider the trace distance which is defined by  T(ρ, σ) = 1  2∥ρ − σ∥tr  (146)  where the norm indicates the trace norm defined by ∥A∥tr = tr  �√  A†A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It has the property that for  any POVM {Ei} the outcome probabilities  pi = tr(Eiρ),  qi = tr(Eiσ)  (147)  the total variation distance between the probability vectors pi and qi satisfy  1  2  �  i  |pi − qi| ≤ T(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (148)  The Helmstrom measurement gives the optimal discrimination probability of two states and has success  probability  psuccess = 1  2 + 1  2T(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (149)  For many applications the fidelity is a more useful distance measure to obtain optimal bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It is  defined by  √  F(ρ, σ) = tr  ��  ρ  1  2 σρ  1  2  �  = ∥ρ  1  2 σ  1  2 ∥tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (150)  Some authors instead call the square of this expression the fidelity and to clarify our convention we  added the square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' As suggested by the notation we set F =  √  F  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We collect some properties of  the fidelity that we will use frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  For a density matrix ρ and a pure state ψ the fidelity is given by  √  F(|ψ⟩ ⟨ψ| , ρ) =  �  ⟨ψ, ρψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (151)  31  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  For any density matrices ρ, σ and a quantum channel E the following bound holds  √  F(ρ, σ) ≤  √  F(E(ρ), E(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (152)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  If the quantum channel E acts by a unitary matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', E(ρ) = UρU † then  √  F(ρ, σ) =  √  F(E(ρ), E(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (153)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The fidelity is strongly concave  √  F(  �  i  piρi,  �  i  qiσi) ≥  �  i  √piqi  √  F(ρi, σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (154)  This directly implies concavity  √  F(  �  i  piρi,  �  i  piσi) ≥  �  i  pi  √  F(ρi, σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (155)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Fidelity and trace distance are related by  1 −  √  F(ρ, σ) ≤ T(ρ, σ) ≤  �  1 − F(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (156)  Those properties can be proved using Uhlmann’s Theorem which states that  √  F(ρ, σ) = max  ϕ,ψ ⟨ϕ, ψ⟩  (157)  where the maximum is over all purifications ψ and ϕ of ρ and σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We use this result to  bound the fidelity change of certain quantum operations (see Lemma 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  B  Auxiliary lemmas  Here we include simple mostly algebraic lemmas that are used in the proof of Theorem 8 and in the  proofs in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The first lemma is a simple Cauchy-Schwarz estimate that in addition exploits  invariant subspaces of an operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It is used in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let O be a unitary operator and P a self-adjoint orthogonal projections such that (1 −  P)O = 1 − P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', O acts trivially on the orthogonal complement of the projection P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then, for any  vector |ϕ⟩ and density matrix σ the bound  ���  ϕ  ���  OσO† − σ  �  ϕ  ��� ≤ 2∥Pϕ∥∥ϕ∥  �  tr  �  OσO† − σ  �2� 1  2  (158)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We define Q = Id − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we have QO = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' By assumption we can decompose  σ − OσO† = (P + Q)(σ − OσO†)(P + Q)  = (σ − OσO†)P + P(σ − OσO†)Q  (159)  where we used QOσO†Q = QσQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Using Cauchy-Schwarz for the Hilbert-Schmidt scalar product we  can bound for M = M †  |⟨ϕ1, Mϕ2⟩| = |Tr(|ϕ2⟩ ⟨ϕ1| M)| ≤  �  Tr(|ϕ2⟩ ⟨ϕ1|ϕ1⟩ ⟨ϕ2|) Tr M 2� 1  2 = ∥ϕ1∥ · ∥ϕ2∥  �  Tr M 2� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (160)  32  Using (159) and (160) we can continue to estimate  ���  ϕ  ���  OσO† − σ  �  ϕ  ��� ≤ (∥Pϕ∥ ∥ϕ∥ + ∥Pϕ∥ ∥Qϕ∥)  �  tr  �  OσO† − σ  �2� 1  2 ≤ 2∥Pϕ∥  �  tr  �  OσO† − σ  �2� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (161)  This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The next lemma states two simple algebraic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For p, q ∈ [c, 1 − c] the following bounds hold  �  (1 − p)(1 − q) + √pq ≥ 1 − |p − q|2  4c(1 − c)  (162)  ���  �  (1 − p)q −  �  (1 − q)p  ��� ≤  |p − q|  2  �  c(1 − c)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (163)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We first consider the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We note that |√x−√y| ≤ |x−y|/(√x+√y) and thus  ���  �  (1 − p)q −  �  (1 − q)p  ��� ≤  |p − q|  �  p(1 − q) +  �  q(1 − p)  ≤  |p − q|  2 4�  p(1 − q)q(1 − p)  ≤  |p − q|  2  �  c(1 − c)  (164)  where we used the arithmetic geometric mean inequality in the middle step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To prove the first bound  we note that  ��  (1 − p)(1 − q) + √pq  �2  +  ��  (1 − p)q −  �  (1 − q)p  �2  = 1  (165)  This implies  �  (1 − p)(1 − q) + √pq =  �  1 −  ��  (1 − p)q −  �  (1 − q)p  �2  ≥ 1 −  ��  (1 − p)q −  �  (1 − q)p  �2  ≥ 1 − |p − q|2  4c(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (166)  The following simple lemma provides a lower bound on the square root that is used in the proof  of Lemma 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let s, t be real numbers such that s + t ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the bound  √  1 + s + t ≥ 1 − |s| + t  2 − t2  2  (167)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' First we note that elementary manipulations show that for all t ∈ R the bound  �  max(1 + t, 0) ≥ 1 + t  2 − t2  2  (168)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' First we consider s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' In this case, we can conclude  √  1 + s + t ≥  �  max(1 + t, 0) ≥ 1 + t  2 − t2  2 ≥ 1 + t  2 − t2  2 − |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (169)  33  Note that x − y = (√x − √y)(√x + √y) > (√x − √y) if x > 1 and x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This implies for s < 0 and  t ≥ 0 that  √  1 + t + s ≥  √  1 + t − |s| ≥ 1 + t  2 − t2  2 − |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (170)  Finally, we consider the case t, s < 0 where we get from (168)  √  1 + t + s ≥ 1 + t + s  2  − (t + s)2  2  = 1 + t  2 − t2  2 + s  �1 − t − s  2  �  ≥ 1 + t  2 − t2  2 − |s|  (171)  using −t − s ≤ 1 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Finally, we state a simple fact on the relation of partial trace and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let ρ be an operator on the system Q ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let O be a linear operator on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then  trR((O × Id)ρ) = O trR(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (172)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' By linearity it is sufficient to consider ρ = S ⊗ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' But then  trR((O × Id)ρ) = trR(OS × T) = tr(T)OS = O trR(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (173)  C  Fidelity loss of oracle calls  In this section we bound the loss in fidelity when applying the oracles Fp  i and Fq  i to density matrices  ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Recall that Fp  i (ρ) = (1 − p)ρ + pOiρO†  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Fidelity bound for invariant operators  Here we discuss a lemma that is useful to bound the loss  in fidelity  √  F(ρ, σ) −  √  F(OρO†, σ) when it is known that Oψ = ψ for many states ψ (the eigenvalue  1 has large multiplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that in terms of the oracles Fp  i this corresponds to the case p = 1 and  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let O be a unitary operator and P a hermitian projection such that  P(O − Id) = (O − Id),  (174)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', 1 − P projects on a subspace of the eigenspace of eigenvalue 1 of O and [P, O] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let ρ, σ be  two density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the bound  √  F(ρ, σ) −  √  F(ρ, OσO†) ≤ 2  �  tr(Pρ) tr(Pσ)  (175)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Call the system on which ρ, σ act Q Let R be a copy of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let ϕ and ψ be purifications of ρ  and σ on the system QR such that  √  F(ρ, σ) = ⟨ϕ, ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then (O ⊗ Id)ψ is a purification of σ and we  get  √  F(ρ, OσO†) ≥ ⟨ϕ, (O ⊗ Id)ψ⟩ = ⟨ϕ, ψ⟩ − ⟨ϕ, ((O − Id) ⊗ Id)ψ⟩  =  √  F(ρ, σ) − ⟨ϕ, (P(O − Id) ⊗ Id)ψ⟩  (176)  34  We now bound using [P, O] = 0 and Lemma 6  ⟨ϕ, (P(O − Id) ⊗ Id)ψ⟩ ≤ ⟨(P ⊗ Id)ϕ, (P ⊗ Id)((O − Id) ⊗ Id)ψ⟩  ≤ ∥(P ⊗ Id)ϕ∥ · ∥((O − Id) ⊗ Id)(P ⊗ Id)ψ∥  ≤ 2  �  tr trR((P ⊗ Id) |ϕ⟩ ⟨ϕ| (P ⊗ Id)†) tr trR((P ⊗ Id) |ψ⟩ ⟨ψ| (P ⊗ Id)†)  � 1  2  ≤ 2  �  tr(Pρ) tr(Pσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (177)  Proof of Lemma 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Call the system on which ρ, σ act Q and we denote ¯ρ = Fp  i (ρ) and ¯σ = Fq  i (σ) Let R be a copy  of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let ϕ and ψ be purifications of ρ and σ on the system QR such that  √  F(ρ, σ) = ⟨ϕ, ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We  use the shorthand ¯Oi = Oi ⊗ IdR in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let S be a system consisting of a single qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We  consider the following state on the system QRS  ω =  �  1 − p |ϕ, 0⟩ + √p  �� ¯Oiϕ, 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (178)  It is easy to check that ω is a purification of ¯ρ  trQR |ω⟩ ⟨ω| = trQ  �  (1 − p) |ϕ⟩ ⟨ϕ| + p  �� ¯Oiϕ  � � ¯Oiϕ  ���  = (1 − p)ρ + pOiρO†  i = ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (179)  To obtain a purification of ¯σ we define for an angle α the state  ζ =  �  1 − q cos(α) |ψ, 0⟩ + √q sin(α)  �� ¯Oiψ, 0  �  −  �  1 − q sin(α) |ψ, 1⟩ + √q cos(α)  �� ¯Oiψ, 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (180)  It is easy to check that ∥ζ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now check that this is a purification of ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that the cross  terms |ψ⟩  � ¯Oiψ  �� and  �� ¯Oiψ  �  ⟨ψ| cancel and thus  trS |ζ⟩ ⟨ζ| = (1 − q) cos2(α) |ψ⟩ ⟨ψ| + q sin2(ω)  �� ¯Oiψ  � � ¯Oiψ  ��  + (1 − q) sin2(α) |ψ⟩ ⟨ψ| + q cos2(α)  �� ¯Oiψ  � � ¯Oiψ  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  = (1 − q) |ψ⟩ ⟨ψ| + q  �� ¯Oiψ  � � ¯Oiψ  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (181)  We calculate  ⟨ω|ζ⟩ =  �  (1 − p)(1 − q) cos(α) ⟨ϕ|ψ⟩ +  �  (1 − p)q sin(α)  �  ϕ  �� ¯Oiψ  �  −  �  p(1 − q) sin(α)  � ¯Oiϕ  ��ψ  �  + √pq cos(α)  � ¯Oiϕ  �� ¯Oiψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (182)  Using that Oi is self-adjoint and unitary we obtain  ⟨η|ζ⟩ =  ��  (1 − p)(1 − q) + √pq  �  cos(α) ⟨ϕ|ψ⟩ +  ��  (1 − p)q −  �  p(1 − q)  �  sin(α)  �  ϕ  �� ¯Oiψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (183)  Now we set  sin(α) =  �  (1 − p)q −  �  p(1 − q)  (184)  cos(α) =  �  (1 − p)(1 − q) + √pq  (185)  and obtain  ⟨ω|ζ⟩ = cos2(α) ⟨ϕ|ψ⟩ + sin(α)2 � ¯Oiϕ  ��ψ  �  = ⟨ϕ|ψ⟩ + sin(α)2 � ¯Oiϕ − ϕ  ��ψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (186)  35  We conclude that  √  F(¯ρ, ¯σ) ≥ | ⟨η|ζ⟩ | ≥  √  F(ρ, σ) − | sin2(α)  � ¯Oiϕ − ϕ  ��ψ  �  |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (187)  As above we write ¯Pi = Pi ⊗ Id and P−i = Id − Pi and we get  ¯Oϕ − ϕ = ( ¯Pi + ¯P−i)( ¯Oiϕ − ϕ) = ¯Pi( ¯Oiϕ − ϕ) + ¯P−i( ¯Oiϕ − ϕ) = ¯Pi( ¯Oiϕ − ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (188)  Then we control using Lemma 6  ��� ¯Oiϕ − ϕ  ��ψ  ���2 =  ���  ( ¯Oiϕ − ϕ)  �� ¯Piψ  ���2 ≤ ∥ ¯Pi( ¯Oiϕ − ϕ)∥2 ∥ ¯Piψ∥2  = tr ¯Pi  ��� ¯Oiϕ − ϕ  � � ¯Oiϕ − ϕ  �� ¯P †  i  �  tr  ��� ¯Piψ  � � ¯Piψ  ���  = tr trR  �  ( ¯Oi − Id) ¯Pi |ϕ⟩ ⟨ϕ| ¯P †  i ( ¯O†  i − Id)  �  tr trR  �  ¯Pi |ψ⟩ ⟨ψ| ¯P †  i  �  ≤ 4 tr (Piρ) tr (Piσ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (189)  Using the bound (163) from Lemma 4 implies  | sin(α)| =  �  (1 − p)q −  �  p(1 − q) ≤  |p − q|  2  �  η(1 − η)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (190)  From (187), (189), and (190) we conclude that  √  F(¯ρ, ¯σ) ≥ | ⟨η|ζ⟩ | ≥  √  F(ρ, σ) − (p − q)2  �  tr (Piρ) tr (Piσ)  2η(1 − η)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (191)  Proof of Lemma 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' First, simple algebra shows ∥ ¯ψ0∥2 = 1 − q′ and  Ep  U(ρ) = ¯ρ0 + ¯ρ1 = (1 − p)ρ0 + pρ1  (192)  Eq  U(|ψ⟩ ⟨ψ|) =  �� ¯ψ0  � � ¯ψ0  �� +  �� ¯ψ1  � � ¯ψ1  �� = (1 − q′) |ψ0⟩ ⟨ψ0| + q′ |ψ1⟩ ⟨ψ1| ,  (193)  in particular (82) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Strong concavity of the fidelity implies the first bound of (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now  address the second estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We can express,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' using that U = U † is self adjoint  �  (1 − p)(1 − q′)  √  F(ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ψ0) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)(1 − q′) ⟨ψ0| ρ |ψ0⟩ =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� ¯ψ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�� ρ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�� ¯ψ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − q) cos2(α) ⟨ψ| ρ |ψ⟩ + q sin2(α) ⟨ψ| UρU † |ψ⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='q(1 − q) cos(α) sin(α)Re(⟨ψ| ρU |ψ⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − q) cos2(α) + q sin2(α) + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='q(1 − q) cos(α) sin(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='S2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='+ q sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='� 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(194)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='Now we calculate using the definition of α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='1 − q cos(α) + √q sin(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)(1 − q)(  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)(1 − q) + √pq) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)q(  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − p)q −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='(1 − q)p)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='= (1 − p)(1 − q) + (1 − p)q = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (195)  36  Then we obtain  �  (1 − p)(1 − q′)  √  F(ρ0, ψ0)  = (1 − p)S  �  1 + q sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ + 2  �  q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  (1 − p)S2  � 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (196)  Similarly the second term can be expressed as  �  pq′√  F(ρ1, ψ1) = √p  � �  q cos2(α) + (1 − q) sin2(α) − 2  �  q(1 − q) cos(α) sin(α)  �  S2  + (1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ − 2  �  q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  � 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (197)  We can calculate  √p  �√q cos(α) −  �  1 − q sin(α)  �  = √p  �√q(  �  (1 − p)(1 − q) + √pq) −  �  1 − q(  �  (1 − p)q −  �  (1 − q)p  �  = pq + (1 − q)p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (198)  we get  �  pq′√  F(ρ1, ψ1) =  = pS  �  1 + (1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ − 2  �  q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  pS2  � 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (199)  We continue to estimate the square root terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that by first order Taylor expansion the mixed  last terms of the two expressions (196) and (199) cancel and only the first term and higher order  corrections remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To make this rigorous we use Lemma 5 in Appendix B which states that for  s + t ≥ −1 the bound  √  1 + s + t ≥ 1 + t  2 − t2  2 − |s|  (200)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We apply this to (196) and (199) where s corresponds to the term involving ⟨ψ| (UρU † − ρ) |ψ⟩  and t corresponds to the term involving Re(⟨ψ| (ρU − ρ) |ψ⟩)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we get from (196)  �  (1 − p)(1 − q′)  √  F(ρ0, ψ0)  ≥ (1 − p)S  �  1 + −q sin2(α)| ⟨ψ| (UρU † − ρ) |ψ⟩ | +  �  q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  (1 − p)S2  − q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)2  2(1 − p)2S4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (201)  From (199) we get similarly  �  pq′√  F(ρ1, ψ1) =  ≥ pS  �  1 + −(1 − q) sin2(α) ⟨ψ| (UρU † − ρ) |ψ⟩ −  �  q(1 − q) cos(α) sin(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  pS2  − q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ) |ψ⟩)  2p2S4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (202)  37  We notice that the linear terms in Re(⟨ψ| (ρU − ρ) |ψ⟩) cancel and we get  �  (1 − p)(1 − q′)  √  F(ρ0, ψ0) +  �  pq′√  F(ρ1, ψ1)  ≥ S − sin2(α)| ⟨ψ| (UρU † − ρ) |ψ⟩ |  S  −  �1  p +  1  1 − p  � q(1 − q) cos2(α) sin2(α)Re(⟨ψ| (ρU − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' |ψ⟩)2  2S3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (203)  Finally we can estimate using the assumption p, q ∈ [η, 1 − η] and Lemma 4  �  (1 − p)(1 − q′)  √  F(ρ0, ψ0) +  �  pq′√  F(ρ1, ψ1)  ≥ S − |p − q|2 · | ⟨ψ| (UρU † − ρ) |ψ⟩ |  4η(1 − η)S  −  �1  η +  1  1 − η  � |p − q|2Re(⟨ψ| (ρU − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' |ψ⟩)2  32η(1 − η)S3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (204)  This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We now sketch the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Unfortunately we cannot directly derive the result but  we need to slightly modify the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The proof proceeds exactly in as the proof of Lemma 1 with the following minor  modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We have to insert an additional term p⟨ ¯ψ0, σ ¯ψ0⟩ in (194) and a term −(1 − p)⟨ ¯ψ1, σ ¯ψ1⟩  in (197).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We carry those terms and when we estimate the square-root terms we add them to the  s part in the bound (200), thus we end up with an additional term −p|⟨ ¯ψ0, σ ¯ψ0⟩|/S in (201) and  −(1 − p)|⟨ ¯ψ1, σ ¯ψ1⟩|/S in (202) and thus their sum appears in (204).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  D  Proof of Theorem 4  Here we prove the lower bound in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us first give a precise statement of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We  consider the same probability vectors pi as introduced at the beginning of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' First, we note that  the result does not hold for fixed oracles Oi with reward vector pi because the algorithm could exploit  the specific structure of the oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' There could be, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', a state ω0 such that O(pi) |j⟩ |ω0⟩ |0⟩ =  |j⟩ |ω0⟩ |δij⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then the problem reduces to the unstructured search problem when ω0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus,  the result only holds when we assume that Oi is a random oracle with the fixed reward vector pi,  emulating the situation where we have no additional information about the oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We consider for  a reward vector r ∈ {0, 1}|HA|·|HP | the oracle Or acting as in (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=',  Or |i⟩ |ω⟩ |c⟩ = |i⟩ |ω⟩ |c + ri(ω)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (205)  We consider random reward distribution r(i) where r(i) is the uniform distribution over all reward  vectors with mean reward vector pi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', |HP |−1 �  ω r(i)j(ω) = (pi)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then a more precise version of  Theorem 4 reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that pj are as before with pi ∈ [η, 1 − η] for some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Any  algorithm that identifies the best arm with probability at least 1 − δ given an oracle Or(i) where r(i) is  distributed as above requires at least  T ≥ 1  2η(1 − 2  �  δ(1 − δ))  � n  �  i=2  ∆−2  i  � 1  2  ≥ c(δ, η)  �  H(p1)  (206)  calls to the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  This is still true even when it is known that the vector of mean rewards is  {p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' , pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  38  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The proof is close to the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume we are given any algorithm acting  by (EO ⊗ Id) ◦ EUT ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ (EO ⊗ Id) ◦ EU1 where Ut are arbitrary unitary maps where O denotes the  given oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Assume that the initial state is a fixed density matrix ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We denote the state using the  oracle Or(i) before the t + 1-th invocation of the oracle by ρr(i)  t  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', ρr(i)  0  = EU1(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We introduce the  notation Ei when we average over the reward distribution r(i) and we write  ρi  t = Eiρr(i)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (207)  By assumption we can identify i given ρi  T with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This implies as in (26), (57)  for i > 1  2  �  δ(1 − δ) ≥  √  F(ρi  T , ρ0  T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (208)  One main ingredient of the proof is to define a suitable coupling of the random variables r(0) and  r(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that its reward vectors are p0 and pi such that (pi)j = pj = (p0)j for j ̸= i and (pi)i = p0,  (p0)i = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We now consider a coupling where r(0)j = r(i)j for i ̸= j and r(i)i ≥ r(0)i and the  distribution of r(i)i ∈ {0, 1}|HP | is uniform over all rewards under this constraint for a fixed r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Note that this entails that the distribution of r(0) for a fixed r(i) is also uniform over all rewards  with the right mean reward satisfying r(0) ≤ r(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' It is straightforward to see that such a coupling  exists (a possible explicit construction is to draw i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' random numbers ui(ω) and set r(0)i(ω) = 1 iff  ui(ω) is in the pi-th quantile of the numbers ui(·) and similarly for r(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We denote this coupling by  pi(r(i), r(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Observe that for this coupling satisfies, for all ω,  P(r(i)j(ω) = 1|r(0)j(ω) = 0) = P(r(i)j = 1|r(0)j = 0) =  �  0  for j ̸= i  p0−pi  1−pi =  ∆i  1−pi  for j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (209)  Similarly, we get for all ω  P(r(0)j(ω) = 0|r(i)j(ω) = 1) = P(r(0)j = 0|r(i)j = 1) =  �  0  for j ̸= i  p0−pi  p0  = ∆i  p0  for j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (210)  We can bound using the concavity of the fidelity  √  F(ρi  t, ρ0  t) ≥  �  r(i),r(0)  pi(r(i), r(0))  √  F(ρr(i)  t  , ρr(0)  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (211)  Now we lower bound the fidelity terms on the right hand side based on our construction of the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  By construction we have r(i) ≥ r(0) which implies that r(i) − r(0) ∈ {0, 1}|HA|·|HP |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that  Or(i)Or(0) |j, ω, c⟩ = |j, ω, c + (r(i))j(ω) + (r(0))j(ω)⟩ = |j, ω, c + (r(i))j(ω) − (r(0))j(ω)⟩  = Or(i)−r(0) |j, ω, c⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (212)  Let us introduce the self adjoint projection P r(i),r(0) given by  P r(i),r(0) |j, ω, c⟩ = 1(r(i))j(ω)̸=(r(0))j(ω) |j, ω, c⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (213)  Note that then  (Id − P r(i),r(0))Or(i)−r(0) |j, ω, c⟩ = (Id − P r(i),r(0)) |j, ω, c⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (214)  We get, using the invariance of the fidelity under unitary transformations, (Or(i))2 = Id, and (212)  √  F(ρr(i)  t+1, ρr(0)  t+1 ) =  √  F(EOr(i)(ρr(i)  t  ), EOr(0)(ρr(0)  t  )) =  √  F(ρr(i)  t  , EOr(i) ◦ EOr(0)(ρr(0)  t  ))  =  √  F(ρr(i)  t  , EOr(i)−r(0)(ρr(0)  t  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (215)  39  Next we apply Lemma 7 to Or(i)−r(0) and P r(i),r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The assumptions of the lemma are satisfied  because (214) and all P r(i),r(0) and Or commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Lemma 7 together with the last display imply  √  F(ρr(i)  t+1, ρr(0)  t+1 ) ≥  √  F(ρr(i)  t  , ρr(0)  t  ) − 2  �  tr  �  P r(i),r(0)ρr(i)  t  �  tr  �  P r(i),r(0)ρr(0)  t  �  (216)  We obtain  √  F(ρi  T , ρ0  T ) ≥ 1 − 2  �  r(i),r(0)  �  t  pi(r(i), r(0))  �  tr  �  P r(i),r(0)ρr(i)  t  �  tr  �  P r(i),r(0)ρr(0)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (217)  Using (208) followed by Cauchy-Schwarz we get  1  2 −  �  δ(1 − δ) ≤  �  r(i),r(0)  �  t  pi(r(i), r(0))  �  tr  �  P r(i),r(0)ρr(i)  t  �  tr  �  P r(i),r(0)ρr(0)  t  �  ≤  �  �  �  t,r(i),r(0)  pi(r(i), r(0)) tr  �  P r(i),r(0)ρr(i)  t  �  �  �  1  2 �  �  �  t,r(i),r(0)  pi(r(i), r(0)) tr  �  P r(i),r(0)ρr(0)  t  �  �  �  1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (218)  We observe that by construction of the coupling and (209) we have  �  r(i)  pi(r(i), r(0))P r(i),r(0) |j, ω, c⟩ =  �  r(i)  pi(r(i), r(0))1(r(i))j(ω)̸=(r(0))j(ω) |j, ω, c⟩  = p(r(0))P((r(i))j(ω) = 1|r(0)) |j, ω, c⟩  = p(r(0))1i=j1r(0)i(ω)=0  ∆i  1 − pi  |j, ω, c⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (219)  And similarly, using (210)  �  r(0)  pi(r(i), r(0))P r(i),r(0) |j, ω, c⟩ = p(r(i))1i=j1r(i)i(ω)=1  ∆i  p0  |j, ω, c⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (220)  As before, we define Pi to be the projection on arm i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=', Pi |j, ω, c⟩ = δij |j, ω, c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We conclude that  �  r(i)  pi(r(i), r(0)) tr  �  P r(i),r(0)ρr(0)  t  �  ≤ p(r(0))∆i  η tr  �  Piρr(0)  t  �  ,  (221)  �  r(0)  pi(r(i), r(0)) tr  �  P r(i),r(0)ρr(i)  t  �  ≤ p(r(i))∆i  η tr  �  Piρr(i)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (222)  Combining this with (218) and (207) we obtain  1  2 −  �  δ(1 − δ) ≤  �  �∆i  η  �  t,r(i)  p(r(i)) tr  �  Piρr(i)  t  �  �  �  1  2 �  �∆i  η  �  t,r(0)  p(r(0)) tr  �  Piρr(0)  t  �  )  �  �  1  2  = ∆i  η  ��  t  tr  �  Piρi  t  �  � 1  2 ��  t  tr  �  Piρ0  t  �  � 1  2  ≤ ∆i  √  T  η  ��  t  tr  �  Piρ0  t  �  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (223)  We square this relation divide by ∆2  i and sum over i > 1 and get  �1  2 −  �  δ(1 − δ)  �2 �  i>1  ∆−2  i  ≤ T  η2  �  i  �  t  tr  �  Piρ0  t  �  = T  η2  �  t  tr  �  ρ0  t  �  = T 2  η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (224)  40  This implies  T ≥ 1  2η(1 − 2  �  δ(1 − δ))  ��  i>1  ∆−2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (225)  E  Proof of Theorem 9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume that we work on a Hilbert space H = HI ⊗ HS ⊗ HA where HI is the input space,  HS is a single qubit state space and HA consists of T ancilla qubits where T = O(  √  N/p) will be  defined below in (234).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We assume that the initial state of HI is |s⟩ =  √  N  −1 � |i⟩ and all remaining  qubits are in state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The algorithm consists of applying in turn the three operators F, a controlled  Grover-diffusion  Uω = id ⊗ |0⟩ ⟨0| ⊗ id + (2 |s⟩ ⟨s| − id) ⊗ |1⟩ ⟨1| ⊗ id,  (226)  and a swap operation St that swaps the state qubit with the t-th ancilla qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We can rewrite this  concisely as  (EST ◦ EUω ◦ F) ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' ◦ (ES1 ◦ EUω ◦ F)(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (227)  We denote by ˜Oi the standard oracle on HI and denote by G = (2 |s⟩ ⟨s|−id) ˜Oi the map from Grover’s  algorithm acting on HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We claim that the final state of the algorithm is  ρT =  �  x∈{0,1}T  p|x|1(1 − p)T −|x|1  ���G|x|1s, 0, x, 0A−T � �  G|x|1s, 0, x, 0A−T ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (228)  The proof of this is by induction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' we note  (EST ◦ EUω ◦ E)(  ���G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���)  = (EST ◦ EUω)  �  p  ��� ˜OG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  ˜OG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���  +(1 − p)  ���G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���  �  = EST  �  p  ���GG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  GG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���  +(1 − p)  ���G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���  �  = p  ���GG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T −1� �  GG|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T −1���  + (1 − p)  ���G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T � �  G|x|1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' 0A−T ���  (229)  which implies (228).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The reduced density matrix on HI is  ρI  T =  T  �  k=0  pk(1 − p)T −k ��Gks  � �  Gks  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (230)  The classical analysis of the Grover algorithm proves  Gk |s⟩ = cos  �2k + 1  2  θ  �  |s′⟩ + sin  �2k + 1  2  θ  �  |i⟩  (231)  41  where s′ =  √  N − 1  − 1  2 �  j̸=i |i⟩ and  θ = 2 arccos  ��  (N − 1)/N  �  = 2 arcsin  �√  N  −1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (232)  We remark that θ/2 ≈  √  N  −1 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Note that for  k ∈ I = (π/(4θ) − 1/2, 3π/(4θ) − 1/2)  (233)  we have sin  � 2k+1  2  θ  �2 ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let  T = ⌊π/(2θp)⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (234)  It remains to be shown that with probability at least 1/2 a Bin(T, p) distributed variable is contained  in the interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Using that the variance of X ∼ Bin(T, p) is Tp(1 − p) we can bound  P(|X − pT| > pT/8) ≤ 64E(|X − pT|2  p2T 2  ≤ 64  pT ≤ 256θ  π  ≤ 1  2  (235)  for N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Moreover, |X − pT| < pT/8 implies  X ∈  �3pT  8 , 5pT  8  �  ⊂  �3π  8θ − 1, 5π  8θ  �  ⊂  � π  4θ − 1/2, 3π  4θ − 1/2  �  (236)  for N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  F  Analysis of reusable oracles  Here we sketch a proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The proof essentially relies on the algorithm given in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  The only building block that needs to be changed is the gapped amplitude estimation (Corollary 2 in  [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let us explain this algorithm along with the replacement based on oracles as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Gapped  amplitude estimation assumes we have access to an oracle Op and its adjoint acting via  Op |0⟩ = √p |1⟩ +  �  1 − p |0⟩ = |coinp⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (237)  Then the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 8 (Corollary 2 in [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For ε > 0, l ∈ [0, 1] and δ > 0 there is a unitary procedure with  O(ε−1 ln(δ)) queries to Op that prepares the state  |coinp⟩ (α0 |0⟩ |ψ1⟩ + α1 |1⟩ |ψ2⟩  (238)  with α0, α1 ∈ [0, 1] and α1 ≤ δ if p ≤ l − 2ε and α0 ≤ δ if p ≥ l − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We replace this by classical estimation of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Using tail bounds of random variables we  obtain the following simple lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Let Sk be the sum of k independent random variables with distribution Ber(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For ε > 0  the bounds  P(Sk > k(p + ε)) ≤ e−2ε2k,  P(Sk < k(p − ε)) ≤ e−2ε2k  (239)  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  42  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This is just Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  We obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Given access to oracles as in (7) we can construct for any ε, δ > 0 and l ∈ [0, 1] an  algorithm A that maps with probability at least 1 − δ for all 1 ≤ i ≤ n  A |i⟩ |0⟩ = |i⟩ |c⟩  (240)  where c = 1 if pi < l − 2ε and c = 0 if pi > l − ε and A requires ε−2 oracle calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Set k = ⌈2ε−2 ln(n/δ)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Then we consider the sequence  |i⟩ |0⟩ |0⟩ → |i⟩  ������  k  �  j=0  Xt  i  �  |0⟩ → |i⟩  ������  k  �  j=0  Xt  i  � ���1�k  j=0 Xt  i <k(l−3ε/2)  �  → |i⟩ |0⟩  ���1�k  j=0 Xt  i <k(l−3ε/2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (241)  Where we in the first step sum the rewards using the oracles OXt, then set the flag register conditional  on the sum and then uncompute the work register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This requires 2k oracle calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Using Lemma 9 we  conclude that for pi > l − ε  P  �  �  k  �  j=0  Xt  i < l − 3ε/2  �  � ≤ e−2k(ε/2)2 ≤ e− ln(n/δ) = δ  n  (242)  and a similar statement holds for pi < l − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The union bound implies the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Now we consider Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Giving a full proof would require a large amount of notation that is  not worth the effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Thus, we give a very brief sketch of the argument and leave the details to the  reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Their proof relies on variable time algorithms [4] which we also do not introduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' For  readers not familiar with them, the following proof shall merely serve as a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Sketch of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' We apply the algorithm constructed in [34] but replace the gapped  amplitude estimation by the algorithm constructed in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' The main strategy used in their  proof is to construct a variable time algorithm based on the gapped amplitude estimate that flags all  arms whose reward is smaller than a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This allows to construct algorithms to count  the number of flagged arms and rotate on the subspace of flagged arms using variable time amplitude  amplification [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' Those sub-routines can be used to first estimate p1 and p2 and then identify the  best arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Their variable time algorithm based on the gapped amplitude estimate has query complexity on  arm i is at most ∆−1  i  log(1/a) with a polynomial in ∆1n and the l2 averaged run-time thus amounts  to  t2  av ≤ C 1  n  n  �  i=2  ∆−2  i  ln2(1/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (243)  When relying on the algorithm from Corollary 5 we obtain the query complexity ∆−2  i  log(n/δ′) for  arm i where δ′ denotes the bound on the failure probability for the algorithm and the l2 averaged  run-time then amounts to  t2  av ≤ C 1  n  n  �  i=2  ∆−4  i  ln2(nδ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (244)  43  Then the variable time amplitude amplification gives an algorithm with success probability more than  1/2 and query complexity tav/√psucc ln(tmax) where psucc denotes the success probability and tmax  the maximal complexity of the initial variable algorithm (there is another term tmax ln(tmax) which is  smaller in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  Since the initial success probability is n−1 we obtain the query complexity bound  T ≤ tav  √n ln(tm) ≤ C  � n  �  i=2  ∆−4  i  � 1  2  ln(n/δ′) ln(tmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  (245)  We need to apply a total of O(ln  �  ∆−1  1  �  such amplified variable time algorithms (essentially we perform  binary search to find lL < lR such that p2 < lL − ∆2/4, lR + ∆2/4 < p1, and lL + ∆2/4 < lR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' To  ensure that all constructed oracles as in Corollary 5 succeed with probability more than 1 − δ it is  thus sufficient to pick δ′ = cδ/ ln  �  ∆−1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content=' This together with tmax = ˜O(∆−2  1 ) ends the sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
+page_content='  44' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQfZR1S/content/2301.08544v1.pdf'}
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+1 
+ 
+Mathematical theory of diffusion in solids: solutions in the semi-infinite body and solution to a 
+diffusion problem with a variable boundary condition. 
+ 
+Guglielmo Macrelli (*) 
+ 
+(*) Isoclima SpA – R&D Dept. Via A.Volta 14, 35042 Este (PD) Italy 
+guglielmomacrelli@hotmail.com 
+ 
+Abstract: 
+ 
+A review of solutions of solid-state diffusion problems in infinite and semi-infinite bodies is 
+presented. Based on the identified solutions for the semi-infinite body a two-step diffusion problem 
+is discussed in detail with the first step characterized by a Dirichlet constant concentration condition 
+and the second step by a Neumann condition.  
+ 
+I. 
+Introduction 
+ 
+In solid state diffusion problems it is common to have diffusing elements in a solid body that can be 
+externally injected from deposited layers, coming from other phases either liquid or gaseous or 
+internally  moving or redistributed from internal locations in  the solid matrix. When the diffusion 
+length D is very small compared to the sample dimension (usually the thickness d) through which 
+diffusion occurs (D << d) than we can approximate the solid to a semi-infinite half body. The main 
+purpose of this study is to provide a detailed proof and justification for a variable boundary condition 
+problem which solution has been provided by the author in the literature1,2 without a proof. The reason 
+for not providing the proof was mainly related to the length and complexity of the proof itself. There 
+can be found mentions of this problem in the literature3,4 that for different reasons do not fully cover 
+the purpose of this study. The solution provided by Malkovich3 inspired the approach used in this 
+study but it is limited to a situation where the two steps diffusion processes are run with the same 
+diffusion coefficient (same temperatures). The solution provided by Kennedy and Morley4 is 
+according to the Green functions approach but it contains presumably a typo in the final solution. In 
+this study we have considered worth to preliminary review the matter of solutions to diffusion 
+problems in the infinite and semi-infinite bodies as a necessary step before approaching the more 
+complex variable boundary diffusion problem. We consider worth to have provided a detailed proof 
+of this solution first to cover a missing gap second because the constructional approach to the proof 
+may be beneficial for similar problems in solid state diffusion. This is a contribution to the 
+mathematical theory of solid state diffusion covering some gaps of past literature.  
+ 
+II. 
+Solution in the infinite and semi-infinite body 
+ 
+In this study we deal with the differential equation for the concentration of diffusing elements into a 
+solid body. We consider the mono-dimensional version of the problem. The starting point is the Fick 
+equation for the flux J of the diffusing elements defining a diffusion coefficient D: 
+ 
+
+−
+=
+
+( , )
+( , )
+c x t
+J x t
+D
+x
+.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(1) 
+Together with the continuity equation: 
+ 
+
+
++
+=
+
+
+( , )
+( , )
+0
+c x t
+J x t
+t
+x
+  
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(2) 
+ 
+
+2 
+ 
+it results the diffusion equation for concentration:  
+  
+
+
+
+
+
+=
+
+
+
+
+
+
+
+( , )
+( , )
+c x t
+c x t
+D
+t
+x
+x
+. 
+ 
+ 
+ 
+                      
+ 
+ 
+(3) 
+ 
+When diffusion coefficient D can be considered reasonably constant, the second member of (3) is just 
+diffusion coefficient times the second derivative of concentration. A similar diffusion equation can 
+be written for the flux J(x,t).  If the concentration function is such that derivative for space and time 
+can be interchanged than, because of (3), the time derivative of Flux for constant D is: 
+ 
+(
+)
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
+−
+= −
+= −
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+= −
+=
+
+
+
+
+
+
+
+2
+2
+2
+2
+2
+2
+( , )
+( , )
+( , )
+( , )
+( , )
+( , )
+J x t
+c x t
+c x t
+c x t
+D
+D
+D
+D
+t
+t
+x
+x
+t
+x
+x
+c x t
+D
+D
+D
+J x t
+x
+x
+x
+ 
+. 
+ 
+(4) 
+ 
+From (4) it is clear that, for flux function, we have the same differential equation that we have for 
+concentration. If flux is known than concentration can be calculated from (1): 
+ 
+
+=
+
++
+
+1
+( , )
+( , )
+( , )
+x
+c x t
+c
+t
+J x t dx
+D
+ . 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(5) 
+The usual condition for the value at infinite of concentration is c(∞,t)=0. 
+ 
+ 
+II.1 – General solution for the infinite body. 
+ 
+We deal here with the diffusion equation for the diffusing elements concentration c(x,t) with constant 
+diffusion coefficient D, written in the time and mono-dimensional space domains (t,x,) where 0 ≤  t  
+≤ ∞ and -∞ ≤ x ≤ +∞. 
+ 
+
+
+=
+
+
+2
+2
+( , )
+( , )
+c x t
+c x t
+D
+t
+x
+ . 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(6) 
+ 
+When the space variable domain is the full real axis we will name it infinite body domain. The 
+solution to equation (6) exists and it is unique providing that we set an initial condition: 
+ 
+   ( ,0)
+( )
+c x
+f x
+=
+  ,   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(7) 
+A solution to this problem has been indicated in the fundamental treatise of Carslaw and Jaeger5 : 
+
++
+−
+−
+−
+=
+
+2
+(
+)
+4
+1
+( , )
+( )
+2
+x y
+Dt
+c x t
+e
+f y dy
+Dt
+    .  
+ 
+ 
+ 
+ 
+ 
+ 
+(8) 
+ 
+A heuristic derivation of this result can be based on the general solution of (6) by the separation of 
+variables: 
+( , )
+( ) ( )
+c x t
+X x T t
+=
+ . 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(9) 
+Inserting position (9) in (6) leads to: 
+
+3 
+ 
+=
+2
+2
+( )
+( )
+( )
+( )
+dT t
+d X x
+X x
+DT t
+dt
+dx
+ , 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(10) 
+that can be split in two equations: 
+
+= −
+2
+1
+( )
+( )
+dT t
+DT t
+dt
+ 
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(11a) 
+
+= −
+2
+2
+2
+1
+( )
+( )
+d X x
+X x
+dx
+ , 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(11b) 
+with general solutions: 
+
+
+−
+=
+2
+( )
+Dt
+T t
+e
+  
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(12a) 
+
+
+
+
+=
++
+( )
+cos(
+)
+sin(
+)
+X x
+x
+x   
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+(12b)          
+According to (9) we can write the generic solution: 
+
+
+2
+( , )
+( )cos(
+)
+( )sin(
+)
+Dt
+c x t
+e
+A
+x
+B
+x
+
+
+
+
+
+−
+=
++
+ . 
+ 
+ 
+ 
+ 
+(13) 
+Because of the linearity of equation (6) the general solution is the one obtained for all possible  
+values generated by the boundary conditions on the considered domain. The values of  depend on 
+the boundary conditions that will be used, in particular for bounded domains the values of  are 
+discrete, leading to: 
+2
+1
+( , )
+(
+)cos(
+)
+(
+)sin(
+)
+j Dt
+j
+j
+j
+j
+j
+c x t
+e
+A
+x
+B
+x
+
+
+
+
+
+
+−
+=
+
+
+=
++
+
+
+
+, 
+ 
+ 
+ 
+(14) 
+while for unbounded domains they are continuous leading to: 
+
+
+2
+( , )
+( )cos(
+)
+( )sin(
+)
+Dt
+c x t
+e
+A
+x
+B
+x d
+
+
+
+
+
+
++
+−
+−
+=
++
+
+ . 
+ 
+ 
+ 
+(15) 
+We consider here solution (15) because we have an unbounded domain. A() and B() functions 
+can be determined by considering the initial condition (7): 
+
+
+( ,0)
+( )
+( )cos(
+)
+( )sin(
+)
+c x
+f x
+A
+x
+B
+x d
+
+
+
+
+
++
+−
+=
+=
++
+
+ . 
+ 
+ 
+ 
+(16) 
+Considering the Fourier theorem6 for the function f(x): 
+(
+)
+(
+)
+1
+( )
+( )cos (
+)
+2
+1
+1
+( )cos
+cos(
+)
+( )sin
+sin(
+)
+2
+2
+f x
+f
+x d
+d
+f
+d
+x
+f
+d
+x
+d
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
++
+−
+−
++
++
++
+−
+−
+−
+
+
+=
+−
+=
+
+
+
+
+
+
+
+
+
+
+
+
+=
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+,(17) 
+And comparing equations (17) and (16) we have: 
+
+4 
+ 
+1
+( )
+( )cos(
+)
+2
+A
+f
+d
+
+
+
+
+
++
+−
+=
+
+ 
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+(18a) 
+1
+( )
+( )sin(
+)
+2
+B
+f
+d
+
+
+
+
+
++
+−
+=
+
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+(18b) 
+ 
+From (15) and (18a) and (18b) the general solution is: 
+ 
+
+
+2
+2
+1
+( , )
+( ) cos(
+)cos(
+)
+sin(
+)sin(
+)
+2
+1
+( )
+cos (
+)
+2
+Dt
+Dt
+c x t
+e
+f
+x
+x
+d
+d
+f
+e
+x d
+d
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
++
++
+−
+−
+−
++
++
+−
+−
+−
+
+
+=
++
+=
+
+
+
+
+
+
+=
+−
+
+
+
+
+
+
+
+
+ 
+(19) 
+ 
+The integral in square brackets is from Budak and Fomin6 page 573: 
+ 
+(
+)
+2
+2
+4
+cos (
+)
+x
+Dt
+Dt
+e
+x d
+e
+Dt
+
+
+
+ 
+
+
+
+−
+
+
++
+−
+
+
+−
+
+
+−
+−
+=
+
+ 
+. 
+ 
+ 
+ 
+ 
+(20) 
+ 
+Equation (14) finally results in equation (8): 
+ 
+(
+)
+2
+4
+1
+( , )
+( )
+2
+x
+Dt
+c x t
+f
+e
+d
+Dt
+
+
+
+
+−
++
+−
+−
+=
+
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+(21) 
+ 
+The same result, namely equation (21), is achieved in a number of different ways as indicated also in 
+Carslaw and Jaeger5, we prefer this heuristic derivation to stress connections to boundary value 
+problems in either bounded and unbounded domains. 
+ 
+II.2 General solution for the semi-infinite body 
+ 
+In many scientifical and technological problems of interest in materials science we can assume, with 
+a good approximations, that the solid body under diffusion of externally delivered elements or 
+impurities is a semi-infinite body extended in the x ≥ 0 spatial dimension. In this case we can 
+conveniently use the solution for the infinite body (16) by extending the initial condition towards the 
+negative half space assuming an indeterminate function c1(x,0) (x<0) and defining the following 
+initial condition: 
+ 
+( )
+( ,0)
+f
+c
+
+
+=
+ ;  >   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(22a) 
+ 
+1
+( )
+( ,0)
+f
+c
+
+
+=
+ ;  <   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(22b) 
+ 
+With these positions solution (21) results: 
+ 
+
+5 
+ 
+(
+)
+(
+)
+2
+2
+0
+4
+4
+1
+0
+1
+( , )
+( ,0)
+( ,0)
+2
+x
+x
+Dt
+Dt
+c x t
+c
+e
+d
+c
+e
+d
+Dt
+
+
+
+
+
+
+
+−
+−
+
+−
+−
+−
+
+
+
+
+=
++
+
+
+
+
+
+
+  
+. 
+ 
+(23) 
+ 
+Taking the first integral in (23) and putting y=- than 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+2
+2
+2
+2
+0
+4
+1
+0
+4
+4
+4
+1
+1
+1
+0
+0
+( ,0)
+(
+,0)
+(
+,0)
+(
+,0)
+x
+Dt
+y x
+x y
+x
+Dt
+Dt
+Dt
+c
+e
+d
+c
+y
+e
+dy
+c
+y
+e
+dy
+c
+e
+d
+
+
+
+
+
+
+−
+−
+−
+− −
++
++
+
+
+−
+−
+−
++
+=
+= −
+−
+=
+−
+=
+−
+
+
+
+
+ 
+, 
+(24) 
+ 
+Hence: 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+2
+2
+2
+2
+4
+4
+1
+0
+0
+4
+4
+1
+0
+1
+( , )
+(
+,0)
+( ,0)
+2
+1
+( ,0)
+(
+,0)
+2
+x
+x
+Dt
+Dt
+x
+x
+Dt
+Dt
+c x t
+c
+e
+d
+c
+e
+d
+Dt
+c
+e
+d
+c
+e
+d
+Dt
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+−
+
+
+−
+−
+−
++
+
+−
+−
+
+
+
+
+=
+−
++
+=
+
+
+
+
+
+
+
+
+=
++
+−
+
+
+
+
+
+
+
+ 
+. 
+ 
+(25) 
+ 
+ 
+II.3 Semi-infinite body with reflecting boundary (SI-RB) 
+ 
+Reflecting boundary condition for the semi-infinite body means that no flow of matter is assumed at 
+x=0. This can be conveniently defined by using the flux equation and putting it equal to zero at the 
+half plane interface: 
+ 
+0
+(
+0, )
+0
+x
+c
+J x
+t
+D x
+=
+
+=
+= −
+=
+
+ 
+ 
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+(26) 
+ 
+Boundary condition (22) can be introduced by differentiating (25): 
+ 
+(
+)
+(
+)
+2
+2
+4
+4
+1
+0
+( , )
+1
+1
+(
+) ( ,0)
+(
+) (
+,0)
+2
+2
+x
+x
+Dt
+Dt
+c x t
+x
+c
+e
+d
+x c
+e
+d
+x
+Dt
+Dt
+
+
+
+
+
+
+
+
+
+−
++
+
+−
+−
+
+
+
+
+
+=
+−
+−
++
+−
+
+
+
+
+
+
+, (27) 
+ 
+and putting this to zero (26): 
+ 
+
+
+2
+4
+1
+0
+0
+( , )
+1
+1
+( ,0)
+(
+,0)
+0
+2
+2
+Dt
+x
+c x t
+c
+c
+e
+d
+x
+Dt
+Dt
+
+
+
+
+
+
+
+−
+=
+
+=
+−
+−
+=
+
+
+ 
+. 
+ 
+(28) 
+ 
+Condition (28) is satisfied only if: 
+ 
+1
+( ,0)
+(
+,0)
+c
+c
+
+
+=
+−
+ . 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(29) 
+
+6 
+ 
+ 
+Condition (29) in (25) leads to the general solution for the SI-RB problem: 
+ 
+(
+)
+(
+)
+2
+2
+4
+4
+0
+1
+( , )
+( ,0)
+2
+x
+x
+Dt
+Dt
+c x t
+c
+e
+e
+d
+Dt
+
+
+
+
+
+−
++
+
+−
+−
+
+
+
+
+=
++
+
+
+
+
+
+ 
+. 
+ 
+ 
+ 
+(30) 
+ 
+II.4 Semi-infinite body with capturing boundary (SI-CB) 
+ 
+ 
+The capturing boundary condition is expressed as follows: 
+ 
+(0, )
+0
+c
+t =
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(31) 
+ 
+This condition in (25) leads to: 
+ 
+
+
+( )
+2
+4
+1
+0
+1
+(0, )
+( ,0)
+(
+,0)
+2
+Dt
+c
+t
+c
+c
+e
+d
+Dt
+
+
+
+
+
+
+−
+=
++
+−
+
+  
+, 
+ 
+ 
+ 
+(32) 
+ 
+hence: 
+ 
+1
+( ,0)
+(
+,0)
+c
+c
+
+
+= −
+−
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(33) 
+ 
+and the general solution for the semi infinite capturing boundary (SI-CB) is: 
+ 
+(
+)
+(
+)
+2
+2
+4
+4
+0
+1
+( , )
+( ,0)
+2
+x
+x
+Dt
+Dt
+c x t
+c
+e
+e
+d
+Dt
+
+
+
+
+
+−
++
+
+−
+−
+
+
+
+
+=
+−
+
+
+
+
+
+ 
+. 
+ 
+ 
+ 
+(34) 
+ 
+A particular application of solution (34) can be considered for the following initial and boundary 
+conditions: 
+ 
+( ,0)
+0
+c x
+=
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(35a) 
+ 
+0
+( , )
+S
+x
+c x t
+C
+= =
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(35b) 
+ 
+Let’s introduce the auxiliary function: 
+ 
+*( , )
+( , )
+s
+c x t
+C
+c x t
+=
+−
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(36) 
+ 
+The diffusion problem for the auxiliary function with conditions (35a) and (35b) is: 
+ 
+
+
+=
+
+
+*
+2 *
+2
+( , )
+( , )
+c x t
+c x t
+D
+t
+x
+; 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(37a) 
+ 
+*(0, )
+0
+c
+t =
+.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(37b) 
+
+7 
+ 
+ 
+Solution to (37a) and (37b) is the one for the SI-CB 
+ 
+(
+)
+(
+)
+2
+2
+*
+*
+4
+4
+0
+1
+( , )
+( ,0)
+2
+x
+x
+Dt
+Dt
+c x t
+c
+e
+e
+d
+Dt
+
+
+
+
+
+−
++
+
+−
+−
+
+
+
+
+=
+−
+
+
+
+
+
+ 
+ 
+ 
+ 
+ 
+(38) 
+*( ,0)
+( ,0)
+s
+s
+c
+C
+c
+C
+
+
+=
+−
+=
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(39) 
+ 
+and: 
+ 
+(
+)
+(
+)
+2
+2
+*
+4
+4
+0
+( , )
+2
+2
+x
+x
+s
+Dt
+Dt
+S
+C
+x
+c x t
+e
+e
+d
+C erf
+Dt
+Dt
+
+
+
+
+−
++
+
+−
+−
+
+
+
+
+
+
+=
+−
+=
+
+
+
+
+
+
+
+
+
+ , 
+ 
+ 
+(40) 
+ 
+Finally: 
+ 
+*
+( , )
+( , )
+1
+2
+2
+s
+S
+s
+x
+x
+c x t
+C
+c x t
+C
+erf
+C erfc
+Dt
+Dt
+
+
+
+
+
+
+=
+−
+=
+−
+=
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+. 
+ 
+(41) 
+ 
+Solution (41) is a well-known and, in some ways, popular solution in the theory of diffusion in solids 
+in a semi-infinite body5,7,8,9.   
+ 
+III. 
+A variable boundary condition diffusion problem 
+ 
+Now we deal with a diffusion problem with a variable boundary condition. Such type of problems 
+are often encountered when diffusion species are introduced and let diffuse in a solid body by different 
+sequential mechanisms like for example diffusion from a continuous source like a bath or a vapour 
+environment than stopped and followed by a thermal treatment at different temperature. We are 
+typically in front of a two steps process characterized by different boundary condition for each step. 
+They can be modelled as constant source up to a certain time than changed as a limited source 
+diffusing in the body without incoming flux through the surface. In this case our problem can be split 
+into two steps separated boundary value problems for the diffusion equation (6): 
+ 
+ (0, )
+s
+c
+t
+C
+=
+ for 0 ≤ t ≤   with a diffusion coefficient D0  
+ 
+ 
+ 
+() 
+ 
+0
+( , )
+0
+x
+c x t
+x
+=
+
+=
+
+, for t≥ t  with a diffusion coefficient D 
+. 
+ 
+ 
+ 
+(43) 
+ 
+The solution up to t= is easily found in equation (41) that at t= becomes itself the initial condition 
+for problem with boundary condition (43): 
+ 
+0
+( , )
+2
+s
+x
+c x
+C erfc
+D
+
+
+
+
+=
+
+
+
+
+
+
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+(44) 
+ 
+Let’s rewrite problem (6), (43) and (44) in terms of flux: 
+ 
+
+8 
+ 
+(
+)
+
+
+=
+
+
+2
+2
+( , )
+( , )
+J x t
+D
+J x t
+t
+x
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(45) 
+ 
+(0, )
+0
+J
+t =
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(46) 
+ 
+Looking to (45) and (46) we recognize a SIB-CB problem for the flux function J(x,t) and the solution 
+reads (38): 
+(
+)
+(
+)
+( )
+2
+2
+2
+2
+4
+4
+0
+4
+4
+0
+1
+( , )
+( ,0)
+2
+( ,0)
+sinh 2
+2
+x
+x
+Dt
+Dt
+x
+Dt
+Dt
+J x t
+J
+e
+e
+d
+Dt
+e
+x
+J
+e
+d
+Dt
+Dt
+
+
+
+
+
+
+
+
+
+
+−
++
+
+−
+−
+−
+
+−
+
+
+
+
+=
+−
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+, 
+ 
+ 
+ 
+(47) 
+ 
+This is obtained by the development of the squares in the arguments of the exponentials and because 
+(
+)
+1
+sinh( )
+2
+z
+z
+z
+e
+e−
+=
+−
+. Additionally, the initial condition for the flux is: 
+2
+0
+4
+0
+0
+( , )
+1
+( ,0)
+D
+s
+t
+c
+t
+J
+D
+C D
+e
+D
+
+
+
+
+
+
+−
+=
+
+= −
+=
+
+, 
+ 
+ 
+ 
+ 
+(48) 
+ 
+finally the solution for the flux function reads: 
+ 
+2
+2
+0
+2
+2
+0
+0
+1
+1
+2
+4
+4
+4
+0
+0
+4
+4
+0
+0
+( , )
+sinh 2
+sinh 2
+x
+Dt
+D
+Dt
+s
+Dt D
+x
+DD t
+s
+Dt
+D
+e
+x
+J x t
+C
+e
+d
+D
+Dt
+Dt
+C
+D
+x
+e
+e
+d
+D
+Dt
+t
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+
+
+
+−
++
+
+
+
+
+
+
++
+
+−
+
+
+−
+
+
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+
+ 
+. 
+ 
+(49) 
+ 
+ 
+Let’s make the following positions: 
+ 
+(
+)
+(
+)
+0
+0
+0
+0
+0
+0
+2
+2
+0
+0
+;
+4
+4
+4
+4
+2
+4
+Dt
+D
+x
+a
+b
+DD t
+Dt
+x D
+DD t
+b
+x
+Dt
+Dt
+D
+a
+Dt Dt
+D
+x D
+b
+a
+Dt Dt
+D
+
+
+
+
+
+
+
+
++
+=
+= −
+= −
+= −
++
++
+=
++
+ 
+, 
+ 
+ 
+ 
+ 
+(50) 
+ 
+with these positions the solution for flux (49) results: 
+ 
+2
+2
+2
+2
+4
+0
+0
+1
+( , )
+2
+x
+a
+b
+b
+s
+Dt
+C
+D
+J x t
+e
+e
+e
+e
+d
+D
+t
+
+
+
+
+
+
+
+−
+−
+−
+
+
+=
+−
+
+
+
+ 
+ 
+ 
+ 
+ 
+(51) 
+
+9 
+ 
+ 
+The integral term in (51) can be evaluated as follows: 
+ 
+2
+2
+2
+2
+0
+1
+1
+( , )
+2
+4
+b
+a
+b
+b
+a
+b
+I a b
+e
+e
+e
+d
+e erf
+a
+a
+
+
+
+
+
+
+−
+−
+
+
+
+
+=
+−
+=
+
+
+
+
+
+
+
+ 
+ 
+ 
+ 
+(52) 
+ 
+This is coming from Abramowitz and Stegun10 7.4.2: 
+ 
+ 
+(
+)
+2
+2
+2
+0
+1
+2
+b
+ac
+a
+b
+c
+a
+b
+e
+d
+e
+erf
+a
+a
+
+
+
+
+
+−
+−
++
++
+
+
+=
+
+
+
+
+
+  
+, 
+ 
+ 
+ 
+ 
+(53) 
+ 
+and from the complementary error function erfc(x)=1-erf(x) properties such that: 
+ 
+ 
+( )
+(
+)
+2
+( )
+erfc z
+erfc
+z
+erf z
+−
+−
+= −
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+(54) 
+ 
+Developing (52) with positions (50) results: 
+ 
+(
+)
+2
+0
+0
+0
+4
+(
+)
+0
+0
+0
+4
+( , )
+2
+x D
+Dt Dt D
+x D
+DD t
+I a b
+erf
+e
+Dt
+D
+Dt Dt
+D
+
+
+
+
+
+
+
++
+
+
+
+
+=
+
+
++
++
+
+
+ 
+. 
+ 
+ 
+(55) 
+ 
+Finally flux solution (51) is: 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+2
+0
+0
+2
+0
+1
+4
+0
+0
+0
+0
+0
+2
+4
+0
+0
+0
+( , )
+2
+2
+D
+x
+Dt
+Dt D
+s
+x
+Dt D
+s
+x D
+C
+DD t
+D
+J x t
+e
+erf
+D
+Dt
+D
+t
+Dt Dt
+D
+x D
+D
+C
+e
+erf
+Dt
+D
+Dt Dt
+D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+
+
++
+
+
+−
++
+
+
+
+
+=
+=
+
+
++
++
+
+
+
+
+
+
+=
+
+
++
++
+
+
+ . 
+(56) 
+ 
+From the flux solution represented by equation (56) we can obtain the corresponding concentration 
+solution by equation (5) that is integrating equation (56). Let’s fix the following positions: 
+ 
+(
+)
+(
+)
+2
+2
+0
+0
+0
+2
+2
+0
+0
+0
+4
+;
+'
+;
+'
+2
+'
+'
+;
+'
+4
+2
+D
+D
+Dt
+k
+Dt
+x
+x
+dx
+dx
+D
+Dtdx
+D
+x
+x
+x
+kx
+Dt
+D
+Dt
+D
+Dt
+
+
+
+
+
+
+
+
+
+=
++
+=
+=
+=
+=
++
+=
+=
++
++
+  
+ 
+ 
+, 
+ 
+ 
+(57) 
+ 
+integrating (56) in x according to (5), because c(∞,t)=0 , considering positions (57) for change in 
+variables it results: 
+
+10 
+ 
+ 
+2'
+/
+( , )
+2
+(
+')
+'
+x
+s
+x
+C
+c x t
+e
+erf kx dx
+
+
+
+−
+=
+
+  
+ 
+ 
+, 
+ 
+ 
+ 
+(58) 
+x’ is an integration variable that we can change in y so we write the final solution: 
+ 
+2
+/
+0
+0
+( , )
+2
+(
+)
+;
+2
+;
+y
+s
+x
+C
+c x t
+e
+erf ky dy
+D
+D
+Dt k
+Dt
+
+
+
+
+
+
+−
+=
+=
++
+=
+
+  
+ 
+ 
+, 
+ 
+ 
+ 
+(59) 
+where D0 is the diffusion coefficient of the first diffusion problem (42) up to time t= and D is the 
+diffusion coefficient of problem (43) up to time t. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+11 
+ 
+ 
+References 
+ 
+ 
+1. G.Macrelli, A.K.Varshneya, J.C.Mauro, Ion exchange in silicate glasses: physics of concentration, 
+residual 
+stress 
+and 
+refractive 
+index 
+profiles, 
+arXiv:2002.08016v2[cond-matrl-sci] 
+https://doi.org/10.48550/arXiv.200208016, 2020.  
+ 
+ 
+2. G.Macrelli,A.K. Varshneya, J.C. Mauro. Thermal treatment of ion-exchanged glass. Int J Appl Glass 
+Sci.2023;14:7–17.https://doi.org/10.1111/ijag.16590 
+ 
+3. R.Sh.Malkovich, Impurity diffusion from a deposited layer, Fiz.metal.metalloved., 15, No6, 880-
+884,1963 
+ 
+4. D.P.Kennedy, P.C.Murley, Impurity Atom Distribution from a two step diffusion process, Proceedings 
+of the IEEE, 52, 623-624, 1964 
+ 
+5. H.S. Carslaw, J.C.Yaeger, Conduction of Heat in Solids, 2nd Edition, Oxford, 1959, Clarendon Press 
+ 
+ 
+6. B.M.Budak, S.V.Fomin, Multiple Integrals, field theory and series, Moscow 1973, Mir 
+publishers. 
+ 
+7. J.Crank, The Mathematics of Diffusion, 2nd Edition, Oxford, 1975, Clarendon Press 
+ 
+8. A.V.Luikov, Analytical heat diffusion theory, New York, 1968, Academic Press 
+ 
+9. R.Ghez, Diffusion Phenomena, New York, 2010, Kluwer Academic/Plenum Publisher 
+ 
+10. M.Abramowitz, I.Stegun, Handbook of mathematical functions, National Bureau of 
+Standards, Applied Mathematics Series, tenth printing 1972. 
+ 
+
diff --git a/fNE4T4oBgHgl3EQfqQ2c/content/tmp_files/load_file.txt b/fNE4T4oBgHgl3EQfqQ2c/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2b93c950ed46ab170b7af9aa814f9917ded05200
--- /dev/null
+++ b/fNE4T4oBgHgl3EQfqQ2c/content/tmp_files/load_file.txt
@@ -0,0 +1,181 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf,len=180
+page_content='1  Mathematical theory of diffusion in solids: solutions in the semi-infinite body and solution to a diffusion problem with a variable boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  Guglielmo Macrelli ( )  (*) Isoclima SpA – R&D Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Via A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Volta 14, 35042 Este (PD) Italy guglielmomacrelli@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='com  Abstract:  A review of solutions of solid-state diffusion problems in infinite and semi-infinite bodies is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Based on the identified solutions for the semi-infinite body a two-step diffusion problem is discussed in detail with the first step characterized by a Dirichlet constant concentration condition and the second step by a Neumann condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  Introduction  In solid state diffusion problems it is common to have diffusing elements in a solid body that can be externally injected from deposited layers, coming from other phases either liquid or gaseous or internally  moving or redistributed from internal locations in  the solid matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' When the diffusion length \uf06cD is very small compared to the sample dimension (usually the thickness d) through which diffusion occurs (\uf06cD << d) than we can approximate the solid to a semi-infinite half body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The main purpose of this study is to provide a detailed proof and justification for a variable boundary condition problem which solution has been provided by the author in the literature1,2 without a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The reason for not providing the proof was mainly related to the length and complexity of the proof itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' There can be found mentions of this problem in the literature3,4 that for different reasons do not fully cover the purpose of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The solution provided by Malkovich3 inspired the approach used in this study but it is limited to a situation where the two steps diffusion processes are run with the same diffusion coefficient (same temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The solution provided by Kennedy and Morley4 is according to the Green functions approach but it contains presumably a typo in the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  In this study we have considered worth to preliminary review the matter of solutions to diffusion problems in the infinite and semi-infinite bodies as a necessary step before approaching the more complex variable boundary diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' We consider worth to have provided a detailed proof of this solution first to cover a missing gap second because the constructional approach to the proof may be beneficial for similar problems in solid state diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' This is a contribution to the mathematical theory of solid state diffusion covering some gaps of past literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Solution in the infinite and semi-infinite body  In this study we deal with the differential equation for the concentration of diffusing elements into a solid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' We consider the mono-dimensional version of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The starting point is the Fick equation for the flux J of the diffusing elements defining a diffusion coefficient D:  \uf0b6  −  =  \uf0b6  ( , )  ( , )  c x t  J x t  D  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (1) Together with the continuity equation:  \uf0b6  \uf0b6  +  =  \uf0b6  \uf0b6  ( , )  ( , )  0  c x t  J x t  t  x  ,  (2)  2  it results the diffusion equation for concentration:  \uf0b6  \uf0b6  \uf0b6  \uf0e6  \uf0f6  =  \uf0e7  \uf0f7  \uf0b6  \uf0b6  \uf0b6  \uf0e8  \uf0f8  ( , )  ( , )  c x t  c x t  D  t  x  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (3)  When diffusion coefficient D can be considered reasonably constant, the second member of (3) is just diffusion coefficient times the second derivative of concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' A similar diffusion equation can be written for the flux J(x,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' If the concentration function is such that derivative for space and time can be interchanged than, because of (3), the time derivative of Flux for constant D is:  (  )  \uf0e6  \uf0f6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0e6  \uf0f6  \uf0e6  \uf0f6  =  −  = −  = −  =  \uf0e7  \uf0f7  \uf0e7  \uf0f7  \uf0e7  \uf0f7  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0b6  \uf0e8  \uf0f8  \uf0e8  \uf0f8  \uf0e8  \uf0f8  \uf0b6  \uf0b6  \uf0b6  \uf0e6  \uf0f6  = −  =  \uf0e7  \uf0f7  \uf0b6  \uf0b6  \uf0b6  \uf0e8  \uf0f8  2  2  2  2  2  2  ( , )  ( , )  ( , )  ( , )  ( , )  ( , )  J x t  c x t  c x t  c x t  D  D  D  D  t  t  x  x  t  x  x  c x t  D  D  D  J x t  x  x  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (4)  From (4) it is clear that, for flux function, we have the same differential equation that we have for concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' If flux is known than concentration can be calculated from (1):  \uf0a5  =  \uf0a5  +  \uf0f2  1  ( , )  ( , )  ( , )  x  c x t  c  t  J x t dx  D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (5) The usual condition for the value at infinite of concentration is c(∞,t)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='1 – General solution for the infinite body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  We deal here with the diffusion equation for the diffusing elements concentration c(x,t) with constant diffusion coefficient D, written in the time and mono-dimensional space domains (t,x,) where 0 ≤  t ≤ ∞ and -∞ ≤ x ≤ +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  \uf0b6  \uf0b6  =  \uf0b6  \uf0b6  2  2  ( , )  ( , )  c x t  c x t  D  t  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (6)  When the space variable domain is the full real axis we will name it infinite body domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The solution to equation (6) exists and it is unique providing that we set an initial condition:  ( ,0)  ( )  c x  f x  =  ,  (7) A solution to this problem has been indicated in the fundamental treatise of Carslaw and Jaeger5 : \uf070 +\uf0a5 − − −\uf0a5 = \uf0f2 2 ( ) 4 1 ( , ) ( ) 2 x y Dt c x t e f y dy Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (8)  A heuristic derivation of this result can be based on the general solution of (6) by the separation of variables: ( , ) ( ) ( ) c x t X x T t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (9) Inserting position (9) in (6) leads to:  3  =  2  2  ( )  ( )  ( )  ( )  dT t  d X x  X x  DT t  dt  dx  ,  (10) that can be split in two equations: \uf06c = − 2 1 ( ) ( ) dT t DT t dt  ,  (11a)  \uf06c  = −  2  2  2  1  ( )  ( )  d X x  X x  dx  ,  (11b)  with general solutions:  \uf06c  \uf067  −  =  2  ( )  Dt  T t  e  ,  (12a)  \uf061  \uf06c  \uf062  \uf06c  =  +  ( )  cos(  )  sin(  )  X x  x  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (12b) According to (9) we can write the generic solution: \uf05b \uf05d 2 ( , ) ( )cos( ) ( )sin( ) Dt c x t e A x B x \uf06c \uf06c \uf06c \uf06c \uf06c − = + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (13) Because of the linearity of equation (6) the general solution is the one obtained for all possible \uf06c values generated by the boundary conditions on the considered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' The values of \uf06c depend on the boundary conditions that will be used, in particular for bounded domains the values of \uf06c are discrete, leading to: 2 1 ( , ) ( )cos( ) ( )sin( ) j Dt j j j j j c x t e A x B x \uf06c \uf06c \uf06c \uf06c \uf06c \uf0a5 − = \uf0e9 \uf0f9 = + \uf0eb \uf0fb \uf0e5 ,  (14) while for unbounded domains they are continuous leading to: \uf05b \uf05d 2 ( , ) ( )cos( ) ( )sin( ) Dt c x t e A x B x d \uf06c \uf06c \uf06c \uf06c \uf06c \uf06c +\uf0a5 − −\uf0a5 = + \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (15) We consider here solution (15) because we have an unbounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' A(\uf06c) and B(\uf06c) functions can be determined by considering the initial condition (7): \uf05b \uf05d ( ,0) ( ) ( )cos( ) ( )sin( ) c x f x A x B x d \uf06c \uf06c \uf06c \uf06c \uf06c +\uf0a5 −\uf0a5 = = + \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (16) Considering the Fourier theorem6 for the function f(x): ( ) ( ) 1 ( ) ( )cos ( ) 2 1 1 ( )cos cos( ) ( )sin sin( ) 2 2 f x f x d d f d x f d x d \uf068 \uf06c \uf068 \uf068 \uf06c \uf070 \uf068 \uf06c\uf068 \uf068 \uf06c \uf068 \uf06c\uf068 \uf068 \uf06c \uf06c \uf070 \uf070 +\uf0a5 +\uf0a5 −\uf0a5 −\uf0a5 +\uf0a5 +\uf0a5 +\uf0a5 −\uf0a5 −\uf0a5 −\uf0a5 \uf0e9 \uf0f9 = − = \uf0ea \uf0fa \uf0eb \uf0fb \uf0ec \uf0fc \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ef \uf0ef = + \uf0ed \uf0fd \uf0ea \uf0fa \uf0ea \uf0fa \uf0ef \uf0ef \uf0eb \uf0fb \uf0eb \uf0fb \uf0ee \uf0fe \uf0f2 \uf0f2 \uf0f2 \uf0f2 \uf0f2 ,(17) And comparing equations (17) and (16) we have:  4  1  ( )  ( )cos(  )  2  A  f  d  \uf06c  \uf068  \uf06c\uf068  \uf068  \uf070  +\uf0a5  −\uf0a5  =  \uf0f2  ,  (18a)  1  ( )  ( )sin(  )  2  B  f  d  \uf06c  \uf068  \uf06c\uf068  \uf068  \uf070  +\uf0a5  −\uf0a5  =  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (18b)  From (15) and (18a) and (18b) the general solution is:  \uf05b  \uf05d  2  2  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  ( ) cos(  )cos(  )  sin(  )sin(  )  2  1  ( )  cos (  )  2  Dt  Dt  c x t  e  f  x  x  d  d  f  e  x d  d  \uf06c  \uf06c  \uf068  \uf06c\uf068  \uf06c  \uf06c\uf068  \uf06c  \uf068  \uf06c  \uf070  \uf068  \uf06c \uf068  \uf06c  \uf068  \uf070  +\uf0a5  +\uf0a5  −  −\uf0a5  −\uf0a5  +\uf0a5  +\uf0a5  −  −\uf0a5  −\uf0a5  \uf0e9  \uf0f9  =  +  =  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0e9  \uf0f9  =  −  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  \uf0f2  \uf0f2  \uf0f2  (19)  The integral in square brackets is from Budak and Fomin6 page 573:  (  )  2  2  4  cos (  )  x  Dt  Dt  e  x d  e  Dt  \uf068  \uf06c  \uf070  \uf06c \uf068  \uf06c  \uf0e9  \uf0f9  −  \uf0ea  \uf0fa  +\uf0a5  −  \uf0ea  \uf0fa  −  \uf0eb  \uf0fb  −\uf0a5  −  =  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (20)  Equation (14) finally results in equation (8):  (  )  2  4  1  ( , )  ( )  2  x  Dt  c x t  f  e  d  Dt  \uf068  \uf068  \uf068  \uf070  −  +\uf0a5  −  −\uf0a5  =  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (21)  The same result, namely equation (21), is achieved in a number of different ways as indicated also in Carslaw and Jaeger5, we prefer this heuristic derivation to stress connections to boundary value problems in either bounded and unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='2 General solution for the semi-infinite body  In many scientifical and technological problems of interest in materials science we can assume, with a good approximations, that the solid body under diffusion of externally delivered elements or impurities is a semi-infinite body extended in the x ≥ 0 spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' In this case we can conveniently use the solution for the infinite body (16) by extending the initial condition towards the negative half space assuming an indeterminate function c1(x,0) (x<0) and defining the following initial condition:  ( )  ( ,0)  f  c  \uf068  \uf068  =  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' \uf068 > \uf030  \uf02c  (22a)  1 ( ) ( ,0) f c \uf068 \uf068 = ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' \uf068 <  \uf030  \uf02e  (22b)  With these positions solution (21) results:  5  (  )  (  )  2  2  0  4  4  1  0  1  ( , )  ( ,0)  ( ,0)  2  x  x  Dt  Dt  c x t  c  e  d  c  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf070  −  −  \uf0a5  −  −  −\uf0a5  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  +  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (23)  Taking the first integral in (23) and putting y=-\uf068 than\uf03a  (  )  (  )  (  )  (  )  2  2  2  2  0  4  1  0  4  4  4  1  1  1  0  0  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  x  Dt  y x  x y  x  Dt  Dt  Dt  c  e  d  c  y  e  dy  c  y  e  dy  c  e  d  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  −  −  −\uf0a5  − −  +  +  \uf0a5  \uf0a5  −  −  −  +\uf0a5  =  = −  −  =  −  =  −  \uf0f2  \uf0f2  \uf0f2  \uf0f2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (24)  Hence:  (  )  (  )  (  )  (  )  2  2  2  2  4  4  1  0  0  4  4  1  0  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  2  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  2  x  x  Dt  Dt  x  x  Dt  Dt  c x t  c  e  d  c  e  d  Dt  c  e  d  c  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf070  \uf068  \uf068  \uf068  \uf068  \uf070  +  −  \uf0a5  \uf0a5  −  −  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  −  +  =  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  +  −  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  \uf0f2  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (25)  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='3 Semi-infinite body with reflecting boundary (SI-RB)  Reflecting boundary condition for the semi-infinite body means that no flow of matter is assumed at x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' This can be conveniently defined by using the flux equation and putting it equal to zero at the half plane interface:  0  (  0, )  0  x  c  J x  t  D x  =  \uf0b6  =  = −  =  \uf0b6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (26)  Boundary condition (22) can be introduced by differentiating (25):  (  )  (  )  2  2  4  4  1  0  ( , )  1  1  (  ) ( ,0)  (  ) (  ,0)  2  2  x  x  Dt  Dt  c x t  x  c  e  d  x c  e  d  x  Dt  Dt  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf068  \uf070  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0b6  \uf0ea  \uf0fa  =  −  −  +  −  \uf0b6  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  , (27)  and putting this to zero (26):  \uf05b  \uf05d  2  4  1  0  0  ( , )  1  1  ( ,0)  (  ,0)  0  2  2  Dt  x  c x t  c  c  e  d  x  Dt  Dt  \uf068  \uf068  \uf068  \uf068  \uf068  \uf070  \uf0a5  −  =  \uf0b6  =  −  −  =  \uf0b6  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (28)  Condition (28) is satisfied only if:  1  ( ,0)  (  ,0)  c  c  \uf068  \uf068  =  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (29)  6  Condition (29) in (25) leads to the general solution for the SI-RB problem:  (  )  (  )  2  2  4  4  0  1  ( , )  ( ,0)  2  x  x  Dt  Dt  c x t  c  e  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf070  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  +  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (30)  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='4 Semi-infinite body with capturing boundary (SI-CB)  The capturing boundary condition is expressed as follows:  (0, )  0  c  t =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (31)  This condition in (25) leads to:  \uf05b  \uf05d  ( )  2  4  1  0  1  (0, )  ( ,0)  (  ,0)  2  Dt  c  t  c  c  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf070  \uf0a5  −  =  +  −  \uf0f2  ,  (32)  hence:  1  ( ,0)  (  ,0)  c  c  \uf068  \uf068  = −  −  ,  (33)  and the general solution for the semi infinite capturing boundary (SI-CB) is:  (  )  (  )  2  2  4  4  0  1  ( , )  ( ,0)  2  x  x  Dt  Dt  c x t  c  e  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf070  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  −  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (34)  A particular application of solution (34) can be considered for the following initial and boundary conditions:  ( ,0)  0  c x  =  (35a)  0  ( , )  S  x  c x t  C  = =  (35b)  Let’s introduce the auxiliary function:  ( , ) ( , ) s c x t C c x t = −  (36)  The diffusion problem for the auxiliary function with conditions (35a) and (35b) is:  \uf0b6  \uf0b6  =  \uf0b6  \uf0b6 2 2  ( , )  ( , )  c x t  c x t  D  t  x  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (37a)  (0, ) 0 c t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (37b)  7  Solution to (37a) and (37b) is the one for the SI-CB  (  )  (  )  2  2 4  4  0  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  2  x  x  Dt  Dt  c x t  c  e  e  d  Dt  \uf068  \uf068  \uf068  \uf068  \uf070  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0ea  \uf0fa  =  −  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0f2  (38) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0)  s  s  c  C  c  C  \uf068  \uf068  =  −  =  (39)  and:  (  )  (  )  2  2 4  4  0  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  2  2  x  x  s  Dt  Dt  S  C  x  c x t  e  e  d  C erf  Dt  Dt  \uf068  \uf068  \uf068  \uf070  −  +  \uf0a5  −  −  \uf0e9  \uf0f9  \uf0e6  \uf0f6  \uf0ea  \uf0fa  =  −  =  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0eb  \uf0fb  \uf0f2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (40)  Finally:  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' ) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' ) 1 2 2 s S s x x c x t C c x t C erf C erfc Dt Dt \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 = − = − = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (41)  Solution (41) is a well-known and, in some ways, popular solution in the theory of diffusion in solids in a semi-infinite body5,7,8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' A variable boundary condition diffusion problem  Now we deal with a diffusion problem with a variable boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Such type of problems are often encountered when diffusion species are introduced and let diffuse in a solid body by different sequential mechanisms like for example diffusion from a continuous source like a bath or a vapour environment than stopped and followed by a thermal treatment at different temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' We are typically in front of a two steps process characterized by different boundary condition for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' They can be modelled as constant source up to a certain time than changed as a limited source diffusing in the body without incoming flux through the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' In this case our problem can be split into two steps separated boundary value problems for the diffusion equation (6):  (0, ) s c t C = for 0 ≤ t ≤ \uf074  with a diffusion coefficient D0 \uf02c  (\uf034\uf032)  0 ( , ) 0 x c x t x = \uf0b6 = \uf0b6 , for t≥ t  with a diffusion coefficient D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (43)  The solution up to t=\uf074 is easily found in equation (41) that at t=\uf074 becomes itself the initial condition for problem with boundary condition (43):  0  ( , )  2  s  x  c x  C erfc  D  \uf074  \uf074  \uf0e6  \uf0f6  =  \uf0e7  \uf0f7  \uf0e7  \uf0f7  \uf0e8  \uf0f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (44)  Let’s rewrite problem (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' (43) and (44) in terms of flux:  8  (  )  \uf0b6  \uf0b6  =  \uf0b6  \uf0b6  2  2  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  J x t  D  J x t  t  x  (45)  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  0  J  t =  (46)  Looking to (45) and (46) we recognize a SIB-CB problem for the flux function J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='t) and the solution reads (38): ( ) ( ) ( ) 2 2 2 2 4 4 0 4 4 0 1 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' ) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0) 2 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0) sinh 2 2 x x Dt Dt x Dt Dt J x t J e e d Dt e x J e d Dt Dt \uf068 \uf068 \uf068 \uf068 \uf068 \uf070 \uf068 \uf068 \uf068 \uf070 − + \uf0a5 − − − \uf0a5 − \uf0e9 \uf0f9 \uf0ea \uf0fa = − = \uf0ea \uf0fa \uf0eb \uf0fb \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0f2 \uf0f2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (47)  This is obtained by the development of the squares in the arguments of the exponentials and because ( ) 1 sinh( ) 2 z z z e e− = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' the initial condition for the flux is: 2 0 4 0 0 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' ) 1 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='0) D s t c t J D C D e D \uf068 \uf074 \uf068 \uf068 \uf068 \uf070\uf074 − = \uf0b6 = − = \uf0b6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (48)  finally the solution for the flux function reads:  2  2  0  2  2  0  0  1  1  2  4  4  4  0  0  4  4  0  0  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  sinh 2  sinh 2  x  Dt  D  Dt  s  Dt D  x  DD t  s  Dt  D  e  x  J x t  C  e  d  D  Dt  Dt  C  D  x  e  e  d  D  Dt  t  \uf068  \uf074  \uf074  \uf068  \uf074  \uf068  \uf068  \uf070\uf074  \uf070  \uf068  \uf068  \uf070  \uf074  −  \uf0e6  \uf0f6  \uf0a5  −  +  \uf0e7  \uf0f7  \uf0e8  \uf0f8  \uf0e6  \uf0f6  +  \uf0a5  −  \uf0e7  \uf0f7  −  \uf0e8  \uf0f8  \uf0e9  \uf0f9  \uf0e6  \uf0f6  =  =  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0eb  \uf0fb  \uf0e9  \uf0f9  \uf0e6  \uf0f6  =  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0eb  \uf0fb  \uf0f2  \uf0f2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (49)  Let’s make the following positions:  (  )  (  )  0  0  0  0  0  0  2  2  0  0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  4  4  4  4  2  4  Dt  D  x  a  b  DD t  Dt  x D  DD t  b  x  Dt  Dt  D  a  Dt Dt  D  x D  b  a  Dt Dt  D  \uf074  \uf074  \uf074  \uf074  \uf074  \uf074  \uf074  \uf074  +  =  = −  = −  = −  +  +  =  +  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (50)  with these positions the solution for flux (49) results:  2  2  2  2  4  0  0  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  2  x  a  b  b  s  Dt  C  D  J x t  e  e  e  e  d  D  t  \uf068  \uf068  \uf068  \uf068  \uf070  \uf074  \uf0a5  −  −  −  \uf0e9  \uf0f9  =  −  \uf0eb  \uf0fb  \uf0f2  (51)  9  The integral term in (51) can be evaluated as follows:  2  2  2  2  0  1  1  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' )  2  4  b  a  b  b  a  b  I a b  e  e  e  d  e erf  a  a  \uf068  \uf068  \uf068  \uf070  \uf068  \uf0a5  −  −  \uf0e6  \uf0f6  \uf0e9  \uf0f9  =  −  =  \uf0e7  \uf0f7  \uf0eb  \uf0fb  \uf0e8  \uf0f8  \uf0f2  (52)  This is coming from Abramowitz and Stegun10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='2:  (  )  2  2  2  0  1  2  b  ac  a  b  c  a  b  e  d  e  erf  a  a  \uf068  \uf068  \uf070  \uf068  \uf0a5  −  −  +  +  \uf0e6  \uf0f6  =  \uf0e7  \uf0f7  \uf0e8  \uf0f8  \uf0f2  ,  (53)  and from the complementary error function erfc(x)=1-erf(x) properties such that:  ( )  (  )  2  ( )  erfc z  erfc  z  erf z  −  −  = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (54)  Developing (52) with positions (50) results:  (  )  2  0  0  0  4  (  )  0  0  0  4  ( , )  2  x D  Dt Dt D  x D  DD t  I a b  erf  e  Dt  D  Dt Dt  D  \uf074  \uf074  \uf074  \uf070  \uf074  \uf074  \uf074  +  \uf0e6  \uf0f6  \uf0e7  \uf0f7  =  \uf0e7  \uf0f7  +  +  \uf0e8  \uf0f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (55)  Finally flux solution (51) is:  (  )  (  )  (  )  (  )  2  0  0  2  0  1  4  0  0  0  0  0  2  4  0  0  0  ( , )  2  2  D  x  Dt  Dt D  s  x  Dt D  s  x D  C  DD t  D  J x t  e  erf  D  Dt  D  t  Dt Dt  D  x D  D  C  e  erf  Dt  D  Dt Dt  D  \uf074  \uf074  \uf074  \uf074  \uf070  \uf074  \uf074  \uf070  \uf074  \uf074  \uf074  \uf070  \uf074  \uf074  \uf0e6  \uf0f6  −  −  \uf0e7  \uf0f7  +  \uf0e8  \uf0f8  −  +  \uf0e6  \uf0f6  \uf0e7  \uf0f7  =  =  \uf0e7  \uf0f7  +  +  \uf0e8  \uf0f8  \uf0e6  \uf0f6  \uf0e7  \uf0f7  =  \uf0e7  \uf0f7  +  +  \uf0e8  \uf0f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  (56)  From the flux solution represented by equation (56) we can obtain the corresponding concentration solution by equation (5) that is integrating equation (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' Let’s fix the following positions:  (  )  (  )  2  2  0  0  0  2  2  0  0  0  4  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content="  '  ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content="  '  2  '  '  ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content="  '  4  2  D  D  Dt  k  Dt  x  x  dx  dx  D  Dtdx  D  x  x  x  kx  Dt  D  Dt  D  Dt  \uf074  \uf067  \uf074  \uf067  \uf074  \uf067  \uf074  \uf074  \uf074  =  +  =  =  =  =  +  =  =  +  +  ,  (57)  integrating (56) in x according to (5), because c(∞,t)=0 , considering positions (57) for change in variables it results:  10  2'  /  ( , )  2  (  ')  '  x  s  x  C  c x t  e  erf kx dx  \uf067  \uf070  \uf0a5  −  =  \uf0f2  ,  (58) x’ is an integration variable that we can change in y so we write the final solution:  2  /  0  0  ( , )  2  (  )  ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  y  s  x  C  c x t  e  erf ky dy  D  D  Dt k  Dt  \uf067  \uf070  \uf074  \uf067  \uf074  \uf0a5  −  =  =  +  =  \uf0f2  ,  (59) where D0 is the diffusion coefficient of the first diffusion problem (42) up to time t=\uf074 and D is the diffusion coefficient of problem (43) up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  11  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='200208016, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Crank, The Mathematics of Diffusion, 2nd Edition, Oxford, 1975, Clarendon Press  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Luikov, Analytical heat diffusion theory, New York, 1968, Academic Press  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Ghez, Diffusion Phenomena, New York, 2010, Kluwer Academic/Plenum Publisher  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Abramowitz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
+page_content='Stegun, Handbook of mathematical functions, National Bureau of Standards, Applied Mathematics Series, tenth printing 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfqQ2c/content/2301.05199v1.pdf'}
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+    
+THE ROLE OF HOSTING CAPACITY STUDY IN POWER SYSTEM 
+ADVANCEMENTS: A REVIEW 
+Utkarsh Singh 
+Address: Depsys SA, Route du Verney 20B, 1070 Puidoux, Switzerland 
+Email: utkarsh.singh@ieee.org   
+Abstract: 
+The fast depletion of conventional energy sources due to increased energy demands 
+and environmental concern has motivated power utilities to integrate more 
+renewable energy sources (RESs) into their power systems. Due to the intermittent 
+nature and low or non-existent inertial response of these sources, a high penetration 
+of RESs can lead to various issues in the operation of power systems such as 
+oscillations in power system’s voltage and frequency, increased harmonic 
+distortion, failure of protective equipment, overloading of transformers and feeders, 
+and increased line losses. Such problems arise when the hosting capacity (HC), 
+defined as the maximum RES capacity that can be installed without having any 
+technical and operational problems, of the network exceeds its limit due to the 
+increased integration of RESs to the existing network. This paper reviews the 
+progress made in HC assessment and enhancement of electrical networks research 
+and development since its inception. Attempts are also made to highlight the current 
+and future issues involved in HC technology for the development of an affordable, 
+in-exhaustive, clean and reliable power supply for longer term benefits.  
+Keywords: Hosting capacity, performance index, renewable energy source, wind 
+energy, solar energy. 
+List of Notations and Abbreviations: 
+BESS  
+Battery Energy Storage System 
+CSP 
+ 
+Concentrated Solar Power 
+C-CG  
+Column-and-constraint Generation 
+DG  
+ 
+Distributed Generator 
+DER 
+ 
+Distributed Energy Resources 
+DSO 
+ 
+Distribution System Operator 
+DVR  
+Dynamic Voltage Restorer 
+DSTATCOM Distribution Static Compensator (fix the rest) 
+ES 
+ 
+energy storage 
+ED 
+ 
+economic dispatch 
+
+ 
+2 
+ELM  
+extreme learning machine 
+HC 
+ 
+hosting capacity 
+HCC  
+hosting Capacity Coefficient 
+LHC 
+ 
+locational hosting capacity 
+OLTC  
+on load tap changer 
+PV 
+ 
+photovoltaic 
+PSI  
+ 
+power supply industry 
+PVIS  
+photovoltaic inverter system 
+PPF 
+ 
+probabilistic power flow 
+PI 
+ 
+prediction interval 
+PSO 
+ 
+particle swarm optimization 
+RES   
+renewable energy source 
+SLHC  
+stream-lined hosting capacity 
+SCUC  
+security-constrained unit commitment 
+TSO 
+ 
+transmission system operator 
+ UC 
+ 
+unit commitment 
+VaR 
+ 
+value-at-risk measures 
+WG  
+ 
+wind generator 
+WPF  
+wind power forecast 
+1. 
+INTRODUCTION 
+The ever-increasing concerns over global climate change, caused by the excessive 
+use of fossil fuels to meet the global energy demands, have encouraged intensive 
+research for green power plants with advanced technology. In this context, since 
+past few years, the use of renewable energy sources, such as wind, tidal, micro-
+hydro, biomass, geothermal in general, and solar for generation of electrical energy 
+has increased tremendously [1]-[4]. This has become possible, only because of the 
+power system reforms. The reasons behind power sector reforms /deregulation 
+worldwide have either been regulatory failures, political reforms, high tariffs, 
+inefficient management, poor efficiency, or global economic crisis. A significant 
+feature of such restructuring is to allow for competition among generators and to 
+create market conditions in the industry, which are considered necessary to reduce 
+the cost of energy production and distribution, eliminate certain inefficiencies, shed 
+manpower and increase customer choice [5]-[10]. Authors in [5] have presented a 
+detailed review on the progress of the movement to privatize and liberalize the 
+
+ 
+3 
+power sector in developing countries. Authors have clearly spelled out that a full-
+scale power reform program generally consists of four main elements: (i) formation 
+and approval of a power policy by the government that provides broad guidelines 
+for the reform program (ii) development of a transparent regulatory framework for 
+the energy market, (iii) unbundling of the electric power supply chain to enable the 
+introduction of competition to improve sector performance in terms of efficiency, 
+customer response, innovation, and viability, and (iv) focus on government's role 
+on policy formation and execution, while divesting the power of state ownership at 
+least in most of the generation and distribution. In [6], a detailed discussion on 
+achievement of power sector deregulation in Latin America, problems encountered 
+in the development of all three sectors of power industry i.e., generation, 
+transmission and distribution are presented. New challenges in deregulation of 
+electrical sectors are also presented. Authors have stressed that more emphasis 
+should be given on the regulating conduct rather than industry structure, free access 
+to international networks, transparent bidding process to award contracts, and the 
+development of price signals as the base for market development and for the linkage 
+between different segments. Enough flexibility should be provided to the regulators 
+to make changes as and when required in agreement with the energy sector and 
+agents of the market. Authors in [7] have thrown light on power sector reforms in 
+the developing world. In this, authors have a detailed discussion on nine important 
+points; (i) meaning of power sector reform and its necessity, (ii) spread of power 
+sector reform in developing world, (iii) effect of political economy on uptake of 
+power sector reform, (iv) works undertaken to restructure utilities and improve 
+governance, (v) contribution made by the private power sector after reform, (vi) 
+whether countries have established the meaningful regulatory frameworks, (vii) 
+progress made by the wholesale power markets, (viii) improvement in efficiency 
+and cost recovery, and (ix) outcome of power sector reform. In the last, the authors 
+have concluded that the link between power sector reforms and final sector 
+outcomes is much weaker, despite some evidence that private sector participation 
+has made a positive contribution. In [8], authors have made a comparative study of 
+the distributed and centralized technique for controlling the distribution network 
+voltages in terms of the capacity of RES that could be integrated within the existing 
+networks as well as contrasting them with the reactive power control approach, 
+using optimal power flow analysis. It is concluded that both distributed and 
+centralized voltage control methods offer significant gains in absorption capacity, 
+particularly in rural networks. It is also inferred that the consequent losses increase 
+substantially, and hence the financial implications of increased losses must be 
+carefully assessed. Authors in [9] have described a cloud shadow model to recreate 
+the variable output power of both distributed and large centralized PVs at various 
+locations on a feeder. For this purpose, a time series load flow analysis is used using 
+an actual EPRI test feeder. The feeder was monitored at all buses, and PV induced 
+voltage quality was measured, including its impacts on voltage control devices. 
+Authors [10] have presented the motivation behind RES/DG integration into the 
+electrical network and the key issues concerning this. A brief discussion on the main 
+
+ 
+4 
+challenges that must be overcome in the integration of DG into the supply systems 
+is also provided. Authors have also emphasized on new grid code and standards to 
+be suitable for contribution of large scale RESs.  
+However, there always exists a conflict of interest amongst the RES investors and 
+distribution system operators (DSOs), as the RES owners are interested in 
+integration of more amount of RES into the existing electrical networks, while 
+DSOs are more concerned about the technical problems caused by excessive 
+penetration level of RESs [11]. High penetration of RESs, predominantly wind 
+power can lead to complexity in the operation of power systems due to the 
+intermittent nature of these sources and frequency stability problems due to the 
+decoupling of the RESs from AC grid using power converters [12], [13]. RESs 
+typically have low (in case of variable speed wind turbines) or non-existent (in case 
+of solar power plants) inertial responses. Attempts have, however been constantly 
+made to mitigate the arising issues in RESs integration and to address various 
+problems. 
+This paper deals with a state-of-the-art discussion on holding capacity of the power 
+network, highlighting the analytical and technical considerations as well as various 
+issues addressed in the literature towards the practical realization of this technology 
+for better utilization of RESs, at reduced cost and high efficiency. One hundred and 
+twenty-two publications [1]-[122] are reviewed and classified in 5 parts. 
+ 
+2.  Hosting Capacity Concept 
+Hosting capacity (HC) is defined as the amount of RESs/DGs that can be integrated 
+into a given power system network, while keeping its performance within an 
+acceptable range and without having any change in the existing power system 
+infrastructure [14]-[16]. The concept of HC was initially conceived by the computer 
+engineers to define the capacity of a web server to host many access requests. Andre 
+Even, for the first time, introduced this terminology for electric power applications 
+to identify the effects of high RES into the power distribution network. The concept 
+was later refined and clearly defined by Math Bollen and Fainan Hassan as the 
+maximum capacity of RES that can be integrated into the system within the 
+acceptable performance indices [11], [15], [17], [18]. Figure 1 shows an example of 
+a HC limit [15] where a performance index is considered. With the increase in RES, 
+an acceptable deterioration is defined. When the amount of new RES generation 
+increases, the performance index will pass a limit, after which the deterioration is 
+unacceptable. 
+ 
+
+ 
+5 
+ 
+Figure 1. Hosting capacity (HC) limit and increase in RES integration [15] 
+ 
+Authors in [14] have presented four methods such as deterministic, stochastic, 
+optimization-based and stream-lined to calculate HC of the distribution networks. It 
+is shown that each method has its own advantages and disadvantages. It is also 
+concluded that, if the generation profile is known or forecasted, deterministic 
+method is found to be most suitable for sizing a single RES at specific location. 
+Whereas stochastic technique can be used for sensitivity study and forecasting. In 
+[15], HC technique is clearly explained. This approach uses the existing power 
+system as a starting point and considers the way, in which distributed generation 
+changes the performance of the system, without having to reinforce the operational 
+equipment (additional conductors, exchange of transformer etc.). For this, a set of 
+performance parameters is needed. Authors in [16] have presented the three 
+different methods named deterministic, stochastic and time series for quantifying 
+the solar PV HC of low voltage distribution grids. Authors have also elaborated the 
+merits and demerits of all the three methods. It is inferred that the deterministic 
+method is fast, and accuracy of this method depend on the model and method used 
+to calculate the voltage rise, whereas stochastic and time series techniques require 
+large number of simulations and computational time, which is a serious issue. In 
+[17], three different decentralised voltage control methods, using reactive power by 
+PV inverters, are compared for their capabilities to limit the voltage rise within a 
+balanced low voltage system. Authors have shown that the static reactive power 
+supply methods, as per German guidelines for generators connected to the low 
+voltage distribution network, can increase the absorption capacity of low voltage 
+network without having any change in the existing power network infrastructure. 
+Existing level
+Amount of generation
+Limit
+Hosting 
+Capacity
+Unacceptable deterioration
+Acceptable deterioration
+Performance index
+Improvement
+
+ 
+6 
+Authors in [18] have applied HC approach to a real network in order to calculate 
+the absorption capacity for integration of new RES. In this case study, two limits 
+overvoltage and overcurrent setting the HC were evaluated. The important findings 
+of these works [11], [14]-[18] are summarized as: 
+ 
+For a given distribution network, there is no single value for HC, as it depends 
+upon the pre-defined limiting factors to calculate it. 
+ 
+Inclusion of so many limiting parameters makes the HC analysis extremely 
+complicated. 
+ 
+It is difficult to point at a single method for quantifying the HC as the most 
+suitable one. It requires an exhaustive application of the methods to many low-
+voltage distribution network and a qualitative comparison of the results. 
+ 
+Deterministic methods use traditional power flow analysis and assume that 
+inputs are fixed and known. This technique neither considers the varying power 
+production due to the change in wind speed in case of wind energy and 
+irradiation in case of solar energy nor varying consumers’ power consumption. 
+ 
+Stochastic and time series methods are more suitable because both methods 
+include variable input parameters i.e., uncertainties due to change in wind and 
+solar power production and customers’ power consumption in the analysis. 
+ 
+In case of stochastic method, computational burden is more due to the inclusion 
+of uncertainties in customer consumption, grid and wind and solar power 
+generation. 
+ 
+Time series method is better than stochastic method as it considers all the time-
+dependencies and correlations. This method can be successfully applied for time 
+varying assessment of hosting capacity and the response of protection /control 
+devices. 
+ 
+If the location, size and number of RESs is known, optimization method will 
+prove to be the best choice, since it can improve overall performance of the 
+distribution network by reducing the losses and/ or cost. 
+ 
+The selection of HC calculation method will always depend on the objective of 
+the study.  
+ 
+Any HC study needs three major inputs: performance index, a corresponding 
+limit, and a method to calculate the performance index as a function of new 
+power production or consumption. 
+ 
+3. Impact of Wind Energy Integration  
+RES technologies are not yet economically competitive with conventional thermal 
+generation, as high-power penetration impacts the power supply industry (PSI) 
+technically and economically both. From the technical point of view, PSI faces a 
+
+ 
+7 
+variety of problems and challenges such as frequency and voltage regulation, power 
+quality issues, available transmission and distribution capacities to accommodate 
+RES plants, monitoring and control, operational practices, ancillary services, 
+connection interfaces, etc. Large wind generator (WG) integration will also impact 
+energy balances and generation mix, electricity markets, and emissions, etc. [18]-
+[22]. Most of the technical challenges are related to volatile nature of wind. Wind 
+power generation fluctuates based on wind speed, which depends on the 
+geographical location. Since the amount of wind may vary from time to time, the 
+variable wind power causes other conventional generator to operate in sub-optimal 
+manner [23]. Power systems must incorporate, for the first time, a source of high 
+uncertainty, high volatility and low predictability [24]. This uncertainty includes 
+input data, power curve, weather conditions and prediction algorithms. Forecasting 
+of wind generation is very challenging and extensive research has been done on this 
+topic [25]-[28]. Authors in [19] have detailed discussion on the impact of high wind 
+power integration on PSI, technical challenges and solutions required. Apart from 
+the technical issues, large wind power integration will have severe impact on PSI 
+economics and affects the other market participants. Authors have stressed that a 
+detailed analysis should be carried out on the important issues like inter-TSOs 
+energy trading, impact on generation mix, energy cost, energy balance, reliability 
+and security of supply. In conclusion, authors have recommended for the design and 
+development of new grid codes and market regulations to ensure the security of 
+supply in terms of system security and generation capacity adequacy. In [20], 
+authors have reported that the integration of RES will change the generation mix 
+against conventional power plants and will also reduce their market share. This 
+impact may be significant for the new market entrants.  Due to this, targets for 
+market opening and large-scale RES integration may conflict each other. Therefore, 
+policies and regulations should be carefully designed to benefit both conventional 
+and RES sectors to mutually achieve the said goals. Works reported in [21] is about 
+fault ride through, grid voltage control, system monitoring and protection as well as 
+retrofitting of old units. In the same paper, authors have discussed the new 
+requirements, defined in accordance with the new developments in wind turbine 
+technologies, which should be utilized in future to meet the grid requirement. 
+Monitoring and system protection is defined under the aspect of sustainability of the 
+measures introduced. In [22], authors have developed a new control scheme for 
+variable speed wind turbine that enables power supply of islanded parts of a 
+distribution network. Models and strategies for fast control of frequency and voltage 
+during islanding are also derived. Based on the simulation results, authors have 
+concluded that wind turbine can operate as conventional generator units supplying 
+power to islanded parts of the distribution grid, while maintain the voltage and 
+frequency close to their normal values. It is further inferred that energy storage (ES) 
+is not required for power balance control, as the frequency loop of the control 
+scheme is very fast, ensuring stable operation in islanded mode. Authors in [23] 
+have presented the wind power variability and its impacts on power systems, 
+covering classification of variability, aggregation effect, and wind power 
+
+ 
+8 
+forecasting error. The effects of wind energy on different operational time fames of 
+conventional power plants, and the cost of balancing requirements to accommodate 
+high wind power integration levels are also discussed in this paper. Authors have 
+concluded that though the wind power balancing cost is highly system dependent, 
+but in general, it increases with the increase in penetration level. The flexibility of 
+existing dispatchable generation units and the available transmission capacity to 
+neighbouring areas play an important role in reducing the balancing costs. Authors 
+in [24] have reported a detailed discussion on integrating wind energy into the 
+electric power system. This includes topics like commercial wind generation 
+technology, development of wind technology, reactive power supply and voltage 
+control, active power regulation, frequency control, wind forecasting, forecast 
+accuracy, effect of spatial spread, and power balancing issues. It is concluded that 
+the stalemate in transmission development is coming to an end, with the new 
+transmission planning paradigm being implemented. Study embodied in [25] 
+presents the detailed investigation on wind power forecast (WPF) uncertainty in unit 
+commitment (UC) and economic dispatch (ED) and analyses the impact of different 
+reserve requirements and UC policies on system operations. It is demonstrated that 
+despite the inherent uncertainty and risks, it is possible to adopt a quantified method 
+to deal accurately with wind power generation. Authors conclude that stochastic 
+decision methods can be successfully adopted to reduce the costs and risks, in the 
+presence of large and unavoidable wind power prediction errors, particularly in 
+power system network dominated by thermal power plants. Authors have further 
+suggested for the use of adaptive reserve requirements, which are the function of 
+wind power forecast. Paper [26] presents the impact of wind forecast error statistics 
+upon unit commitment for a large wind integration test system. Authors further 
+report that variance has the most impact and suggest that if skewness is included in 
+the evaluation of error information, kurtosis should also be included to reduce the 
+system cost. In the same work, it is inferred that the interactions of variance, 
+skewness, and kurtosis changes the utilization and commitment of units, and the 
+representation of variance, skewness and kurtosis can affect the dependency of 
+commitment upon flexible units and the way it is used. Researchers in [27] presents 
+a state-of-the-art review on probabilistic forecasting of wind power generation and 
+a brief discussion on three different representations of wind power uncertainty. 
+Three different forecasting methods, in terms of uncertainty representation, i.e., 
+probabilistic forecasts (parametric and non-parametric), risk index forecasts and 
+space-time scenario forecasts have also been discussed. In the last, authors have 
+concluded that uncertainty forecasting techniques can reduce the economic and 
+technical risk caused by wind power uncertainty and have recommended that there 
+is need of more research on wind power uncertainty forecasting. A new hybrid 
+intelligent algorithm approach to produce prediction intervals (PIs) of wind power 
+generation, based on the extreme learning machine (ELM) and particle swarm 
+optimization (PSO) is presented in [28]. Optimal PIs have been obtained without 
+prior knowledge, statistical inference or distribution assumption of forecasting 
+errors that are required in most of the traditional probabilistic techniques. It is 
+
+ 
+9 
+concluded that the proposed hybrid scheme provides a general framework of 
+probabilistic wind power forecasting with high flexibility for reserve determination 
+by transmission system operator (TSO) to meet the load, and to economically 
+operate the systems. A two-stage stochastic unit commitment model integrated with 
+value-at-risk measures (VaR), including non-generation resources like demand 
+response and energy storage systems, to balance between cost and system reliability 
+due to the fluctuation in variable power generation is presented in [29]. To solve the 
+VaR-based model and to reduce the computational time, modified Benders’ 
+decomposition algorithm is used. Sensitivity analysis is carried out for evaluation 
+of reliability parameters to reduce the generation costs. Numerical experimentation 
+is also carried out to find optimal unit commitment solutions and to compare the 
+effect of risk of non-generation resources on power generation. Li et. al. [30] have 
+proposed a stochastic dynamic model to formulate the spatial and temporal 
+correlation between the atmospheric and near-surface wind fields of geographically 
+distributed wind farms. It is inferred that the model can provide competitive interval 
+forecasts with conventional statistical based models. In [31], a robust optimization-
+based analysis is carried out by combining the concentrated solar power (CSP) 
+plants with wind farms to reduce the overall uncertainty in the joint power output. 
+A new approach, based on nested column-and-constraint generation (C-CG) 
+method, is developed to solve the multilevel optimization with mixed-integer 
+recourse problem. The proposed technique is found to be suitable in identifying 
+robust yet narrow intervals in a reasonable amount of time. It is concluded that the 
+combination of different types of renewable resources is very effective in reducing 
+the generation uncertainty. Hu et. al [32] have thrown light on how variant robust 
+security-constrained unit commitment (SCUC) models in terms of different worst-
+case definitions could impact operational security and economics of power systems 
+under uncertainties. In addition, what is the proper robust SCUC model that can 
+fulfil the specific market operational requirements of independent system operators 
+/regional transmission owners for effectively operating the system. Four different 
+robust SCUC models are studied by the authors. In [33], various techniques for 
+inertia and frequency control developed for variable speed wind turbine and solar 
+PV generators are systematically reviewed. Authors [34] have presented a more 
+tractable and adaptive distributionally robust unit commitment (DRUC-dW) 
+formulation using distance-based data aggregation, and an efficient cutting plane 
+algorithm, which solves the two-stage problem quite efficiently by leveraging the 
+extremal distributions constructed. It was found that UC solution, yielded from 
+DRUC-dW model, achieves a reasonable balance between the robustness and cost 
+efficiency. In [35], a non-anticipative robust unit commitment models (NRUCs), 
+where determining the dispatch policy is delayed until the uncertainty decreases is 
+developed. The proposed NRUC features three decision-making problems 
+sequentially solved under different degrees of uncertainty. At first, the decision-
+making problem is formulated as an intractable three-stage robust optimization 
+problem. Then, a suboptimal approach is developed, where a constraint is imposed 
+on the dispatch policy so that the transmission capacity constraint is met irrespective 
+
+ 
+10 
+of the dispatch level. The significant findings of the studies may be summarized as 
+follows [18]-[35]: 
+ 
+Wind turbine generators are not equipped with exciter and voltage control and 
+have no ability to provide high short-circuit currents to reduce the voltage dip. 
+Hence, if voltage drops below a certain level, WGs trip. 
+ 
+The impact of wind power integration on any power network depends on the 
+penetration level and system flexibility. Increase in the penetration level 
+increases the impacts perceived by the power network, whereas system having 
+more flexibility can accommodate more power without perceiving unwanted 
+effects caused by wind generation units. 
+ 
+Impacts of wind power integration are classified as short-term and long-term 
+impact. Short-term impacts deal with operational time scale, such as system 
+balance issues, which are represented by the requirements and cost caused by 
+the fluctuating wind power. Long-term impacts are related to planning for peak 
+load periods. 
+ 
+Fluctuating wind power causes the other conventional power units to operate in 
+sub-optimal manner with reduced efficiency. This problem can be reduced by 
+accurate wind generation forecasting and prediction. 
+ 
+Depending on the penetration level, wind power can increase or decrease 
+system (transmission and distribution) losses. 
+ 
+Fluctuation in power output leads to voltage variation in case of fixed-speed 
+wind turbine. However, the variable speed turbines, such as doubly fed 
+induction generators, can provide reactive power to network. 
+ 
+As for as the impact of wind power integration on distribution network under 
+normal operation is concerned, there may appear slow voltage variations due to 
+change in power flows and short duration voltage dips during switching 
+(on/off), fast voltage variations due to changes in wind speed that may cause 
+‘flickering’ effects, and voltage distortion due to harmonics. 
+ 
+During network faults, integration of WG can lead to: increased stress of circuit 
+breakers since the short circuits are additionally fed by the wind generators, 
+malfunctioning of protection equipment, since it is designed to operate within 
+strictly ‘radial structure’ of the MV network, and islanding of parts of the grid, 
+fed solely by wind generators, which may cause failure of the end-use 
+equipment and also accidents to the utility ‘s personnel. It is more possible in 
+the presence of high capacitance equipment such as cables. 
+ 
+High penetration of wind power needs expansion of more complex transmission 
+networks resulting in more transmission losses. 
+ 
+The long distance between the wind forms and load requires long transmission 
+lines to be laid down and can lead to transmission congestion. 
+
+ 
+11 
+ 
+With the increase in wind power integration, the amount of reserve needs to be 
+increased over long period of times to maintain the system reliability. 
+ 
+High integration of wind power leads to frequency deviation from the normal 
+range, because maintaining the balance between supply and load demand in 
+case of high volatility of supply is challenging and more difficult. 
+ 
+The increase in penetration level of wind power results in an equivalent decrease 
+in conventional generation units, and thus system rotational inertia becomes 
+very low. This can lead to serious effects on the frequency deviation. 
+ 
+4. Impact of Solar Energy Integration  
+Solar power is the conversion of sunlight into electricity, either directly using 
+photovoltaic (PV) or indirectly through concentrated solar power (CSP). Solar 
+power in its various forms such as solar photovoltaic, solar heat, solar thermal 
+electricity and solar fuels offer a clean, climate-friendly, very abundant, and in-
+exhaustive energy source to mankind [36]. Solar power generation is one of the 
+most advanced technologies for renewable energy production. Solar energy has 
+delivered more new capacities than nuclear and fossil fuels [37]. Worldwide growth 
+of solar power is extremely dynamic and varies strongly by country. By the end of 
+2019, a cumulative amount of 629 GW of solar power was installed throughout the 
+world [38]. By early 2020, the leading country for solar power was China with 208 
+GW [39, 40], accounting for one-third of global installed solar capacity. As of 2020, 
+there are at least 37 countries around the world with a cumulative PV capacity of 
+more than one GW. The countries in the list of top installers of 2016 through 2019 
+were China, United States, and India [41, 42]. The top 10 countries by added solar 
+PV capacity in 2019 [43] and Top 10 countries by cumulative solar PV capacity in 
+2019 is shown [44]. The solar PV capacity by country and territory (MW) and share 
+of total electricity consumption is given [45]-[48]. 
+Solar-grid integration is another important technology as it optimizes the energy 
+balance leading to improvement in economics of PV system, reduction in operating 
+cost, and provides added value to utility and consumers. Solar-grid integration 
+technology consists of advanced inverter technology, grid-plant protection 
+technology, anti-islanding technology, smart grid technology and solar-grid 
+forecasting technology [49], [50]. Authors in [49] have presented a technical report 
+on ‘Treatment of solar generation in electric utility resource planning’, jointly 
+prepared by the National Renewable Energy Laboratory (NREL) and Solar Electric 
+Power Association (SEPA). A detailed study was conducted on inclusion of solar 
+in long-term resource planning processes through interviews and a questionnaire. 
+Table 1 includes the benefits and challenges of solar, based on utility interviews. 
+With these benefits and challenges in mind, utilities can more accurately incorporate 
+solar generation into their long-term planning processes. Some of the leading, 
+utility-identified best-practices and analysis that needs attention are: (i) analyse and 
+
+ 
+12 
+assign appropriate capacity values to solar resources, (ii) analyse solar individually, 
+to get more accurate aggregate results, (iii) improve modelling assumptions and 
+methods, (iv) pursue sub-hourly sensitivities, (v) evaluate whether to treat 
+distributed generation as a resource, and (vi) utility-identified analysis needs. In 
+[50], a detailed study on the effect and challenges of integration, the current solar-
+grid integration technology, its benefit, solar system characteristic for integration, 
+and issues and compatibility of both the systems have been carried out by Nwaigwe 
+et al. Authors further conclude that solar power integration can reduce the 
+transmission and distribution losses, increase grid resilience, lower generation cost, 
+and reduce requirements to invest in new utility generation capacity. 
+ 
+Table 1 Benefits and challenges of solar power integration [49] 
+Benefits of solar power integration 
+Challenges 
+of 
+solar 
+power 
+integration 
+• 
+Meet 
+renewable 
+standard 
+requirements 
+• 
+Fuel diversification 
+• 
+Cost stability 
+• 
+Geographic dispersal benefits 
+and modularity  
+• 
+Partial correlation with peak 
+demand 
+• 
+Mitigation 
+of 
+environmental 
+compliance risks 
+• 
+Avoid line losses (typically DG 
+only) 
+• 
+Variable 
+and 
+uncertain 
+output 
+• 
+Ramping issues 
+• 
+Economics 
+• 
+Lack of current capacity 
+need 
+• 
+Cross-subsidization 
+concerns (DG) 
+• 
+Reduced capacity benefit 
+over time with increasing solar 
+penetration 
+ 
+Photovoltaic (PV) technology has presently become a significant form of power 
+generation on many power system networks. Impact of high PV integration has 
+drawn attention towards the issues of grid management, operation and planning, 
+particularly where there is variability in PV system output due to cloud cover. 
+Variability in PV irradiance is considered as a major challenge to high levels of PV 
+penetration into the existing power networks, and this variability in PV generation 
+can have a negative impact on the local electricity network. Network level problems 
+occur, where intermittent changes in PV generation on a power network are unable 
+to be accommodated by the base load generation over the time frame of the change. 
+The potential impacts of intermittent generation occur at different time scales, and 
+the associated considerations include [51]: (i) rapid changes in network demand or 
+
+ 
+13 
+in PV generation can lead to power quality issues such voltage flicker and harmonic 
+distortion, (ii) the spinning reserve of system generation is required to have 
+sufficient total capacity and ramping capability to meet the short term changes in 
+network demand or in supporting PV generation, otherwise it may  lead to power 
+quality and system outage issues, and (iii) generation planning requires sufficient 
+generation to be available at any time to fulfil the projected network demand and 
+sufficient operating reserve.  
+The integration of distributed generation (DG), particularly solar power can 
+significantly impact the flow of power and voltage conditions at customers and 
+utility equipment. Depending on the nature of DG and operating characteristics of 
+distribution system, these impacts may be either positive or negative. The positive 
+impacts [52] are: 
+ 
+Voltage support and improved power quality. 
+ 
+Loss reduction. 
+ 
+Improved utility system reliability. 
+ 
+Transmission and distribution capacity release. 
+ 
+Deferments of new or upgraded transmission and distribution infrastructure. 
+ 
+Reduction of emissions. 
+ 
+The above listed benefits are, in practice, much difficult to achieve. The DG sources 
+must be reliable, dispatchable, of the proper size and installed at the proper locations 
+and must also meet various other operating criteria. Since many DGs will not be 
+utility owned or due to intermittent nature of solar and wind power, it is difficult to 
+satisfy all these requirements. In fact, power system operations may be adversely 
+affected by the integration of DG, if certain minimum standards for control, 
+installation and placement are not maintained. The major negative impacts are: 
+voltage variation and unbalance, current and voltage harmonics, stress on 
+transformer, grid-islanding and power quality issues [53], [54]. The severity of these 
+impacts depends on the penetration level of PV, its location and configuration of 
+distribution network. The negative impacts can be summarized [55] as: 
+ 
+Voltage fluctuation in feeder, resulting in voltage rise or fall and unbalanced 
+voltage 
+ 
+Malfunctioning of on-load tap changer, line voltage regulators and capacitor 
+banks. 
+ 
+Possibility of overload in distribution feeders. 
+ 
+Variation of reactive power flow caused by malfunctioning of capacitor bank 
+devices. 
+
+ 
+14 
+ 
+Malfunctioning of overcurrent and overvoltage devices. 
+ 
+Islanding operation and detection in case of grid disconnection 
+ 
+Reliability and security of the distribution networks. 
+ 
+The electric grid -an interconnected power network shown in Figure 2 maintains an 
+instantaneous balance between the supply and demand (generation and load), while 
+transferring electricity from generation source to customer [56]. If the power 
+produced by the DER (solar PV) is more than power consumed by the load, voltage 
+level of that load bus increases, and can cause damage to various equipment 
+connected to that feeder. On the other hand, there will be drop in voltage, if power 
+demand by the load is more than the power produced. In [57], authors have 
+presented a detailed study on voltage fluctuation in power networks with 
+photovoltaic energy sources, and a flickermeter model has been used for the 
+evaluation of flicker assessment under climate change (sunny and cloudy 
+situations). In the last, authors conclude that the flickermeter model fulfils the 
+requirements defined in IEC 61000-4-15 standard, and it has been tested under 
+additional tests defined in CIGRE/CIRED/UIE test protocol. Authors have also 
+recommended the use of flickermeter for evaluating the flicker severity and voltage 
+fluctuations produced by the photovoltaic energy sources. The voltage unbalance is 
+a major power quality problem in low voltage distribution networks due to the 
+random location and rating of single-phase rooftop photovoltaic cells (PV). In [58], 
+Wang et al have presented a detailed study on voltage issues at the point of common 
+coupling caused by the integration of photovoltaic system. For this purpose, a 7.2 
+kW grid connected PV system on a radial LV distribution network has been set up, 
+and the probability density of voltage rise, and voltage unbalanced factors were 
+derived from the test data. Short-term and long-term voltage flicker indexes were 
+also calculated to evaluate the severity of flicker. It is stated that the intermittent PV 
+power output can introduce a maximum short-term flicker of 2.0 and long-term 
+flickers of 0.78. It is also shown that 1% of the voltage unbalance can produce 6-10 
+times current unbalance. To improve voltage unbalance, authors in [59] have 
+proposed a system with converter topology and control algorithm, based on the 
+application of series dynamic voltage restorer (DVR) and parallel distribution static 
+compensator (DSTATCOM) custom power devices. A state feedback control, based 
+on pole-shift technique, has been developed to regulate the DSTATCOM and DVR 
+converters output voltage for voltage balancing in the network. 
+Integration of PV systems in the distribution network, harmonic distortion of 
+voltage and current waveform takes place, resulting in poor power quality. PV 
+inverters are the main source of harmonic injection. The maximum penetration level 
+of grid connected identical photovoltaic inverter system (PVIS) that can be installed 
+is based on the acceptable voltage distortion levels within the distribution network 
+as determined in [60]. To mitigate harmonics in the network, Inductor-Capacitor-
+Inductor (LCL) filters with robust control techniques are currently used. For new 
+
+ 
+15 
+interface converters, their EMI/EMC related impact should be considered since its 
+early stage of design. It is not enough to mitigate the problem in final design, but to 
+go to the root of the problem so that harmful emission does not occur [61]. New 
+more complex converter topologies and modulation schemes are being studied to 
+solve the power quality issues, such as harmonics, voltage imbalances etc. [62], 
+[63].  
+ 
+ 
+ 
+Figure 2.  Structure of an AC microgrid [56], showing multiple generation units and 
+load. To maintain a demand-supply balance and reliable operation in such a hosting 
+capacity analysis is crucial.  
+ 
+The major findings of the studies may be summarized as follows [36]-[63]: 
+ 
+ Some renewable energy technologies provide power only when the resource is 
+available. These resources are often contracted as ‘must-take’ generators, where 
+their output is always used, when it is available. However, it is difficult to 
+integrate a large amount of ‘must-take’ generation into the grid, because its 
+availability is uncertain and constantly changing. 
+ 
+Photovoltaics (PV) may be centrally located in large plants or distributed on 
+rooftops. Distributed PV has benefits, such as low land use and no transmission 
+needs. Distributed and central PV both are usually ‘must-take’ generators. 
+ 
+Storing large amounts of power is difficult, while storing thermal energy is 
+relatively easy. Since, concentrated solar power (CSP) plants collect and 
+convert thermal energy into electricity, they can collect and store thermal energy 
+for later conversion into electricity. CSP plants with thermal energy storage 
+provide assurance that the generator will be available, when needed. These CSP 
+plants are dispatchable and can meet the intermediate and baseload demands. 
+
+PCC
+Solar PV
+Energy storage
+Small hydro
+Converter
+synchronous DG
+System
+system
+AC
+DC7
+Relay
+AC
+AC
+S3
+RDG-2
+Static Switch
+个口
+S2
+DC
+RDG-1
+R
+AC
+F2
+S1
+Distribution
+RDG-5 
+R1
+transformer
+口
+7
+R8
+R4
+R2
+AC
+RDG-6
+R12
+Load2
+RB
+I RDG.3
+Flywheel
+Load1
+AC
+Utility grid
+R6
+c
+Fuel cell
+Distribution 
+S4
+F3
+个□R11
+F1
+R5
+R7
+R10
+RpcC
+R9
+RDG-4
+S
+PCC
+DC
+R13
+S6
+ grid
+Wind Farm
+R14
+S7
+Load3
+RS
+Ry
+F4
+0
+Load4
+R15
+R16
+Load5
+Microgrid 
+16 
+ 
+Although PV deployment depends on various integration issues, most CSP 
+plants respond more slowly to changing weather, particularly equipped with 
+thermal energy storage system, and output from these plants is easier to forecast 
+and integrate into the electric grid. 
+ 
+PV generation without energy storage system does not provide all the 
+characteristics required for stable grid operation. 
+ 
+Careful integration of distributed generation and deployment of utility-scale 
+generation will be needed to provide the mix of power and reliability required 
+for a clean and healthy power supply as renewables contribute an increasingly 
+larger share of energy needs. 
+ 
+Static synchronous compensator (STATCOM) in conjunction with energy 
+storage system is likely to play an effective role in mitigating the voltage issues 
+without curtailing any renewable energy and can provide active power support 
+during the grid contingency. 
+ 
+Distribution system designs and operating practices are normally based on 
+radial power flows, and this creates a special challenge to the successful 
+introduction of distributed generation. Some important issues such as voltage 
+regulation and losses, voltage flicker, harmonics etc. must be considered to 
+ensure that DG will not degrade the distribution system power quality, safety or 
+reliability. 
+ 
+The fault contribution from a single small DG unit is not large, however the 
+aggregate contributions of many small units, or a few large units, can alter the 
+short circuit levels enough to cause fuse-breaker miscoordination. 
+ 
+Distributed generation must be applied with a transformer configuration and 
+grounding arrangement compatible with the utility system to which it is to be 
+connected. Otherwise, voltage swells and over voltages may be imposed on the 
+utility system that may damage the utility or customer equipment. 
+ 
+Islanding can occur only if the generator(s) can self-excite and sustain the load 
+in the islanded section. In most cases, it is not desirable for a DG to island with 
+any part of the utility system, because this can lead to safety and power quality 
+problems that will affect the utility system and load both. 
+ 
+The implementation of DG can increase the reliability of electric service if units 
+are configured to provide ‘backup- islands’ during upstream utility source 
+outages. To be effective, this requires reliable DG units and careful coordination 
+of utility sectionalizing and protection equipment. 
+ 
+To avoid costly impact studies for all but those applications that actually need 
+them, proposed DG can be screened based on factors such as: utility system 
+fault levels at the point of DG interconnection, size of DG unit, its intended 
+mode of operation and expected output fluctuations, aggregate capacity of DG, 
+
+ 
+17 
+secondary configuration of the DG site (including presence of adjacent 
+customers), feeder voltage regulation practice, transformer and grounding 
+compatibility of the system, size of generation relative to load at the 
+interconnection point and type of interfacing power converter. 
+ 
+The main cause of new solar panel failure is poor design and defects during 
+manufacturing, contact defects in junction boxes, glass breakage, burst frames, 
+breakage of cell interconnections and problems with the diode associated with 
+a higher rate of cell degradations and interconnectors, fuse boxes, charge 
+controllers and cabling as well as issues with grounding. 
+ 
+The variance created by the installation of a further dispersed PV inputs into 
+power grids can end up being very similar to step-change ‘noise’ variance, 
+which currently occurs in the network. 
+ 
+Analysis of the size, number, and spatial diversity to optimise PV input into the 
+grid should be undertaken with a view to determining the marginal benefit of 
+additional diversity and/or the extent to which the benefits of diversity diminish, 
+if the separation of systems gets too great. 
+ 
+Under-generation and over-generation both by solar PV cause instability on the 
+grid. Effective solution to this involves use of better forecasting tools for more 
+accurate prediction of when solar generation might decline to minimum 
+penetration capacity, installing solar panels across a large geographic area to 
+minimize any impact of generation variability due to local cloud cover, shifting 
+power supply and storing excess energy for later use and encouraging customers 
+to use power, when it is more readily available. 
+ 
+Prior to deployment of solar panels, advanced integration technologies should 
+be considered. Optimized forecasting is essential for proper system stability. 
+Also, due to economic viability and robustness of the system, solar technology 
+can be treated as a major guideline for sustainable development. 
+ 
+Integrating CSP plants with wind power can reduce the generation uncertainty 
+and improve the capacity factor of the combined plant, because of the negative 
+correlation of wind and solar power, and the availability of high-efficiency 
+thermal energy storage systems. 
+ 
+5. Assessment of Hosting Capacity (HC) 
+The HC concept has been most widely used to evaluate the benefits of different 
+voltage regulation techniques, amount of solar and wind energy that a grid can 
+accommodate without violating the acceptable performance indices of the existing 
+power network. Various power system phenomena and related performance indices 
+are given in [64], and the same is reproduced in Table 2. Distributed generation 
+(DG) such as solar and wind will impact the performance of the grid, and this sets 
+a limit to the amount of such RES that can be integrated. New communication 
+
+ 
+18 
+schemes together with energy storage system and grid control strategies allow 
+integration of increased amounts of renewables into existing power networks, 
+without unacceptable effects on users and grid performance. The first work on the 
+hosting capacity of electrical grids was reported by Math Bollen and Hager in 2004 
+[65]. Later, many methods have been applied to examine the capacity of existing 
+distribution grids to accept DG [66]. However, some important differences do exist 
+between methods. Most of the statistical approaches proposed in the literature aim 
+at defining the optimal DG location and sizing [67]. It is very essential to define 
+suitable performance indices before calculating the amount of DG that can be 
+integrated into the existing network, as the hosting capacity is based on this. The 
+choice of performance indices and limit have a big influence on the amount of 
+distributed generation that can be accepted. The hosting capacity is a tool that 
+provides trustworthy and secure platform for fair and open discussion between the 
+different stakeholders (network operators, owner of DG, owner of the existing large 
+power plants and other customers) and a transparent balancing between their 
+interests, for example, acceptable reliability and voltage quality for all customers, 
+no unreasonable barriers against new generation and acceptable costs for the 
+network operator [68], [69]. The integration of DG should not result in a 
+deterioration of the supply reliability, but certain deterioration in quality of the 
+supply should be acceptable for most customers. Reliability is quantified by several 
+indices. The steps to be followed to calculate HC is summarized below [15]: 
+i. 
+Choose a phenomenon and one or more performance indices, 
+ii. 
+Determine a suitable limit or limits, 
+iii. 
+Calculate the performance indices as a function of the amount of generation, 
+iv. 
+Obtain the hosting capacity. 
+ 
+Table 2 
+Power system phenomena and related performance indices [64] 
+S.N. Phenomena 
+Performance Indices 
+1 
+Overloading from wind power 
+Maximum hourly value of current 
+through transformer 
+2 
+Frequency variation 
+99% interval of 3 s average of 
+frequency 
+3 
+Overvoltage from roof top solar 
+photovoltaic cells 
+Highest 10 min average of voltage 
+4 
+Undervoltage from fast charging of 
+electric vehicles 
+Lowest 10 min. average of voltage 
+
+ 
+19 
+5 
+Protection mal-trip 
+Lowest recorded current causing 
+interruption 
+6 
+Harmonics 
+10 min. average of voltage and 
+currents 
+ 
+In [70], authors have presented the hosting capacity approach and explained some 
+important developments, such as uncertainty in location and size of production 
+units, and curtailment to connect more production than according to the initial 
+hosting capacity. Authors have given the new HC terminology as:  
+i. 
+HC uncertainty,  
+ii. 
+HC Coefficient,  
+iii. 
+Stochastic HC (SHC), 
+iv. 
+Locational HC (LHC).  
+Uncertainty in the estimation of HC may occur due to volatile and intermittent 
+nature of DG output powers, unknown rating of DG units and their locations, load 
+variability and insufficient data, when performing power system calculation. Hence, 
+HC will never be a single value, but multiple values will be obtained depending on 
+the uncertainty percentage. Since randomness is involved in the calculation of HC, 
+it cannot be termed as deterministic approach, instead it should be viewed as a 
+probabilistic approach, where level of accuracy and uncertainty are considered. 
+Basically, HC is location-based concept; the maximum HC occurs at the maximum 
+loading conditions and minimum generation, whereas minimum HC is obtained at 
+minimum loading conditions and maximum generation [11], [71].  
+Curtailment of power production is defined as reducing the active power output 
+from certain energy resources at times to increase the HC. The ability to curtail the 
+production will allow for a larger installed capacity of distributed generation. The 
+increase in installed capacity for a given percentage of permitted overloading varies 
+with the type of energy source. The gain in hosting capacity will depend on 
+characteristics of the network and performance index being considered [72]. This 
+approach can be extended to cover different voltage levels and hosting capacity 
+limits determined by very different performance indices (and therefore different 
+power system phenomena). For an objective comparison, the Hosting Capacity 
+Coefficient (HCC) is defined as the ratio between the curtailed energy and the 
+installed capacity above the initial HC as given in Eq. (1). In the same work, it is 
+inferred that a low HCC corresponds to an unfavourable location for curtailment. A 
+high value of HCC corresponds to a high ability to accommodate DG using 
+curtailment. However, it is possible that a high coefficient corresponds to a low HC 
+limit. 
+
+ 
+20 
+HCC =
+Curtailed Energy
+Installed Capacity−Initial HC 
+ 
+ 
+(1) 
+ 
+Integration of DG units into the power network has many unknown variables, such 
+as size and location of DG, number of customers who will utilize DG. The 
+consumption profiles and output of DG are also intermittent in nature. These 
+unknown variables have an impact on the HC. Considering the randomicity of some 
+of these unknown variables, Probabilistic Power Flow (PPF) analysis is carried out. 
+In PPF, several load flow calculations are performed for thousands of random cases 
+of number, location and/or size of DG units. Networks’ variables such as voltage, 
+current, losses etc. are recorded against the performance limits for the determination 
+of HC [73]. The stochastic HC estimation have been reported by many authors [74]-
+[80]. In [74], authors have proposed a stochastic multi-objective optimization model 
+to maximize HC of the distribution network for wind power and minimize the 
+energy procurement costs in a wind integrated power system, while considering 
+technical and economic aspects. Cost of the purchased energy from upstream 
+network, and operation and maintenance cost of wind farms are taken as objective 
+functions. Analysis was carried out on a standard radial 69 bus distribution feeder 
+and a practical 152 bus distribution system.  
+A detailed report on feeder modelling, analysis, and evaluation of issues to assess 
+the impact of integration of distributed solar PV and to determine the hosting 
+capacity prepared by EPRI is given in [75], [76]. An EPRI Technical Update report 
+on developing a screening methodology, which effectively evaluates the new 
+interconnection requests, while considering PV and feeder-specific factors is 
+presented in [77]. This methodology has considered the peak load levels along with 
+other critical factors, including PV location, aggregate PV effects, and most 
+importantly specific feeder characteristics such as voltage class, voltage regulation 
+schemes, and operating criteria. Authors in [78] have used a stochastic approach to 
+simulate various DG deployment scenarios. Various limits such as over-voltages, 
+voltage deviations and voltage unbalance are considered, while calculating HC. Le 
+Baut et. al [ref] have presented a three-step probabilistic HC assessment technique 
+in [79]. In [80], a high-resolution approach is presented to assess the DG and storage 
+systems connection into the distribution network. Authors have used a stochastic 
+demand model for this work. 
+Since HC is highly location based, integration of new DG can be accepted at some 
+locations, but not at other places. The voltage profile of the feeders is one of the key 
+criteria in defining the locational HC (LHC). Rylander et al [81] presented the 
+concept of stream-lined methodology to calculate HC termed as stream-lined HC 
+(SLHC). This method was developed by EPRI to calculate the HC of feeder 
+considering size and location of DER, integration technology and physical 
+characteristic of the feeders. Application of the results include improved screening 
+tools, visualization of HC constrained feeders, and identification of problematic 
+locations within a feeder. High penetration of DER/DG into the electrical networks 
+
+ 
+21 
+adversely affects various performance indices. The primary issues used to identify 
+hosting capacity include overvoltage, voltage deviation, element fault current, 
+breaker reduction of reach, sympathetic breaker tripping, breaker/fuse coordination. 
+Anti-islanding and large-scale PV protection issues with grounded wye-delta 
+interconnect transformer have significant impact on HC, whereas voltage 
+imbalance, overloads and harmonics have low impact on hosting capacity.  
+Determination of hosting capacity of distribution network can be broadly 
+categorised as utility-based and customer-based. In utility-based, problem can be 
+defined as an optimization problem with an objective to maximize the integration 
+of DERs without technical violation in the distribution networks. In customer-based, 
+stochastic methods are mostly used for assessment of HC, since in this case, utilities 
+have no control over the number, locations, and size of DERs. There are four 
+techniques used of the estimation of HC: (i) deterministic, (ii) stochastic, (iii) 
+optimization-based, and (iv) stream-lined. Though, all these methods follow the 
+same general procedure, but their implementation is quite different. In all the four 
+methodologies, power flow analysis is carried out to find voltages and currents in 
+the distribution networks. For power system simulation and analysis, a wide range 
+of commercial and non-commercial software tools are also available; for examples, 
+PSS Sincal Integrated Capacity Analysis Module [106], DigSILENT PowerFactory 
+[107], NEPLAN [108], Synergi Electric [109], and CYME [110]. A list of these 
+software programs is given in [111]. A brief and precise discussion on the 
+methodology used for the assessment of HC and software tools is provided in [14] 
+and references therein. A summary of the studies made by various researcher [82]-
+[105] is presented in Table 3. 
+ 
+Table 3 
+Studies that utilize various performance indices for assessment of hosting capacity  
+Author/ 
+Reference 
+Year 
+DER 
+Tech. 
+Performance 
+Indices 
+Objective 
+and 
+Technique used 
+Remarks 
+Sakar et al, 
+2017        
+[82]          
+PV 
+Overvoltage, under 
+voltage, 
+current 
+capacity of power 
+lines, and harmonic 
+distortion 
+of 
+the 
+system 
+HC estimation of 
+a 
+distorted 
+distribution 
+system 
+with 
+Photovoltaic 
+(PV)-based DG 
+units, 
+using 
+optimization-
+based technique 
+Lower 
+HC 
+with 
+higher 
+distortion in 
+distribution 
+networks. 
+
+ 
+22 
+Sakar et al, 
+2018        
+[83] 
+PV 
+Bus voltage limits, 
+line ampacities, and 
+harmonic distortion 
+limits 
+To find system's 
+HC 
+and 
+parameters 
+of 
+the 
+proposed 
+filter, 
+by 
+an 
+optimization 
+algorithm. 
+HC decreases 
+with 
+the 
+increase 
+in 
+utility side’s 
+background 
+voltage 
+distortion and 
+load 
+side’s 
+nonlinearity 
+values. 
+Braga et al, 
+2018        
+[84] 
+PV 
+Harmonic 
+distortion, 
+and 
+location  
+To 
+find 
+harmonic 
+hosting capacity 
+of the IEEE 13 
+bus network at 
+four 
+locations 
+through 
+sensitivity 
+analysis 
+on 
+Matlab 
+and 
+OpenDSS.  
+Presence 
+of 
+capacitors 
+lead to higher 
+value 
+of 
+voltage total 
+harmonic 
+distortion 
+THDv due to 
+the 
+parallel 
+and 
+series 
+resonance. 
+Mirbagheri 
+et al, 2018   
+[85] 
+DG 
+Steady-state voltage 
+variations, 
+rapid 
+voltage 
+changes, 
+and thermal limits 
+of 
+lines 
+and 
+transformers  
+To 
+determine 
+HC 
+of 
+distribution 
+grids 
+even 
+in 
+case 
+of 
+uncertainties in 
+grid parameters 
+or lack of data, 
+by using a novel 
+method 
+called 
+‘Bricks 
+approach. 
+The detailed 
+topology 
+of 
+the grid and 
+power profile 
+for 
+all 
+the 
+nodes are not 
+required. 
+Method 
+is 
+fast. 
+Chathurang
+i et al, 2018    
+[86] 
+PV 
+Voltage limits and 
+network loss 
+To find technical 
+impacts caused 
+by 
+higher 
+penetration 
+of 
+roof top solar 
+PVs 
+on 
+the 
+operating 
+performance of 
+LV distribution 
+Further 
+connections 
+of solar will 
+cause 
+violation 
+of 
+acceptable 
+voltage limits 
+and increase 
+network 
+
+ 
+23 
+networks, 
+by 
+modelling 
+the 
+detailed network 
+in DIgSILENT 
+PowerFactory 
+simulation 
+platform. 
+losses due to 
+the 
+reverse 
+power flow. 
+Network 
+already 
+has 
+40% 
+solar 
+penetration 
+level (based 
+on 
+transformer 
+capacity)  
+Faishal 
+Fuad et al, 
+2018 
+                 
+[87] 
+PV 
+Voltage violation, 
+current 
+violation 
+and 
+power 
+flow 
+direction 
+To find HC of 
+the distribution 
+network 
+under 
+different loading 
+conditions, by an 
+iterative method. 
+Study reveals 
+that 
+voltage 
+violation 
+occurs 
+only 
+when the load 
+is less than 
+48% of the 
+average 
+value, while 
+reverse 
+power 
+flow 
+occurs 
+in 
+most of the 
+cases. 
+Al-Saadi et 
+al, 
+2018         
+[88] 
+PV 
+Overvoltage 
+and 
+over loading 
+Assessment 
+of 
+HC 
+by 
+conducting 
+stochastic 
+simulations, 
+in 
+which 
+the 
+uncertainties of 
+the 
+solar 
+irradiance 
+and 
+load 
+variation 
+are considered.  
+Hourly 
+assessment 
+will help to 
+serve 
+the 
+automation 
+strategies, 
+where 
+corrective 
+actions can be 
+taken for HC 
+maximization
+. Inclusion of 
+more 
+performance 
+index, such as 
+frequency 
+variation 
+index, 
+may 
+
+ 
+24 
+further 
+improve the 
+method.  
+Peppanen 
+et al, 2018    
+[89] 
+DER 
+Thermal 
+limit, 
+steady-state voltage 
+and 
+voltage 
+deviation 
+To find HC of 
+three real North 
+American 
+residential 
+distribution 
+systems. 
+Load 
+variability 
+and 
+unbalance 
+are 
+also analysed. 
+Results 
+showed that 
+the 
+most 
+significant 
+limiting 
+factor 
+is 
+voltage 
+deviation. 
+Alturki 
+et 
+al, 
+2018         
+[90] 
+DG 
+Overvoltage 
+and 
+overloading 
+To estimate HC 
+in 
+distribution 
+grids, by using 
+an optimization-
+based method. 
+Low 
+computationa
+l 
+time 
+as 
+compared to 
+traditional 
+methods, 
+while 
+offering 
+comparable 
+results. 
+Abad et al, 
+2018         
+[91] 
+DG  
+Voltage deviation, 
+load 
+growth, 
+network 
+structure 
+and type of DG 
+Probabilistic 
+model based on a 
+two-step 
+linearization 
+algorithm 
+is 
+presented 
+to 
+linearize the HC 
+model. 
+HC can be 
+approximated 
+by 
+a 
+Gaussian-
+shape 
+distribution. 
+Al-Saffar et 
+al, 
+2019         
+[92] 
+PV 
+Overvoltage 
+HC of three real 
+circuits 
+in 
+Alberta, Canada 
+are 
+evaluated 
+using 
+Monte 
+Carlo simulation 
+based 
+probabilistic 
+power 
+flow 
+method.  
+The use of 
+stochastic 
+approach can 
+overcome the 
+shortcomings 
+of 
+deterministic 
+methods, and 
+can, 
+in 
+principle, 
+handle 
+circuits 
+of 
+
+ 
+25 
+any 
+complexity. 
+Duwadi et 
+al, 
+2019         
+[93] 
+PV 
+Overvoltage, 
+undervoltage, 
+and 
+unbalance between 
+the phases  
+To find HC of a 
+feeder 
+using 
+Monte 
+Carlo 
+simulation. 
+A 
+parallel 
+algorithm 
+was 
+developed using 
+Message Passing 
+Interface 
+to 
+decrease 
+the 
+simulation time. 
+PV 
+penetration 
+was found to 
+be limited to 
+10%, on days 
+when 
+solar 
+irradiance 
+was high. PV 
+penetration 
+may go upto 
+50% on days 
+with 
+low 
+energy 
+demand. 
+Steyn et al, 
+2019         
+[94] 
+Roof-
+top 
+PV 
+Voltage 
+increase 
+and 
+equipment 
+overload 
+To find impact 
+on integration of 
+distributed 
+rooftop 
+PV 
+systems on three 
+distribution 
+networks 
+(residential, 
+commercial, and 
+industrial) 
+in 
+Cape 
+Town, 
+South Africa. 
+Integration of 
+PV 
+systems 
+decreases the 
+line 
+and 
+transformer 
+loading, but 
+after a certain 
+level, 
+the 
+effect 
+of 
+integration 
+start 
+to 
+reverse due to 
+increase 
+in 
+PV 
+penetration 
+level 
+and 
+reverse 
+power flow. 
+Lillebo 
+et 
+al, 
+2019         
+[95] 
+EV 
+Overloading 
+To explore the 
+impacts 
+of 
+increasing 
+EV 
+penetration level 
+in a Norwegian 
+distribution grid, 
+by 
+using 
+real 
+power 
+measurements 
+Applying 
+a 
+fast 
+charger 
+in the grid 
+with standard 
+power factor 
+of 0.98 lag 
+caused 
+significant 
+voltage 
+
+ 
+26 
+obtained 
+from 
+household smart 
+meters in load 
+flow analyses. 
+deviations at 
+several 
+locations, the 
+worst 
+of 
+which 
+reached close 
+to 0.03 p.u 
+Soukaina et 
+al, 
+2019         
+[96]   
+DG 
+Overvoltage, 
+line 
+overloading 
+and 
+transformer 
+overloading 
+A new method to 
+calculate HC of 
+DG, considering 
+the variety of 
+distribution line 
+configurations 
+and 
+cross 
+admittance, 
+is 
+presented. 
+The 
+proposed 
+method was also 
+applied 
+to 
+an 
+underground 
+medium voltage 
+grid. Simulation 
+was carried out 
+using 
+Matlab 
+and Etap. 
+Based 
+on 
+obtained 
+results, it is 
+shown 
+that 
+including line 
+capacitances 
+in calculation 
+increases the 
+maximum 
+power of DG 
+unit. 
+Ismael 
+et 
+al, 
+2019         
+[97] 
+PV 
+Bus voltage, line 
+thermal 
+capacity, 
+power factor, and 
+individual and total 
+harmonic distortion 
+To 
+find 
+probabilistic 
+hosting capacity 
+(PHC) due to 
+high penetration 
+of PV units in a 
+non-sinusoidal 
+power 
+distribution 
+network, 
+by 
+using 
+Monte 
+Carlo 
+simulation, 
+considering 
+various 
+uncertain 
+parameters, such 
+as 
+intermittent 
+Results 
+shows 
+that 
+deterministic 
+hosting 
+capacity 
+(DHC) 
+studies, 
+which ignore 
+the 
+uncertainty of 
+electrical 
+parameters, 
+result 
+in 
+optimistic 
+results 
+that 
+cause 
+a 
+noticeable 
+underestimati
+
+ 
+27 
+output power of 
+DGs, 
+background 
+voltage 
+harmonics, load 
+alteration, 
+and 
+filter 
+parameters’ 
+variations. 
+Hybrid PSO and 
+gravitational 
+search algorithm 
+(PSOGSA) has 
+been used for 
+optimal 
+design 
+of the proposed 
+filter. 
+on to the HC 
+levels that are 
+achieved 
+from 
+the 
+probabilistic 
+studies. 
+Abideen et 
+al, 
+2019,        
+[98] 
+PV 
+Bus 
+overvoltage 
+and 
+distribution 
+network losses 
+A new algorithm 
+based on gradual 
+increase in the 
+PV penetration 
+level 
+is 
+introduced 
+to 
+estimate 
+the 
+maximum 
+PV 
+penetration level 
+in 
+LV 
+distribution 
+networks using 
+Matlab. 
+This 
+technique can 
+be applied to 
+any 
+distribution 
+network, 
+if 
+the 
+power 
+demand and 
+the 
+solar 
+irradiation 
+data 
+are 
+available.  
+Li 
+et 
+al, 
+2019 
+                 
+[99] 
+DER 
+ 
+A 
+valuation 
+scheme 
+to 
+quantify 
+the 
+value of DER in 
+active 
+distribution 
+network (AND) 
+is developed. A 
+two-tier scheme 
+is proposed to 
+value 
+and 
+compensate 
+DER 
+portfolio 
+The proposed 
+scheme will 
+not 
+only 
+encourage the 
+proactive 
+investment of 
+DERs 
+in 
+AND, 
+but 
+also 
+help 
+enhance 
+the 
+role of DERs 
+in 
+offering 
+reliable, 
+
+ 
+28 
+proposed 
+by 
+customers 
+and 
+independent 
+third parties. 
+resilient, 
+affordable 
+and 
+sustainable 
+power supply 
+to customers.  
+Essackjee 
+et al, 2019 
+[100] 
+PV 
+System harmonics 
+A detailed study 
+on 
+maximum 
+rooftop PV HC 
+is presented. The 
+level 
+of 
+harmonic 
+pollution 
+was 
+observed 
+with 
+increasing 
+penetration level 
+of photovoltaic 
+units and using 
+different sizes of 
+such units. 
+The findings 
+can be used 
+by 
+an 
+electrical 
+utility to limit 
+the number of 
+rooftops 
+photovoltaic 
+being 
+connected so 
+as 
+not 
+to 
+degrade 
+the 
+quality 
+of 
+supply. 
+Sadeghian 
+et al, 2020  
+[101] 
+PV 
+Voltage 
+violation 
+and reverse power 
+flow 
+ 
+An 
+impact-
+assessment 
+framework, 
+based on Monte 
+Carlo technique, 
+to 
+assess 
+the 
+impacts of two 
+different types of 
+distributed solar 
+photovoltaic 
+(DPV) 
+installation 
+(customer based 
+and 
+utility 
+based) 
+on 
+a 
+realistic 
+distribution 
+network. 
+To 
+achieve 
+higher 
+penetration 
+ratios without 
+reverse 
+power 
+flow 
+and 
+other 
+negative 
+impacts, 
+utility-aided 
+installation is 
+necessary. 
+Mulenga  et 
+al, 
+2021       
+[102] 
+PV 
+Overvoltage 
+A 
+stochastic 
+algorithm 
+for 
+estimation of HC 
+of 
+low-voltage 
+network for solar 
+It is shown 
+that 
+performance 
+indices used 
+in 
+
+ 
+29 
+PV 
+is 
+introduced. Two 
+types 
+of 
+uncertainties 
+aleatory 
+and 
+epistemic 
+are 
+distinctively 
+considered. 
+deterministic 
+and 
+time-
+series 
+HC 
+studies need 
+to 
+be 
+complemente
+d 
+with 
+a 
+planning risk 
+percentile for 
+stochastic 
+study. 
+Du et al, 
+2021 
+               
+[103] 
+PV- 
+Wind- 
+Load  
+Voltage violation 
+A 
+stochastic 
+framework 
+for 
+HC 
+based 
+on 
+wind-PV-load 
+temporal 
+characteristic is 
+presented from a 
+probabilistic 
+view. 
+A 
+discretization- 
+aggregation 
+method 
+is 
+introduced 
+to 
+generate 
+filter 
+extreme 
+combinations. 
+Study 
+helps 
+in 
+making 
+better use of 
+available 
+renewable 
+resources and 
+promoting 
+the 
+application of 
+distributed 
+hybrid power 
+generation in 
+the 
+power 
+grid. 
+Paudyal et 
+al, 
+2021      
+[104] 
+EV 
+Voltage limit, 
+thermal limit 
+Study on EV HC 
+by focusing on 
+extreme 
+fast 
+charging (xFC) 
+of EV charging 
+loads is carried 
+out to identify 
+representative 
+feeders of the 
+utility 
+distribution 
+network 
+in 
+a 
+certain region. 
+Study 
+can 
+help utilities 
+plan 
+for 
+optimal 
+system 
+upgrades 
+to 
+facilitate 
+future needs 
+and 
+greatly 
+reduce 
+the 
+cost of EV 
+integration. 
+Mulenga et 
+al, 
+2021 
+[105] 
+EV 
+Undervoltage, 
+transformer 
+and 
+feeder overload 
+A 
+stochastic 
+methodology to 
+single-phase and 
+The method 
+can 
+be 
+applied as an 
+
+ 
+30 
+three-phase EV 
+charge 
+hosting 
+capacity 
+for 
+distribution 
+networks 
+is 
+developed. The 
+method includes 
+two 
+types 
+of 
+uncertainties, 
+aleatory 
+and 
+epistemic. 
+extension to 
+find HC as a 
+function 
+of 
+time of day, 
+week, or year, 
+and the best 
+periods 
+for 
+charging can 
+be identified 
+and used in 
+designing 
+smart 
+charging 
+mechanisms. 
+  
+Enhancing hosting capacity of the network is considered as one of the important 
+goals for distribution system operator (DSO). The technical solutions to enhance 
+systems’ HC have been grouped in three categories, such as DSO solutions, 
+prosumer solution, and interactive solutions. The effectiveness of these solutions 
+depends on various factors, such as technology readiness, investment cost, and 
+impact on congestion and compliance with the applied grid codes. The various 
+methods used are: reactive power control [112], [113], voltage control using on load 
+tap changer (OLTC) [114], [115], [116], active power curtailment [117], [118], 
+energy storage technologies [119], [120], network reconfiguration and 
+reinforcement [121], and harmonic mitigation techniques [83], [122]. A brief study 
+on the various techniques used for HC enhancement is presented in [11] and 
+references therein. Authors in [112], [113] have reported that using reactive power 
+control and energy storage system, DG penetration can be increased, and the system 
+losses can be decreased. It is concluded that the model can be successfully utilized 
+in smart grids, where integration of large-scale DG sources is preferred. Navarro et 
+al [114] carried out a techno-economic study on the use of OLTC to mitigate 
+overvoltage problem in UK caused by high PV penetration. Authors examined the 
+local and remote voltage control methods and compared it with conventional 
+reinforcement solutions. Rauma et al [115, 116] have studied the advanced control 
+of OLTC transformers and its application in increasing HC. Authors developed an 
+analytical model for a set of 631 real LV electrical systems in France, based on 
+actual voltage measurements recorded by advanced metering tools, and concluded 
+that the use of OLTC can enhance the HC. In [117], [118], authors presented a study 
+on the role of active power curtailment and dynamic line rating in enhancing HC 
+for LV and MV distribution networks. The authors have categorized the active 
+power curtailment into soft and hard curtailments. Poulios in [119] studied the 
+optimal size, location and economic aspects of battery energy storage systems 
+(BESSs) for increasing the system HC and have stressed that the prices of BESS 
+
+ 
+31 
+need to be greatly reduced in order to make it competitive to other available HC 
+enhancement solutions. Authors in [120] proposed a cost-based multi-objective 
+optimization tool built on Matlab platform to evaluate the BESS capacity. Authors 
+examined the role of BESS for voltage regulation, network loss reduction and peak 
+load reduction. Ismael et al [121] have examined the problem of selecting the 
+optimal conductor for a real radial distribution network in Egypt and presented a 
+novel feeder enforcement index to assist network planners and DSOs to recognize 
+the feeders that need to be reinforced first. Bollen et al in [122] have discussed the 
+HC assessment considering over and under voltage performance limits. Authors 
+have also studied the impact of inter-harmonics and super-harmonics on HC 
+estimation.  
+The key findings of the studies are summarized as follows [64]-[122]: 
+ 
+Higher the percentage of time during which curtailment is acceptable, higher is 
+the amount of production capacity that can be integrated to the network. 
+However, even though the hosting capacity is increased, the proportion of 
+additional energy that must be curtailed increases quickly making additional 
+increase in hosting capacity less and less attractive.  
+ 
+Hosting capacities for each feeder in the residential and commercial PV 
+scenario and utility-scale PV scenario are different due to the possible PV 
+locations. In the utility-scale analysis, the deployed PV could be located close 
+to start-of-circuit at the substation or in feeder extremities. In the residential and 
+commercial deployed PV scenario analysis, the PV location depends to a greater 
+degree upon the customer location. 
+ 
+The process of developing the baseline model and methodology for running the 
+simulation, includes hundreds of different decisions, which can significantly 
+shape the outcome, both in terms of final hosting capacity figure, and its 
+accuracy, when compared to real life conditions. 
+ 
+The streamlined methodology was significantly faster to run (from a computing 
+standpoint), however it had accuracy issues (both over and underestimating 
+hosting capacity) in a not-insignificant number of cases, particularly when it 
+came to application on complex circuits and with respect to two of the four 
+power system criteria that were evaluated: power quality/voltage and 
+protection. 
+ 
+The iterative methodology results were found to be sufficiently accurate, but 
+running the model was computationally intense, and thus would require more 
+resources to deploy and may not be able to be run as often as needed for the 
+type of scenario analysis that may be used for planning. 
+ 
+Based on the method for PV deployment, voltage imbalance typically 
+decreases, and overloads seldom occur. This is strongly influenced by limiting 
+the PV size to the customer peak load in the small-scale analysis. For balance 
+
+ 
+32 
+large-scale PV, voltage imbalance seldom occurs, however the PV systems do 
+have a slight potential to cause overloads. 
+ 
+Deterministic method uses few input parameters that are readily available. It is 
+fast and easily implementable and presents a quick overview of grid 
+performance. It assumes fixed value of parameters and does not consider 
+intermittent nature of RES. It does not consider uncertainties. HC obtained by 
+deterministic method is an estimate of the worst-case scenario and not the true 
+value. The impact, in this method, is overestimated and the hosting capacity 
+underestimated.  
+ 
+Stochastic technique considers uncertainties in the network and presents a 
+realistic overview of the grid performance under renewable energy sources 
+integration based on probability distribution functions or possibility theory. 
+This method simulates realistic network scenarios and is less time consuming 
+than time-series method. It accommodates all probabilistic distribution 
+functions and is comparatively easy to execute. 
+ 
+Stochastic method needs large computational time with the increase in 
+uncertainties considered in large distribution networks. It does not assess the 
+time-related operation of control elements and network performance, and 
+requires use of probability distribution functions, which can affect the accuracy 
+of the result. In this method, complexity increases with the increase in the 
+number of uncertainty types, and the evaluation and interpretation of HC 
+quantification becomes a difficult task.  
+ 
+Time-Series method considers time correlation in the network, power 
+consumption and production. Time varying impacts of renewable energy 
+sources on the network and operation of control elements are considered in HC 
+determination. The method presents realistic overview of the network 
+performance, based on time varying nature of power consumption and 
+production. This method can give information related to the ‘when’ and ‘how’ 
+of the HC quantification. 
+ 
+Time-Series method requires many measurements data (time series data), and a 
+lot of simulation may be needed. Some performance indices may require very 
+low resolution and pose a computational challenge. Also, method is very time 
+consuming for high resolution simulations. 
+ 
+Harmonic issues are strongly influenced by load and existing resonance rather 
+than PV penetration. 
+ 
+By injecting reactive power, larger loads like a fast charger or a large EV 
+household charger might be installed in weaker parts of a power grid. 
+ 
+Voltage deviation constraint does not affect the HC probability curve 
+significantly, which implies that distributing DGs over the system decreases the 
+importance of voltage deviation constraint. 
+
+ 
+33 
+ 
+The voltage deviation constraint often limits the locational HC, but not the HC. 
+ 
+Voltage unbalance factor (VUF), which is an important index for unbalanced 
+systems does not constrain HC of the test system. 
+ 
+DG technology has a great effect on probability curve of HC. This is probably, 
+because the PV capacity factor is higher than wind capacity factor in the test 
+system. 
+ 
+Stochastic hosting capacity methods can also be applied for addressing the 
+potential overload due to large amount of solar power. 
+ 
+For single-phase units, installation of distributed energy storage can reduce the 
+unacceptable voltage rise with the customer. 
+ 
+Energy storage can provide reserves, change net load shape to minimize 
+ramping requirements, and shift supply of variable generation to periods of 
+increased net load. 
+ 
+Voluntary load reduction or load shifting can provide multiple benefits to 
+integrating solar and reducing curtailment, including reducing the dependence 
+on partially loaded synchronous generators for providing frequency stability 
+and operating reserves and changing the shape of net load, which can reduce 
+ramp rates, better align solar supply with demand, and reduce peak capacity 
+needs. 
+ 
+Balancing supply and demand over larger areas reduces the net variability of 
+load and renewable resources such as PV, owing to greater spatial diversity of 
+variable generation resources. 
+ 
+Changing the way the grid is scheduled and dispatched, including changes to 
+market rules, does not require new technologies and often represents the ‘least 
+cost’ way to aid vehicle to grid (VG) integration. 
+ 
+Inverter-based solar and wind plants can provide the grid’s frequency response 
+needs as these plants become a larger proportion of the generation fleet and new 
+mechanisms are developed. 
+ 
+Generators can respond better to net load shape created by additional PV via 
+increased ramp rates and ranges as well as the ability to start and stop more 
+frequently. 
+ 
+Due to the integration of distributed generation, the level of power quality 
+disturbances may increase beyond what is acceptable for other customers. 
+ 
+6. Conclusion 
+The electricity technology sector has gone through a marked change from its 
+traditional architecture of large-scale centralized electric power supply systems that 
+
+ 
+34 
+take advantage of significant economies of scale. Renewable energy resources, 
+particularly wind and solar energy in form of distributed generation certainly fit this 
+trend. Thus, the traditional cost comparisons, based on large bulk energy market, 
+may be very misleading. Distributed generation is likely to pioneer the development 
+of a new power market, in which the new energy technology does not simply supply 
+energy but must instead meet the demand of services like energy management, 
+emergency or back-up power, environmental improvements and fuel diversity. 
+Driven by the need for a more sustainable energy system, this new power production 
+is often of the renewable type, connected through a power electronic converter 
+interface. The excessive integration of distributed generation systems into the 
+existing electrical networks may adversely affect the system’s performance, and 
+may lead to problems and operational limit violations, when the system exceeds its 
+hosting capacity. Hence, it has now become essential to know the hosting capacity 
+of the network to accept any further integration request. In this paper, a brief review 
+on the concept of hosting capacity, techniques being used for its assessment and 
+enhancement and operational limits is presented. Substantial progress has been 
+made in HC assessment and enhancement covering analysis, simulation, and the 
+hardware development and testing for efficiency maximization and cost 
+minimization. However, many problems and issues, especially those related to the 
+development of affordable, in-exhaustible and clean renewable energy technologies 
+for huge longer-term benefits, and broad range of policies, market regulations, new 
+grid codes and standards needed to be addressed for appropriate system planning 
+and operation of the power system to supply a reliable and good quality electric 
+power. 
+7. ACKNOWLEDGEMENTS 
+This work is a part of project ARMOUR supported by the European Union’s 
+Horizon 2020 program under the Marie Skłodowska-Curie grant agreement no. 
+890844. 
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+page_content='singh@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='org  Abstract:  The fast depletion of conventional energy sources due to increased energy demands  and environmental concern has motivated power utilities to integrate more  renewable energy sources (RESs) into their power systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Due to the intermittent  nature and low or non-existent inertial response of these sources, a high penetration  of RESs can lead to various issues in the operation of power systems such as  oscillations in power system’s voltage and frequency, increased harmonic  distortion, failure of protective equipment, overloading of transformers and feeders,  and increased line losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Such problems arise when the hosting capacity (HC),  defined as the maximum RES capacity that can be installed without having any  technical and operational problems, of the network exceeds its limit due to the  increased integration of RESs to the existing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This paper reviews the  progress made in HC assessment and enhancement of electrical networks research  and development since its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Attempts are also made to highlight the current  and future issues involved in HC technology for the development of an affordable,  in-exhaustive, clean and reliable power supply for longer term benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Keywords: Hosting capacity, performance index, renewable energy source, wind  energy, solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='List of Notations and Abbreviations:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='BESS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Battery Energy Storage System  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='CSP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Concentrated Solar Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='C-CG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Column-and-constraint Generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='DG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Distributed Generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='DER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Distributed Energy Resources  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='DSO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Distribution System Operator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='DVR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Dynamic Voltage Restorer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='DSTATCOM Distribution Static Compensator (fix the rest)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='ES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='energy storage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='ED  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='economic dispatch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='ELM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='extreme learning machine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='HC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='hosting capacity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='HCC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='hosting Capacity Coefficient  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='LHC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='locational hosting capacity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='OLTC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='on load tap changer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='photovoltaic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PSI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='power supply industry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PVIS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='photovoltaic inverter system  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PPF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='probabilistic power flow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='prediction interval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='PSO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='particle swarm optimization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='RES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='renewable energy source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='SLHC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='stream-lined hosting capacity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='SCUC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='security-constrained unit commitment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='TSO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='transmission system operator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='UC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='unit commitment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='VaR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='value-at-risk measures  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='WG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='wind generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='WPF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='wind power forecast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  INTRODUCTION  The ever-increasing concerns over global climate change, caused by the excessive  use of fossil fuels to meet the global energy demands, have encouraged intensive  research for green power plants with advanced technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In this context, since  past few years, the use of renewable energy sources, such as wind, tidal, micro-  hydro, biomass, geothermal in general, and solar for generation of electrical energy  has increased tremendously [1]-[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This has become possible, only because of the  power system reforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The reasons behind power sector reforms /deregulation  worldwide have either been regulatory failures, political reforms, high tariffs,  inefficient management, poor efficiency, or global economic crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A significant  feature of such restructuring is to allow for competition among generators and to  create market conditions in the industry, which are considered necessary to reduce  the cost of energy production and distribution, eliminate certain inefficiencies, shed  manpower and increase customer choice [5]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [5] have presented a  detailed review on the progress of the movement to privatize and liberalize the  3  power sector in developing countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have clearly spelled out that a full-  scale power reform program generally consists of four main elements: (i) formation  and approval of a power policy by the government that provides broad guidelines  for the reform program (ii) development of a transparent regulatory framework for  the energy market,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (iii) unbundling of the electric power supply chain to enable the  introduction of competition to improve sector performance in terms of efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  customer response,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' innovation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' and viability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=" and (iv) focus on government's role  on policy formation and execution," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' while divesting the power of state ownership at  least in most of the generation and distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [6], a detailed discussion on  achievement of power sector deregulation in Latin America, problems encountered  in the development of all three sectors of power industry i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=', generation,  transmission and distribution are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' New challenges in deregulation of  electrical sectors are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have stressed that more emphasis  should be given on the regulating conduct rather than industry structure, free access  to international networks, transparent bidding process to award contracts, and the  development of price signals as the base for market development and for the linkage  between different segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Enough flexibility should be provided to the regulators  to make changes as and when required in agreement with the energy sector and  agents of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [7] have thrown light on power sector reforms in  the developing world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In this, authors have a detailed discussion on nine important  points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (i) meaning of power sector reform and its necessity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (ii) spread of power  sector reform in developing world,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (iii) effect of political economy on uptake of  power sector reform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (iv) works undertaken to restructure utilities and improve  governance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (v) contribution made by the private power sector after reform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (vi)  whether countries have established the meaningful regulatory frameworks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (vii)  progress made by the wholesale power markets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (viii) improvement in efficiency  and cost recovery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' and (ix) outcome of power sector reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the last, the authors  have concluded that the link between power sector reforms and final sector  outcomes is much weaker, despite some evidence that private sector participation  has made a positive contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [8], authors have made a comparative study of  the distributed and centralized technique for controlling the distribution network  voltages in terms of the capacity of RES that could be integrated within the existing  networks as well as contrasting them with the reactive power control approach,  using optimal power flow analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is concluded that both distributed and  centralized voltage control methods offer significant gains in absorption capacity,  particularly in rural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is also inferred that the consequent losses increase  substantially, and hence the financial implications of increased losses must be  carefully assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [9] have described a cloud shadow model to recreate  the variable output power of both distributed and large centralized PVs at various  locations on a feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For this purpose, a time series load flow analysis is used using  an actual EPRI test feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The feeder was monitored at all buses, and PV induced  voltage quality was measured, including its impacts on voltage control devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Authors [10] have presented the motivation behind RES/DG integration into the  electrical network and the key issues concerning this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A brief discussion on the main  4  challenges that must be overcome in the integration of DG into the supply systems  is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have also emphasized on new grid code and standards to  be suitable for contribution of large scale RESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  However, there always exists a conflict of interest amongst the RES investors and  distribution system operators (DSOs), as the RES owners are interested in  integration of more amount of RES into the existing electrical networks, while  DSOs are more concerned about the technical problems caused by excessive  penetration level of RESs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' High penetration of RESs, predominantly wind  power can lead to complexity in the operation of power systems due to the  intermittent nature of these sources and frequency stability problems due to the  decoupling of the RESs from AC grid using power converters [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' RESs  typically have low (in case of variable speed wind turbines) or non-existent (in case  of solar power plants) inertial responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Attempts have, however been constantly  made to mitigate the arising issues in RESs integration and to address various  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  This paper deals with a state-of-the-art discussion on holding capacity of the power  network, highlighting the analytical and technical considerations as well as various  issues addressed in the literature towards the practical realization of this technology  for better utilization of RESs, at reduced cost and high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' One hundred and  twenty-two publications [1]-[122] are reviewed and classified in 5 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hosting Capacity Concept  Hosting capacity (HC) is defined as the amount of RESs/DGs that can be integrated  into a given power system network, while keeping its performance within an  acceptable range and without having any change in the existing power system  infrastructure [14]-[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The concept of HC was initially conceived by the computer  engineers to define the capacity of a web server to host many access requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Andre  Even, for the first time, introduced this terminology for electric power applications  to identify the effects of high RES into the power distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The concept  was later refined and clearly defined by Math Bollen and Fainan Hassan as the  maximum capacity of RES that can be integrated into the system within the  acceptable performance indices [11], [15], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Figure 1 shows an example of  a HC limit [15] where a performance index is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' With the increase in RES,  an acceptable deterioration is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' When the amount of new RES generation  increases, the performance index will pass a limit, after which the deterioration is  unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hosting capacity (HC) limit and increase in RES integration [15]  Authors in [14] have presented four methods such as deterministic, stochastic,  optimization-based and stream-lined to calculate HC of the distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It  is shown that each method has its own advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is also  concluded that, if the generation profile is known or forecasted, deterministic  method is found to be most suitable for sizing a single RES at specific location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Whereas stochastic technique can be used for sensitivity study and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In  [15], HC technique is clearly explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This approach uses the existing power  system as a starting point and considers the way, in which distributed generation  changes the performance of the system, without having to reinforce the operational  equipment (additional conductors, exchange of transformer etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For this, a set of  performance parameters is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [16] have presented the three  different methods named deterministic, stochastic and time series for quantifying  the solar PV HC of low voltage distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have also elaborated the  merits and demerits of all the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is inferred that the deterministic  method is fast, and accuracy of this method depend on the model and method used  to calculate the voltage rise, whereas stochastic and time series techniques require  large number of simulations and computational time, which is a serious issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In  [17], three different decentralised voltage control methods, using reactive power by  PV inverters, are compared for their capabilities to limit the voltage rise within a  balanced low voltage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have shown that the static reactive power  supply methods, as per German guidelines for generators connected to the low  voltage distribution network, can increase the absorption capacity of low voltage  network without having any change in the existing power network infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Existing level  Amount of generation  Limit  Hosting  Capacity  Unacceptable deterioration  Acceptable deterioration  Performance index  Improvement  6  Authors in [18] have applied HC approach to a real network in order to calculate  the absorption capacity for integration of new RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In this case study, two limits  overvoltage and overcurrent setting the HC were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The important findings  of these works [11], [14]-[18] are summarized as: For a given distribution network, there is no single value for HC, as it depends  upon the pre-defined limiting factors to calculate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Inclusion of so many limiting parameters makes the HC analysis extremely  complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is difficult to point at a single method for quantifying the HC as the most  suitable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It requires an exhaustive application of the methods to many low-  voltage distribution network and a qualitative comparison of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Deterministic methods use traditional power flow analysis and assume that  inputs are fixed and known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This technique neither considers the varying power  production due to the change in wind speed in case of wind energy and  irradiation in case of solar energy nor varying consumers’ power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Stochastic and time series methods are more suitable because both methods  include variable input parameters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=', uncertainties due to change in wind and  solar power production and customers’ power consumption in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In case of stochastic method, computational burden is more due to the inclusion  of uncertainties in customer consumption, grid and wind and solar power  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Time series method is better than stochastic method as it considers all the time-  dependencies and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This method can be successfully applied for time  varying assessment of hosting capacity and the response of protection /control  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' If the location, size and number of RESs is known, optimization method will  prove to be the best choice, since it can improve overall performance of the  distribution network by reducing the losses and/ or cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The selection of HC calculation method will always depend on the objective of  the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Any HC study needs three major inputs: performance index, a corresponding  limit, and a method to calculate the performance index as a function of new  power production or consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Impact of Wind Energy Integration  RES technologies are not yet economically competitive with conventional thermal  generation, as high-power penetration impacts the power supply industry (PSI)  technically and economically both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' From the technical point of view, PSI faces a  7  variety of problems and challenges such as frequency and voltage regulation, power  quality issues, available transmission and distribution capacities to accommodate  RES plants, monitoring and control, operational practices, ancillary services,  connection interfaces, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Large wind generator (WG) integration will also impact  energy balances and generation mix, electricity markets, and emissions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' [18]-  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Most of the technical challenges are related to volatile nature of wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Wind  power generation fluctuates based on wind speed, which depends on the  geographical location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Since the amount of wind may vary from time to time, the  variable wind power causes other conventional generator to operate in sub-optimal  manner [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Power systems must incorporate, for the first time, a source of high  uncertainty, high volatility and low predictability [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This uncertainty includes  input data, power curve, weather conditions and prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Forecasting  of wind generation is very challenging and extensive research has been done on this  topic [25]-[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [19] have detailed discussion on the impact of high wind  power integration on PSI, technical challenges and solutions required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Apart from  the technical issues, large wind power integration will have severe impact on PSI  economics and affects the other market participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have stressed that a  detailed analysis should be carried out on the important issues like inter-TSOs  energy trading, impact on generation mix, energy cost, energy balance, reliability  and security of supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In conclusion, authors have recommended for the design and  development of new grid codes and market regulations to ensure the security of  supply in terms of system security and generation capacity adequacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [20],  authors have reported that the integration of RES will change the generation mix  against conventional power plants and will also reduce their market share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This  impact may be significant for the new market entrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Due to this, targets for  market opening and large-scale RES integration may conflict each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Therefore,  policies and regulations should be carefully designed to benefit both conventional  and RES sectors to mutually achieve the said goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Works reported in [21] is about  fault ride through, grid voltage control, system monitoring and protection as well as  retrofitting of old units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the same paper, authors have discussed the new  requirements, defined in accordance with the new developments in wind turbine  technologies, which should be utilized in future to meet the grid requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Monitoring and system protection is defined under the aspect of sustainability of the  measures introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [22], authors have developed a new control scheme for  variable speed wind turbine that enables power supply of islanded parts of a  distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Models and strategies for fast control of frequency and voltage  during islanding are also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Based on the simulation results, authors have  concluded that wind turbine can operate as conventional generator units supplying  power to islanded parts of the distribution grid, while maintain the voltage and  frequency close to their normal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is further inferred that energy storage (ES)  is not required for power balance control, as the frequency loop of the control  scheme is very fast, ensuring stable operation in islanded mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [23]  have presented the wind power variability and its impacts on power systems,  covering classification of variability, aggregation effect, and wind power  8  forecasting error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The effects of wind energy on different operational time fames of  conventional power plants, and the cost of balancing requirements to accommodate  high wind power integration levels are also discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have  concluded that though the wind power balancing cost is highly system dependent,  but in general, it increases with the increase in penetration level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The flexibility of  existing dispatchable generation units and the available transmission capacity to  neighbouring areas play an important role in reducing the balancing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors  in [24] have reported a detailed discussion on integrating wind energy into the  electric power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This includes topics like commercial wind generation  technology, development of wind technology, reactive power supply and voltage  control, active power regulation, frequency control, wind forecasting, forecast  accuracy, effect of spatial spread, and power balancing issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is concluded that  the stalemate in transmission development is coming to an end, with the new  transmission planning paradigm being implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Study embodied in [25]  presents the detailed investigation on wind power forecast (WPF) uncertainty in unit  commitment (UC) and economic dispatch (ED) and analyses the impact of different  reserve requirements and UC policies on system operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is demonstrated that  despite the inherent uncertainty and risks, it is possible to adopt a quantified method  to deal accurately with wind power generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors conclude that stochastic  decision methods can be successfully adopted to reduce the costs and risks, in the  presence of large and unavoidable wind power prediction errors, particularly in  power system network dominated by thermal power plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have further  suggested for the use of adaptive reserve requirements, which are the function of  wind power forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Paper [26] presents the impact of wind forecast error statistics  upon unit commitment for a large wind integration test system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors further  report that variance has the most impact and suggest that if skewness is included in  the evaluation of error information, kurtosis should also be included to reduce the  system cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the same work, it is inferred that the interactions of variance,  skewness, and kurtosis changes the utilization and commitment of units, and the  representation of variance, skewness and kurtosis can affect the dependency of  commitment upon flexible units and the way it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Researchers in [27] presents  a state-of-the-art review on probabilistic forecasting of wind power generation and  a brief discussion on three different representations of wind power uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Three different forecasting methods, in terms of uncertainty representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=',  probabilistic forecasts (parametric and non-parametric), risk index forecasts and  space-time scenario forecasts have also been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the last, authors have  concluded that uncertainty forecasting techniques can reduce the economic and  technical risk caused by wind power uncertainty and have recommended that there  is need of more research on wind power uncertainty forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A new hybrid  intelligent algorithm approach to produce prediction intervals (PIs) of wind power  generation, based on the extreme learning machine (ELM) and particle swarm  optimization (PSO) is presented in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Optimal PIs have been obtained without  prior knowledge, statistical inference or distribution assumption of forecasting  errors that are required in most of the traditional probabilistic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is  9  concluded that the proposed hybrid scheme provides a general framework of  probabilistic wind power forecasting with high flexibility for reserve determination  by transmission system operator (TSO) to meet the load, and to economically  operate the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A two-stage stochastic unit commitment model integrated with  value-at-risk measures (VaR), including non-generation resources like demand  response and energy storage systems, to balance between cost and system reliability  due to the fluctuation in variable power generation is presented in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To solve the  VaR-based model and to reduce the computational time, modified Benders’  decomposition algorithm is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Sensitivity analysis is carried out for evaluation  of reliability parameters to reduce the generation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Numerical experimentation  is also carried out to find optimal unit commitment solutions and to compare the  effect of risk of non-generation resources on power generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Li et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' [30] have  proposed a stochastic dynamic model to formulate the spatial and temporal  correlation between the atmospheric and near-surface wind fields of geographically  distributed wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is inferred that the model can provide competitive interval  forecasts with conventional statistical based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [31], a robust optimization-  based analysis is carried out by combining the concentrated solar power (CSP)  plants with wind farms to reduce the overall uncertainty in the joint power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  A new approach, based on nested column-and-constraint generation (C-CG)  method, is developed to solve the multilevel optimization with mixed-integer  recourse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The proposed technique is found to be suitable in identifying  robust yet narrow intervals in a reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is concluded that the  combination of different types of renewable resources is very effective in reducing  the generation uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' al [32] have thrown light on how variant robust  security-constrained unit commitment (SCUC) models in terms of different worst-  case definitions could impact operational security and economics of power systems  under uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In addition, what is the proper robust SCUC model that can  fulfil the specific market operational requirements of independent system operators  /regional transmission owners for effectively operating the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Four different  robust SCUC models are studied by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [33], various techniques for  inertia and frequency control developed for variable speed wind turbine and solar  PV generators are systematically reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors [34] have presented a more  tractable and adaptive distributionally robust unit commitment (DRUC-dW)  formulation using distance-based data aggregation, and an efficient cutting plane  algorithm, which solves the two-stage problem quite efficiently by leveraging the  extremal distributions constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It was found that UC solution, yielded from  DRUC-dW model, achieves a reasonable balance between the robustness and cost  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [35], a non-anticipative robust unit commitment models (NRUCs),  where determining the dispatch policy is delayed until the uncertainty decreases is  developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The proposed NRUC features three decision-making problems  sequentially solved under different degrees of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' At first, the decision-  making problem is formulated as an intractable three-stage robust optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Then, a suboptimal approach is developed, where a constraint is imposed  on the dispatch policy so that the transmission capacity constraint is met irrespective  10  of the dispatch level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The significant findings of the studies may be summarized as  follows [18]-[35]: Wind turbine generators are not equipped with exciter and voltage control and  have no ability to provide high short-circuit currents to reduce the voltage dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Hence, if voltage drops below a certain level, WGs trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The impact of wind power integration on any power network depends on the  penetration level and system flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Increase in the penetration level  increases the impacts perceived by the power network, whereas system having  more flexibility can accommodate more power without perceiving unwanted  effects caused by wind generation units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Impacts of wind power integration are classified as short-term and long-term  impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Short-term impacts deal with operational time scale, such as system  balance issues, which are represented by the requirements and cost caused by  the fluctuating wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Long-term impacts are related to planning for peak  load periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Fluctuating wind power causes the other conventional power units to operate in  sub-optimal manner with reduced efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This problem can be reduced by  accurate wind generation forecasting and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Depending on the penetration level, wind power can increase or decrease  system (transmission and distribution) losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Fluctuation in power output leads to voltage variation in case of fixed-speed  wind turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' However, the variable speed turbines, such as doubly fed  induction generators, can provide reactive power to network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' As for as the impact of wind power integration on distribution network under  normal operation is concerned, there may appear slow voltage variations due to  change in power flows and short duration voltage dips during switching  (on/off), fast voltage variations due to changes in wind speed that may cause  ‘flickering’ effects, and voltage distortion due to harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' During network faults, integration of WG can lead to: increased stress of circuit  breakers since the short circuits are additionally fed by the wind generators,  malfunctioning of protection equipment, since it is designed to operate within  strictly ‘radial structure’ of the MV network, and islanding of parts of the grid,  fed solely by wind generators, which may cause failure of the end-use  equipment and also accidents to the utility ‘s personnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is more possible in  the presence of high capacitance equipment such as cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' High penetration of wind power needs expansion of more complex transmission  networks resulting in more transmission losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The long distance between the wind forms and load requires long transmission  lines to be laid down and can lead to transmission congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  11 With the increase in wind power integration, the amount of reserve needs to be  increased over long period of times to maintain the system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' High integration of wind power leads to frequency deviation from the normal  range, because maintaining the balance between supply and load demand in  case of high volatility of supply is challenging and more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The increase in penetration level of wind power results in an equivalent decrease  in conventional generation units, and thus system rotational inertia becomes  very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This can lead to serious effects on the frequency deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Impact of Solar Energy Integration  Solar power is the conversion of sunlight into electricity, either directly using  photovoltaic (PV) or indirectly through concentrated solar power (CSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Solar  power in its various forms such as solar photovoltaic, solar heat, solar thermal  electricity and solar fuels offer a clean, climate-friendly, very abundant, and in-  exhaustive energy source to mankind [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Solar power generation is one of the  most advanced technologies for renewable energy production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Solar energy has  delivered more new capacities than nuclear and fossil fuels [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Worldwide growth  of solar power is extremely dynamic and varies strongly by country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' By the end of  2019, a cumulative amount of 629 GW of solar power was installed throughout the  world [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' By early 2020, the leading country for solar power was China with 208  GW [39, 40], accounting for one-third of global installed solar capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' As of 2020,  there are at least 37 countries around the world with a cumulative PV capacity of  more than one GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The countries in the list of top installers of 2016 through 2019  were China, United States, and India [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The top 10 countries by added solar  PV capacity in 2019 [43] and Top 10 countries by cumulative solar PV capacity in  2019 is shown [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The solar PV capacity by country and territory (MW) and share  of total electricity consumption is given [45]-[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Solar-grid integration is another important technology as it optimizes the energy  balance leading to improvement in economics of PV system, reduction in operating  cost, and provides added value to utility and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Solar-grid integration  technology consists of advanced inverter technology, grid-plant protection  technology, anti-islanding technology, smart grid technology and solar-grid  forecasting technology [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [49] have presented a technical report  on ‘Treatment of solar generation in electric utility resource planning’, jointly  prepared by the National Renewable Energy Laboratory (NREL) and Solar Electric  Power Association (SEPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A detailed study was conducted on inclusion of solar  in long-term resource planning processes through interviews and a questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Table 1 includes the benefits and challenges of solar, based on utility interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  With these benefits and challenges in mind, utilities can more accurately incorporate  solar generation into their long-term planning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Some of the leading,  utility-identified best-practices and analysis that needs attention are: (i) analyse and  12  assign appropriate capacity values to solar resources, (ii) analyse solar individually,  to get more accurate aggregate results, (iii) improve modelling assumptions and  methods, (iv) pursue sub-hourly sensitivities, (v) evaluate whether to treat  distributed generation as a resource, and (vi) utility-identified analysis needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In  [50], a detailed study on the effect and challenges of integration, the current solar-  grid integration technology, its benefit, solar system characteristic for integration,  and issues and compatibility of both the systems have been carried out by Nwaigwe  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors further conclude that solar power integration can reduce the  transmission and distribution losses, increase grid resilience, lower generation cost,  and reduce requirements to invest in new utility generation capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Table 1 Benefits and challenges of solar power integration [49]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Benefits of solar power integration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Challenges  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='solar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='integration Meet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='renewable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='standard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='requirements Fuel diversification Cost stability Geographic dispersal benefits  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='and modularity Partial correlation with peak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='demand Mitigation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='environmental  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='compliance risks Avoid line losses (typically DG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='only) Variable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='uncertain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='output Ramping issues Economics Lack of current capacity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='need Cross-subsidization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='concerns (DG) Reduced capacity benefit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='over time with increasing solar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='penetration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='Photovoltaic (PV) technology has presently become a significant form of power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='generation on many power system networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Impact of high PV integration has  drawn attention towards the issues of grid management, operation and planning,  particularly where there is variability in PV system output due to cloud cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Variability in PV irradiance is considered as a major challenge to high levels of PV  penetration into the existing power networks, and this variability in PV generation  can have a negative impact on the local electricity network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Network level problems  occur, where intermittent changes in PV generation on a power network are unable  to be accommodated by the base load generation over the time frame of the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The potential impacts of intermittent generation occur at different time scales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' and  the associated considerations include [51]: (i) rapid changes in network demand or  13  in PV generation can lead to power quality issues such voltage flicker and harmonic  distortion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (ii) the spinning reserve of system generation is required to have  sufficient total capacity and ramping capability to meet the short term changes in  network demand or in supporting PV generation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' otherwise it may  lead to power  quality and system outage issues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' and (iii) generation planning requires sufficient  generation to be available at any time to fulfil the projected network demand and  sufficient operating reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The integration of distributed generation (DG), particularly solar power can  significantly impact the flow of power and voltage conditions at customers and  utility equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Depending on the nature of DG and operating characteristics of  distribution system, these impacts may be either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The positive  impacts [52] are: Voltage support and improved power quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Loss reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Improved utility system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Transmission and distribution capacity release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Deferments of new or upgraded transmission and distribution infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Reduction of emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The above listed benefits are, in practice, much difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The DG sources  must be reliable, dispatchable, of the proper size and installed at the proper locations  and must also meet various other operating criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Since many DGs will not be  utility owned or due to intermittent nature of solar and wind power, it is difficult to  satisfy all these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In fact, power system operations may be adversely  affected by the integration of DG, if certain minimum standards for control,  installation and placement are not maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The major negative impacts are:  voltage variation and unbalance, current and voltage harmonics, stress on  transformer, grid-islanding and power quality issues [53], [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The severity of these  impacts depends on the penetration level of PV, its location and configuration of  distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The negative impacts can be summarized [55] as: Voltage fluctuation in feeder, resulting in voltage rise or fall and unbalanced  voltage Malfunctioning of on-load tap changer, line voltage regulators and capacitor  banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Possibility of overload in distribution feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Variation of reactive power flow caused by malfunctioning of capacitor bank  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  14 Malfunctioning of overcurrent and overvoltage devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Islanding operation and detection in case of grid disconnection Reliability and security of the distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The electric grid -an interconnected power network shown in Figure 2 maintains an  instantaneous balance between the supply and demand (generation and load), while  transferring electricity from generation source to customer [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' If the power  produced by the DER (solar PV) is more than power consumed by the load, voltage  level of that load bus increases, and can cause damage to various equipment  connected to that feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' On the other hand, there will be drop in voltage, if power  demand by the load is more than the power produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [57], authors have  presented a detailed study on voltage fluctuation in power networks with  photovoltaic energy sources, and a flickermeter model has been used for the  evaluation of flicker assessment under climate change (sunny and cloudy  situations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the last, authors conclude that the flickermeter model fulfils the  requirements defined in IEC 61000-4-15 standard, and it has been tested under  additional tests defined in CIGRE/CIRED/UIE test protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have also  recommended the use of flickermeter for evaluating the flicker severity and voltage  fluctuations produced by the photovoltaic energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The voltage unbalance is  a major power quality problem in low voltage distribution networks due to the  random location and rating of single-phase rooftop photovoltaic cells (PV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [58],  Wang et al have presented a detailed study on voltage issues at the point of common  coupling caused by the integration of photovoltaic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For this purpose, a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='2  kW grid connected PV system on a radial LV distribution network has been set up,  and the probability density of voltage rise, and voltage unbalanced factors were  derived from the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Short-term and long-term voltage flicker indexes were  also calculated to evaluate the severity of flicker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is stated that the intermittent PV  power output can introduce a maximum short-term flicker of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='0 and long-term  flickers of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is also shown that 1% of the voltage unbalance can produce 6-10  times current unbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To improve voltage unbalance, authors in [59] have  proposed a system with converter topology and control algorithm, based on the  application of series dynamic voltage restorer (DVR) and parallel distribution static  compensator (DSTATCOM) custom power devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A state feedback control, based  on pole-shift technique, has been developed to regulate the DSTATCOM and DVR  converters output voltage for voltage balancing in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Integration of PV systems in the distribution network, harmonic distortion of  voltage and current waveform takes place, resulting in poor power quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' PV  inverters are the main source of harmonic injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The maximum penetration level  of grid connected identical photovoltaic inverter system (PVIS) that can be installed  is based on the acceptable voltage distortion levels within the distribution network  as determined in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To mitigate harmonics in the network, Inductor-Capacitor-  Inductor (LCL) filters with robust control techniques are currently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For new  15  interface converters, their EMI/EMC related impact should be considered since its  early stage of design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is not enough to mitigate the problem in final design, but to  go to the root of the problem so that harmful emission does not occur [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' New  more complex converter topologies and modulation schemes are being studied to  solve the power quality issues, such as harmonics, voltage imbalances etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' [62],  [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Structure of an AC microgrid [56], showing multiple generation units and  load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To maintain a demand-supply balance and reliable operation in such a hosting  capacity analysis is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The major findings of the studies may be summarized as follows [36]-[63]: Some renewable energy technologies provide power only when the resource is  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' These resources are often contracted as ‘must-take’ generators, where  their output is always used, when it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' However, it is difficult to  integrate a large amount of ‘must-take’ generation into the grid, because its  availability is uncertain and constantly changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Photovoltaics (PV) may be centrally located in large plants or distributed on  rooftops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distributed PV has benefits, such as low land use and no transmission  needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distributed and central PV both are usually ‘must-take’ generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Storing large amounts of power is difficult, while storing thermal energy is  relatively easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Since, concentrated solar power (CSP) plants collect and  convert thermal energy into electricity, they can collect and store thermal energy  for later conversion into electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' CSP plants with thermal energy storage  provide assurance that the generator will be available, when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' These CSP  plants are dispatchable and can meet the intermediate and baseload demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  PCC  Solar PV  Energy storage  Small hydro  Converter  synchronous DG  System  system  AC  DC7  Relay  AC  AC  S3  RDG-2  Static Switch  个口  S2  DC  RDG-1  R  AC  F2  S1  Distribution  RDG-5  R1  transformer  口  7  R8  R4  R2  AC  RDG-6  R12  Load2  RB  I RDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='3  Flywheel  Load1  AC  Utility grid  R6  c  Fuel cell  Distribution  S4  F3  个□R11  F1  R5  R7  R10  RpcC  R9  RDG-4  S  PCC  DC  R13  S6  grid  Wind Farm  R14  S7  Load3  RS  Ry  F4  0  Load4  R15  R16  Load5  Microgrid  16 Although PV deployment depends on various integration issues, most CSP  plants respond more slowly to changing weather, particularly equipped with  thermal energy storage system, and output from these plants is easier to forecast  and integrate into the electric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' PV generation without energy storage system does not provide all the  characteristics required for stable grid operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Careful integration of distributed generation and deployment of utility-scale  generation will be needed to provide the mix of power and reliability required  for a clean and healthy power supply as renewables contribute an increasingly  larger share of energy needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Static synchronous compensator (STATCOM) in conjunction with energy  storage system is likely to play an effective role in mitigating the voltage issues  without curtailing any renewable energy and can provide active power support  during the grid contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distribution system designs and operating practices are normally based on  radial power flows, and this creates a special challenge to the successful  introduction of distributed generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Some important issues such as voltage  regulation and losses, voltage flicker, harmonics etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' must be considered to  ensure that DG will not degrade the distribution system power quality, safety or  reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The fault contribution from a single small DG unit is not large, however the  aggregate contributions of many small units, or a few large units, can alter the  short circuit levels enough to cause fuse-breaker miscoordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distributed generation must be applied with a transformer configuration and  grounding arrangement compatible with the utility system to which it is to be  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Otherwise, voltage swells and over voltages may be imposed on the  utility system that may damage the utility or customer equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Islanding can occur only if the generator(s) can self-excite and sustain the load  in the islanded section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In most cases, it is not desirable for a DG to island with  any part of the utility system, because this can lead to safety and power quality  problems that will affect the utility system and load both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The implementation of DG can increase the reliability of electric service if units  are configured to provide ‘backup- islands’ during upstream utility source  outages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To be effective, this requires reliable DG units and careful coordination  of utility sectionalizing and protection equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' To avoid costly impact studies for all but those applications that actually need  them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' proposed DG can be screened based on factors such as: utility system  fault levels at the point of DG interconnection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' size of DG unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' its intended  mode of operation and expected output fluctuations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' aggregate capacity of DG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  17  secondary configuration of the DG site (including presence of adjacent  customers),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' feeder voltage regulation practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' transformer and grounding  compatibility of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' size of generation relative to load at the  interconnection point and type of interfacing power converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The main cause of new solar panel failure is poor design and defects during  manufacturing, contact defects in junction boxes, glass breakage, burst frames,  breakage of cell interconnections and problems with the diode associated with  a higher rate of cell degradations and interconnectors, fuse boxes, charge  controllers and cabling as well as issues with grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The variance created by the installation of a further dispersed PV inputs into  power grids can end up being very similar to step-change ‘noise’ variance,  which currently occurs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Analysis of the size, number, and spatial diversity to optimise PV input into the  grid should be undertaken with a view to determining the marginal benefit of  additional diversity and/or the extent to which the benefits of diversity diminish,  if the separation of systems gets too great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Under-generation and over-generation both by solar PV cause instability on the  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Effective solution to this involves use of better forecasting tools for more  accurate prediction of when solar generation might decline to minimum  penetration capacity, installing solar panels across a large geographic area to  minimize any impact of generation variability due to local cloud cover, shifting  power supply and storing excess energy for later use and encouraging customers  to use power, when it is more readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Prior to deployment of solar panels, advanced integration technologies should  be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Optimized forecasting is essential for proper system stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Also, due to economic viability and robustness of the system, solar technology  can be treated as a major guideline for sustainable development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Integrating CSP plants with wind power can reduce the generation uncertainty  and improve the capacity factor of the combined plant, because of the negative  correlation of wind and solar power, and the availability of high-efficiency  thermal energy storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Assessment of Hosting Capacity (HC)  The HC concept has been most widely used to evaluate the benefits of different  voltage regulation techniques, amount of solar and wind energy that a grid can  accommodate without violating the acceptable performance indices of the existing  power network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Various power system phenomena and related performance indices  are given in [64], and the same is reproduced in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distributed generation  (DG) such as solar and wind will impact the performance of the grid, and this sets  a limit to the amount of such RES that can be integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' New communication  18  schemes together with energy storage system and grid control strategies allow  integration of increased amounts of renewables into existing power networks,  without unacceptable effects on users and grid performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The first work on the  hosting capacity of electrical grids was reported by Math Bollen and Hager in 2004  [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Later, many methods have been applied to examine the capacity of existing  distribution grids to accept DG [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' However, some important differences do exist  between methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Most of the statistical approaches proposed in the literature aim  at defining the optimal DG location and sizing [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is very essential to define  suitable performance indices before calculating the amount of DG that can be  integrated into the existing network, as the hosting capacity is based on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The  choice of performance indices and limit have a big influence on the amount of  distributed generation that can be accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The hosting capacity is a tool that  provides trustworthy and secure platform for fair and open discussion between the  different stakeholders (network operators, owner of DG, owner of the existing large  power plants and other customers) and a transparent balancing between their  interests, for example, acceptable reliability and voltage quality for all customers,  no unreasonable barriers against new generation and acceptable costs for the  network operator [68], [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The integration of DG should not result in a  deterioration of the supply reliability, but certain deterioration in quality of the  supply should be acceptable for most customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Reliability is quantified by several  indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The steps to be followed to calculate HC is summarized below [15]:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Choose a phenomenon and one or more performance indices,  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Determine a suitable limit or limits,  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Calculate the performance indices as a function of the amount of generation,  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Obtain the hosting capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Table 2  Power system phenomena and related performance indices [64]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Phenomena  Performance Indices  1  Overloading from wind power  Maximum hourly value of current  through transformer  2  Frequency variation  99% interval of 3 s average of  frequency  3  Overvoltage from roof top solar  photovoltaic cells  Highest 10 min average of voltage  4  Undervoltage from fast charging of  electric vehicles  Lowest 10 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' average of voltage  19  5  Protection mal-trip  Lowest recorded current causing  interruption  6  Harmonics  10 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' average of voltage and  currents  In [70], authors have presented the hosting capacity approach and explained some  important developments, such as uncertainty in location and size of production  units, and curtailment to connect more production than according to the initial  hosting capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have given the new HC terminology as:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  HC uncertainty,  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  HC Coefficient,  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Stochastic HC (SHC),  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Locational HC (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Uncertainty in the estimation of HC may occur due to volatile and intermittent  nature of DG output powers, unknown rating of DG units and their locations, load  variability and insufficient data, when performing power system calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hence,  HC will never be a single value, but multiple values will be obtained depending on  the uncertainty percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Since randomness is involved in the calculation of HC,  it cannot be termed as deterministic approach, instead it should be viewed as a  probabilistic approach, where level of accuracy and uncertainty are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Basically, HC is location-based concept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' the maximum HC occurs at the maximum  loading conditions and minimum generation, whereas minimum HC is obtained at  minimum loading conditions and maximum generation [11], [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Curtailment of power production is defined as reducing the active power output  from certain energy resources at times to increase the HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The ability to curtail the  production will allow for a larger installed capacity of distributed generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The  increase in installed capacity for a given percentage of permitted overloading varies  with the type of energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The gain in hosting capacity will depend on  characteristics of the network and performance index being considered [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This  approach can be extended to cover different voltage levels and hosting capacity  limits determined by very different performance indices (and therefore different  power system phenomena).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For an objective comparison, the Hosting Capacity  Coefficient (HCC) is defined as the ratio between the curtailed energy and the  installed capacity above the initial HC as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the same work, it is  inferred that a low HCC corresponds to an unfavourable location for curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A  high value of HCC corresponds to a high ability to accommodate DG using  curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' However, it is possible that a high coefficient corresponds to a low HC  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  20  HCC =  Curtailed Energy  Installed Capacity−Initial HC  (1)  Integration of DG units into the power network has many unknown variables, such  as size and location of DG, number of customers who will utilize DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The  consumption profiles and output of DG are also intermittent in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' These  unknown variables have an impact on the HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Considering the randomicity of some  of these unknown variables, Probabilistic Power Flow (PPF) analysis is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  In PPF, several load flow calculations are performed for thousands of random cases  of number, location and/or size of DG units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Networks’ variables such as voltage,  current, losses etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' are recorded against the performance limits for the determination  of HC [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The stochastic HC estimation have been reported by many authors [74]-  [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [74], authors have proposed a stochastic multi-objective optimization model  to maximize HC of the distribution network for wind power and minimize the  energy procurement costs in a wind integrated power system, while considering  technical and economic aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Cost of the purchased energy from upstream  network, and operation and maintenance cost of wind farms are taken as objective  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Analysis was carried out on a standard radial 69 bus distribution feeder  and a practical 152 bus distribution system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  A detailed report on feeder modelling, analysis, and evaluation of issues to assess  the impact of integration of distributed solar PV and to determine the hosting  capacity prepared by EPRI is given in [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' An EPRI Technical Update report  on developing a screening methodology, which effectively evaluates the new  interconnection requests, while considering PV and feeder-specific factors is  presented in [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This methodology has considered the peak load levels along with  other critical factors, including PV location, aggregate PV effects, and most  importantly specific feeder characteristics such as voltage class, voltage regulation  schemes, and operating criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [78] have used a stochastic approach to  simulate various DG deployment scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Various limits such as over-voltages,  voltage deviations and voltage unbalance are considered, while calculating HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Le  Baut et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' al [ref] have presented a three-step probabilistic HC assessment technique  in [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [80], a high-resolution approach is presented to assess the DG and storage  systems connection into the distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors have used a stochastic  demand model for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Since HC is highly location based, integration of new DG can be accepted at some  locations, but not at other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The voltage profile of the feeders is one of the key  criteria in defining the locational HC (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Rylander et al [81] presented the  concept of stream-lined methodology to calculate HC termed as stream-lined HC  (SLHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This method was developed by EPRI to calculate the HC of feeder  considering size and location of DER, integration technology and physical  characteristic of the feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Application of the results include improved screening  tools, visualization of HC constrained feeders, and identification of problematic  locations within a feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' High penetration of DER/DG into the electrical networks  21  adversely affects various performance indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The primary issues used to identify  hosting capacity include overvoltage, voltage deviation, element fault current,  breaker reduction of reach, sympathetic breaker tripping, breaker/fuse coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Anti-islanding and large-scale PV protection issues with grounded wye-delta  interconnect transformer have significant impact on HC, whereas voltage  imbalance, overloads and harmonics have low impact on hosting capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Determination of hosting capacity of distribution network can be broadly  categorised as utility-based and customer-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In utility-based, problem can be  defined as an optimization problem with an objective to maximize the integration  of DERs without technical violation in the distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In customer-based,  stochastic methods are mostly used for assessment of HC, since in this case, utilities  have no control over the number, locations, and size of DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' There are four  techniques used of the estimation of HC: (i) deterministic, (ii) stochastic, (iii)  optimization-based, and (iv) stream-lined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Though, all these methods follow the  same general procedure, but their implementation is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In all the four  methodologies, power flow analysis is carried out to find voltages and currents in  the distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For power system simulation and analysis, a wide range  of commercial and non-commercial software tools are also available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' for examples,  PSS Sincal Integrated Capacity Analysis Module [106], DigSILENT PowerFactory  [107], NEPLAN [108], Synergi Electric [109], and CYME [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A list of these  software programs is given in [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A brief and precise discussion on the  methodology used for the assessment of HC and software tools is provided in [14]  and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A summary of the studies made by various researcher [82]-  [105] is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Table 3  Studies that utilize various performance indices for assessment of hosting capacity  Author/  Reference  Year  DER  Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Performance  Indices  Objective  and  Technique used  Remarks  Sakar et al,  2017  [82]  PV  Overvoltage, under  voltage,  current  capacity of power  lines, and harmonic  distortion  of  the  system  HC estimation of  a  distorted  distribution  system  with  Photovoltaic  (PV)-based DG  units,  using  optimization-  based technique  Lower  HC  with  higher  distortion in  distribution  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content="  22  Sakar et al,  2018  [83]  PV  Bus voltage limits,  line ampacities, and  harmonic distortion  limits  To find system's  HC  and  parameters  of  the  proposed  filter,  by  an  optimization  algorithm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  HC decreases  with  the  increase  in  utility side’s  background  voltage  distortion and  load  side’s  nonlinearity  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Braga et al,  2018  [84]  PV  Harmonic  distortion,  and  location  To  find  harmonic  hosting capacity  of the IEEE 13  bus network at  four  locations  through  sensitivity  analysis  on  Matlab  and  OpenDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Presence  of  capacitors  lead to higher  value  of  voltage total  harmonic  distortion  THDv due to  the  parallel  and  series  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Mirbagheri  et al, 2018  [85]  DG  Steady-state voltage  variations,  rapid  voltage  changes,  and thermal limits  of  lines  and  transformers  To  determine  HC  of  distribution  grids  even  in  case  of  uncertainties in  grid parameters  or lack of data,  by using a novel  method  called  ‘Bricks  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The detailed  topology  of  the grid and  power profile  for  all  the  nodes are not  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Method  is  fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Chathurang  i et al, 2018  [86]  PV  Voltage limits and  network loss  To find technical  impacts caused  by  higher  penetration  of  roof top solar  PVs  on  the  operating  performance of  LV distribution  Further  connections  of solar will  cause  violation  of  acceptable  voltage limits  and increase  network  23  networks,  by  modelling  the  detailed network  in DIgSILENT  PowerFactory  simulation  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  losses due to  the  reverse  power flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Network  already  has  40%  solar  penetration  level (based  on  transformer  capacity)  Faishal  Fuad et al,  2018  [87]  PV  Voltage violation,  current  violation  and  power  flow  direction  To find HC of  the distribution  network  under  different loading  conditions, by an  iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Study reveals  that  voltage  violation  occurs  only  when the load  is less than  48% of the  average  value, while  reverse  power  flow  occurs  in  most of the  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Al-Saadi et  al,  2018  [88]  PV  Overvoltage  and  over loading  Assessment  of  HC  by  conducting  stochastic  simulations,  in  which  the  uncertainties of  the  solar  irradiance  and  load  variation  are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Hourly  assessment  will help to  serve  the  automation  strategies,  where  corrective  actions can be  taken for HC  maximization  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Inclusion of  more  performance  index, such as  frequency  variation  index,  may  24  further  improve the  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Peppanen  et al, 2018  [89]  DER  Thermal  limit,  steady-state voltage  and  voltage  deviation  To find HC of  three real North  American  residential  distribution  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Load  variability  and  unbalance  are  also analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Results  showed that  the  most  significant  limiting  factor  is  voltage  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Alturki  et  al,  2018  [90]  DG  Overvoltage  and  overloading  To estimate HC  in  distribution  grids, by using  an optimization-  based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Low  computationa  l  time  as  compared to  traditional  methods,  while  offering  comparable  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Abad et al,  2018  [91]  DG  Voltage deviation,  load  growth,  network  structure  and type of DG  Probabilistic  model based on a  two-step  linearization  algorithm  is  presented  to  linearize the HC  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  HC can be  approximated  by  a  Gaussian-  shape  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Al-Saffar et  al,  2019  [92]  PV  Overvoltage  HC of three real  circuits  in  Alberta, Canada  are  evaluated  using  Monte  Carlo simulation  based  probabilistic  power  flow  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The use of  stochastic  approach can  overcome the  shortcomings  of  deterministic  methods, and  can,  in  principle,  handle  circuits  of  25  any  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Duwadi et  al,  2019  [93]  PV  Overvoltage,  undervoltage,  and  unbalance between  the phases  To find HC of a  feeder  using  Monte  Carlo  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  A  parallel  algorithm  was  developed using  Message Passing  Interface  to  decrease  the  simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  PV  penetration  was found to  be limited to  10%, on days  when  solar  irradiance  was high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' PV  penetration  may go upto  50% on days  with  low  energy  demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Steyn et al,  2019  [94]  Roof-  top  PV  Voltage  increase  and  equipment  overload  To find impact  on integration of  distributed  rooftop  PV  systems on three  distribution  networks  (residential,  commercial, and  industrial)  in  Cape  Town,  South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Integration of  PV  systems  decreases the  line  and  transformer  loading, but  after a certain  level,  the  effect  of  integration  start  to  reverse due to  increase  in  PV  penetration  level  and  reverse  power flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Lillebo  et  al,  2019  [95]  EV  Overloading  To explore the  impacts  of  increasing  EV  penetration level  in a Norwegian  distribution grid,  by  using  real  power  measurements  Applying  a  fast  charger  in the grid  with standard  power factor  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='98 lag  caused  significant  voltage  26  obtained  from  household smart  meters in load  flow analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  deviations at  several  locations, the  worst  of  which  reached close  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='03 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='u  Soukaina et  al,  2019  [96]  DG  Overvoltage,  line  overloading  and  transformer  overloading  A new method to  calculate HC of  DG, considering  the variety of  distribution line  configurations  and  cross  admittance,  is  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The  proposed  method was also  applied  to  an  underground  medium voltage  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Simulation  was carried out  using  Matlab  and Etap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Based  on  obtained  results, it is  shown  that  including line  capacitances  in calculation  increases the  maximum  power of DG  unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Ismael  et  al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  2019  [97]  PV  Bus voltage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' line  thermal  capacity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  power factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' and  individual and total  harmonic distortion  To  find  probabilistic  hosting capacity  (PHC) due to  high penetration  of PV units in a  non-sinusoidal  power  distribution  network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  by  using  Monte  Carlo  simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  considering  various  uncertain  parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' such  as  intermittent  Results  shows  that  deterministic  hosting  capacity  (DHC)  studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  which ignore  the  uncertainty of  electrical  parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  result  in  optimistic  results  that  cause  a  noticeable  underestimati  27  output power of  DGs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  background  voltage  harmonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' load  alteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  and  filter  parameters’  variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Hybrid PSO and  gravitational  search algorithm  (PSOGSA) has  been used for  optimal  design  of the proposed  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  on to the HC  levels that are  achieved  from  the  probabilistic  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Abideen et  al,  2019,  [98]  PV  Bus  overvoltage  and  distribution  network losses  A new algorithm  based on gradual  increase in the  PV penetration  level  is  introduced  to  estimate  the  maximum  PV  penetration level  in  LV  distribution  networks using  Matlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  This  technique can  be applied to  any  distribution  network,  if  the  power  demand and  the  solar  irradiation  data  are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Li  et  al,  2019  [99]  DER  A  valuation  scheme  to  quantify  the  value of DER in  active  distribution  network (AND)  is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A  two-tier scheme  is proposed to  value  and  compensate  DER  portfolio  The proposed  scheme will  not  only  encourage the  proactive  investment of  DERs  in  AND,  but  also  help  enhance  the  role of DERs  in  offering  reliable,  28  proposed  by  customers  and  independent  third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  resilient,  affordable  and  sustainable  power supply  to customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Essackjee  et al, 2019  [100]  PV  System harmonics  A detailed study  on  maximum  rooftop PV HC  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The  level  of  harmonic  pollution  was  observed  with  increasing  penetration level  of photovoltaic  units and using  different sizes of  such units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The findings  can be used  by  an  electrical  utility to limit  the number of  rooftops  photovoltaic  being  connected so  as  not  to  degrade  the  quality  of  supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Sadeghian  et al, 2020  [101]  PV  Voltage  violation  and reverse power  flow  An  impact-  assessment  framework,  based on Monte  Carlo technique,  to  assess  the  impacts of two  different types of  distributed solar  photovoltaic  (DPV)  installation  (customer based  and  utility  based)  on  a  realistic  distribution  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  To  achieve  higher  penetration  ratios without  reverse  power  flow  and  other  negative  impacts,  utility-aided  installation is  necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Mulenga  et  al,  2021  [102]  PV  Overvoltage  A  stochastic  algorithm  for  estimation of HC  of  low-voltage  network for solar  It is shown  that  performance  indices used  in  29  PV  is  introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Two  types  of  uncertainties  aleatory  and  epistemic  are  distinctively  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  deterministic  and  time-  series  HC  studies need  to  be  complemente  d  with  a  planning risk  percentile for  stochastic  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Du et al,  2021  [103]  PV-  Wind-  Load  Voltage violation  A  stochastic  framework  for  HC  based  on  wind-PV-load  temporal  characteristic is  presented from a  probabilistic  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  A  discretization-  aggregation  method  is  introduced  to  generate  filter  extreme  combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Study  helps  in  making  better use of  available  renewable  resources and  promoting  the  application of  distributed  hybrid power  generation in  the  power  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Paudyal et  al,  2021  [104]  EV  Voltage limit,  thermal limit  Study on EV HC  by focusing on  extreme  fast  charging (xFC)  of EV charging  loads is carried  out to identify  representative  feeders of the  utility  distribution  network  in  a  certain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Study  can  help utilities  plan  for  optimal  system  upgrades  to  facilitate  future needs  and  greatly  reduce  the  cost of EV  integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Mulenga et  al,  2021  [105]  EV  Undervoltage,  transformer  and  feeder overload  A  stochastic  methodology to  single-phase and  The method  can  be  applied as an  30  three-phase EV  charge  hosting  capacity  for  distribution  networks  is  developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The  method includes  two  types  of  uncertainties,  aleatory  and  epistemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  extension to  find HC as a  function  of  time of day,  week, or year,  and the best  periods  for  charging can  be identified  and used in  designing  smart  charging  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Enhancing hosting capacity of the network is considered as one of the important  goals for distribution system operator (DSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The technical solutions to enhance  systems’ HC have been grouped in three categories, such as DSO solutions,  prosumer solution, and interactive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The effectiveness of these solutions  depends on various factors, such as technology readiness, investment cost, and  impact on congestion and compliance with the applied grid codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The various  methods used are: reactive power control [112], [113], voltage control using on load  tap changer (OLTC) [114], [115], [116], active power curtailment [117], [118],  energy storage technologies [119], [120], network reconfiguration and  reinforcement [121], and harmonic mitigation techniques [83], [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' A brief study  on the various techniques used for HC enhancement is presented in [11] and  references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [112], [113] have reported that using reactive power  control and energy storage system, DG penetration can be increased, and the system  losses can be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is concluded that the model can be successfully utilized  in smart grids, where integration of large-scale DG sources is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Navarro et  al [114] carried out a techno-economic study on the use of OLTC to mitigate  overvoltage problem in UK caused by high PV penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors examined the  local and remote voltage control methods and compared it with conventional  reinforcement solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Rauma et al [115, 116] have studied the advanced control  of OLTC transformers and its application in increasing HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors developed an  analytical model for a set of 631 real LV electrical systems in France, based on  actual voltage measurements recorded by advanced metering tools, and concluded  that the use of OLTC can enhance the HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In [117], [118], authors presented a study  on the role of active power curtailment and dynamic line rating in enhancing HC  for LV and MV distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The authors have categorized the active  power curtailment into soft and hard curtailments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Poulios in [119] studied the  optimal size, location and economic aspects of battery energy storage systems  (BESSs) for increasing the system HC and have stressed that the prices of BESS  31  need to be greatly reduced in order to make it competitive to other available HC  enhancement solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors in [120] proposed a cost-based multi-objective  optimization tool built on Matlab platform to evaluate the BESS capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors  examined the role of BESS for voltage regulation, network loss reduction and peak  load reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Ismael et al [121] have examined the problem of selecting the  optimal conductor for a real radial distribution network in Egypt and presented a  novel feeder enforcement index to assist network planners and DSOs to recognize  the feeders that need to be reinforced first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Bollen et al in [122] have discussed the  HC assessment considering over and under voltage performance limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Authors  have also studied the impact of inter-harmonics and super-harmonics on HC  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  The key findings of the studies are summarized as follows [64]-[122]: Higher the percentage of time during which curtailment is acceptable, higher is  the amount of production capacity that can be integrated to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  However, even though the hosting capacity is increased, the proportion of  additional energy that must be curtailed increases quickly making additional  increase in hosting capacity less and less attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hosting capacities for each feeder in the residential and commercial PV  scenario and utility-scale PV scenario are different due to the possible PV  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the utility-scale analysis, the deployed PV could be located close  to start-of-circuit at the substation or in feeder extremities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In the residential and  commercial deployed PV scenario analysis, the PV location depends to a greater  degree upon the customer location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The process of developing the baseline model and methodology for running the  simulation, includes hundreds of different decisions, which can significantly  shape the outcome, both in terms of final hosting capacity figure, and its  accuracy, when compared to real life conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The streamlined methodology was significantly faster to run (from a computing  standpoint), however it had accuracy issues (both over and underestimating  hosting capacity) in a not-insignificant number of cases, particularly when it  came to application on complex circuits and with respect to two of the four  power system criteria that were evaluated: power quality/voltage and  protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The iterative methodology results were found to be sufficiently accurate, but  running the model was computationally intense, and thus would require more  resources to deploy and may not be able to be run as often as needed for the  type of scenario analysis that may be used for planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Based on the method for PV deployment, voltage imbalance typically  decreases, and overloads seldom occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This is strongly influenced by limiting  the PV size to the customer peak load in the small-scale analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For balance  32  large-scale PV, voltage imbalance seldom occurs, however the PV systems do  have a slight potential to cause overloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Deterministic method uses few input parameters that are readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It is  fast and easily implementable and presents a quick overview of grid  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It assumes fixed value of parameters and does not consider  intermittent nature of RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It does not consider uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' HC obtained by  deterministic method is an estimate of the worst-case scenario and not the true  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The impact, in this method, is overestimated and the hosting capacity  underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Stochastic technique considers uncertainties in the network and presents a  realistic overview of the grid performance under renewable energy sources  integration based on probability distribution functions or possibility theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  This method simulates realistic network scenarios and is less time consuming  than time-series method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It accommodates all probabilistic distribution  functions and is comparatively easy to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Stochastic method needs large computational time with the increase in  uncertainties considered in large distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' It does not assess the  time-related operation of control elements and network performance, and  requires use of probability distribution functions, which can affect the accuracy  of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In this method, complexity increases with the increase in the  number of uncertainty types, and the evaluation and interpretation of HC  quantification becomes a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Time-Series method considers time correlation in the network, power  consumption and production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Time varying impacts of renewable energy  sources on the network and operation of control elements are considered in HC  determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The method presents realistic overview of the network  performance, based on time varying nature of power consumption and  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This method can give information related to the ‘when’ and ‘how’  of the HC quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Time-Series method requires many measurements data (time series data), and a  lot of simulation may be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Some performance indices may require very  low resolution and pose a computational challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Also, method is very time  consuming for high resolution simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Harmonic issues are strongly influenced by load and existing resonance rather  than PV penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' By injecting reactive power, larger loads like a fast charger or a large EV  household charger might be installed in weaker parts of a power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Voltage deviation constraint does not affect the HC probability curve  significantly, which implies that distributing DGs over the system decreases the  importance of voltage deviation constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  33 The voltage deviation constraint often limits the locational HC, but not the HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Voltage unbalance factor (VUF), which is an important index for unbalanced  systems does not constrain HC of the test system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' DG technology has a great effect on probability curve of HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' This is probably,  because the PV capacity factor is higher than wind capacity factor in the test  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Stochastic hosting capacity methods can also be applied for addressing the  potential overload due to large amount of solar power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' For single-phase units, installation of distributed energy storage can reduce the  unacceptable voltage rise with the customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Energy storage can provide reserves, change net load shape to minimize  ramping requirements, and shift supply of variable generation to periods of  increased net load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Voluntary load reduction or load shifting can provide multiple benefits to  integrating solar and reducing curtailment, including reducing the dependence  on partially loaded synchronous generators for providing frequency stability  and operating reserves and changing the shape of net load, which can reduce  ramp rates, better align solar supply with demand, and reduce peak capacity  needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Balancing supply and demand over larger areas reduces the net variability of  load and renewable resources such as PV, owing to greater spatial diversity of  variable generation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Changing the way the grid is scheduled and dispatched, including changes to  market rules, does not require new technologies and often represents the ‘least  cost’ way to aid vehicle to grid (VG) integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Inverter-based solar and wind plants can provide the grid’s frequency response  needs as these plants become a larger proportion of the generation fleet and new  mechanisms are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Generators can respond better to net load shape created by additional PV via  increased ramp rates and ranges as well as the ability to start and stop more  frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Due to the integration of distributed generation, the level of power quality  disturbances may increase beyond what is acceptable for other customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Conclusion  The electricity technology sector has gone through a marked change from its  traditional architecture of large-scale centralized electric power supply systems that  34  take advantage of significant economies of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Renewable energy resources,  particularly wind and solar energy in form of distributed generation certainly fit this  trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Thus, the traditional cost comparisons, based on large bulk energy market,  may be very misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Distributed generation is likely to pioneer the development  of a new power market, in which the new energy technology does not simply supply  energy but must instead meet the demand of services like energy management,  emergency or back-up power, environmental improvements and fuel diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  Driven by the need for a more sustainable energy system, this new power production  is often of the renewable type, connected through a power electronic converter  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' The excessive integration of distributed generation systems into the  existing electrical networks may adversely affect the system’s performance, and  may lead to problems and operational limit violations, when the system exceeds its  hosting capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hence, it has now become essential to know the hosting capacity  of the network to accept any further integration request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In this paper, a brief review  on the concept of hosting capacity, techniques being used for its assessment and  enhancement and operational limits is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Substantial progress has been  made in HC assessment and enhancement covering analysis, simulation, and the  hardware development and testing for efficiency maximization and cost  minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' However, many problems and issues, especially those related to the  development of affordable, in-exhaustible and clean renewable energy technologies  for huge longer-term benefits, and broad range of policies, market regulations, new  grid codes and standards needed to be addressed for appropriate system planning  and operation of the power system to supply a reliable and good quality electric  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  This work is a part of project ARMOUR supported by the European Union’s  Horizon 2020 program under the Marie Skłodowska-Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  890844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Merlo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Hosting Capacity  Evaluation in Networks with Parameter Uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' In Proceedings of the 2018  18th International Conference on Harmonics and Quality of Power (ICHQP),  Ljubljana, Slovenia, 13–16 May 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' Jayatunga, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' Rathnayake, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' Wickramasinghe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content='  Agalgaonkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' Perera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' In Proceedings of the 2018  18th International Conference on Harmonics and Quality of Power (ICHQP),  Ljubljana, Slovenia, 13–16 May 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' In Proceedings of the 2018  5th International Conference on Information Technology, Computer, and  Electrical Engineering (ICITACEE), Semarang, Indonesia, 27–28 September  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' In Proceedings of the  43  2018 IEEE International Conference on Probabilistic Methods Applied to Power  Systems (PMAPS), Boise, ID, USA, 24–28 June 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+page_content=' Ronnberg, ‘Hosting capacity of the power grid for renewable  electricity production and new consumption equipment’, Energies 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
+page_content=' 10:  1325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQf3gvo/content/2301.04765v1.pdf'}
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+LA-UR-23-20515
+Auxiliary field diffusion Monte Carlo calculations of magnetic moments
+of light nuclei with chiral EFT interactions
+J. D. Martin,1 S. J. Novario,1 D. Lonardoni,2, 1 J. Carlson,1 S. Gandolfi,1 and I. Tews1
+1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
+2XCP-2: Eulerian Codes Group, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
+We calculate the magnetic moments of light nuclei (A < 20) using the auxiliary field diffusion
+Monte Carlo method and local two- and three-nucleon forces with electromagnetic currents from
+chiral effective field theory. For all nuclei under consideration, we also calculate the ground-state
+energies and charge radii. We generally find a good agreement with experimental values for all
+of these observables. For the electromagnetic currents, we explore the impact of employing two
+different power countings, and study theoretical uncertainties stemming from the truncation of the
+chiral expansion order-by-order for select nuclei within these two approaches. We find that it is
+crucial to employ consistent power countings for interactions and currents to achieve a systematic
+order-by-order convergence.
+Introduction— Electromagnetic (EM) phenomena are
+of great importance in nuclear physics both as external
+probes into the structure of atomic nuclei and to un-
+derstand certain internal observables. From high-energy
+electron scattering that explores nuclear distributions [1–
+3] to EM transition strengths that reveal details of nu-
+clear structure [4–6], it is crucial to have a robust theoret-
+ical description of both strong and EM forces in nuclear-
+physics systems. One instance of their union manifests
+in the magnetic dipole moment of an atomic nucleus.
+Magnetic moments are fundamental properties of nuclei
+which interact with atomic electrons and give rise to the
+hyper-fine structure in electronic spectra which can be
+used as a powerful tool to test quantum electrodynam-
+ics and nuclear structure. Additionally, nuclear magnetic
+moments are also a compelling test of both nuclear many-
+body methods and the construction of nuclear interac-
+tions and EM currents in a low-energy framework.
+The nuclear magnetic dipole moment is the vector that
+defines the strength and direction of the magnetic field
+created by an atomic nucleus. In a simple independent-
+particle model, the magnetic moment can be computed
+from the sum of individual nucleon magnetic moments
+and contributions from nonzero orbital angular momenta
+of protons. For a more realistic description, this model is
+greatly complicated in two ways: correlations in the nu-
+clear wave function and inter-nucleon EM currents. The
+former can be handled by quantum many-body methods
+such as the quantum Monte-Carlo (QMC) [7] method,
+while the latter can be handled in a consistent way by
+including higher-order contributions from an EM current
+derived from chiral effective field theory (EFT)[8–13].
+Chiral EFT provides a framework for modeling the in-
+teractions both among constituent nucleons and with ex-
+ternal probes in terms of an expansion in powers of ratios
+of the relevant momentum scale (Q) to the chiral break-
+down scale (Λ). Retaining a finite number of terms in
+the chiral expansion allows one to study nuclear systems
+using a systematically improvable yet tractable model of
+the interactions. Additionally, theoretical uncertainties
+can be systematically estimated by analyzing the order-
+by-order convergence of the expansion [15, 16]. In this
+expansion, the lowest order term in Q/Λ is referred to
+as the leading order (LO) term, the second lowest order
+term as next-to-leading order (NLO), the third lowest
+order term as next-to-next-to-leading order (N2LO), and
+so on.
+In this letter, we employ local interactions and EM cur-
+rents from chiral EFT and use them to compute the mag-
+netic dipole moments of light nuclei with the auxiliary
+2H
+3H
+3He
+6Li
+7Li
+8Li
+15N
+15O
+17O
+17F
+−2
+−1
+0
+1
+2
+3
+4
+5
+µ (µN)
+Exp.
+LO
+NLO
+N2LO
+N3LO (A)
+N3LO (B)
+FIG. 1.
+AFDMC results for the magnetic moments for all
+nuclei studied in this work for the N2LOE1 interaction with
+R0 = 1.0 fm [14]. We show the experimental values (green
+lines) and results with the EM current at LO (black points),
+NLO (purple stars), N2LO (yellow diamonds), and N3LO
+(blue squares for scheme A and red triangles for scheme B; see
+text for details). The indicated uncertainties are statistical
+resulting from the Monte Carlo estimation. The uncertain-
+ties for each experimental value and for the MC statistics are
+indiscernible on this scale.
+arXiv:2301.08349v1  [nucl-th]  19 Jan 2023
+
+2
+field diffusion Monte Carlo (AFDMC) method [7, 17, 18].
+In Fig. 1, we compare the calculated magnetic moments
+to experimental data and generally find good agreement.
+To bolster these results, we also compute the correspond-
+ing ground-state energies and charge radii. We then ex-
+plore different power counting (PC) schemes for the chi-
+ral expansion for EM currents and address the consis-
+tency between these and chiral interactions. Low-energy
+constants introduced by the EM currents at N3LO are
+constrained by two different fits to data. Additionally,
+we compute the order-by-order results and use those to
+estimate theoretical uncertainties in both PC schemes.
+Methods— We treat nuclei as a collection of A point-
+like interacting nucleons of average mass m described via
+the non-relativistic intrinsic Hamiltonian
+H =
+�
+i
+− ∇2
+i
+2m +
+�
+i<j
+Vij +
+�
+i<j<k
+Vijk .
+(1)
+Here, the first term describes the kinetic contribution
+to the Hamiltonian, and Vij and Vijk are the nucleon-
+nucleon (NN) and three-nucleon (3N) potentials, respec-
+tively, the former of which also includes the Coulomb
+force.
+The NN interactions we use in this work were
+derived in Refs. [19, 20] and the 3N interactions in
+Refs. [14, 21] and are based on a local formulation of chi-
+ral EFT. Chiral EFT is a systematic theory for nuclear
+forces describing interactions in terms of a systematic
+momentum expansion [22, 23]. It enables an improve-
+ment of interactions order-by-order and enables theoreti-
+cal uncertainty estimates [15, 16]. Local interaction from
+chiral EFT [24] have been successfully applied to QMC
+calculations of various nuclear systems [25]. Here, we per-
+form AFDMC calculations of the magnetic moment us-
+ing local chiral NN and 3N interactions with cutoff R0 =
+1.0 fm up to next-to-next-to-leading order using the E1
+parametrization for the 3N forces (N2LOE1) [14]. These
+interactions have been used to study the ground-state
+properties of various atomic nuclei up to 16O [14, 18, 26–
+31], few neutron systems [32, 33], and neutron-star mat-
+ter [14, 34–39].
+With the Hamiltonian in hand, as the first step, we
+optimize the variational trial wave function of the form
+|ΨT⟩ = [FC + F2 + F3]|Φ⟩Jπ,T ,
+(2)
+where FC
+accounts for all the spin- and isospin-
+independent correlations, and F2 and F3 are NN and 3N
+correlations linear in spin- and isospin-pairs as described
+in Ref. [7]. The term |Φ⟩J,T is taken to be a shell-model-
+like state with total angular momentum J, parity π, and
+total isospin T describing the target nucleus. Its wave
+function consists of a sum of Slater determinants (D)
+constructed using single-particle orbitals:
+⟨RS|Φ⟩Jπ,T =
+�
+n
+cn
+��
+CJT D
+�
+φα(ri, si)
+��
+Jπ,T , (3)
+where ri are the spatial coordinates of the nucleons and
+si represent their spins. Each single-particle orbital φα
+consists of a radial function ϕ(r) coupled to the spin
+and isospin states. The determinants are coupled with
+Clebsch-Gordan coefficients (CJT ) to total J and T, and
+the cn are variational parameters multiplying different
+components having the same quantum numbers. The ra-
+dial functions ϕ(r) are obtained by solving for the eigen-
+functions of a Wood-Saxon well, and all parameters are
+chosen by minimizing the variational energy as described
+in Ref. [40].
+Then, using the AFDMC algorithm, the
+ground-state wave function is projected out by evolving
+the variational wave function with the time-propagation
+operator in imaginary time,
+|Ψ (τ)⟩ = lim
+τ→∞ e−(H−E0)τ|ΨT⟩ .
+(4)
+More details on the AFDMC method for nuclear systems
+are given in Refs. [7, 17, 25].
+Here, we will use the AFDMC method to calculate the
+magnetic moments of the nuclei 2H, 3H, 3He, 6Li, 7Li,
+8Li, 15N, 15O, 17O, and 17F. For these nuclei, we show
+the binding energies per nucleon and the charge radii
+without theoretical uncertainties in Fig. 2. Overall, our
+calculations follow the trends of the experimental data.
+For the binding energies, we observe a slight underbind-
+ing consistent with the results of Ref. [28] for which all
+results agree with experiment within theoretical uncer-
+tainties.
+For the charge radii, on the other hand, we
+find a very good description of experimental values. The
+magnetic moment can be calculated from the mixed ex-
+pectation value of both VMC and DMC wave functions
+(see Ref. [18] for more details) for the magnetic form fac-
+tor FM,
+FM(q; τ) ≈ −i2m
+q
+�
+2⟨ΨT|jy(qˆx)|Ψ(τ)⟩
+⟨ΨT|Ψ(τ)⟩
+− ⟨ΨT|jy(qˆx)|ΨT⟩
+⟨ΨT|ΨT⟩
+�
+,
+(5)
+by taking the limit of zero external momentum, µ =
+FM(0; τ).
+Electromagnetic
+current
+contributions
+order-by-
+order— We will study magnetic moments by employing
+two different PCs for EM currents: the Pisa [12, 41] and
+Bochum [13] PCs. In the Pisa PC, ratios of momentum
+to the nucleon mass are included at the same order as
+ratios of the momentum to the chiral symmetry breaking
+scale (Λ) so that
+Q
+m ∼ Q
+Λ . The leading-order (LO) and
+next-to-leading order (NLO) contributions to the EM
+currents for the Pisa PC are given in coordinate space in
+Eqs. (2.1) and (2.3) of Ref. [41] and and in momentum
+space in Ref. [12]. In contrast, the Bochum PC counts
+terms proportional to the inverse of the nucleon mass
+as one order suppressed relative to the Pisa PC so that
+
+3
+2H
+3H 3He 6Li
+7Li
+8Li 15N 15O 17O 17F
+−8
+−7
+−6
+−5
+−4
+−3
+−2
+−1
+0
+E/A (MeV)
+Exp.
+AFDMC
+2H
+3H 3He 6Li
+7Li
+8Li 15N 15O 17O 17F
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+�
+⟨r2
+ch⟩ (fm)
+Exp.
+AFDMC
+FIG. 2.
+Binding energies per nucleon (left) and nuclear charge radii (right) estimated using AFDMC (black diamonds) and
+the experimentally determined values (green lines denoted “Exp.”) for all nuclei studied in this work. Binding energies are
+calculated from a transient unconstrained path evolution for the N2LOE1 interaction with R0 = 1.0 fm, while the charge radii
+are estimated using AFDMC using a constrained-path extrapolation. (See [18] for discussion on (un)constrained estimates.)
+There is no available experimental data for charge radii 15O and 17F. The indicated errors represent the standard error arising
+from the statistical uncertainty of the Monte Carlo estimate.
+Q
+m ∼
+Q2
+Λ2 . This leads the lowest-order contributions to
+the electromagnetic vector current operators appearing
+at NLO in the Bochum PC. The total contribution to the
+EM current operators at NLO is identical between the
+Bochum and the Pisa PCs and is equal to the sum of the
+LO single-nucleon operators and the one-pion-exchange
+(OPE) NLO contributions of Ref. [41].
+At N2LO, there are several differences between both
+PCs. First, relativistic corrections proportional to 1/m2
+appear at N2LO in the Pisa PC while the same con-
+tribution appears at N4LO in the Bochum PC. Second,
+the Pisa PC explicitly includes intermediate excitations
+of the ∆ isobar. The corresponding N2LO operator is
+proportional to an equivalent operator contribution at
+N3LO in a ∆-less PC. The term at N2LO is assumed to
+be significantly larger than that at N3LO which is subse-
+quently omitted in Ref. [41]. Because we employ a ∆-less
+Hamiltonian, we omit this structure at N2LO, and restore
+the saturated operator structure at N3LO. As a result,
+Eq. (2.9) of Ref. [41] is modified: first, the term propor-
+tional to τz,i appears at N3LO (weighted by the πN LEC
+d8), and the operator proportional to (τi × τj)z appears
+at N3LO but with a different multiplicative factor. We
+therefore retain the term proportional to (τi × τj)z at
+N3LO and make the substitution of its prefactor
+gAhAµ∆Nm4
+π
+36πm m∆Nf 2π
+→
+g4
+Am4
+π
+512π3f 4π
+We discuss our treatment of the ∆-saturated N3LO term
+proportional to τi,z in the next section.
+Finally, we employ the phenomenologically determined
+Sachs form factors instead of EM currents arising from
+the chiral expansion of these form factors which is only
+slowly converging. As a consequence of this, there is no
+EM current contribution at N2LO in the Bochum PC.
+Two
+nucleon,
+pion-nucleon,
+and
+electromagnetic
+LECs—
+At N3LO in both the Bochum and Pisa PCs,
+there appear three types of low-energy constants (LECs).
+First, there are NN contact LECs which already appear
+in the NLO nuclear interaction (C2, C4, C5, and C7).
+These LECs are associated with the particular parame-
+terization of the nuclear interaction used in Ref. [8] which
+includes both local and nonlocal operators in momen-
+tum space.
+Here, instead, the Hamiltonian employed
+for the imaginary time propagation includes a different
+set of purely local operators expressed in configuration
+space [20], described by a different set of NN contact
+LECs.
+We can employ the Fierz rearrangement free-
+dom [42] in order to relate the LECs appearing in the
+EM currents (Ci) to our interaction LECs (C′
+i):
+C2 = −4 (C′
+2 + 3 C′
+4 + C′
+7) ,
+(6a)
+C4 = −4 (C′
+2 − C′
+4 − C′
+7) ,
+(6b)
+C5 = −C′
+5 ,
+(6c)
+C7 = −8 C′
+7 .
+(6d)
+The C′
+i were determined from fits to NN scattering
+data [20].
+Details on the mapping between the LECs
+identified via the Fierz rearrangement are provided in
+the Supplemental Material [43].
+Next, there are contributions to the EM currents
+from the N3LO and N4LO pion-nucleon (πN) Lagrangian
+which are proportional to the di and li LECs, respec-
+tively.
+All except three terms proportional to the πN
+
+4
+LECs vanish in the q → 0 limit, and do not contribute
+to the evaluation of nuclear magnetic moments.
+The
+three which contribute are proportional to d8, d9 and
+d21. Here, we chose to exclude all operators proportional
+to di, as these LECs are of higher order compared to the
+employed nuclear Hamiltonians.
+Finally, there arise two unknown purely EM LECs that
+must be fit to experimental data. These are called L1
+and L2 in Ref. [13] or dS
+1 and dV
+1 in Ref. [41], respec-
+tively. We fit these two unknown EM LECs using two
+prescriptions, hereafter denoted scheme A and scheme
+B. In scheme A, we have fit the unknown LECs to repro-
+duce the experimental values of the magnetic moments
+of 3H and 3He. For scheme B we have performed a linear
+least squares fit of the two LECs to the entire data set
+for the magnetic moments. Using the naming convention
+of Ref. [41], we find the LEC values for scheme A to be
+dS
+1 = −0.09596 ± 0.0018 and dV
+1 = −0.11085 ± 0.00099
+with a χ2 per degree-of-freedom (DOF) of 0.48078. For
+scheme B we find dS
+1 = −0.03958 ± 0.0035 and dV
+1 =
+−0.06420 ± 0.0028 with a χ2 per DOF of 1.5035.
+Results— In Fig. 1, we show AFDMC results for the
+magnetic moments at each order in the chiral expansion
+for the EM currents as expressed in Ref. [41].
+For all
+calculations, we have used the N2LOE1 interaction and
+included all contributions to the EM currents up to the
+specified order. None of these results include theoreti-
+cal uncertainty estimates. Generally, we observe a good
+order-by-order convergence for all nuclei and reproduce
+experimental data well. As a simple measure of the con-
+vergence, we consider the absolute difference between the
+estimated magnetic moment and the experimental value
+averaged across all nuclei at each order in the EM cur-
+rents. This difference is 0.18 at LO, 0.13 at NLO, 0.13
+at N2LO, and 0.08 (0.09) at N3LO in scheme A (B).
+Next we note that N3LO contributions seem to be
+larger than expected based on lower-order contributions.
+To highlight this, in Fig. 3 we show the total contribution
+of each order of the chiral expansion to the total magnetic
+moment using the Pisa PC. One feature of the magnetic
+moment immediately noted is the nearly universal large
+size of the N3LO contribution, particularly with respect
+to the relativistic corrections at N2LO, with most nuclei
+displaying O(µNLO) ∼ O(µN3LO). This feature does not
+depend on the choice of the fitting scheme.
+Next, we study the systematic error arising from the
+truncation of the chiral expansion for the magnetic mo-
+ments in both PCs. For this, we use the simple prescrip-
+tion introduced in Ref. [15] and perform AFDMC calcu-
+lations of the magnetic moments order-by-order in the
+chiral expansion for both the nuclear interaction Hamil-
+tonian and the electromagnetic current operators. In this
+work, we performed this uncertainty estimate for 6Li, 7Li,
+8Li, and 15N. For the Pisa PC, the theoretical uncertainty
+at each order is found using the following recursive ex-
+2H
+3H
+3He
+6Li
+7Li
+8Li
+15N
+15O
+17O
+17F
+µLO
+µNLO
+µN2LO
+µN3LO
+� µn
+0.821 2.463 -1.696 0.794 2.988 1.394 -0.244 0.630 -1.822 4.665
+0.000 0.199 -0.197 -0.004 0.179 0.167 -0.167 0.146 0.090 -0.143
+0.005 0.006 0.007 0.006 0.017 0.019 0.016 0.003 0.009 0.044
+-0.021 0.165 -0.211 -0.039 0.177 0.146 0.005 -0.009 -0.103 0.066
+0.806 2.832 -2.096 0.757 3.360 1.727 -0.391 0.770 -1.826 4.632
+−5
+−4
+−3
+−2
+−1
+0
+log10(|µn|)
+FIG. 3.
+Contribution to the magnetic moment at each order
+in the chiral expansion of the nuclear EM current for each
+nucleus considered. We show results for the Pisa PC [41] but
+the sum of the LO and NLO results is equivalent to the NLO
+contribution in the Bochum PC. The N2LO row represents
+only the 1/m2 relativistic correction while the N3LO results
+are obtained by using the LECs fit in scheme B. The number
+appearing in each cell represents the MC estimate of the con-
+tribution, while the color of the cell represents the logarithm
+of the magnitude of the contribution to guide the eye.
+pressions:
+δ(µLO
+LO) = Q|µLO
+LO| ,
+(7)
+δ(µN2LO
+NLO ) = max
+�
+Q|µN2LO
+NLO − µLO
+LO|, Q2δ(µLO
+LO)
+�
+,
+(8)
+δ(µN3LO
+N2LO) = max
+�
+Q|µN3LO
+N2LO − µNLO
+NLO|, Qδ(µNLO
+NLO)
+�
+,
+(9)
+where we take Q = mπ/Λ, and Λ = 500 MeV. In
+the above expressions superscripts denote the highest in-
+cluded order in the expansion of the currents, and sub-
+scripts denote the highest included order in the nuclear
+interaction Hamiltonian. At leading order ((Q/Λ)−2 in
+the Pisa PC), the magnetic moment is calculated using
+both the Hamiltonian and the EM currents at LO. At
+NLO ((Q/Λ)−1 in both PCs), the magnetic moment is
+calculated using both the NLO Hamiltonian and EM cur-
+rents through N2LO. This is because the Hamiltonian at
+NLO is two powers of Q/Λ greater than the Hamiltonian
+at NLO. For consistency, we include all current opera-
+tors which are less than or equal to 2 powers of Q/Λ
+greater than the operators at LO. Finally, at N2LO, the
+magnetic moment is calculated by employing the N2LO
+Hamiltonian and the sum of the EM currents up to, and
+including, N3LO.
+Similarly,
+we estimate the uncertainty using the
+Bochum PC for the EM currents. This PC is more consis-
+tent with the employed interaction Hamiltonian. For the
+equivalent theoretical uncertainty estimation in the con-
+sistent power counting, only the Bochum group definition
+of the NLO current is employed, as the LO and N2LO
+contributions to the magnetic moments vanish. Then, in
+this PC, differences in the magnetic moments arise due
+to the nuclear interactions order-by-order. The theoreti-
+cal uncertainty estimate using the Bochum group PC is
+
+5
+LO
+NLO N2LO
+LO
+NLO N2LO
+LO
+NLO N2LO
+LO
+NLO N2LO
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+µ/µexp.
+6Li
+7Li
+8Li
+15N
+Bochum
+Pisa (Scheme A)
+Pisa (Scheme B)
+FIG. 4.
+Order-by-order contributions to the magnetic mo-
+ment, relative to the experimentally determined value, for the
+indicated nuclei. The abscissa labels indicate the highest or-
+der in the chiral expansion utilized in the nuclear interaction
+Hamiltonian.
+given by
+δ(µNLO
+NLO) = Q|µNLO
+NLO| ,
+(10)
+δ(µNLO
+N2LO) = max
+�
+Q|µNLO
+N2LO − µNLO
+NLO|, Qδ(µNLO)
+�
+. (11)
+The ratio of the QMC estimate of the magnetic mo-
+ment for the selected nuclei to the experimentally deter-
+mined value is shown in Fig. 4 for both the Pisa and
+Bochum PCs. For all nuclei, we observe a systematic be-
+havior within the Bochum PC. However, for 15N, the cal-
+culations seem to predict a magnetic moment 40% above
+the experimental value. For the Pisa PC, we observe a
+systematic convergence for 6Li, 7Li but for 8Li and 15N,
+in contrast, the order-by-order results seem to be less
+systematic. In particular, the theoretical uncertainty is
+largest at NLO in this PC. This might point to a sys-
+tematic underestimate of the characteristic momentum
+scale Q for these nuclei, and a more detailed investigation
+into the one- and two-nucleon momentum distributions
+in these nuclei is warranted. In addition, this might show
+that consistent PCs for interactions and currents need to
+be employed. The choice of fitting scheme for the N3LO
+EM LECs also produces some variance in the estimated
+values of the magnetic moments. However, this variance
+is of the same order as the theoretical uncertainty esti-
+mated using the NLO EM currents.
+Discussion and conclusions— We performed AFDMC
+calculations of the magnetic moments of several light nu-
+clei with nuclear interactions and EM currents derived
+from chiral EFT and found agreement with experiment.
+These results are supported by comparing supplementary
+AFDMC calculations of the ground-state binding ener-
+gies and charge-radii to experiment as well. Addition-
+ally, we explored different power countings of the chiral
+EM currents compared to the power counting of the nu-
+clear interaction and the order-by-order convergence of
+the magnetic moments.
+While our accurate magnetic moment results are valu-
+able in verifying the different components of these ab ini-
+tio calculations, our uncertainty analysis indicates that
+the consistency between employed nuclear interactions
+and EM currents is crucial. Our results for the consis-
+tent power counting shows a more natural order-by-order
+convergence pattern.
+Future work to derive consistent
+currents for the interactions used in AFDMC is impor-
+tant. Furthermore, given the importance of uncertainty
+quantification, more work needs to be done to ensure the
+proper estimates for momentum scales are being used.
+This is difficult using the simple uncertainty estimate
+used in this work [15], but employing uncertainty quan-
+tification tools based on Gaussian Processes [16] might
+be beneficial. Finally, given the prominence of the N3LO
+EM contributions to the calculated magnetic moments in
+this work, the convergence up to N3LO and fulfilling the
+corresponding continuity equation are crucial [13].
+We thank Alex Gnech, Rocco Schiavilla, Saori Pas-
+tore, Hermann Krebs, and Ronen Weiss for useful dis-
+cussions. The work of J.D.M., S.N., J.C., S.G. and I.T.
+was supported by the U.S. Department of Energy, Of-
+fice of Science, Office of Nuclear Physics, under contract
+No. DE-AC52-06NA25396. The work of D.L., J.C., S.G.
+and I.T. was supported by the U.S. Department of En-
+ergy, Office of Science, Office of Advanced Scientific Com-
+puting Research, Scientific Discovery through Advanced
+Computing (SciDAC) NUCLEI program. The work of
+J.D.M., S.N. and S.G. was also supported by the De-
+partment of Energy Early Career Award Program. Com-
+puter time was provided by the Los Alamos National
+Laboratory Institutional Computing Program, which is
+supported by the U.S. Department of Energy National
+Nuclear Security Administration under Contract No.
+89233218CNA000001, and by the National Energy Re-
+search Scientific Computing Center (NERSC), which is
+supported by the U.S. Department of Energy, Office of
+Science, under contract No. DE-AC02-05CH11231.
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+
+LA-UR-23-20515
+Supplemental Material: Auxiliary field diffusion Monte Carlo calculations of magnetic
+moments of light nuclei with chiral EFT interactions
+J. D. Martin,1 S. J. Novario,1 D. Lonardoni,2, 1 J. Carlson,1 S. Gandolfi,1 and I. Tews1
+1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
+2XCP-2: Eulerian Codes Group, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
+FIERZ TRANSFORMATION OF NN CONTACT
+INTERACTIONS ENTERING EM CURRENTS
+The
+quantum
+states
+which
+describe
+nuclei
+are
+fermionic many-body states and are therefore antisym-
+metric. In chiral EFT, there are 14 sub-leading contact
+operators at NLO and N2LO. Because of the fermionic
+antisymmetry requirement, out of the 14 operators only
+7 are linearly independent [1]. Hence, the choice of oper-
+ators to include in the NN potential is not unique, and
+one parameterization of the potential may be related to
+a different parameterization through a Fierz transforma-
+tion. Therefore, given a particular choice of 7 operators
+and associated LECs, one can antisymmetrize the NN
+potential and choose a different set of operators weighted
+by consistent linear combinations of the original LECs.
+The electromagnetic (EM) current operators utilized
+in this work were derived in chiral effective field the-
+ory from a set of nuclear interactions which implement a
+two-nucleon (NN) isospin-independent nonlocal poten-
+tial at N2LO in the chiral expansion. This chiral poten-
+tial (Eq. 2.4 of Ref. [2]) is
+v(q, k) =C(i)
+1 q2 + C2k2
++ (C3q2 + C4k2)σ1 · σ2
++ iC5
+2 (σ1 + σ2) · k × q + C6σ1 · qσ2 · q
++ C7σ1 · kσ2 · k .
+(1)
+In this and following expressions, k = (p′ + p)/2 and
+q = p′ − p where p and p′ denote the incoming and
+outgoing nucleon momenta, respectively.
+In contrast, the chiral potentials employed for the
+AFDMC imaginary time propagation in configuration
+space are isospin-dependent and local. The momentum
+space representation of this potential (Eq. 22 of Ref. [1])
+is
+V (q, k) =C′
+1q2 + C′
+2q2τ1 · τ2
++ C′
+3q2σ1 · σ2
++ C′
+4q2σ1 · σ2τ1 · τ2
++ iC′
+5
+2 (σ1 + σ2) · q × k
++ C6(σ1 · q)(σ2 · q)
++ C7(σ1 · q)(σ2 · q)τ1 · τ2 .
+(2)
+To ensure that the correct NN contact LECs enter the
+employed EM currents, we determine the linear trans-
+formation between the Ci LECs appearing in the Pisa
+parameterization of the EM currents and their NN po-
+tential, and the C′
+i LECs appearing in our interactions.
+The antisymmetrization of an NN potential is ob-
+tained by
+vas = 1
+2(v − A[v]) ,
+(3)
+where
+A[v] =1
+4 (1 + σ12 + τ12 + σ12τ12)
+×v(q → −2k, k → −1
+2q) ,
+(4)
+and σ12 = σ1 · σ2 and τ12 = τ1 · τ2.
+After performing the antisymmetrization of the Pisa
+potential (Eq. 1) and collecting all operators, we find
+vas(q, k) = 1
+2
+� �
+C1 − 1
+16(C2 + 3C4 + C7)
+�
+q2 + (C2 − (C1 + 3C3 + C6)) k2
+− 1
+16(C2 + 3C4 + C7)q2τ12 − (C1 + 3C3 + C6)k2τ12
++
+�
+C3 − 1
+16(C2 − C4 − C7)
+�
+q2σ12 + (C4 − (C1 − C3 − C6)) k2σ12
+− 1
+16(C2 − C4 − C7)q2σ12τ12 − (C1 − C3 − C6)k2σ12τ12
++ i3C5
+4 (σ1 + σ2) · (k × q) + iC5
+4 (σ1 + σ2) · (k × q)τ12
+arXiv:2301.08349v1  [nucl-th]  19 Jan 2023
+
+2
++
+�
+C6 − 1
+8C7
+�
+(σ1 · q)(σ2 · q) + (C7 − 2C6)(σ1 · k)(σ2 · k)
+− 1
+8C7(σ1 · q)(σ2 · q)τ12 − 2C6(σ1 · k)(σ2 · k)τ12
+�
+.
+(5)
+A different parameterization of the momentum-space potential is employed in the Norfolk interactions, and the
+subsequent linear mapping between the LECs relevant for the EM currents is provided in Ref. [3].
+We similarly antisymmetrize our NN potential (Eq. 2) and find
+Vas(q, k) = 1
+2
+�
+C′
+1q2 − (C′
+1 + 3C′
+2 + 3C′
+3 + 9C′
+4 + C′
+6 + 3C′
+7)k2
++ C′
+2q2τ12 − (C′
+1 − C′
+2 + 3C′
+3 − 3C′
+4 + C′
+6 − C′
+7)k2τ12
++ C′
+3q2σ12 − (C′
+1 + 3C′
+2 − C′
+3 − 3C′
+4 − C′
+6 − 3C′
+7)k2σ12
++ C′
+4q2σ12τ12 − (C′
+1 − C′
+2 − C′
+3 + C′
+4 − C′
+6 + C′
+7)k2σ12τ12
++ i3C′
+5
+4 (σ1 + σ2) · (q × k) + iC′
+5
+4 (σ1 + σ2) · (q × k)τ12
++ C′
+6(σ1 · q)(σ2 · q) − (2C′
+6 + 6C′
+7)(σ1 · k)(σ2 · k)
++ C′
+7(σ1 · q)(σ2 · q)τ12 − (2C′
+6 − 2C′
+7)(σ1 · k)(σ2 · k)τ12
+�
+.
+(6)
+We can immediately see that C′
+5 = −C5.
+To find
+the relationship between the other 6 LECs, we choose
+to match the operator set
+{q2, q2τ12, q2σ12, q2σ12τ12, (σ1·q)(σ2·q), (σ1·q)(σ2·q)τ12} .
+From these operators, we may relate each primed LEC
+to a linear combination of the unprimed LECs. Inverting
+this permits us to express the unprimed LECs in terms
+of the primed LECs.
+1
+16
+�
+�
+�
+�
+�
+�
+�
+�
+16 −1
+0
+−3
+0
+−1
+0
+−1
+0
+−3
+0
+−1
+0
+−1 16
+1
+0
+1
+0
+−1
+0
+1
+0
+1
+0
+0
+0
+0
+16 −2
+0
+0
+0
+0
+0
+−2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+C1
+C2
+C3
+C4
+C6
+C7
+�
+�
+�
+�
+�
+�
+�
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+1
+1
+1
+1
+1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+6
+C′
+7
+�
+�
+�
+�
+�
+�
+�
+�
+(7)
+Denoting the matrix in parenthesis, without the 1/16
+prefactor, as M, Eq. 7 can be more compactly written
+as M ⃗C = 16⃗C′.
+Inverting this linear system permits
+us to write the Pisa (unprimed) LECs in terms of linear
+combinations of our (primed) LECs. For the inversion,
+we find that
+M−1 = 1
+16
+�
+�
+�
+�
+�
+�
+�
+�
+1 −1 0
+0
+0
+0
+0 −4 0 −12 0 −4
+0
+0
+1
+−1
+0
+0
+0 −4 0
+4
+0
+4
+0
+0
+0
+0
+1 −1
+0
+0
+0
+0
+0 −8
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(8)
+Thus, we obtain
+�
+�
+�
+�
+�
+�
+�
+�
+C1
+C2
+C3
+C4
+C6
+C7
+�
+�
+�
+�
+�
+�
+�
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+1 −1 0
+0
+0
+0
+0 −4 0 −12 0 −4
+0
+0
+1
+−1
+0
+0
+0 −4 0
+4
+0
+4
+0
+0
+0
+0
+1 −1
+0
+0
+0
+0
+0 −8
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+6
+C′
+7
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(9)
+We have further validated that different choices for the
+set of operators in the LEC matching procedure leads to
+the same mapping between LECs for the two different
+parameterizations of the N2LO chiral potentials.
+
+3
+POWER COUNTING COMPARISON
+The power counting (PC) schemes employed by the
+Pisa and Bochum groups in their respective derivation
+of the electromagnetic currents differ in how they count
+inverse powers of the nucleon mass. In table I we show
+the highest order included at each order in the chiral
+expansion for the Hamiltonian we employ, as well as the
+expansion order for the current operators derived by the
+Pisa and Bochum groups.
+Order H JPisa JBochum
+LO
+Q0 Q−2
+Q−3
+NLO Q2 Q−1
+Q−1
+N2LO Q3
+Q0
+Q0
+N2LO Q4
+Q1
+Q1
+TABLE I. Highest order term in the chiral expansion of the
+Hamiltonian employed in this work (H column), the EM
+current operators derived by the Pisa group (JPisa column),
+and the EM current operators derived by the Bochum group
+(JBochum).
+The total summed contribution to the EM currents is
+identical between the Pisa and Bochum PC’s. However,
+the LO contribution to the EM vector current is 0 in the
+Bochum PC. This is because the term which appears at
+LO in the Pisa PC is proportional to Q/m (the ratio of
+momentum to the nucleon mass) which is counted as one
+order higher in the Bochum PC.
+INDIVIDUAL CONTRIBUTIONS TO THE EM
+CURRENT
+[1] A.
+Gezerlis,
+I.
+Tews,
+E.
+Epelbaum,
+S.
+Gandolfi,
+K. Hebeler, A. Nogga, and A. Schwenk, Phys. Rev. Lett.
+111, 032501 (2013).
+[2] S. Pastore, L. Girlanda, R. Schiavilla, M. Viviani,
+and
+R. B. Wiringa, Phys. Rev. C 80, 034004 (2009).
+[3] A. Gnech and R. Schiavilla, Phys. Rev. C 106, 044001
+(2022), arXiv:2207.05528 [nucl-th].
+
+4
+2H
+3H
+3He
+6Li
+7Li
+8Li
+15N
+15O
+17O
+17F
+µN3LO
+NM(dS
+1 )
+µN3LO
+NM(dV
+1 )
+� µn
+0.035
+0.041
+0.040
+0.060
+0.052
+0.068 -0.015 -0.015 0.031
+0.032
+-0.000 0.122 -0.121 -0.000 0.156
+0.114 -0.077 0.075 -0.092 0.099
+0.806
+2.832 -2.096 0.757
+3.360
+1.727 -0.391 0.770 -1.826 4.632
+B
+µN3LO
+NM(dS
+1 )
+µN3LO
+NM(dV
+1 )
+� µn
+0.084
+0.098
+0.096
+0.147
+0.125
+0.166 -0.035 -0.036 0.076
+0.078
+-0.000 0.211 -0.208 -0.000 0.269
+0.197 -0.133 0.129 -0.158 0.171
+0.855
+2.979 -2.128 0.843
+3.547
+1.907 -0.468 0.803 -1.849 4.750
+A
+µLO
+µNLO
+µN2LO
+RC
+µN2LO
+∆ (r.w.)
+µN3LO
+TPE
+µN3LO
+MIN
+µN3LO
+CT
+0.821
+2.463 -1.696 0.794
+2.988
+1.394 -0.244 0.630 -1.822 4.665
+0.000
+0.199 -0.197 -0.004 0.179
+0.167 -0.167 0.146
+0.090 -0.143
+0.005
+0.006
+0.007
+0.006
+0.017
+0.019
+0.016
+0.003
+0.009
+0.044
+0.000
+0.003 -0.003 0.000
+0.003
+0.002 -0.001 0.001 -0.001 0.001
+-0.000 0.021 -0.021 0.003
+0.029
+0.014
+0.021 -0.019 -0.050 0.065
+-0.055 -0.077 -0.052 -0.102 -0.130 -0.108 0.106 -0.079 0.041 -0.167
+-0.000 0.055 -0.055 -0.000 0.068
+0.055 -0.030 0.028 -0.033 0.035
+−6
+−5
+−4
+−3
+−2
+−1
+0
+1
+log10(|µn|)
+FIG. 1. Evaluated contribution to the magnetic moment for
+each term in chiral expansion of the EM current operator at
+N2LO in the Pisa PC. The top panel are the values common
+to both EM LECs fitting schemes. The first two rows of the
+middle panel are the contributions to the magnetic moment
+from the terms proportional to the unknown EM LECs fit
+with scheme A, and the final row is the total evaluated mag-
+netic moment including all terms of the first panel and first
+two rows of the middle panel. The bottom panel is similar to
+the middle panel, but the top two rows represent those terms
+proportional to the unknown EM LECs fit with scheme B.
+
diff --git a/h9E_T4oBgHgl3EQf3xz2/content/tmp_files/load_file.txt b/h9E_T4oBgHgl3EQf3xz2/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..52377157a7bf76c2e55395c4c317fa8aa85d3ac4
--- /dev/null
+++ b/h9E_T4oBgHgl3EQf3xz2/content/tmp_files/load_file.txt
@@ -0,0 +1,1343 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf,len=1342
+page_content='LA-UR-23-20515  Auxiliary field diffusion Monte Carlo calculations of magnetic moments  of light nuclei with chiral EFT interactions  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Martin,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Novario,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni,2, 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi,1 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tews1  1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA  2XCP-2: Eulerian Codes Group, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA  We calculate the magnetic moments of light nuclei (A < 20) using the auxiliary field diffusion  Monte Carlo method and local two- and three-nucleon forces with electromagnetic currents from  chiral effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For all nuclei under consideration, we also calculate the ground-state  energies and charge radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We generally find a good agreement with experimental values for all  of these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For the electromagnetic currents, we explore the impact of employing two  different power countings, and study theoretical uncertainties stemming from the truncation of the  chiral expansion order-by-order for select nuclei within these two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We find that it is  crucial to employ consistent power countings for interactions and currents to achieve a systematic  order-by-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Introduction— Electromagnetic (EM) phenomena are  of great importance in nuclear physics both as external  probes into the structure of atomic nuclei and to un-  derstand certain internal observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' From high-energy  electron scattering that explores nuclear distributions [1–  3] to EM transition strengths that reveal details of nu-  clear structure [4–6], it is crucial to have a robust theoret-  ical description of both strong and EM forces in nuclear-  physics systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' One instance of their union manifests  in the magnetic dipole moment of an atomic nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Magnetic moments are fundamental properties of nuclei  which interact with atomic electrons and give rise to the  hyper-fine structure in electronic spectra which can be  used as a powerful tool to test quantum electrodynam-  ics and nuclear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Additionally, nuclear magnetic  moments are also a compelling test of both nuclear many-  body methods and the construction of nuclear interac-  tions and EM currents in a low-energy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The nuclear magnetic dipole moment is the vector that  defines the strength and direction of the magnetic field  created by an atomic nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In a simple independent-  particle model, the magnetic moment can be computed  from the sum of individual nucleon magnetic moments  and contributions from nonzero orbital angular momenta  of protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For a more realistic description, this model is  greatly complicated in two ways: correlations in the nu-  clear wave function and inter-nucleon EM currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The  former can be handled by quantum many-body methods  such as the quantum Monte-Carlo (QMC) [7] method,  while the latter can be handled in a consistent way by  including higher-order contributions from an EM current  derived from chiral effective field theory (EFT)[8–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Chiral EFT provides a framework for modeling the in-  teractions both among constituent nucleons and with ex-  ternal probes in terms of an expansion in powers of ratios  of the relevant momentum scale (Q) to the chiral break-  down scale (Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Retaining a finite number of terms in  the chiral expansion allows one to study nuclear systems  using a systematically improvable yet tractable model of  the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Additionally, theoretical uncertainties  can be systematically estimated by analyzing the order-  by-order convergence of the expansion [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In this  expansion, the lowest order term in Q/Λ is referred to  as the leading order (LO) term, the second lowest order  term as next-to-leading order (NLO), the third lowest  order term as next-to-next-to-leading order (N2LO), and  so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  In this letter, we employ local interactions and EM cur-  rents from chiral EFT and use them to compute the mag-  netic dipole moments of light nuclei with the auxiliary  2H  3H  3He  6Li  7Li  8Li  15N  15O  17O  17F  −2  −1  0  1  2  3  4  5  µ (µN)  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  LO  NLO  N2LO  N3LO (A)  N3LO (B)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  AFDMC results for the magnetic moments for all  nuclei studied in this work for the N2LOE1 interaction with  R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0 fm [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We show the experimental values (green  lines) and results with the EM current at LO (black points),  NLO (purple stars), N2LO (yellow diamonds), and N3LO  (blue squares for scheme A and red triangles for scheme B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' see  text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The indicated uncertainties are statistical  resulting from the Monte Carlo estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The uncertain-  ties for each experimental value and for the MC statistics are  indiscernible on this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='08349v1  [nucl-th]  19 Jan 2023  2  field diffusion Monte Carlo (AFDMC) method [7, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 1, we compare the calculated magnetic moments  to experimental data and generally find good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  To bolster these results, we also compute the correspond-  ing ground-state energies and charge radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We then ex-  plore different power counting (PC) schemes for the chi-  ral expansion for EM currents and address the consis-  tency between these and chiral interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Low-energy  constants introduced by the EM currents at N3LO are  constrained by two different fits to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Additionally,  we compute the order-by-order results and use those to  estimate theoretical uncertainties in both PC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Methods— We treat nuclei as a collection of A point-  like interacting nucleons of average mass m described via  the non-relativistic intrinsic Hamiltonian  H =  �  i  − ∇2  i  2m +  �  i<j  Vij +  �  i<j<k  Vijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (1)  Here, the first term describes the kinetic contribution  to the Hamiltonian, and Vij and Vijk are the nucleon-  nucleon (NN) and three-nucleon (3N) potentials, respec-  tively, the former of which also includes the Coulomb  force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The NN interactions we use in this work were  derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [19, 20] and the 3N interactions in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [14, 21] and are based on a local formulation of chi-  ral EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Chiral EFT is a systematic theory for nuclear  forces describing interactions in terms of a systematic  momentum expansion [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' It enables an improve-  ment of interactions order-by-order and enables theoreti-  cal uncertainty estimates [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Local interaction from  chiral EFT [24] have been successfully applied to QMC  calculations of various nuclear systems [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Here, we per-  form AFDMC calculations of the magnetic moment us-  ing local chiral NN and 3N interactions with cutoff R0 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0 fm up to next-to-next-to-leading order using the E1  parametrization for the 3N forces (N2LOE1) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' These  interactions have been used to study the ground-state  properties of various atomic nuclei up to 16O [14, 18, 26–  31], few neutron systems [32, 33], and neutron-star mat-  ter [14, 34–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  With the Hamiltonian in hand, as the first step, we  optimize the variational trial wave function of the form  |ΨT⟩ = [FC + F2 + F3]|Φ⟩Jπ,T ,  (2)  where FC  accounts for all the spin- and isospin-  independent correlations, and F2 and F3 are NN and 3N  correlations linear in spin- and isospin-pairs as described  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The term |Φ⟩J,T is taken to be a shell-model-  like state with total angular momentum J, parity π, and  total isospin T describing the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Its wave  function consists of a sum of Slater determinants (D)  constructed using single-particle orbitals:  ⟨RS|Φ⟩Jπ,T =  �  n  cn  ��  CJT D  �  φα(ri, si)  ��  Jπ,T , (3)  where ri are the spatial coordinates of the nucleons and  si represent their spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Each single-particle orbital φα  consists of a radial function ϕ(r) coupled to the spin  and isospin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The determinants are coupled with  Clebsch-Gordan coefficients (CJT ) to total J and T, and  the cn are variational parameters multiplying different  components having the same quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The ra-  dial functions ϕ(r) are obtained by solving for the eigen-  functions of a Wood-Saxon well, and all parameters are  chosen by minimizing the variational energy as described  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Then, using the AFDMC algorithm, the  ground-state wave function is projected out by evolving  the variational wave function with the time-propagation  operator in imaginary time,  |Ψ (τ)⟩ = lim  τ→∞ e−(H−E0)τ|ΨT⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (4)  More details on the AFDMC method for nuclear systems  are given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [7, 17, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Here, we will use the AFDMC method to calculate the  magnetic moments of the nuclei 2H, 3H, 3He, 6Li, 7Li,  8Li, 15N, 15O, 17O, and 17F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For these nuclei, we show  the binding energies per nucleon and the charge radii  without theoretical uncertainties in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Overall, our  calculations follow the trends of the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  For the binding energies, we observe a slight underbind-  ing consistent with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [28] for which all  results agree with experiment within theoretical uncer-  tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  For the charge radii, on the other hand, we  find a very good description of experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The  magnetic moment can be calculated from the mixed ex-  pectation value of both VMC and DMC wave functions  (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [18] for more details) for the magnetic form fac-  tor FM,  FM(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' τ) ≈ −i2m  q  �  2⟨ΨT|jy(qˆx)|Ψ(τ)⟩  ⟨ΨT|Ψ(τ)⟩  − ⟨ΨT|jy(qˆx)|ΨT⟩  ⟨ΨT|ΨT⟩  �  ,  (5)  by taking the limit of zero external momentum, µ =  FM(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Electromagnetic  current  contributions  order-by-  order— We will study magnetic moments by employing  two different PCs for EM currents: the Pisa [12, 41] and  Bochum [13] PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In the Pisa PC, ratios of momentum  to the nucleon mass are included at the same order as  ratios of the momentum to the chiral symmetry breaking  scale (Λ) so that  Q  m ∼ Q  Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The leading-order (LO) and  next-to-leading order (NLO) contributions to the EM  currents for the Pisa PC are given in coordinate space in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='3) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41] and and in momentum  space in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In contrast, the Bochum PC counts  terms proportional to the inverse of the nucleon mass  as one order suppressed relative to the Pisa PC so that  3  2H  3H 3He 6Li  7Li  8Li 15N 15O 17O 17F  −8  −7  −6  −5  −4  −3  −2  −1  0  E/A (MeV)  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  AFDMC  2H  3H 3He 6Li  7Li  8Li 15N 15O 17O 17F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='8  �  ⟨r2  ch⟩ (fm)  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  AFDMC  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Binding energies per nucleon (left) and nuclear charge radii (right) estimated using AFDMC (black diamonds) and  the experimentally determined values (green lines denoted “Exp.”) for all nuclei studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Binding energies are  calculated from a transient unconstrained path evolution for the N2LOE1 interaction with R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0 fm, while the charge radii  are estimated using AFDMC using a constrained-path extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' (See [18] for discussion on (un)constrained estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=')  There is no available experimental data for charge radii 15O and 17F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The indicated errors represent the standard error arising  from the statistical uncertainty of the Monte Carlo estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Q  m ∼  Q2  Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This leads the lowest-order contributions to  the electromagnetic vector current operators appearing  at NLO in the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The total contribution to the  EM current operators at NLO is identical between the  Bochum and the Pisa PCs and is equal to the sum of the  LO single-nucleon operators and the one-pion-exchange  (OPE) NLO contributions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  At N2LO, there are several differences between both  PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' First, relativistic corrections proportional to 1/m2  appear at N2LO in the Pisa PC while the same con-  tribution appears at N4LO in the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Second,  the Pisa PC explicitly includes intermediate excitations  of the ∆ isobar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The corresponding N2LO operator is  proportional to an equivalent operator contribution at  N3LO in a ∆-less PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The term at N2LO is assumed to  be significantly larger than that at N3LO which is subse-  quently omitted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Because we employ a ∆-less  Hamiltonian, we omit this structure at N2LO, and restore  the saturated operator structure at N3LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' As a result,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='9) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41] is modified: first, the term propor-  tional to τz,i appears at N3LO (weighted by the πN LEC  d8), and the operator proportional to (τi × τj)z appears  at N3LO but with a different multiplicative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We  therefore retain the term proportional to (τi × τj)z at  N3LO and make the substitution of its prefactor  gAhAµ∆Nm4  π  36πm m∆Nf 2π  →  g4  Am4  π  512π3f 4π  We discuss our treatment of the ∆-saturated N3LO term  proportional to τi,z in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Finally, we employ the phenomenologically determined  Sachs form factors instead of EM currents arising from  the chiral expansion of these form factors which is only  slowly converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' As a consequence of this, there is no  EM current contribution at N2LO in the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Two  nucleon,  pion-nucleon,  and  electromagnetic  LECs—  At N3LO in both the Bochum and Pisa PCs,  there appear three types of low-energy constants (LECs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  First, there are NN contact LECs which already appear  in the NLO nuclear interaction (C2, C4, C5, and C7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  These LECs are associated with the particular parame-  terization of the nuclear interaction used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [8] which  includes both local and nonlocal operators in momen-  tum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Here, instead, the Hamiltonian employed  for the imaginary time propagation includes a different  set of purely local operators expressed in configuration  space [20], described by a different set of NN contact  LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  We can employ the Fierz rearrangement free-  dom [42] in order to relate the LECs appearing in the  EM currents (Ci) to our interaction LECs (C′  i):  C2 = −4 (C′  2 + 3 C′  4 + C′  7) ,  (6a)  C4 = −4 (C′  2 − C′  4 − C′  7) ,  (6b)  C5 = −C′  5 ,  (6c)  C7 = −8 C′  7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (6d)  The C′  i were determined from fits to NN scattering  data [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Details on the mapping between the LECs  identified via the Fierz rearrangement are provided in  the Supplemental Material [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Next, there are contributions to the EM currents  from the N3LO and N4LO pion-nucleon (πN) Lagrangian  which are proportional to the di and li LECs, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  All except three terms proportional to the πN  4  LECs vanish in the q → 0 limit, and do not contribute  to the evaluation of nuclear magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The  three which contribute are proportional to d8, d9 and  d21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Here, we chose to exclude all operators proportional  to di, as these LECs are of higher order compared to the  employed nuclear Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Finally, there arise two unknown purely EM LECs that  must be fit to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' These are called L1  and L2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [13] or dS  1 and dV  1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41], respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We fit these two unknown EM LECs using two  prescriptions, hereafter denoted scheme A and scheme  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In scheme A, we have fit the unknown LECs to repro-  duce the experimental values of the magnetic moments  of 3H and 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For scheme B we have performed a linear  least squares fit of the two LECs to the entire data set  for the magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Using the naming convention  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41], we find the LEC values for scheme A to be  dS  1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='09596 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0018 and dV  1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='11085 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='00099  with a χ2 per degree-of-freedom (DOF) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='48078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For  scheme B we find dS  1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='03958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0035 and dV  1 =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='06420 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0028 with a χ2 per DOF of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='5035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Results— In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 1, we show AFDMC results for the  magnetic moments at each order in the chiral expansion  for the EM currents as expressed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  For all  calculations, we have used the N2LOE1 interaction and  included all contributions to the EM currents up to the  specified order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' None of these results include theoreti-  cal uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Generally, we observe a good  order-by-order convergence for all nuclei and reproduce  experimental data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' As a simple measure of the con-  vergence, we consider the absolute difference between the  estimated magnetic moment and the experimental value  averaged across all nuclei at each order in the EM cur-  rents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='18 at LO, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='13 at NLO, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='13  at N2LO, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='09) at N3LO in scheme A (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Next we note that N3LO contributions seem to be  larger than expected based on lower-order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  To highlight this, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 3 we show the total contribution  of each order of the chiral expansion to the total magnetic  moment using the Pisa PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' One feature of the magnetic  moment immediately noted is the nearly universal large  size of the N3LO contribution, particularly with respect  to the relativistic corrections at N2LO, with most nuclei  displaying O(µNLO) ∼ O(µN3LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This feature does not  depend on the choice of the fitting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Next, we study the systematic error arising from the  truncation of the chiral expansion for the magnetic mo-  ments in both PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For this, we use the simple prescrip-  tion introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [15] and perform AFDMC calcu-  lations of the magnetic moments order-by-order in the  chiral expansion for both the nuclear interaction Hamil-  tonian and the electromagnetic current operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In this  work, we performed this uncertainty estimate for 6Li, 7Li,  8Li, and 15N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For the Pisa PC, the theoretical uncertainty  at each order is found using the following recursive ex-  2H  3H  3He  6Li  7Li  8Li  15N  15O  17O  17F  µLO  µNLO  µN2LO  µN3LO  � µn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='821 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='463 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='988 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='394 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='822 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='665  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='832 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='757 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='360 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='727 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='770 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='826 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='632  −5  −4  −3  −2  −1  0  log10(|µn|)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Contribution to the magnetic moment at each order  in the chiral expansion of the nuclear EM current for each  nucleus considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' We show results for the Pisa PC [41] but  the sum of the LO and NLO results is equivalent to the NLO  contribution in the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The N2LO row represents  only the 1/m2 relativistic correction while the N3LO results  are obtained by using the LECs fit in scheme B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The number  appearing in each cell represents the MC estimate of the con-  tribution, while the color of the cell represents the logarithm  of the magnitude of the contribution to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  pressions:  δ(µLO  LO) = Q|µLO  LO| ,  (7)  δ(µN2LO  NLO ) = max  �  Q|µN2LO  NLO − µLO  LO|, Q2δ(µLO  LO)  �  ,  (8)  δ(µN3LO  N2LO) = max  �  Q|µN3LO  N2LO − µNLO  NLO|, Qδ(µNLO  NLO)  �  ,  (9)  where we take Q = mπ/Λ, and Λ = 500 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In  the above expressions superscripts denote the highest in-  cluded order in the expansion of the currents, and sub-  scripts denote the highest included order in the nuclear  interaction Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' At leading order ((Q/Λ)−2 in  the Pisa PC), the magnetic moment is calculated using  both the Hamiltonian and the EM currents at LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' At  NLO ((Q/Λ)−1 in both PCs), the magnetic moment is  calculated using both the NLO Hamiltonian and EM cur-  rents through N2LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This is because the Hamiltonian at  NLO is two powers of Q/Λ greater than the Hamiltonian  at NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For consistency, we include all current opera-  tors which are less than or equal to 2 powers of Q/Λ  greater than the operators at LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Finally, at N2LO, the  magnetic moment is calculated by employing the N2LO  Hamiltonian and the sum of the EM currents up to, and  including, N3LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Similarly,  we estimate the uncertainty using the  Bochum PC for the EM currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This PC is more consis-  tent with the employed interaction Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For the  equivalent theoretical uncertainty estimation in the con-  sistent power counting, only the Bochum group definition  of the NLO current is employed, as the LO and N2LO  contributions to the magnetic moments vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Then, in  this PC, differences in the magnetic moments arise due  to the nuclear interactions order-by-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The theoreti-  cal uncertainty estimate using the Bochum group PC is  5  LO  NLO N2LO  LO  NLO N2LO  LO  NLO N2LO  LO  NLO N2LO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='6  µ/µexp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  6Li  7Li  8Li  15N  Bochum  Pisa (Scheme A)  Pisa (Scheme B)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Order-by-order contributions to the magnetic mo-  ment, relative to the experimentally determined value, for the  indicated nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The abscissa labels indicate the highest or-  der in the chiral expansion utilized in the nuclear interaction  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  given by  δ(µNLO  NLO) = Q|µNLO  NLO| ,  (10)  δ(µNLO  N2LO) = max  �  Q|µNLO  N2LO − µNLO  NLO|, Qδ(µNLO)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' (11)  The ratio of the QMC estimate of the magnetic mo-  ment for the selected nuclei to the experimentally deter-  mined value is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 4 for both the Pisa and  Bochum PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For all nuclei, we observe a systematic be-  havior within the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' However, for 15N, the cal-  culations seem to predict a magnetic moment 40% above  the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For the Pisa PC, we observe a  systematic convergence for 6Li, 7Li but for 8Li and 15N,  in contrast, the order-by-order results seem to be less  systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In particular, the theoretical uncertainty is  largest at NLO in this PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This might point to a sys-  tematic underestimate of the characteristic momentum  scale Q for these nuclei, and a more detailed investigation  into the one- and two-nucleon momentum distributions  in these nuclei is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In addition, this might show  that consistent PCs for interactions and currents need to  be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The choice of fitting scheme for the N3LO  EM LECs also produces some variance in the estimated  values of the magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' However, this variance  is of the same order as the theoretical uncertainty esti-  mated using the NLO EM currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Discussion and conclusions— We performed AFDMC  calculations of the magnetic moments of several light nu-  clei with nuclear interactions and EM currents derived  from chiral EFT and found agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  These results are supported by comparing supplementary  AFDMC calculations of the ground-state binding ener-  gies and charge-radii to experiment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Addition-  ally, we explored different power countings of the chiral  EM currents compared to the power counting of the nu-  clear interaction and the order-by-order convergence of  the magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  While our accurate magnetic moment results are valu-  able in verifying the different components of these ab ini-  tio calculations, our uncertainty analysis indicates that  the consistency between employed nuclear interactions  and EM currents is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Our results for the consis-  tent power counting shows a more natural order-by-order  convergence pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Future work to derive consistent  currents for the interactions used in AFDMC is impor-  tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Furthermore, given the importance of uncertainty  quantification, more work needs to be done to ensure the  proper estimates for momentum scales are being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  This is difficult using the simple uncertainty estimate  used in this work [15], but employing uncertainty quan-  tification tools based on Gaussian Processes [16] might  be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Finally, given the prominence of the N3LO  EM contributions to the calculated magnetic moments in  this work, the convergence up to N3LO and fulfilling the  corresponding continuity equation are crucial [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  We thank Alex Gnech, Rocco Schiavilla, Saori Pas-  tore, Hermann Krebs, and Ronen Weiss for useful dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' DE-AC52-06NA25396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Department of En-  ergy, Office of Science, Office of Advanced Scientific Com-  puting Research, Scientific Discovery through Advanced  Computing (SciDAC) NUCLEI program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' was also supported by the De-  partment of Energy Early Career Award Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Com-  puter time was provided by the Los Alamos National  Laboratory Institutional Computing Program, which is  supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Department of Energy National  Nuclear Security Administration under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  89233218CNA000001, and by the National Energy Re-  search Scientific Computing Center (NERSC), which is  supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Department of Energy, Office of  Science, under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' de Forest Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Walecka, Advances in Physics 15,  1 (1966), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='1080/00018736600101254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Adhikari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Albataineh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Androic, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Aniol, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Armstrong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Averett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ayerbe Gayoso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Barcus,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Bellini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Beminiwattha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Benesch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Bhatt,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Bhatta Pathak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Bhetuwal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Blaikie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Cam-  pagna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Camsonne, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Cates, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Clarke,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Cornejo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Covrig Dusa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Datta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Deshpande,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Dutta, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Feldman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Fuchey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gaskell,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gautam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gericke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ghosh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Rosner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rossi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sabati´e, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Salgado,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schmookler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schumacher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Segarra,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Smith, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sokhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sparveris,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Stepanyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Voskanyan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Tews, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 69, 279 (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='04868  [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lynn, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tews, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gezerlis,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schmidt, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schwenk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lynn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Chen,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Detmold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schwenk, Journal  of Physics G: Nuclear and Particle Physics 47, 045109  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lynn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Schmidt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schwenk, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  120, 122502 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Cruz-Torres,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Li,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Liu,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Liyanage, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Machleidt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Marcucci,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Markowitz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' McClellan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Meekins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Mey-Tal  Beck, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Meziani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Michaels, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Mihoviloviˇc, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ne-  lyubin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Nuruzzaman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Nycz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Obrecht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ol-  son, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Owen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pandey, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pandey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pa-  padopoulou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Park, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Patsyuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Paul, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Petratos,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Piasetzky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pomatsalyuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Premathilake, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Puck-  ett, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Punjabi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ransome, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rashad, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Reimer,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Riordan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Roche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sammarruca, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Santiesteban,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sawatzky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Segarra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schmookler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Shahinyan,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' ˇSirca, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sparveris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Su, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Suleiman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Szumila-  Vance, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tadepalli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tireman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tortorici,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Urciuoli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Viviani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Weinstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wojtsekhowski,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wood, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ye, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Ye, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Zhang, Physics Letters B  797, 134890 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content='  Lonardoni,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Weiss,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Pi-  arulli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Barnea, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Higinbotham, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Piasetzky,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schmidt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Weinstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' Tews, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Reddy, The  Astrophysical Journal 860, 149 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Research 2, 022033 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [39] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Riz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pederiva, Journal of Physics  G: Nuclear and Particle Physics 47, 045106 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Sorella, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' B 64, 024512 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Schiavilla,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Baroni,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Pastore,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Piarulli,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Girlanda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Kievsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lovato, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Marcucci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Pieper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Viviani, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wiringa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' C 99,  034005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [42] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Huth, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tews, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lynn,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schwenk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' C 96, 054003 (2017), arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='03194 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [43] “Supplemental material,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  LA-UR-23-20515  Supplemental Material: Auxiliary field diffusion Monte Carlo calculations of magnetic  moments of light nuclei with chiral EFT interactions  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Martin,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Novario,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lonardoni,2, 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Carlson,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gandolfi,1 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Tews1  1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA  2XCP-2: Eulerian Codes Group, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA  FIERZ TRANSFORMATION OF NN CONTACT  INTERACTIONS ENTERING EM CURRENTS  The  quantum  states  which  describe  nuclei  are  fermionic many-body states and are therefore antisym-  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In chiral EFT, there are 14 sub-leading contact  operators at NLO and N2LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Because of the fermionic  antisymmetry requirement, out of the 14 operators only  7 are linearly independent [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Hence, the choice of oper-  ators to include in the NN potential is not unique, and  one parameterization of the potential may be related to  a different parameterization through a Fierz transforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Therefore, given a particular choice of 7 operators  and associated LECs, one can antisymmetrize the NN  potential and choose a different set of operators weighted  by consistent linear combinations of the original LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The electromagnetic (EM) current operators utilized  in this work were derived in chiral effective field the-  ory from a set of nuclear interactions which implement a  two-nucleon (NN) isospin-independent nonlocal poten-  tial at N2LO in the chiral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This chiral poten-  tial (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [2]) is  v(q, k) =C(i)  1 q2 + C2k2  + (C3q2 + C4k2)σ1 · σ2  + iC5  2 (σ1 + σ2) · k × q + C6σ1 · qσ2 · q  + C7σ1 · kσ2 · k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (1)  In this and following expressions, k = (p′ + p)/2 and  q = p′ − p where p and p′ denote the incoming and  outgoing nucleon momenta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  In contrast, the chiral potentials employed for the  AFDMC imaginary time propagation in configuration  space are isospin-dependent and local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' The momentum  space representation of this potential (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 22 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [1])  is  V (q, k) =C′  1q2 + C′  2q2τ1 · τ2  + C′  3q2σ1 · σ2  + C′  4q2σ1 · σ2τ1 · τ2  + iC′  5  2 (σ1 + σ2) · q × k  + C6(σ1 · q)(σ2 · q)  + C7(σ1 · q)(σ2 · q)τ1 · τ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (2)  To ensure that the correct NN contact LECs enter the  employed EM currents, we determine the linear trans-  formation between the Ci LECs appearing in the Pisa  parameterization of the EM currents and their NN po-  tential, and the C′  i LECs appearing in our interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The antisymmetrization of an NN potential is ob-  tained by  vas = 1  2(v − A[v]) ,  (3)  where  A[v] =1  4 (1 + σ12 + τ12 + σ12τ12)  ×v(q → −2k, k → −1  2q) ,  (4)  and σ12 = σ1 · σ2 and τ12 = τ1 · τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  After performing the antisymmetrization of the Pisa  potential (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 1) and collecting all operators, we find  vas(q, k) = 1  2  � �  C1 − 1  16(C2 + 3C4 + C7)  �  q2 + (C2 − (C1 + 3C3 + C6)) k2  − 1  16(C2 + 3C4 + C7)q2τ12 − (C1 + 3C3 + C6)k2τ12  +  �  C3 − 1  16(C2 − C4 − C7)  �  q2σ12 + (C4 − (C1 − C3 − C6)) k2σ12  − 1  16(C2 − C4 − C7)q2σ12τ12 − (C1 − C3 − C6)k2σ12τ12  + i3C5  4 (σ1 + σ2) · (k × q) + iC5  4 (σ1 + σ2) · (k × q)τ12  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='08349v1  [nucl-th]  19 Jan 2023  2  +  �  C6 − 1  8C7  �  (σ1 · q)(σ2 · q) + (C7 − 2C6)(σ1 · k)(σ2 · k)  − 1  8C7(σ1 · q)(σ2 · q)τ12 − 2C6(σ1 · k)(σ2 · k)τ12  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (5)  A different parameterization of the momentum-space potential is employed in the Norfolk interactions, and the  subsequent linear mapping between the LECs relevant for the EM currents is provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  We similarly antisymmetrize our NN potential (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 2) and find  Vas(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' k) = 1  2  �  C′  1q2 − (C′  1 + 3C′  2 + 3C′  3 + 9C′  4 + C′  6 + 3C′  7)k2  + C′  2q2τ12 − (C′  1 − C′  2 + 3C′  3 − 3C′  4 + C′  6 − C′  7)k2τ12  + C′  3q2σ12 − (C′  1 + 3C′  2 − C′  3 − 3C′  4 − C′  6 − 3C′  7)k2σ12  + C′  4q2σ12τ12 − (C′  1 − C′  2 − C′  3 + C′  4 − C′  6 + C′  7)k2σ12τ12  + i3C′  5  4 (σ1 + σ2) · (q × k) + iC′  5  4 (σ1 + σ2) · (q × k)τ12  + C′  6(σ1 · q)(σ2 · q) − (2C′  6 + 6C′  7)(σ1 · k)(σ2 · k)  + C′  7(σ1 · q)(σ2 · q)τ12 − (2C′  6 − 2C′  7)(σ1 · k)(σ2 · k)τ12  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (6)  We can immediately see that C′  5 = −C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  To find  the relationship between the other 6 LECs, we choose  to match the operator set  {q2, q2τ12, q2σ12, q2σ12τ12, (σ1·q)(σ2·q), (σ1·q)(σ2·q)τ12} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  From these operators, we may relate each primed LEC  to a linear combination of the unprimed LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Inverting  this permits us to express the unprimed LECs in terms  of the primed LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  1  16  �  �  �  �  �  �  �  �  16 −1  0  −3  0  −1  0  −1  0  −3  0  −1  0  −1 16  1  0  1  0  −1  0  1  0  1  0  0  0  0  16 −2  0  0  0  0  0  −2  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  C1  C2  C3  C4  C6  C7  �  �  �  �  �  �  �  �  =  �  �  �  �  �  �  �  �  1  1  1  1  1  1  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  C′  1  C′  2  C′  3  C′  4  C′  6  C′  7  �  �  �  �  �  �  �  �  (7)  Denoting the matrix in parenthesis, without the 1/16  prefactor, as M, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' 7 can be more compactly written  as M ⃗C = 16⃗C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Inverting this linear system permits  us to write the Pisa (unprimed) LECs in terms of linear  combinations of our (primed) LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' For the inversion,  we find that  M−1 = 1  16  �  �  �  �  �  �  �  �  1 −1 0  0  0  0  0 −4 0 −12 0 −4  0  0  1  −1  0  0  0 −4 0  4  0  4  0  0  0  0  1 −1  0  0  0  0  0 −8  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (8)  Thus, we obtain  �  �  �  �  �  �  �  �  C1  C2  C3  C4  C6  C7  �  �  �  �  �  �  �  �  =  �  �  �  �  �  �  �  �  1 −1 0  0  0  0  0 −4 0 −12 0 −4  0  0  1  −1  0  0  0 −4 0  4  0  4  0  0  0  0  1 −1  0  0  0  0  0 −8  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  C′  1  C′  2  C′  3  C′  4  C′  6  C′  7  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  (9)  We have further validated that different choices for the  set of operators in the LEC matching procedure leads to  the same mapping between LECs for the two different  parameterizations of the N2LO chiral potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  3  POWER COUNTING COMPARISON  The power counting (PC) schemes employed by the  Pisa and Bochum groups in their respective derivation  of the electromagnetic currents differ in how they count  inverse powers of the nucleon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' In table I we show  the highest order included at each order in the chiral  expansion for the Hamiltonian we employ, as well as the  expansion order for the current operators derived by the  Pisa and Bochum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Order H JPisa JBochum  LO  Q0 Q−2  Q−3  NLO Q2 Q−1  Q−1  N2LO Q3  Q0  Q0  N2LO Q4  Q1  Q1  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Highest order term in the chiral expansion of the  Hamiltonian employed in this work (H column), the EM  current operators derived by the Pisa group (JPisa column),  and the EM current operators derived by the Bochum group  (JBochum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  The total summed contribution to the EM currents is  identical between the Pisa and Bochum PC’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' However,  the LO contribution to the EM vector current is 0 in the  Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' This is because the term which appears at  LO in the Pisa PC is proportional to Q/m (the ratio of  momentum to the nucleon mass) which is counted as one  order higher in the Bochum PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  INDIVIDUAL CONTRIBUTIONS TO THE EM  CURRENT  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Gezerlis,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Tews,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Epelbaum,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  Gandolfi,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Hebeler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Nogga, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schwenk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  111, 032501 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Pastore, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Girlanda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Schiavilla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Viviani,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Wiringa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' C 80, 034004 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
+page_content=' Gnech and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+page_content=' C 106, 044001  (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E_T4oBgHgl3EQf3xz2/content/2301.08349v1.pdf'}
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+Towards Verifying the Geometric Robustness of Large-scale Neural Networks
+Fu Wang1, Peipei Xu 2*, Wenjie Ruan1†, Xiaowei Huang2
+1 Department of Computer Science, University of Exeter, Exeter, EX4 4QF, UK
+2 Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK
+{fw377, w.ruan}@exeter.ac.uk, {peipei.xu, xiaowei.huang}@liverpool.ac.uk
+† Corresponding Author
+Abstract
+Deep neural networks (DNNs) are known to be vulnerable
+to adversarial geometric transformation. This paper aims to
+verify the robustness of large-scale DNNs against the com-
+bination of multiple geometric transformations with a prov-
+able guarantee. Given a set of transformations (e.g., rotation,
+scaling, etc.), we develop GeoRobust, a black-box robust-
+ness analyser built upon a novel global optimisation strat-
+egy, for locating the worst-case combination of transforma-
+tions that affect and even alter a network’s output. GeoRo-
+bust can provide provable guarantees on finding the worst-
+case combination based on recent advances in Lipschitzian
+theory. Due to its black-box nature, GeoRobust can be de-
+ployed on large-scale DNNs regardless of their architectures,
+activation functions, and the number of neurons. In practice,
+GeoRobust can locate the worst-case geometric transforma-
+tion with high precision for the ResNet50 model on Ima-
+geNet in a few seconds on average. We examined 18 Im-
+ageNet classifiers, including the ResNet family and vision
+transformers, and found a positive correlation between the
+geometric robustness of the networks and the parameter num-
+bers. We also observe that increasing the depth of DNN is
+more beneficial than increasing its width in terms of improv-
+ing its geometric robustness. Our tool GeoRobust is available
+at https://github.com/TrustAI/GeoRobust.
+Introduction
+Although deep neural networks have achieved human-level
+performance, concerns are raised about their safety and re-
+liability (Huang et al. 2020; Ruan et al. 2019; Wang et al.
+2022; Ruan, Yi, and Huang 2021). In computer vision tasks,
+while deep learning models are known to be vulnerable
+to adversarial perturbations in pixel values (Szegedy et al.
+2014; Croce and Hein 2020; Yin, Ruan, and Fieldsend 2022;
+Mu et al. 2021, 2022), Engstrom et al. (2019) show that a
+slight rotation of an input example can also fool DNNs. Al-
+though modern DNNs are believed to be able to learn geo-
+metric information from training data (Bakry et al. 2016),
+they are not yet invariant to simple adversarial geometric
+transformations (Zhang et al. 2020).
+*This work was done when Peipei was visiting the University
+of Exeter.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Although additive adversarial perturbation has received
+tremendous attention, geometric transformations are more
+common and applicable in the physical world but have
+been less studied. There is no efficient solution for search-
+ing the worst-case adversarial transformation with provable
+guarantees for large-scale DNNs. Engstrom et al. (2019)
+showed that, although adversarial geometric transformation
+can be discovered through the random pick, it is a highly
+non-convex task that gradient-ascent-based adversarial at-
+tacks perform poorly. Pei et al. (2017) adopt the exhaustive
+search to find the worst-case transformation that alters the
+target model’s prediction, but its computational complex-
+ity grows exponentially with the dimension of the consid-
+ered transformations. Some researchers have adopted ver-
+ification techniques (Weng et al. 2018; Singh et al. 2019;
+Cohen, Rosenfeld, and Kolter 2019) to analyse geometric
+transformations.Relaxation-based approaches (Weng et al.
+2018; Balunovi´c et al. 2019; Mohapatra et al. 2020) require
+the Lp-norm based constraint on pixel space for an input ex-
+ample. However, as shown in Fig. 1, geometric transforma-
+tion can significantly change the values of the pixels, leading
+to severe violence against the constraint in the pixel space
+while still preserving human imperceptibility. Therefore,
+the scalability of Lp norm-based verification is limited for
+dealing with geometric transformation. Recently, Fischer,
+Baader, and Vechev (2020); Li et al. (2021) showed that
+randomised smoothing could be utilised to verify robust-
+ness against a single geometric transform, but their methods
+cannot handle the combination of multiple geometric trans-
+formations. In addition, current methods can only provide a
+verifiable lower bound for verification purposes but cannot
+identify the worst-case geometric transformation that would
+actually minimise the model’s confidence and potentially al-
+ter its prediction.
+In this paper, we develop a novel black-box evalua-
+tion framework, GeoRobust, to study the geometric robust-
+ness, i.e., the robustness of the model against adversar-
+ial geometric transformations. GeoRobust takes advantage
+of both recent developments in Lipschitzian optimisation
+methods (Jones, Perttunen, and Stuckman 1993; Gablon-
+sky 2001) that provide provable guarantees on locating the
+worst-case transformation and the efficient parallel compu-
+tation on Graphic Processing Units (GPUs). The workflow
+of GeoRobust is presented in Fig. 1. Given a set of geo-
+arXiv:2301.12456v1  [cs.LG]  29 Jan 2023
+
+0
+1
+1
+1.1
+0.9
+Original 
+Parameter space
+United search space
+Normalisation
+Difference 
+Difference 
+Difference 
+First iteration
+Tenth iteration
+Fiftieth iteration
+Initialisation
+Optimisation
+…
+…
+Transformed
+Transformed
+Transformed
+-20o
+20o
+Rotation: [-20o, 20o]
+Scaling: [0.9, 1.1]
+Figure 1: Schematic illustration of GeoRobust framework. After normalising the parameter space to a unit search space, GeoRo-
+bust performs a sequence of space divisions to find the global worst-case transformation.
+metric transformations and an input example, GeoRobust
+converges to the worst-case combination for minimising an
+adversarial loss within a finite number of queries. We en-
+able GeoRobust to better utilise GPUs by easing its sam-
+pling condition. Besides, a lower bound estimation method
+is also introduced to make GeoRobust an anytime verifica-
+tion, which can produce the lower and upper bound of the
+worst-case loss value whenever the algorithm stops.
+In summary, our key contributions lie in three aspects.
+1. We prove the geometric transformation done by spatial
+transformation network (STN) (Jaderberg et al. 2015) is
+Lipschitz continuous. By stacking STN module in front
+of Lipschitz-continuous neural networks, we can analyse
+their geometric robustness with guaranteed convergence.
+2. We develop GeoRobust, a black-box geometric robust-
+ness analyser, by taking advantage of Lipschitzian opti-
+misation theory (Jones, Perttunen, and Stuckman 1993).
+The convergence of GeoRobust is theoretically guaran-
+teed, and it is also highly efficient in practice. In our ex-
+periment, GeoRobust could find the worst-case adversar-
+ial transformations on an ImageNet image to evaluate a
+ResNet50 classifier with desirable precision in seconds.
+3. We use GeoRobust to benchmark the geometric robust-
+ness of state-of-the-art ImageNet classifiers, including
+the ResNet family and vision transformers. There are two
+main takeaways from our experiments: i) the geometric
+robustness of DNNs has a positive correlation with the
+number of parameters; and ii) increasing the number of
+layers seems to be more effective than adding more hid-
+den units in each layer in improving the geometric ro-
+bustness of DNNs.
+Preliminaries
+Lipschitz continuity and Lipschitzian optimisation
+Pre-
+vious studies indicate that the majority of modern DNNs are
+Lipschitz continuous (Szegedy et al. 2014; Ruan, Huang,
+and Kwiatkowska 2018; Virmaux and Scaman 2018). The
+Lipschitz constant for a DNN gives an upper bound on how
+fast its output could change when small perturbations are
+applied to its input (Szegedy et al. 2014; Ruan, Huang, and
+Kwiatkowska 2018). Such a concept is closely related to the
+robustness of DNN, but exactly computing the smallest Lip-
+schitz constant of a DNN is proven to be an NP-hard prob-
+lem (Virmaux and Scaman 2018).
+Relying on the Lipschitz continuity, Lipschitzian opti-
+misation is a query-based optimisation method that uses
+the Lipschitz constant of the objective function to grad-
+ually narrow the search space and locate the global op-
+timum (Piyavskii 1972; Shubert 1972; Ruan, Huang, and
+Kwiatkowska 2018; Xu, Ruan, and Huang 2022; Zhang,
+Ruan, and Xu 2023). While the Lipschitz constant of the
+objective function is necessary for classic Lipschitzian op-
+timisation, a novel Lipschitzian optimisation solution, DI-
+RECT (Jones, Perttunen, and Stuckman 1993; Jones and
+Martins 2021a), does not require the Lipschitz constant to
+find the global optimum. As detailed in the methodology
+section, we improve the DIRECT method for studying the
+geometric robustness of DNNs.
+Geometric transformations
+Geometric transformations
+are element-wise manipulation that can be conducted via
+several physically meaningful parameters (Szeliski 2022).
+Given an image example, x ∈ RH×W ×C with height H,
+width W, and colour channels C, the geometric transfor-
+mation Tθ is carried out on each channel equally. Let xc ∈
+RH×W be any channel of x and the output of Tθ be x′
+c. For a
+pixel in x′
+c with index (x′
+i, y′
+i), its value Vi is mapped to the
+pixel indexed by (xj, yj) in xc via a transformation matrix
+Aθ, i.e.,
+�
+xj
+yj
+�
+= Aθ
+�
+�
+x′
+i
+y′
+i
+1
+�
+� =
+�
+θ11
+θ12
+θ13
+θ21
+θ22
+θ23
+� �
+�
+x′
+i
+y′
+i
+1
+�
+� .
+(1)
+We adopt Spatial Transformation Network (STN) (Jaderberg
+et al. 2015) to conduct the geometric transformation and use
+the bilinear sampling kernel to handle the projection of the
+non-integer index, which gives the transformation result:
+Vi =
+H
+�
+h
+W
+�
+w
+Uhw max (0, 1 − |xi − w|) max (0, 1 − |yi − h|) ,
+(2)
+where Uhw denotes the value of a pixel, indexed by (xh, yw),
+in xc, and the index does not need to be an integer.
+
+Problem formulation
+Given a neural network F : RN → RK, an input example
+x ∈ RN, and its label y ∈ {1, . . . , K}, we aim to find the
+optimal combination of several geometric transformations
+Tθ that can minimise ℓ : Rn → R, i.e.,
+min
+θ∈Θ ℓ(θ; F, x, y),
+(3)
+where Θ is the adversarial space that contains all feasible θ,
+and ℓ denotes the margin loss defined as
+ℓ(θ; F, x, y) = Fy(Tθ(x)) −
+max
+k∈{1,...,K}\{y} Fk(Tθ(x)),
+(4)
+which allow us to determine whether the model F can be
+fooled by Tθ(x) via verifying the lower bound of ℓ. Specif-
+ically, if infθ∈Θ ℓ(θ; F, x, y) > 0 is satisfied, the robustness
+of the model F on the example x would be verified.
+Regarding geometric transformation, we consider rota-
+tion, translation, and isotropic scaling. The corresponding
+transformation matrix Aθ can be written as
+Aθ =
+�
+θ11
+θ12
+θ13
+θ21
+θ22
+θ23
+�
+=
+�
+λ cos γ
+− sin γ
+thor
+sin γ
+λ cos γ
+tvrt
+�
+,
+(5)
+where λ is the scaling factor, γ is the rotation angle, and thor
+and tvrt control the horizontal and vertical translation.
+Methodology
+This section introduces a Lipschitzian optimisation-based
+approach, GeoRobust, to search for the worst-case trans-
+formation. GeoRobust is composed of four components: 1)
+a Lipschitz continuous module for performing geometric
+transformations; 2) a space division procedure; 3) a mech-
+anism to select Potential Optimal (PO) subspaces that are
+more likely to contain the global minimum points than oth-
+ers; 4) and an anytime estimation of the global minimum.
+While the space division procedure and the PO subspace se-
+lection are adopted from the DIRECT algorithm, we extend
+the definition of PO subspace and encourage the algorithm
+to query more subspaces at each iteration. Because all evalu-
+ations at the same iteration can be done parallelly in a single
+forward propagation, our method could reach convergence
+within reduced iterations. Furthermore, GeoRobust can esti-
+mate the model’s Lipschitz constant and produce a reason-
+able lower bound of the given loss function. The pseudocode
+of GeoRobust and related proofs are provided in Appendix1.
+Notations
+Given a matrix P ∈ R2×n, one can define an n-
+dimensional parameter space, in which the upper and lower
+bounds on i-th transformation factors are given by Pi, where
+i ∈ {1, . . . , n}. GeoRobust normalises the parameter space
+into an n-dimensional unit hypercube, namely the search
+space, whose centre point corresponds to the identical trans-
+formation. For a hyperrectangle with index q, denoted by
+Hq, in the search space, the value of the objective function
+at its centre point cq is denoted by ℓ(cq). We denote by lq
+i
+the side length w.r.t. the i-th dimension, and by ei the unit
+vector along i-th dimension. The size of Hq is defined as
+1Appendix
+can
+be
+found
+at
+https://github.com/TrustAI/
+GeoRobust/blob/main/appendix.pdf
+σq = maxi∈{1,...,n}
+1
+2lq
+i , which is the same L∞ norm based
+measurement used by Gablonsky (2001). For all hyperrect-
+angles within the search space, we denote by H the set of
+hyperrectangles’ indexes, and by ℓmin = minp∈H ℓ(cp) the
+current best query result. The Lipschitz constant of the ob-
+jective function w.r.t. the search space is denoted by ˜K.
+GeoRobust computes the slope ˆK between queried points
+w.r.t. the parameter space during optimisation and produces
+ℓ∗
+min, an estimation of the lower bound of ℓmin.
+Geometric transformations module
+The convergence guarantee of GeoRobust is related to the
+Lipschitz continuity of the target model. We give the fol-
+lowing lemma to show that geometric transformations with
+bilinear sampling kernel are Lipschitz continuous, which
+means, as long as a DNN model is Lipschitz continuous,
+stacking a geometric transformation module in front of it
+would not compromise the Lipschitz continuity (Tsuzuku,
+Sato, and Sugiyama 2018).
+Lemma 1.
+Given an input image example x ∈ RH×W ×C
+and the ranges of transformation factors, the first-order
+derivative of geometric transformation with bilinear sam-
+pling w.r.t. each transformation factor is bounded.
+Finding optimal geometric transformation
+GeoRobust first divides the search space into subspaces ac-
+cording to the query results at their centre points. Then some
+subspaces that are more likely to contain the global mini-
+mum than others will be chosen as PO subspaces. GeoRo-
+bust separates selected PO spaces and identifies new ones
+throughout each iteration of the optimisation process till the
+termination criteria are satisfied.
+Space division
+As the only hypercube after initialisation,
+the initial PO subspace is the united search space itself.
+GeoRobust trisects the subspace and assigns the generated
+new subspace according to the query result, where the larger
+hyperrectangles include the better query result. Without loss
+of generality, let Hp be a PO subspace, which is an n-
+dimensional hyperrectangle containing m dimensions with
+long sides of a length 3−d, where m ≤ n, and n−m dimen-
+sions with short sides of a length 3−d−1. GeoRobust ignores
+short sides and queries the value of the object function at
+the points c ± 3−d−1ei, where i ∈ {1, . . . , m}. For each
+dimension with long sides, the best query result is given by
+wi = min
+�
+ℓ(c + 3−d−1ei), ℓ(c − 3−d−1ei)
+�
+.
+(6)
+As GeoRobust performs trisection only during division, the
+sizes of new subspaces are deterministic. By dividing the
+above Hp, GeoRobust creates 2m + 1 new subspaces, in-
+cluding 3 sub-hypercubes with the side length of 3−d−1 and
+m−1 pairs of hyperrectangles, which have 1 to m−1 dimen-
+sions with long sides of length 3−d. The point corresponding
+to the best query result, mini∈{1,...,m} wi, is a centre point
+of a hyperrectangle with m − 1 long sides. A visualisation
+of this space division procedure is presented in Appendix.
+Overall, such a division strategy encourages GeoRobust to
+divide the search space uniformly and further explore the
+
+area around the current best result, which we will detail later
+in the PO subspace selection.
+Identifying potential optimal subspaces
+The space di-
+vision procedure creates new subspaces, and the next step
+is to locate new PO subspaces for further division. Ideally,
+a PO hyperrectangle Hp is expected to satisfy two condi-
+tions (Jones, Perttunen, and Stuckman 1993):
+ℓ (cp) − ˜Kσp ≤ ℓ (cq) − ˜Kσq, ∀q ∈ H,
+(7)
+ℓ (cp) − ˜Kσp ≤ ℓmin − τ |ℓmin| .
+(8)
+Inequation (7) indicates that only the hyperrectangles with
+the potential to improve the current ℓmin can be chosen as
+PO subspaces. Meanwhile, the second condition (8) ensures
+that the possible improvement in the chosen subspaces is
+greater than τ |ℓmin|, where a reasonable choice of τ is be-
+tween 10−3 and 10−7 (Jones and Martins 2021b). Taking
+advantage of DIRECT optimisation, GeoRobust does not
+need to know the Lipschitz constant. The following lemma
+demonstrates how to search for PO hyperrectangles in the
+absence of ˜K.
+Lemma 2. (Gablonsky 2001) Given the index set H and a
+positive tolerance τ > 0. Let ℓmin denote the current best
+query result. Let Hp
+1 = {q ∈ H : σq < σp}, Hp
+2 = {q ∈ H :
+σq > σp} and Hp
+3 = {q ∈ H : σq = σp}. A hyperrectangle
+Hp is said to be potentially optimal if
+ℓ(cp) ≤ ℓ(cq), ∀q ∈ Hp
+3,
+(9)
+and there is a ˜K > 0 such that
+max
+q∈Hp
+1
+ℓ (cp) − ℓ (cq)
+σp − σq
+≤ ˜K ≤ min
+q∈Hp
+2
+ℓ (cq) − ℓ (cp)
+σq − σp
+,
+(10)
+and
+�
+�
+�
+τ ≤
+ℓmin−ℓ(cp)
+|ℓmin|
++
+σp
+|ℓmin| minq∈Hp
+2
+ℓ(cq)−ℓ(cp)
+σq−σp
+,
+if ℓmin ̸= 0,
+ℓ (cp) ≤ σp minq∈Hp
+2
+ℓ(cq)−ℓ(cp)
+σq−σp
+,
+otherwise.
+(11)
+Generalising PO conditions to unleash the power of
+parallel computation
+The conditions in Lemma 2 are meant to select a small num-
+ber of subspaces to reduce the total number of function eval-
+uations, which is typically the most time-consuming proce-
+dure. However, by leveraging modern deep learning frame-
+works, one can easily query multiple examples on a target
+DNN via a single forward propagation on GPUs, where the
+computational time difference between evaluating a single
+sample and a batch of samples is marginal. Therefore, re-
+laxing the constraint given by inequation (9), we define the
+following α candidate set to select more subspaces.
+Definition 1 (α candidate set). Following the same notions
+in Lemma 2, we define the α candidate set as
+Hα =
+�
+∅,
+if maxp∈Hp
+3 sp ≤ 0,
+{p1, . . . , pα′ : max �α′
+j=1 spj},
+otherwise,
+(12)
+where α′ ≤ α and the optimal score sp is given by
+sp = min
+q∈Hp
+2
+ℓ (cq) − ℓ (cp)
+σq − σp
+− max
+q∈Hp
+1
+ℓ (cp) − ℓ (cq)
+σp − σq
+.
+(13)
+Modifying the condition (10) into the score described in
+Eq. (13) allows us to rank the potential minimum contained
+by subspaces with the same size. The proposed α candidate
+set is easy to control. When α = 1, Definition 1 degrades
+to condition (10), while increasing α, GeoRobust explores
+more subspaces in each iteration. As we will describe in the
+next section, all subspaces will get subdivided by GeoRobust
+after several iterations. Instead of only partitioning spaces
+satisfying inequation (7), α candidate set also selects hyper-
+rectangles that would likely satisfy Lemma 2 in the next few
+rounds. While the number of queries would rise, using the
+α candidate set enables GeoRobust to discover the optimal
+subspace quickly. On the other hand, because the function
+evaluations can be done in parallel on GPUs, replacing con-
+dition (10) with α candidate set only has a small influence
+on the computational time cost.
+Stop criteria and convergence analysis
+In practice, GeoRobust is limited by three factors: the max-
+imal number of iteration T, the maximal number of queries
+Q, and the maximal number of trisection along each dimen-
+sion, which is denoted by depth D. The first two stop criteria
+are straightforward. We stop the optimisation once the com-
+putational budget runs out. The limitation on depth is ap-
+plied for two reasons (Gablonsky 2001). On the one hand, it
+puts a limitation on the smallest size of subspaces. By doing
+so, GeoRobust is compelled to halt local search and encour-
+aged to conduct global exploration when the current optimal
+subspace is sufficiently small, which accelerates the conver-
+gence. On the other hand, defining the smallest subspace size
+also sets an upper bound for the number of queries. For an
+n-dimensional search space, GeoRobust could conduct up to
+3nD times of queries, which is equivalent to a grid search.
+Besides, our implementation adopted the L∞ norm to mea-
+sure hyperrectangles’ size (Gablonsky 2001). Such a mea-
+surement simplifies both space division and PO space se-
+lection. For the former, the L∞ norm is more computation-
+ally efficient than the Euclidean norm (Jones and Martins
+2021b). For the latter, hyperrectangles are grouped by their
+longest side length under the L∞ norm, which introduces
+less number of different sizes to consider.
+Convergence analysis
+GeoRobust is guaranteed to con-
+verge to the global minimum within a pre-defined small tol-
+erance after a finite number of queries if the objective func-
+tion is continuous (Jones, Perttunen, and Stuckman 1993).
+This guarantee comes from the following observation shown
+in Remark 1.
+Remark 1. (Gablonsky 2001) Following the same notions
+in Lemma 2, there is at least one hyperrectangle Hp will be
+identified as PO subspace at each iteration, where Hp satis-
+fies Hp
+2 = ∅ and makes inequation (9) holds so that every
+hyperrectangle will be subdivided after finite iterations.
+Note that theoretically proving a specific DNN is Lips-
+chitz continuous is beyond the scope of this paper, and ex-
+isting works demonstrate the Lipschitz continuity of convo-
+lutional neural networks (Ruan, Huang, and Kwiatkowska
+2018) and vision transformers (Vuckovic, Baratin, and des
+
+Combes 2021; Wang and Ruan 2022) used in the image clas-
+sification task. In addition, given by Lemma 1, we prove that
+the geometric transformation is Lipschitz continuous. So, as
+long as the neural network satisfies a Lipschitz condition, the
+objective function (3) is Lipschitz continuous, and GeoRo-
+bust is guaranteed to locate to the global minimum after a
+sufficient number of queries. The convergence complexity
+is described in Theorem 1.
+Theorem 1. Let C be the n-dimensional united search
+space and ˜K be the Lipschitz constant of ℓ w.r.t. C. The gap
+between current minima and global minima after T itera-
+tions can be written as
+ℓmin − min
+c∈C ℓ(c) ≤ ε < ˜K · (T + 1)− 1
+n .
+(14)
+Therefore, to achieve any desired ε, we need up to
+O
+�
+( ˜K/ε)n�
+iterations.
+Estimating the global minimum
+While GeoRobust is guaranteed to find the global minimum
+eventually, we enable it to be an anytime analyser that can
+utilise intermediate query results to estimate the lower bound
+of the global minimum at each iteration.
+Recall that the parameter space is defined via the matrix
+P ∈ R2×n that contains the upper and lower bounds of n
+transformation factors. To divide m dimensions of a hyper-
+rectangle, GeoRobust samples and evaluates new points at
+c ± 3−1 · liei, where i ∈ {1, . . . , m}. The slopes along the
+i-th dimension are given by
+ˆKc+
+i = ∥ℓ(c) − ℓ(c+
+i )∥
+dc
+i
+and
+ˆKc−
+i = ∥ℓ(c) − ℓ(c−
+i )∥
+dc
+i
+,
+where c+
+i and c−
+i are short hands for c ± 3−1 · liei, and we
+denote by dc
+i = 3−1 · liPi ·
+�
+1
+−1
+�
+the distance between
+c and c+/−
+i
+within the parameter space. Then ˆKc, the local
+slope of c, is updated to be the largest local slope, i.e.,
+ˆKc = max{ ˆKc+
+1 , ˆKc−
+1 , . . . , ˆKc+
+m, ˆKc−
+m}.
+GeoRobust updates the local slope after space division so
+that it can estimate the global minimum based on the query
+results at that time. Let co denote the centre of the current
+optimal hyperrectangle Ho that has ℓ(co) = ℓmin and ˆK =
+maxq∈H ˆKcq, the lower bound of global minimum can be
+estimated via
+ℓ∗
+min = ℓ(co) − ˆKmax¯σo,
+(15)
+where we take a relaxation on σo and compute it via the
+Manhattan distance, i.e., ¯σo =
+1
+2
+�n
+i=1 do
+i . Therefore, as
+long as GeoRobust can locate a Ho containing the global
+minimum ˆℓmin, we have ℓ∗
+min ≤ ˆℓmin.
+Experiments
+Our experiments include three parts. First, we compare
+GeoRobust to state-of-the-art baseline methods for verify-
+ing robustness against three geometric transformations, i.e.,
+rotation, translation, and scaling. Then, we take advantage
+of GeoRobust to efficiently benchmark popular large-scale
+networks on ImageNet regarding their robustness over the
+combination of all three transformations. Finally, we con-
+duct an empirical analysis to study the impact of the depth
+and α conditions on the convergence of GeoRobust.
+General setup
+For geometric transformations, we denoted
+by R(γ) the rotation angle between ±γ, by S(λ) the scal-
+ing range between 1±λ, and by T(thor, tvrt) the translation
+that moves an example up to thor and tvrt pixels horizontally
+and vertically, respectively. For the model architectures, the
+MNIST classifier is a network with four convolutional layers
+and three linear layers, and the CIFAR10 classifier’s archi-
+tecture is ResNet101. We adopted the pretrained MNIST and
+CIFAR10 models released by Li et al. (2021) and utilised
+TIMM (Wightman 2019), a model zoo of ImageNet clas-
+sifiers, to investigate the geometric robustness of popular
+large-scale DNNs. Our experiment is performed on a ma-
+chine with an Intel i7-10700KF CPU, an RTX 3090 GPU,
+and 48 gigabytes of memory. More implementation details
+can be found in Appendix.
+Comparison with previous works
+Since GeoRobust only requires querying the target models’
+output, it can be easily deployed on any pretrained neural
+network. We follow the same setup used in TSS (Li et al.
+2021) and GSmooth (Hao et al. 2022) and apply GeoRo-
+bust on their pretrained models to conduct robustness veri-
+fying on MNIST and CIFAR10, where, for GeoRobust, the
+robustness of an example is verified if the corresponding
+lower bound ℓ∗
+min > 0. Because exhaustive search is com-
+putationally infeasible, we implement a grid search with a
+sufficient computational budget to run through the param-
+eter space to test whether the model’s prediction on each
+input example can be altered, which serves as the ground
+truth on the verified accuracy. Please note that GeoRobust
+works on L∞-norm based parameter space, while Li et al.
+(2021) uses L2-norm based constraint on the translation,
+which is currently inapplicable for GeoRobust. Therefore,
+the comparison is done on rotation and scaling transforma-
+tions. Although DeepG (Balunovi´c et al. 2019) and TSS (Li
+et al. 2021) can analyse the geometric robustness of Ima-
+geNet classifiers, but they are time-consuming and ineffi-
+cient when dealing with transformation combinations. Be-
+sides, we failed to properly reload the ImageNet classifiers
+evaluated by TSS (see Appendix for details). Therefore, the
+evaluation on ImageNet is done on a ResNet50 model, the
+same architecture used by Li et al. (2021), against different
+combinations of transformations, and we compare the per-
+formance with grid search and random pick.
+The comparison results on MNIST and CIFAR10 are
+summarised in Tab. 1, where we report the verified accuracy
+determined by each baseline method. GeoRobust outper-
+forms previous methods under all scenarios and reports the
+same verified accuracy as grid search. Such a performance
+demonstrates the effectiveness of GeoRobust in verifying
+the geometric robustness against 1-dimensional transforma-
+tion. The evaluation on ImageNet is summarised in Tab. 2,
+
+Table 1: Comparing with baseline methods on MNIST and CIFAR-10 against rotation and Scaling. We denote by − an unsup-
+ported setting and by 0% a failed verification. Baselines’ performance is adopted from (Li et al. 2021; Hao et al. 2022).
+Dataset
+Geo. Trans.
+GeoRobust
+Gsmooth
+TSS
+DeepG
+Interval
+Semantify-NN
+DistSPT
+TSS attack
+Grid search
+MNIST
+R(50◦)
+98.2%
+95.7%
+97.4%
+≤ 85.8% (R(30◦))
+≤6.0% (R(30◦))
+≤ 92.48%
+82%
+98.2%
+98.2%
+S(0.3)
+99.2%
+95.9%
+97.2%
+85.0%
+16.4%
+-
+-
+99.2%
+99.2%
+CIFAR10
+R(10◦)
+74.8%
+65.6%
+70.6%
+62.5%
+20.2%
+-
+37%
+76.4%
+74.8%
+R(30◦)
+66.4%
+-
+63.6%
+10.6%
+0.0%
+49.37%
+22%
+69.4%
+66.4%
+S(0.3)
+63.4%
+54.3%
+58.8%
+0.0%
+0.0%
+-
+-
+67.0%
+63.4%
+Table 2: Verifying geometric robustness on ImageNet.
+The target model is ResNet50, which vanilla accuracy is
+74%. Geometric transformations are R(20◦), S(0.1), and
+T(22.4, 22.4).
+Methods
+Transformation
+R
+T
+S + T
+R + T + S
+GeoRobust
+58%
+57%
+57%
+46%
+Random pick
+58%
+59%
+60%
+49%
+Grid search
+58%
+59%
+57%
+46%
+Figure 2: Comparing the global minimum found by grid
+search, random pick, and GeoRobust. The geometric trans-
+formations are the same as in Tab. 2. We mark an example as
+a match if its corresponding minimum found by a method is
+equal to or smaller than the minimum found by grid search.
+where we can see that the accuracy verified by GeoRobust
+is comparable to or better than the grid search. At the same
+time, random pick with sufficient queries can achieve a sim-
+ilar performance as grid search on locating geometric adver-
+sarial examples. Still, it tends to perform worse as the di-
+mension of parameter space becomes larger. In addition, as
+the minimum found by grid search is more likely to be the
+ground truth minimum, we mark an example as a match if
+its corresponding minimum found by a method is equal to or
+smaller than the minimum found by grid search. As shown
+in Fig. 2, we can see that the estimated lower bound achieves
+considerable precision with a limited number of queries. The
+performance on verifying only translation is slightly worse
+than other transformations, where the reason might be the
+distortion introduced by bilinear sampling.
+Runtime
+The effectiveness of GeoRobust is highly related
+to the transformation’s dimensions. In Tab. 1, GeoRobust is
+Table 3: Benchmarking the geometric robustness of eighteen
+ImageNet classifiers.
+Models
+Vanilla
+Attack
+Verified
+#Parameters
+Inception v3.
+73.60%
+28.20%
+24.20%
+2.4×107
+Inception v3adv
+75.00%
+30.60%
+27.00%
+2.4×107
+Inception v4
+78.40%
+40.20%
+36.40%
+4.3×107
+ResNet34
+64.40%
+10.60%
+9.00%
+2.2×107
+ResNet50
+78.40%
+54.00%
+31.12%
+2.6×107
+Wide ResNet50.
+81.60%
+49.40%
+40.00%
+6.9×107
+ResNet101
+80.00%
+54.20%
+48.20%
+4.5×107
+ResNet152
+79.40%
+53.80%
+46.20%
+6.0×107
+Vit32×32
+75.60%
+23.40%
+19.00%
+8.8×107
+Vit16×16
+81.40%
+41.20%
+34.20%
+8.6×107
+Large Vit16×16
+83.40%
+49.20%
+40.20%
+3.0×108
+Beit16×16.
+83.80%
+56.40%
+52.00%
+6.5×107
+Large Beit16×16
+85.60%
+65.60%
+58.20%
+2.3×108
+Gmlp
+77.96%
+40.80%
+36.80%
+1.9×107
+Mixer
+72.20%
+27.20%
+23.40%
+6.0×107
+Swin
+80.20%
+34.60%
+13.20%
+8.8×107
+Xcit
+76.80%
+40.40%
+20.60%
+8.4×107
+Pit
+79.40%
+36.60%
+20.00%
+7.4×107
+only performed on 1-dimensional transformation. Its aver-
+age runtime is 0.18 seconds and 0.72 seconds per example
+on MNIST and CIFAR10, respectively. Furthermore, the av-
+erage runtime for analysing the ResNet50 ImageNet clas-
+sifier from 1-dimensional to 4-dimensional transformations
+are 2.4 seconds, 3.6 seconds, 4.0 seconds, and 4.5 seconds.
+In comparison, according to (Li et al. 2021), it takes TSS
+17.7 and 1201.2 seconds, respectively, to analyse an MNIST
+example and an ImageNet example on the same model ar-
+chitectures w.r.t. a 1-dimensional transformation.
+Benchmarking geometric robustness
+In this section, we investigate the robustness of some popular
+DNN classifiers against geometric transformations. The pre-
+vious section demonstrates that GeoRobust can efficiently
+find a worst-case combination of transformations in a black-
+box manner. We utilise such an advantage to test large-
+scale ImageNet classifiers regarding their geometric robust-
+ness against the combination of rotation R(20◦), translation
+T(22.4, 22.4), and scaling S(0.1).
+The results are summarised in Tab. 3, and we can see
+that 1) models with more parameters appear to have bet-
+ter geometric robustness than those with fewer; 2) widening
+a network seems less beneficial than deepening it in terms
+
+0*
+Random Pick
+Xmin
+min
+100
+80
+# Match
+60
+40
+20
+R
+T
+S+T
+R+T+S
+Geometric transformations(a)
+(b)
+Figure 3: Carrying out GeoRobust with different combinations of candidates set size α and depth D on ResNet50. The black
+dot line in 3(a) corresponds to a global minimum found in a grid search with 2.5 × 105 function evaluations.
+of improving the geometric robustness; 3) the large version
+of Beit showed the best geometric robustness, whereas the
+basic Beit model is the second most robust model. This
+phenomenon suggests that bidirectional modelling could be
+helpful for DNNs learning geometric information and ob-
+taining geometric robustness; 4) comparing the performance
+between Inception V3 and Inception V3adv, an adversarially
+trained model, we can see that adversarial training does not
+significantly improve the model’s geometric robustness.
+Empirical analysis
+In Fig. 3, we carry out GeoRobust with different combina-
+tions of candidates set size α and depth D on ResNet50.
+Increasing the size of α candidates set enables GeoRobust
+to be more efficient in exploring the search space and locat-
+ing the optimal subspace. Due to the limitation on the sub-
+spaces’ minimal size, as the depth gets larger, the optimal
+transformation combination found by GeoRobust is closer to
+the ground truth worst-case, and the estimated lower bound
+is closer to the global minimum as well. It can be observed
+that the upper bound remains unchanged after convergence,
+while the estimated lower bound would be updated when-
+ever GeoRobust finds a larger local slope, which is why the
+estimations change in the right side plot of Fig. 3(a). As
+shown in Fig. 3(b), while the impact of depth on computa-
+tional cost is trivial, increasing the α candidates set would
+significantly raise the total number of function queries in
+fixed iterations. The runtime of GeoRobust with α = 1 and
+D = 5 is 6.9 seconds, and the runtime is 16.1 seconds when
+it is carried out at α = 3 and D = 7. We can see that the
+runtime increases sub-linearly with the number of queries
+because the queries are done parallelly on GPUs.
+Related works
+In this paper, we compared GeoRobust to Interval (Singh
+et al. 2019), DeepG (Balunovi´c et al. 2019), Semantify-
+NN (Mohapatra et al. 2020), TSS (Li et al. 2021),
+GSmooth (Hao et al. 2022), and DistSPT (Fischer, Baader,
+and Vechev 2020). DeepG, Semantify-NN, and Interval ex-
+tend verification techniques designed for Lp-norm based ad-
+ditive perturbation. Both Semantify-NN and GSmooth in-
+troduce small networks to simulate the geometric manipula-
+tion, where Semantify-NN adopts a linear relaxation based
+verification (Weng et al. 2018) and GSmooth applies random
+smoothing. DistSPT and TSS are also randomised smooth-
+ing based approaches, where TSS is a black-box analyser
+that is scalable to large DNNs. Besides, although the pa-
+rameter space of control factors for most geometric ma-
+nipulations is continuous, the image pixels’ coordinates are
+bounded integers, which means the possible outcomes for
+a particular set of transformations are finite. Pei et al. (Pei
+et al. 2017) empirically evaluated the robustness of DNNs
+against geometric transformations by enumerating all pos-
+sible values. In contrast, our GeoRobust is a query-based
+black-box analyser that is fundamentally different to the
+above method. We demonstrated that as long as the target
+model is Lipschitz continuous, GeoRobust can verify the ro-
+bustness of large-scale DNNs against a combination of ge-
+ometric transformations in seconds. The collaboration with
+probabilistic approaches (Zhang, Ruan, and Fieldsend 2022)
+will be explored in our future works.
+Conclusion
+In this paper, we propose a black-box analyser, GeoRo-
+bust, to efficiently verify the robustness of large-scale
+DNNs against geometric transformation. Given the bound
+of multiple geometric transformations and an input exam-
+ple, GeoRobust is guaranteed to find the worst-case manipu-
+lation that can minimise an adversarial loss without knowing
+the internal structures of the target model. Theoretically, we
+prove the Lipschitz continuity of geometric transformations
+operated by STN and analyse the convergence complexity of
+the proposed method. On the methodology side, we gener-
+alise the sampling strategy from DIRECT to better leverage
+GPU parallel computation and design an anytime estimation
+method to produce a reasonable lower bound. With GeoRo-
+bust, we systematically benchmark the geometric robustness
+of popular ImageNet classifiers. Our empirical study shows
+that larger neural networks are more robust against geomet-
+ric manipulation. Deepening a network improves its geomet-
+ric robustness better than increasing its width.
+
+α=1,D=5
+α=1,D=6
+α=1,D=7
+α= 2, D= 5
+α= 2, D= 6
+α= 2,D= 7
+α=3, D= 5
+α= 3,D= 6
+α=3, D= 7α=2
+α=3
+α=1
+Computional cost
+5000
+4500
+4000
+#Queries
+3500
+3000
+2500
+2000
+1500
+5
+6
+7
+DepthAcknowledgement.
+This work is supported by Partnership Resource Fund
+of
+ORCA
+Hub
+via
+the
+UK
+EPSRC
+under
+project
+[EP/R026173/1].
+XH has received funding from the Eu-
+ropean Union’s Horizon 2020 research and innovation pro-
+gramme under grant agreement No 956123, and is also sup-
+ported by the UK EPSRC under project [EP/T026995/1].
+FW is funded by the Faculty of Environment, Science and
+Economy at the University of Exeter.
+We would like to thank Haozhe Wang, Anjan Dutta, and
+the anonymous reviewers for their helpful comments and
+Linyi Li for sharing the pretrained models with us.
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+Appendix
+Algorithm pseudocode
+Algorithm 1: GeoRobust
+Input: An input example x, the objective function ℓ, the
+bound of the parameter space P, the number of
+function evaluation Q, the number of iterations T,
+the maximum depth D, the size of candidates set α
+Output: ℓmin with the corresponding solution cmin and an
+estimation of the ground truth minimum ℓ∗
+min
+1 Normalise the parameter space to a unit hypercube with
+centre point c0
+2 t ← 0, q ← 0
+3 Initialise the index set of hyperrectangles H = {0}
+4 Initialise the set of potential optimal space P = {0}
+5 while (t ≤ T) ∩ (q < Q) ∩ (P ̸= ∅) do
+6
+Initialise X = {}
+7
+for each potential optimal hyperrectangle p in P do
+8
+if hyperrectangle size σp = 3−D then
+9
+Continue
+10
+else
+11
+for each dimension i with long edge of
+hyperrectangle p do
+12
+Append(X, cp ± δ
+cp
+j ei)
+13
+Append(H, {q + 1, q + 2})
+14
+q += 2
+/* Conduct function evaluation via a
+single forward propagation
+*/
+15
+Y = ℓ(X)
+/* Space division
+*/
+16
+for each potential optimal hyperrectangle p in P do
+17
+Subdivide hyperrectangle p based on query results
+in Y
+18
+Recording the size σ and local slope ˆK for all new
+generated subspaces
+19
+Update p’s size σp and local slope ˆKcp
+/* Record current best evaluation
+and corresponding solution
+*/
+20
+ℓmin = minq∈H ℓ(cq), and cmin = arg mincq ℓ(cq)
+21
+Estimate the ground truth ℓ∗
+min via Eq. (15)
+/* Select potential optimal
+subspaces
+*/
+22
+Reset P = {}
+23
+for d ∈ {0, 1, . . . , D − 1} do
+24
+Build candidates set Hα from hyperrectangles
+with σ = 1/3d
+25
+for each hyperrectangle q in Hα do
+26
+if q satisfies condition (11) then
+27
+Append(P, q)
+28
+t = t+1
+Proofs
+Proof of Lemma 1
+Lemma 1. Given an input image example x ∈ RH×W ×C
+and the ranges of transformation factors, the first-order
+derivative of geometric transformation with bilinear sam-
+pling w.r.t. each transformation factor is bounded.
+Proof. The derivative of pixel value Vi w.r.t. xi is given by
+∂Vi
+∂xi =
+H
+�
+n
+W
+�
+m
+Unm max(0, 1 − |yi − n|)
+×
+�
+�
+�
+�
+�
+0
+if |m − xi| ≥ 1,
+1
+if |m − xi| < 1 and m ≥ xi,
+−1
+if |m − xi| < 1 and m < xi.
+(16)
+There are only four neighbouring pixels satisfy |m−xi| < 1
+and |yi − n| < 1, so Eq. (16) can then be written as
+∂Vi
+∂xi = U
+¯n ¯
+m · (1 − yi + ¯n) + U¯n ¯
+m · (1 − ¯n + yi)
+− U¯n ¯m · (1 − ¯n + yi) − U
+¯n ¯m · (1 − yi + ¯n)
+= (1 − yi + ¯n)
+�
+U
+¯n ¯
+m − U
+¯n ¯m
+�
++ (1 + yi − ¯n)
+�
+U¯n ¯
+m − U¯n ¯m
+�
+,
+(17)
+where (¯n, ¯m) = (⌈yi⌉, ⌈xi⌉) and (¯n, ¯m) = (⌊yi⌋, ⌊xi⌋). We
+can see that
+(1 − yi + ¯n) + (1 + yi − ¯n) = 1,
+(18)
+which means Eq. (17) is taking a weighted average of the
+difference between two pairs of pixels. Without loss of gen-
+erality, suppose the eligible pixel value is in [0, 1]. We have
+sup
+x∈x(∂Vi
+∂x ) = 1,
+(19)
+and following a similar deduction, the same result can be ob-
+tained for ∂Vi
+∂y . Then, the derivatives of x and y w.r.t. trans-
+formation matrix Aθ are
+∂xi
+∂Aθ
+=
+�
+∂xi
+∂θ11
+∂xi
+∂θ12
+∂xi
+∂θ13
+0
+0
+0
+�
+=
+�
+x′
+i
+y′
+i
+1
+0
+0
+0
+�
+,
+(20)
+and
+∂yi
+∂Aθ
+=
+�
+0
+0
+0
+∂yi
+∂θ21
+∂yi
+∂θ22
+∂yi
+∂θ23
+�
+=
+�
+0
+0
+0
+x′
+i
+y′
+i
+1
+�
+.
+(21)
+For each θ, we have
+sup
+θ∈Aθ
+(∂xi
+∂θ ) = W, and sup
+θ∈Aθ
+(∂yi
+∂θ ) = H.
+(22)
+In the final step, let us take the scaling factor λ as an exam-
+ple. Following the chain rule, the partial derivative is given
+by
+∂Vi
+∂λ = ∂Vi
+∂xi
+∂xi
+∂θ11
+∂θ11
+∂λ + ∂Vi
+∂yi
+∂yi
+∂θ22
+∂θ22
+∂λ .
+(23)
+Let R be the set of all eligible γ, we can substitute Eq. (19)
+and (22) into Eq. (23) and bound the derivative as
+∂Vi
+∂λ ≤ sup
+γ∈R
+(cos γ) · (W + H).
+(24)
+Because there are finite numbers of pixels, the overall
+derivative has an upper bound as well. Similarly, by spec-
+ifying the range of each transformation factor, their deriva-
+tives are upper bound correspondingly, and this completes
+the proof.
+
+𝜎
+Search space
+Satisfy the 
+potential 
+condition in 
+Eqn. (9-11)
+Satisfy the 
+potential 
+condition in 
+Eqn. (10,11)
+Figure 4: A visualisation about potential optimal condi-
+tion (7) and α candidate set (α = 2). A partition of the
+search space is presented in the upper figure, and the rela-
+tionship between the sizes and corresponding function val-
+ues of all subspaces is plotted in the lower figure. GeoRobust
+would select both cp and c′
+p as PO subspaces.
+Proof of Lemma 2
+Lemma 2 ((Gablonsky 2001)). Given the index set H and
+a positive tolerance τ > 0. Let ℓmin denote the current best
+query result. Let Hp
+1 = {q ∈ H : σq < σp}, Hp
+2 = {q ∈ H :
+σq > σp} and Hp
+3 = {q ∈ H : σq = σp}. A hyperrectangle
+Hp is said to be potentially optimal if
+ℓ(cp) ≤ ℓ(cq), ∀q ∈ Hp
+3,
+(9)
+and there is a ˜K > 0 such that
+max
+q∈Hp
+1
+ℓ (cp) − ℓ (cq)
+σp − σq
+≤ ˜K ≤ min
+q∈Hp
+2
+ℓ (cq) − ℓ (cp)
+σq − σp
+,
+(10)
+and
+�
+τ ≤ ℓmin−ℓ(cp)
+|ℓmin|
++
+σp
+|ℓmin| minq∈Hp
+2
+ℓ(cq)−ℓ(cp)
+σq−σp
+,
+if ℓmin ̸= 0,
+ℓ (cp) ≤ σp minq∈Hp
+2
+ℓ(cq)−ℓ(cp)
+σq−σp
+,
+otherwise.
+(11)
+Proof. For a hyperrectangle p, we can group all hyperrect-
+angles into Hp
+1, Hp
+2, and Hp
+3, then inequation (7) can be
+rewritten into three inequalities,
+˜K ≥ ℓ (cj) − ℓ (ci)
+σj − σi
+, ∀i ∈ Hj
+1,
+(25)
+˜K ≤ ℓ (ci) − ℓ (cj)
+σi − σj
+, ∀i ∈ Hj
+2,
+(26)
+and inequation (9). Putting inequalities (25) and (26) to-
+gether gives inequity (10). If a hyperrectangle satisfies in-
+equalities (9) and (10) at the same time, then the PO con-
+dition (7) is satisfied. While we do not know the true ˜K in
+Eq. (8), it can be replaced by an upper bound given in (26).
+Substituting condition (26) into condition (8) gives us in-
+equalities (11). This completes the proof.
+Explanation of Definition 1
+We encourage GeoRobust to select more PO subspace via
+remove inequation (9). A visualisation of α candidate set
+is presented in Fig. 4, where both cp and c′
+p would be se-
+lected and queried by GeoRobust, while DIRECT optimisa-
+tion would only choose cp.
+Explanation of Remark 1
+In every iteration, there is a hyperrectangle p satisfies σp =
+maxq∈H σq and ℓ(cp) = minq∈Hp
+3 ℓ(cq). Please recall that
+Hp
+3 contains the indexes of hyperrectangles with the same
+size as p and Hp
+1 contains the indexes of hyperrectangles
+that are smaller than p, while Hp
+2 is empty because no hy-
+perrectangle larger than p exists. Because Hp
+2 = ∅, PO con-
+dition (10) only produces a lower bound on ˜K, i.e.,
+max
+i∈Hj
+1
+ℓ (cj) − ℓ (ci)
+σj − σi
+≤ ˜K,
+(27)
+which means we can always find a slope that is large enough
+to satisfy PO conditions, making hyperrectangle p a PO
+subspace. Therefore, GeoRobust would identify and parti-
+tion at least one PO hyperrectangle throughout each itera-
+tion. Furthermore, for any hyperrectangle q, there is only
+a finite number of hyperrectangles in its Hq
+2 and Hq
+3. Un-
+der the worst situation, hyperrectangle q will be selected
+as a PO space and get divided in the next iteration when
+Hq
+2 ∪ Hi
+3 = {q}.
+Proof of Theorem 1
+To prove Theorem 1, we need a relationship between the
+depth of the largest subspace and the number of queries,
+which is given in the Theorem 4.2 from (Gablonsky 2001).
+Theorem 4.2. (Gablonsky 2001) Assuming that only one
+hyperrectangle gets divided in every iteration, the number of
+iterations T after which no hyperrectangle of depth d − 1 is
+left is given by
+T = 3n−1�3nd − 1
+3n − 1
+�
+< 3nd − 1.
+(28)
+We can now prove Theorem 1.
+Theorem 1. Let C be the n-dimensional united search space
+and ˜K be the Lipschitz constant of ℓ w.r.t. C. The gap be-
+tween current minima and global minima after T iterations
+can be written as
+ℓmin − min
+c∈C ℓ(c) ≤ ε < ˜K · (T + 1)− 1
+n .
+(14)
+Therefore, to achieve any desired ε, we need up to
+O
+�
+( ˜K/ε)n�
+iterations.
+
+H1H3H2Proof. As the global minima must be contained in one of
+the subspaces, and the objective function is Lipschitz con-
+tinuous in the search space, we have
+ℓmin − min
+c∈C ℓ(c) ≤ ∀q ∈ H, ℓ(cq) − min
+c∈C ℓ(c)
+(29)
+≤ ε ≤ ˜K · 3−d,
+(30)
+where d is the depth of the current largest subspace in the
+unit search space. According to Eq. (28), we have d ≥
+log3(T +1)
+n
+, and substituting it into Eq. (29) gives
+ε ≤ ˜K · 3− log3(T +1)
+n
+= ˜K · (T + 1)− 1
+n ,
+(31)
+which leads to Eq. (14). The relationship between any de-
+sired ε and the number of iterations T is then given by
+T ≤ ( ˜K/ε)n − 1.
+(32)
+We can see that the number of iterations is bound by
+O
+�
+( ˜K/ε)n�
+. This completes the proof.
+Detailed related works
+Numerous studies have been conducted to find the worst-
+case adversarial perturbation. While several adversarial at-
+tacks, such as the projected gradient descent attack (Madry
+et al. 2018), and Auto Attack (Croce and Hein 2020), can
+generate strong adversarial examples, they cannot ensure
+finding the worst-case perturbation (Huang et al. 2020).
+Some complete verification technologies can be used to find
+the worst-case perturbation (Liu et al. 2021), where com-
+pleteness means that a method is guaranteed to find adver-
+sarial examples within a given norm ball unless no adversar-
+ial example exists, but most of them are computationally in-
+efficient and have specific requirements for their target mod-
+els. ExactReach (Xiang, Tran, and Johnson 2017) and Relu-
+Val (Wang et al. 2018), for example, perform layer-by-layer
+propagation through target models with only linear or ReLU
+activations, requiring their target models to be fully accessi-
+ble. Therefore, these methods only work under the white-
+box setting and are unsuitable for large-scale neural net-
+works. Apart from the limitation on scalability, the layer-by-
+layer propagation operation needs a Lp norm based pixel-
+level or element-level bounding box of the input. As illus-
+trated in Fig. 1, it is difficult to establish such a bounding box
+for geometric transformations because even a small transfor-
+mation could affect a huge number of pixels and drastically
+alter their value. DeepGo (Ruan, Huang, and Kwiatkowska
+2018) is a global optimisation based method that operates
+under the grey-box environment, i.e., requiring no knowl-
+edge of the model’s parameters but a pre-estimation of the
+model’s Lipschitz constant, which is difficult to get in real-
+ity. Due to space limitations, we cannot cover all complete
+verification methods here and refer readers to a recent survey
+on verification techniques (Liu et al. 2021).
+On the other hand, there are also some studies on the
+geometric robustness of DNNs, and we summarise the dif-
+ference between our method and related works in Table 4
+on finding the worst-case geometric transformation. Jader-
+berg et al. (2015) proposed a differential module called
+spatial transformer network (STN) to enhance neural net-
+works’ learning ability regarding geometric transformations.
+Although the parameter space of control factors for most ge-
+ometric manipulations is continued, the image pixels’ coor-
+dinates are discrete, bounded integers, which means the pos-
+sible outcomes for a particular set of transformations are fi-
+nite. Pei et al. (Pei et al. 2017) empirically evaluated DNNs’
+resistance toward geometric transformations by enumerat-
+ing all possible values. Similarly, Engstrom et al. (Engstrom
+et al. 2019) employed random pick and grid search to dis-
+cover the adversarial translation and rotation to deceive tar-
+get models. DeepG (Balunovi´c et al. 2019) computes a con-
+vex relaxation of the bounding box for a set of geometric
+transformations and then certifies the robustness property
+via existing robustness verifier (Singh et al. 2019). Mopha-
+patra et al. (Mohapatra et al. 2020) introduced a small net-
+work, called Semantify-NN, to simulate the geometric ma-
+nipulation and adopted existing verifier (Weng et al. 2018)
+to examine the hybrid model composed of Semantify-NN
+and a target network. Because FastLin (Weng et al. 2018)
+and DeepPoly (Singh et al. 2019) are incompleteness ver-
+ifiers, these two works may certify whether a set of geo-
+metric transformations can affect the predictions of a tar-
+get classifier but are unable to determine the worst-case
+transformations precisely. Besides, these two verifiers use
+layer-by-layer propagation, which is computationally inef-
+ficient and limited to small networks in the white-box set-
+ting. DistSPT (Fischer, Baader, and Vechev 2020), TSS (Li
+et al. 2021), and GSmooth (Hao et al. 2022) utilise ran-
+dom smoothing techniques to verify the geometric robust-
+ness. TSS is a black-box verification method that is based
+on random smoothing. DistSPT combines random smooth-
+ing and interval bound propagation together to conduct the
+verification on tasks beyond Lp norm. GSmooth also uses
+an image-to-image network to simulate the geometric trans-
+formation.
+Parallel to geometric transformation, several works (Alai-
+fari, Alberti, and Gauksson 2019; Xiao et al. 2018) inves-
+tigated spatial transformation, which is a spatial distortion
+of the coordinates of pixels. Please note that spatial trans-
+formations performed using vector fields remain pixel-level
+perturbation. Thus, it is fundamentally distinct from geomet-
+ric transformation and beyond the scope of this paper.
+Experiments
+Implementation details within Tab. 1
+To make the comparison, we use GeoRobust to verify the
+same models used in (Li et al. 2021) on the same subsets of
+MNIST, CIFAR-10, and ImageNet. In Tab. 5, we present the
+benign accuracy reported by (Li et al. 2021) and obtained on
+our machine. It can be seen that the reproduced accuracy of
+MNIST and CIFAR-10 models are basically consistent with
+the reported accuracy, while the ImageNet models are much
+less accurate than expected (corresponding code is provided
+for reviewing). Since we failed to load the ImageNet mod-
+els properly, the comparison was only done on MNIST and
+CIFAR-10 datasets in Tab. 1.
+GeoRobust is carried out with D = 5 and α = 2, and
+its computational budget is up to 200 iterations and 2000
+
+Table 4: Comparison of methods for finding the worst-case transformation and providing the lower bound for verification.
+Method
+Approach
+Requirement
+Efficiency
+Scalability
+Guarantee
+Architecture
+Scale
+Lower bound
+Worst-case
+Exhaustive search (Pei et al. 2017)
+Query
+None
+�
+�
+�
+�
+�
+Random pick (Engstrom et al. 2019)
+Query
+None
+�
+�
+�
+�
+�
+DeepG (Balunovi´c et al. 2019)
+Layer-by-layer
+propagation
+Specify transformation
+access all parameters
+�
+�
+�
+�
+�
+Semantify-NN (Mohapatra et al. 2020)
+Surrogate network
+and layer-by-layer
+propagation
+Specify transformation
+access all parameters
+�
+�
+�
+�
+�
+DistSPT (Fischer, Baader, and Vechev 2020)
+Random smoothing
+and Layer-by-layer
+propagation
+Specify transformation
+access all parameters
+�
+�
+�
+�
+�
+TSS (Li et al. 2021)
+Random smoothing
+Specify transformation
+�
+�
+�
+�
+�
+GSmooth (Hao et al. 2022)
+Surrogate network
+and random smoothing
+Specify transformation
+access all parameters
+�
+�
+�
+�
+�
+GeoRobust (ours)
+Query
+None
+�
+�
+�
+�
+�
+Table 5: Benign accuracy of models trained and verified
+by Li et al. (2021). The small CNN used for MNIST clas-
+sification has 4 convolutional layers and 3 fully connected
+layers.
+Model
+Dataset
+Transf.
+Reported acc.
+Reproduced acc.
+small CNN
+MNIST
+R(50◦)
+99.4%
+99.4%
+S(0.3)
+99.4%
+99.4%
+ResNet101
+CIFAR-10
+R(10◦)
+83.2%
+84%
+R(30◦)
+82.6%
+81.2%
+S(0.3)
+79.8%
+80.8%
+ResNet50
+ImageNet
+R(30◦)
+46.2%
+20.8%
+S(0.3)
+50.8%
+26.6%
+queries. In practice, GeoRobust only conducted 244 queries
+on average to verify the 1-dimensional adversarial geomet-
+ric transformations. The average runtime on MNIST and
+CIFAR-10 are 0.18 and 0.74 seconds, respectively. The grid
+search, here, is carried out with 2000 queries, which is suf-
+ficient for exploring 1-dimension transformation.
+Additional experiments on all combinations of geomet-
+ric transformation
+In Tab. 2, we compared GeoRobust
+to random pick and grid search under four combinations
+of geometric transformations, while the comparison on all
+combinations of geometric transformations is summarised in
+Tab. 6. GeoRobust is carried out with D = 6 and α = 2, and
+the computational budget is up to 150 iterations and 2000
+queries. It can be seen from Tab. 6 that GeoRobust is sig-
+nificantly more efficient than random pick and grid search
+under all settings. While three methods perform similarly on
+verifying 1-dimensional geometric transformations, GeoRo-
+bust can achieve the same and sometimes even better perfor-
+mance than grid search when verifying multiple transforma-
+tions together.
+Detailed benchmark on ImageNet classifiers
+We present
+a completed version of Tab. 3, presenting the average num-
+bers of queries and runtime. Here GeoRobust can run up to
+150 iterations and 3000 queries per example. The depth and
+candidate set are set to be D = 6 and α = 2. The geomet-
+ric transformations are R(20◦), S(0.1), and T(22.4, 22.4).
+In Tab. 7, we can see that GeoRobust can conduct super ef-
+ficient analysis on all ImageNet classifiers, and it only takes
+GeoRobust less than 11 seconds to analyse one example on
+the large Vit with 3 × 108 parameters, which is the largest
+model here.
+
+Table 6: Verifying geometric robustness on ImageNet against all combination of three transformations:R(20◦), S(0.1), and
+T(22.4, 22.4). The target model is ResNet50, which achieves 74% classification accuracy. To make a fair comparison on
+efficiency, the random pick and grid search are also implemented on GPU, where the batch size is 128.
+Transformations
+GeoRobust
+Random pick
+Grid search
+Verified Acc.
+#Queries
+Runtime (s)
+Verified Acc.
+#Queries
+Runtime (s)
+Verified Acc.
+#Queries
+Runtime (s)
+R
+58%
+667 ± 29
+2.4
+58%
+2000
+4.8
+58%
+2000
+4.7
+S
+59%
+679 ± 30
+2.4
+59%
+4.7
+59%
+4.7
+R + S
+54%
+1096 ± 151
+3.8
+56%
+4000
+9.6
+56%
+5000
+12.1
+T
+57%
+1046 ± 99
+3.6
+57%
+9.4
+59%
+11.9
+R + T
+46%
+1187 ± 112
+4.1
+51%
+6000
+14.4
+49%
+1 × 104
+29.3
+S + T
+57%
+1170 ± 111
+4.0
+60%
+14.1
+57%
+28.6
+R + S + T
+46%
+1295 ± 117
+4.5
+49%
+8000
+19.1
+46%
+1 × 105
+251.3
+Table 7: A completed version of Tab. 3: Benchmarking Geometric Robustness on ImageNet
+Models
+Clean
+Attack
+Verified
+#Parameters
+# Queries
+Runtime (s)
+Inception v3.
+73.60%
+28.20%
+24.20%
+2.4×107
+1405±205
+4.6±0.5
+Inception v3adv
+75.00%
+30.60%
+27.00%
+2.4×107
+1398±192
+4.6±0.5
+Inception v4
+78.40%
+40.20%
+36.40%
+4.3×107
+1444±256
+6.3±0.9
+ResNet34
+64.40%
+10.60%
+9.00%
+2.2×107
+1475±245
+3.1±0.4
+ResNet50
+78.40%
+54.00%
+31.12%
+2.6×107
+805±110
+4.8±0.6
+Wide ResNet50.
+81.60%
+49.40%
+40.00%
+6.9×107
+1283±125
+6.0±0.5
+ResNet101
+80.00%
+54.20%
+48.20%
+4.5×107
+1291±134
+5.8±0.5
+ResNet152
+79.40%
+53.80%
+46.20%
+6.0×107
+1278±129
+7.3±0.6
+Vit32×32
+75.60%
+23.40%
+19.00%
+8.8×107
+1528±297
+3.3±0.5
+Vit16×16
+81.40%
+41.20%
+34.20%
+8.6×107
+1471±268
+5.0±0.8
+Large Vit16×16
+83.40%
+49.20%
+40.20%
+3.0×108
+1410±244
+10.4±1.6
+Beit16×16.
+83.80%
+56.40%
+52.00%
+6.5×107
+1403±215
+4.9±0.7
+Large Beit16×16
+85.60%
+65.60%
+58.20%
+2.3×108
+1363±190
+10.5±1.3
+Gmlp
+77.96%
+40.80%
+36.80%
+1.9×107
+1661±417
+4.3±0.9
+Mixer
+72.20%
+27.20%
+23.40%
+6.0×107
+1566±337
+4.5±0.9
+Swin
+80.20%
+34.60%
+13.20%
+8.8×107
+1292±136
+5.4±0.5
+Xcit
+76.80%
+40.40%
+20.60%
+8.4×107
+1497±355
+6.5±1.1
+Pit
+79.40%
+36.60%
+20.00%
+7.4×107
+1538±371
+11.0±1.1
+
diff --git a/jNFMT4oBgHgl3EQf5jEL/content/tmp_files/load_file.txt b/jNFMT4oBgHgl3EQf5jEL/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf,len=1297
+page_content='Towards Verifying the Geometric Robustness of Large-scale Neural Networks  Fu Wang1, Peipei Xu 2*, Wenjie Ruan1†, Xiaowei Huang2  1 Department of Computer Science, University of Exeter, Exeter, EX4 4QF, UK  2 Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK  {fw377, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='ruan}@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='uk, {peipei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='xu, xiaowei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='huang}@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='uk  † Corresponding Author  Abstract  Deep neural networks (DNNs) are known to be vulnerable  to adversarial geometric transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' This paper aims to  verify the robustness of large-scale DNNs against the com-  bination of multiple geometric transformations with a prov-  able guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Given a set of transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=', rotation,  scaling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' ), we develop GeoRobust, a black-box robust-  ness analyser built upon a novel global optimisation strat-  egy, for locating the worst-case combination of transforma-  tions that affect and even alter a network’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRo-  bust can provide provable guarantees on finding the worst-  case combination based on recent advances in Lipschitzian  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Due to its black-box nature, GeoRobust can be de-  ployed on large-scale DNNs regardless of their architectures,  activation functions, and the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In practice,  GeoRobust can locate the worst-case geometric transforma-  tion with high precision for the ResNet50 model on Ima-  geNet in a few seconds on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We examined 18 Im-  ageNet classifiers, including the ResNet family and vision  transformers, and found a positive correlation between the  geometric robustness of the networks and the parameter num-  bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We also observe that increasing the depth of DNN is  more beneficial than increasing its width in terms of improv-  ing its geometric robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Our tool GeoRobust is available  at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='com/TrustAI/GeoRobust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Introduction  Although deep neural networks have achieved human-level  performance, concerns are raised about their safety and re-  liability (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, Yi, and Huang 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In computer vision tasks,  while deep learning models are known to be vulnerable  to adversarial perturbations in pixel values (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Croce and Hein 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Yin, Ruan, and Fieldsend 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021, 2022), Engstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2019) show that a  slight rotation of an input example can also fool DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Al-  though modern DNNs are believed to be able to learn geo-  metric information from training data (Bakry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2016),  they are not yet invariant to simple adversarial geometric  transformations (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  This work was done when Peipei was visiting the University  of Exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Although additive adversarial perturbation has received  tremendous attention, geometric transformations are more  common and applicable in the physical world but have  been less studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' There is no efficient solution for search-  ing the worst-case adversarial transformation with provable  guarantees for large-scale DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Engstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2019)  showed that, although adversarial geometric transformation  can be discovered through the random pick, it is a highly  non-convex task that gradient-ascent-based adversarial at-  tacks perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2017) adopt the exhaustive  search to find the worst-case transformation that alters the  target model’s prediction, but its computational complex-  ity grows exponentially with the dimension of the consid-  ered transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Some researchers have adopted ver-  ification techniques (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Cohen, Rosenfeld, and Kolter 2019) to analyse geometric  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='Relaxation-based approaches (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Balunovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020) require  the Lp-norm based constraint on pixel space for an input ex-  ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1, geometric transforma-  tion can significantly change the values of the pixels, leading  to severe violence against the constraint in the pixel space  while still preserving human imperceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore,  the scalability of Lp norm-based verification is limited for  dealing with geometric transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Recently, Fischer,  Baader, and Vechev (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2021) showed that  randomised smoothing could be utilised to verify robust-  ness against a single geometric transform, but their methods  cannot handle the combination of multiple geometric trans-  formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In addition, current methods can only provide a  verifiable lower bound for verification purposes but cannot  identify the worst-case geometric transformation that would  actually minimise the model’s confidence and potentially al-  ter its prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  In this paper, we develop a novel black-box evalua-  tion framework, GeoRobust, to study the geometric robust-  ness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=', the robustness of the model against adversar-  ial geometric transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust takes advantage  of both recent developments in Lipschitzian optimisation  methods (Jones, Perttunen, and Stuckman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Gablon-  sky 2001) that provide provable guarantees on locating the  worst-case transformation and the efficient parallel compu-  tation on Graphic Processing Units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The workflow  of GeoRobust is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Given a set of geo-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='12456v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='LG]  29 Jan 2023  0  1  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9  Original  Parameter space  United search space  Normalisation  Difference  Difference  Difference  First iteration  Tenth iteration  Fiftieth iteration  Initialisation  Optimisation  …  …  Transformed  Transformed  Transformed  20o  20o  Rotation: [-20o, 20o]  Scaling: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1]  Figure 1: Schematic illustration of GeoRobust framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' After normalising the parameter space to a unit search space, GeoRo-  bust performs a sequence of space divisions to find the global worst-case transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  metric transformations and an input example, GeoRobust  converges to the worst-case combination for minimising an  adversarial loss within a finite number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We en-  able GeoRobust to better utilise GPUs by easing its sam-  pling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Besides, a lower bound estimation method  is also introduced to make GeoRobust an anytime verifica-  tion, which can produce the lower and upper bound of the  worst-case loss value whenever the algorithm stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  In summary, our key contributions lie in three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We prove the geometric transformation done by spatial  transformation network (STN) (Jaderberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2015) is  Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' By stacking STN module in front  of Lipschitz-continuous neural networks, we can analyse  their geometric robustness with guaranteed convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We develop GeoRobust, a black-box geometric robust-  ness analyser, by taking advantage of Lipschitzian opti-  misation theory (Jones, Perttunen, and Stuckman 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  The convergence of GeoRobust is theoretically guaran-  teed, and it is also highly efficient in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In our ex-  periment, GeoRobust could find the worst-case adversar-  ial transformations on an ImageNet image to evaluate a  ResNet50 classifier with desirable precision in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We use GeoRobust to benchmark the geometric robust-  ness of state-of-the-art ImageNet classifiers, including  the ResNet family and vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' There are two  main takeaways from our experiments: i) the geometric  robustness of DNNs has a positive correlation with the  number of parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and ii) increasing the number of  layers seems to be more effective than adding more hid-  den units in each layer in improving the geometric ro-  bustness of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Preliminaries  Lipschitz continuity and Lipschitzian optimisation  Pre-  vious studies indicate that the majority of modern DNNs are  Lipschitz continuous (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, Huang,  and Kwiatkowska 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Virmaux and Scaman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The  Lipschitz constant for a DNN gives an upper bound on how  fast its output could change when small perturbations are  applied to its input (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, Huang, and  Kwiatkowska 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Such a concept is closely related to the  robustness of DNN, but exactly computing the smallest Lip-  schitz constant of a DNN is proven to be an NP-hard prob-  lem (Virmaux and Scaman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Relying on the Lipschitz continuity, Lipschitzian opti-  misation is a query-based optimisation method that uses  the Lipschitz constant of the objective function to grad-  ually narrow the search space and locate the global op-  timum (Piyavskii 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Shubert 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, Huang, and  Kwiatkowska 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Xu, Ruan, and Huang 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Zhang,  Ruan, and Xu 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' While the Lipschitz constant of the  objective function is necessary for classic Lipschitzian op-  timisation, a novel Lipschitzian optimisation solution, DI-  RECT (Jones, Perttunen, and Stuckman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Jones and  Martins 2021a), does not require the Lipschitz constant to  find the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As detailed in the methodology  section, we improve the DIRECT method for studying the  geometric robustness of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Geometric transformations  Geometric transformations  are element-wise manipulation that can be conducted via  several physically meaningful parameters (Szeliski 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Given an image example, x ∈ RH×W ×C with height H,  width W, and colour channels C, the geometric transfor-  mation Tθ is carried out on each channel equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let xc ∈  RH×W be any channel of x and the output of Tθ be x′  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For a  pixel in x′  c with index (x′  i, y′  i), its value Vi is mapped to the  pixel indexed by (xj, yj) in xc via a transformation matrix  Aθ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',  �  xj  yj  �  = Aθ  �  �  x′  i  y′  i  1  �  � =  �  θ11  θ12  θ13  θ21  θ22  θ23  � �  �  x′  i  y′  i  1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (1)  We adopt Spatial Transformation Network (STN) (Jaderberg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2015) to conduct the geometric transformation and use  the bilinear sampling kernel to handle the projection of the  non-integer index, which gives the transformation result:  Vi =  H  �  h  W  �  w  Uhw max (0, 1 − |xi − w|) max (0, 1 − |yi − h|) ,  (2)  where Uhw denotes the value of a pixel, indexed by (xh, yw),  in xc, and the index does not need to be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Problem formulation  Given a neural network F : RN → RK, an input example  x ∈ RN, and its label y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , K}, we aim to find the  optimal combination of several geometric transformations  Tθ that can minimise ℓ : Rn → R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',  min  θ∈Θ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' F, x, y),  (3)  where Θ is the adversarial space that contains all feasible θ,  and ℓ denotes the margin loss defined as  ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' F, x, y) = Fy(Tθ(x)) −  max  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',K}\\{y} Fk(Tθ(x)),  (4)  which allow us to determine whether the model F can be  fooled by Tθ(x) via verifying the lower bound of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Specif-  ically, if infθ∈Θ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' F, x, y) > 0 is satisfied, the robustness  of the model F on the example x would be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Regarding geometric transformation, we consider rota-  tion, translation, and isotropic scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The corresponding  transformation matrix Aθ can be written as  Aθ =  �  θ11  θ12  θ13  θ21  θ22  θ23  �  =  �  λ cos γ  − sin γ  thor  sin γ  λ cos γ  tvrt  �  ,  (5)  where λ is the scaling factor, γ is the rotation angle, and thor  and tvrt control the horizontal and vertical translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Methodology  This section introduces a Lipschitzian optimisation-based  approach, GeoRobust, to search for the worst-case trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust is composed of four components: 1)  a Lipschitz continuous module for performing geometric  transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2) a space division procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3) a mech-  anism to select Potential Optimal (PO) subspaces that are  more likely to contain the global minimum points than oth-  ers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 4) and an anytime estimation of the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  While the space division procedure and the PO subspace se-  lection are adopted from the DIRECT algorithm, we extend  the definition of PO subspace and encourage the algorithm  to query more subspaces at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Because all evalu-  ations at the same iteration can be done parallelly in a single  forward propagation, our method could reach convergence  within reduced iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Furthermore, GeoRobust can esti-  mate the model’s Lipschitz constant and produce a reason-  able lower bound of the given loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The pseudocode  of GeoRobust and related proofs are provided in Appendix1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Notations  Given a matrix P ∈ R2×n, one can define an n-  dimensional parameter space, in which the upper and lower  bounds on i-th transformation factors are given by Pi, where  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust normalises the parameter space  into an n-dimensional unit hypercube, namely the search  space, whose centre point corresponds to the identical trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For a hyperrectangle with index q, denoted by  Hq, in the search space, the value of the objective function  at its centre point cq is denoted by ℓ(cq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We denote by lq  i  the side length w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the i-th dimension, and by ei the unit  vector along i-th dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The size of Hq is defined as  1Appendix  can  be  found  at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='com/TrustAI/  GeoRobust/blob/main/appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='pdf  σq = maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',n}  1  2lq  i , which is the same L∞ norm based  measurement used by Gablonsky (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For all hyperrect-  angles within the search space, we denote by H the set of  hyperrectangles’ indexes, and by ℓmin = minp∈H ℓ(cp) the  current best query result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The Lipschitz constant of the ob-  jective function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the search space is denoted by ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  GeoRobust computes the slope ˆK between queried points  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the parameter space during optimisation and produces  ℓ∗  min, an estimation of the lower bound of ℓmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Geometric transformations module  The convergence guarantee of GeoRobust is related to the  Lipschitz continuity of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We give the fol-  lowing lemma to show that geometric transformations with  bilinear sampling kernel are Lipschitz continuous, which  means, as long as a DNN model is Lipschitz continuous,  stacking a geometric transformation module in front of it  would not compromise the Lipschitz continuity (Tsuzuku,  Sato, and Sugiyama 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Given an input image example x ∈ RH×W ×C  and the ranges of transformation factors, the first-order  derivative of geometric transformation with bilinear sam-  pling w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' each transformation factor is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Finding optimal geometric transformation  GeoRobust first divides the search space into subspaces ac-  cording to the query results at their centre points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Then some  subspaces that are more likely to contain the global mini-  mum than others will be chosen as PO subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRo-  bust separates selected PO spaces and identifies new ones  throughout each iteration of the optimisation process till the  termination criteria are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Space division  As the only hypercube after initialisation,  the initial PO subspace is the united search space itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  GeoRobust trisects the subspace and assigns the generated  new subspace according to the query result, where the larger  hyperrectangles include the better query result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Without loss  of generality, let Hp be a PO subspace, which is an n-  dimensional hyperrectangle containing m dimensions with  long sides of a length 3−d, where m ≤ n, and n−m dimen-  sions with short sides of a length 3−d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust ignores  short sides and queries the value of the object function at  the points c ± 3−d−1ei, where i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For each  dimension with long sides, the best query result is given by  wi = min  �  ℓ(c + 3−d−1ei), ℓ(c − 3−d−1ei)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (6)  As GeoRobust performs trisection only during division, the  sizes of new subspaces are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' By dividing the  above Hp, GeoRobust creates 2m + 1 new subspaces, in-  cluding 3 sub-hypercubes with the side length of 3−d−1 and  m−1 pairs of hyperrectangles, which have 1 to m−1 dimen-  sions with long sides of length 3−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The point corresponding  to the best query result, mini∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',m} wi, is a centre point  of a hyperrectangle with m − 1 long sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A visualisation  of this space division procedure is presented in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Overall, such a division strategy encourages GeoRobust to  divide the search space uniformly and further explore the  area around the current best result, which we will detail later  in the PO subspace selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Identifying potential optimal subspaces  The space di-  vision procedure creates new subspaces, and the next step  is to locate new PO subspaces for further division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ideally,  a PO hyperrectangle Hp is expected to satisfy two condi-  tions (Jones, Perttunen, and Stuckman 1993):  ℓ (cp) − ˜Kσp ≤ ℓ (cq) − ˜Kσq, ∀q ∈ H,  (7)  ℓ (cp) − ˜Kσp ≤ ℓmin − τ |ℓmin| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (8)  Inequation (7) indicates that only the hyperrectangles with  the potential to improve the current ℓmin can be chosen as  PO subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Meanwhile, the second condition (8) ensures  that the possible improvement in the chosen subspaces is  greater than τ |ℓmin|, where a reasonable choice of τ is be-  tween 10−3 and 10−7 (Jones and Martins 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Taking  advantage of DIRECT optimisation, GeoRobust does not  need to know the Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The following lemma  demonstrates how to search for PO hyperrectangles in the  absence of ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Gablonsky 2001) Given the index set H and a  positive tolerance τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let ℓmin denote the current best  query result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let Hp  1 = {q ∈ H : σq < σp}, Hp  2 = {q ∈ H :  σq > σp} and Hp  3 = {q ∈ H : σq = σp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A hyperrectangle  Hp is said to be potentially optimal if  ℓ(cp) ≤ ℓ(cq), ∀q ∈ Hp  3,  (9)  and there is a ˜K > 0 such that  max  q∈Hp  1  ℓ (cp) − ℓ (cq)  σp − σq  ≤ ˜K ≤ min  q∈Hp  2  ℓ (cq) − ℓ (cp)  σq − σp  ,  (10)  and  �  �  �  τ ≤  ℓmin−ℓ(cp)  |ℓmin|  +  σp  |ℓmin| minq∈Hp  2  ℓ(cq)−ℓ(cp)  σq−σp  ,  if ℓmin ̸= 0,  ℓ (cp) ≤ σp minq∈Hp  2  ℓ(cq)−ℓ(cp)  σq−σp  ,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (11)  Generalising PO conditions to unleash the power of  parallel computation  The conditions in Lemma 2 are meant to select a small num-  ber of subspaces to reduce the total number of function eval-  uations, which is typically the most time-consuming proce-  dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' However, by leveraging modern deep learning frame-  works, one can easily query multiple examples on a target  DNN via a single forward propagation on GPUs, where the  computational time difference between evaluating a single  sample and a batch of samples is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore, re-  laxing the constraint given by inequation (9), we define the  following α candidate set to select more subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Definition 1 (α candidate set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Following the same notions  in Lemma 2, we define the α candidate set as  Hα =  �  ∅,  if maxp∈Hp  3 sp ≤ 0,  {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , pα′ : max �α′  j=1 spj},  otherwise,  (12)  where α′ ≤ α and the optimal score sp is given by  sp = min  q∈Hp  2  ℓ (cq) − ℓ (cp)  σq − σp  − max  q∈Hp  1  ℓ (cp) − ℓ (cq)  σp − σq  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (13)  Modifying the condition (10) into the score described in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (13) allows us to rank the potential minimum contained  by subspaces with the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The proposed α candidate  set is easy to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' When α = 1, Definition 1 degrades  to condition (10), while increasing α, GeoRobust explores  more subspaces in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As we will describe in the  next section, all subspaces will get subdivided by GeoRobust  after several iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Instead of only partitioning spaces  satisfying inequation (7), α candidate set also selects hyper-  rectangles that would likely satisfy Lemma 2 in the next few  rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' While the number of queries would rise, using the  α candidate set enables GeoRobust to discover the optimal  subspace quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' On the other hand, because the function  evaluations can be done in parallel on GPUs, replacing con-  dition (10) with α candidate set only has a small influence  on the computational time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Stop criteria and convergence analysis  In practice, GeoRobust is limited by three factors: the max-  imal number of iteration T, the maximal number of queries  Q, and the maximal number of trisection along each dimen-  sion, which is denoted by depth D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The first two stop criteria  are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We stop the optimisation once the com-  putational budget runs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The limitation on depth is ap-  plied for two reasons (Gablonsky 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' On the one hand, it  puts a limitation on the smallest size of subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' By doing  so, GeoRobust is compelled to halt local search and encour-  aged to conduct global exploration when the current optimal  subspace is sufficiently small, which accelerates the conver-  gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' On the other hand, defining the smallest subspace size  also sets an upper bound for the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For an  n-dimensional search space, GeoRobust could conduct up to  3nD times of queries, which is equivalent to a grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Besides, our implementation adopted the L∞ norm to mea-  sure hyperrectangles’ size (Gablonsky 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Such a mea-  surement simplifies both space division and PO space se-  lection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For the former, the L∞ norm is more computation-  ally efficient than the Euclidean norm (Jones and Martins  2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For the latter, hyperrectangles are grouped by their  longest side length under the L∞ norm, which introduces  less number of different sizes to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Convergence analysis  GeoRobust is guaranteed to con-  verge to the global minimum within a pre-defined small tol-  erance after a finite number of queries if the objective func-  tion is continuous (Jones, Perttunen, and Stuckman 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  This guarantee comes from the following observation shown  in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Gablonsky 2001) Following the same notions  in Lemma 2, there is at least one hyperrectangle Hp will be  identified as PO subspace at each iteration, where Hp satis-  fies Hp  2 = ∅ and makes inequation (9) holds so that every  hyperrectangle will be subdivided after finite iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Note that theoretically proving a specific DNN is Lips-  chitz continuous is beyond the scope of this paper, and ex-  isting works demonstrate the Lipschitz continuity of convo-  lutional neural networks (Ruan, Huang, and Kwiatkowska  2018) and vision transformers (Vuckovic, Baratin, and des  Combes 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Wang and Ruan 2022) used in the image clas-  sification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In addition, given by Lemma 1, we prove that  the geometric transformation is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' So, as  long as the neural network satisfies a Lipschitz condition, the  objective function (3) is Lipschitz continuous, and GeoRo-  bust is guaranteed to locate to the global minimum after a  sufficient number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The convergence complexity  is described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let C be the n-dimensional united search  space and ˜K be the Lipschitz constant of ℓ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The gap  between current minima and global minima after T itera-  tions can be written as  ℓmin − min  c∈C ℓ(c) ≤ ε < ˜K · (T + 1)− 1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (14)  Therefore, to achieve any desired ε, we need up to  O  �  ( ˜K/ε)n�  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Estimating the global minimum  While GeoRobust is guaranteed to find the global minimum  eventually, we enable it to be an anytime analyser that can  utilise intermediate query results to estimate the lower bound  of the global minimum at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Recall that the parameter space is defined via the matrix  P ∈ R2×n that contains the upper and lower bounds of n  transformation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' To divide m dimensions of a hyper-  rectangle, GeoRobust samples and evaluates new points at  c ± 3−1 · liei, where i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The slopes along the  i-th dimension are given by  ˆKc+  i = ∥ℓ(c) − ℓ(c+  i )∥  dc  i  and  ˆKc−  i = ∥ℓ(c) − ℓ(c−  i )∥  dc  i  ,  where c+  i and c−  i are short hands for c ± 3−1 · liei, and we  denote by dc  i = 3−1 · liPi ·  �  1  −1  �  the distance between  c and c+/−  i  within the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Then ˆKc, the local  slope of c, is updated to be the largest local slope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',  ˆKc = max{ ˆKc+  1 , ˆKc−  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , ˆKc+  m, ˆKc−  m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  GeoRobust updates the local slope after space division so  that it can estimate the global minimum based on the query  results at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let co denote the centre of the current  optimal hyperrectangle Ho that has ℓ(co) = ℓmin and ˆK =  maxq∈H ˆKcq, the lower bound of global minimum can be  estimated via  ℓ∗  min = ℓ(co) − ˆKmax¯σo,  (15)  where we take a relaxation on σo and compute it via the  Manhattan distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=', ¯σo =  1  2  �n  i=1 do  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore, as  long as GeoRobust can locate a Ho containing the global  minimum ˆℓmin, we have ℓ∗  min ≤ ˆℓmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Experiments  Our experiments include three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' First, we compare  GeoRobust to state-of-the-art baseline methods for verify-  ing robustness against three geometric transformations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',  rotation, translation, and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Then, we take advantage  of GeoRobust to efficiently benchmark popular large-scale  networks on ImageNet regarding their robustness over the  combination of all three transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Finally, we con-  duct an empirical analysis to study the impact of the depth  and α conditions on the convergence of GeoRobust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  General setup  For geometric transformations, we denoted  by R(γ) the rotation angle between ±γ, by S(λ) the scal-  ing range between 1±λ, and by T(thor, tvrt) the translation  that moves an example up to thor and tvrt pixels horizontally  and vertically, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For the model architectures, the  MNIST classifier is a network with four convolutional layers  and three linear layers, and the CIFAR10 classifier’s archi-  tecture is ResNet101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We adopted the pretrained MNIST and  CIFAR10 models released by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2021) and utilised  TIMM (Wightman 2019), a model zoo of ImageNet clas-  sifiers, to investigate the geometric robustness of popular  large-scale DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Our experiment is performed on a ma-  chine with an Intel i7-10700KF CPU, an RTX 3090 GPU,  and 48 gigabytes of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' More implementation details  can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Comparison with previous works  Since GeoRobust only requires querying the target models’  output, it can be easily deployed on any pretrained neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We follow the same setup used in TSS (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  2021) and GSmooth (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022) and apply GeoRo-  bust on their pretrained models to conduct robustness veri-  fying on MNIST and CIFAR10, where, for GeoRobust, the  robustness of an example is verified if the corresponding  lower bound ℓ∗  min > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Because exhaustive search is com-  putationally infeasible, we implement a grid search with a  sufficient computational budget to run through the param-  eter space to test whether the model’s prediction on each  input example can be altered, which serves as the ground  truth on the verified accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Please note that GeoRobust  works on L∞-norm based parameter space, while Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (2021) uses L2-norm based constraint on the translation,  which is currently inapplicable for GeoRobust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore,  the comparison is done on rotation and scaling transforma-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Although DeepG (Balunovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019) and TSS (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021) can analyse the geometric robustness of Ima-  geNet classifiers, but they are time-consuming and ineffi-  cient when dealing with transformation combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Be-  sides, we failed to properly reload the ImageNet classifiers  evaluated by TSS (see Appendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore, the  evaluation on ImageNet is done on a ResNet50 model, the  same architecture used by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2021), against different  combinations of transformations, and we compare the per-  formance with grid search and random pick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  The comparison results on MNIST and CIFAR10 are  summarised in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1, where we report the verified accuracy  determined by each baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust outper-  forms previous methods under all scenarios and reports the  same verified accuracy as grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Such a performance  demonstrates the effectiveness of GeoRobust in verifying  the geometric robustness against 1-dimensional transforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The evaluation on ImageNet is summarised in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2,  Table 1: Comparing with baseline methods on MNIST and CIFAR-10 against rotation and Scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We denote by − an unsup-  ported setting and by 0% a failed verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Baselines’ performance is adopted from (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Dataset  Geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  GeoRobust  Gsmooth  TSS  DeepG  Interval  Semantify-NN  DistSPT  TSS attack  Grid search  MNIST  R(50◦)  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2%  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='7%  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4%  ≤ 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8% (R(30◦))  ≤6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='0% (R(30◦))  ≤ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='48%  82%  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='2%  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9%  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2%  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='0%  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='2%  CIFAR10  R(10◦)  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='6%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='8%  R(30◦)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='4%  Table 2: Verifying geometric robustness on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  The target model is ResNet50, which vanilla accuracy is  74%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Geometric transformations are R(20◦), S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1), and  T(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Methods  Transformation  R  T  S + T  R + T + S  GeoRobust  58%  57%  57%  46%  Random pick  58%  59%  60%  49%  Grid search  58%  59%  57%  46%  Figure 2: Comparing the global minimum found by grid  search, random pick, and GeoRobust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The geometric trans-  formations are the same as in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We mark an example as  a match if its corresponding minimum found by a method is  equal to or smaller than the minimum found by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  where we can see that the accuracy verified by GeoRobust  is comparable to or better than the grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' At the same  time, random pick with sufficient queries can achieve a sim-  ilar performance as grid search on locating geometric adver-  sarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Still, it tends to perform worse as the di-  mension of parameter space becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In addition, as  the minimum found by grid search is more likely to be the  ground truth minimum, we mark an example as a match if  its corresponding minimum found by a method is equal to or  smaller than the minimum found by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2, we can see that the estimated lower bound achieves  considerable precision with a limited number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The  performance on verifying only translation is slightly worse  than other transformations, where the reason might be the  distortion introduced by bilinear sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Runtime  The effectiveness of GeoRobust is highly related  to the transformation’s dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1, GeoRobust is  Table 3: Benchmarking the geometric robustness of eighteen  ImageNet classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Its aver-  age runtime is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='18 seconds and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='72 seconds per example  on MNIST and CIFAR10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Furthermore, the av-  erage runtime for analysing the ResNet50 ImageNet clas-  sifier from 1-dimensional to 4-dimensional transformations  are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4 seconds, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6 seconds, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='0 seconds, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  In comparison, according to (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021), it takes TSS  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='7 and 1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2 seconds, respectively, to analyse an MNIST  example and an ImageNet example on the same model ar-  chitectures w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' a 1-dimensional transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Benchmarking geometric robustness  In this section, we investigate the robustness of some popular  DNN classifiers against geometric transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The pre-  vious section demonstrates that GeoRobust can efficiently  find a worst-case combination of transformations in a black-  box manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We utilise such an advantage to test large-  scale ImageNet classifiers regarding their geometric robust-  ness against the combination of rotation R(20◦), translation  T(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4), and scaling S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  The results are summarised in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3, and we can see  that 1) models with more parameters appear to have bet-  ter geometric robustness than those with fewer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2) widening  a network seems less beneficial than deepening it in terms  0*  Random Pick  Xmin  min  100  80  # Match  60  40  20  R  T  S+T  R+T+S  Geometric transformations(a)  (b)  Figure 3: Carrying out GeoRobust with different combinations of candidates set size α and depth D on ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The black  dot line in 3(a) corresponds to a global minimum found in a grid search with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5 × 105 function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  of improving the geometric robustness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3) the large version  of Beit showed the best geometric robustness, whereas the  basic Beit model is the second most robust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' This  phenomenon suggests that bidirectional modelling could be  helpful for DNNs learning geometric information and ob-  taining geometric robustness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 4) comparing the performance  between Inception V3 and Inception V3adv, an adversarially  trained model, we can see that adversarial training does not  significantly improve the model’s geometric robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Empirical analysis  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3, we carry out GeoRobust with different combina-  tions of candidates set size α and depth D on ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Increasing the size of α candidates set enables GeoRobust  to be more efficient in exploring the search space and locat-  ing the optimal subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Due to the limitation on the sub-  spaces’ minimal size, as the depth gets larger, the optimal  transformation combination found by GeoRobust is closer to  the ground truth worst-case, and the estimated lower bound  is closer to the global minimum as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' It can be observed  that the upper bound remains unchanged after convergence,  while the estimated lower bound would be updated when-  ever GeoRobust finds a larger local slope, which is why the  estimations change in the right side plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3(b), while the impact of depth on computa-  tional cost is trivial, increasing the α candidates set would  significantly raise the total number of function queries in  fixed iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The runtime of GeoRobust with α = 1 and  D = 5 is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9 seconds, and the runtime is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1 seconds when  it is carried out at α = 3 and D = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We can see that the  runtime increases sub-linearly with the number of queries  because the queries are done parallelly on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Related works  In this paper, we compared GeoRobust to Interval (Singh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019), DeepG (Balunovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019), Semantify-  NN (Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020), TSS (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021),  GSmooth (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022), and DistSPT (Fischer, Baader,  and Vechev 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DeepG, Semantify-NN, and Interval ex-  tend verification techniques designed for Lp-norm based ad-  ditive perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Both Semantify-NN and GSmooth in-  troduce small networks to simulate the geometric manipula-  tion, where Semantify-NN adopts a linear relaxation based  verification (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018) and GSmooth applies random  smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DistSPT and TSS are also randomised smooth-  ing based approaches, where TSS is a black-box analyser  that is scalable to large DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Besides, although the pa-  rameter space of control factors for most geometric ma-  nipulations is continuous, the image pixels’ coordinates are  bounded integers, which means the possible outcomes for  a particular set of transformations are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Pei  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2017) empirically evaluated the robustness of DNNs  against geometric transformations by enumerating all pos-  sible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In contrast, our GeoRobust is a query-based  black-box analyser that is fundamentally different to the  above method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We demonstrated that as long as the target  model is Lipschitz continuous, GeoRobust can verify the ro-  bustness of large-scale DNNs against a combination of ge-  ometric transformations in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The collaboration with  probabilistic approaches (Zhang, Ruan, and Fieldsend 2022)  will be explored in our future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Conclusion  In this paper, we propose a black-box analyser, GeoRo-  bust, to efficiently verify the robustness of large-scale  DNNs against geometric transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Given the bound  of multiple geometric transformations and an input exam-  ple, GeoRobust is guaranteed to find the worst-case manipu-  lation that can minimise an adversarial loss without knowing  the internal structures of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Theoretically, we  prove the Lipschitz continuity of geometric transformations  operated by STN and analyse the convergence complexity of  the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' On the methodology side, we gener-  alise the sampling strategy from DIRECT to better leverage  GPU parallel computation and design an anytime estimation  method to produce a reasonable lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' With GeoRo-  bust, we systematically benchmark the geometric robustness  of popular ImageNet classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Our empirical study shows  that larger neural networks are more robust against geomet-  ric manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Deepening a network improves its geomet-  ric robustness better than increasing its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  α=1,D=5  α=1,D=6  α=1,D=7  α= 2, D= 5  α= 2, D= 6  α= 2,D= 7  α=3, D= 5  α= 3,D= 6  α=3, D= 7α=2  α=3  α=1  Computional cost  5000  4500  4000  #Queries  3500  3000  2500  2000  1500  5  6  7  DepthAcknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  This work is supported by Partnership Resource Fund  of  ORCA  Hub  via  the  UK  EPSRC  under  project  [EP/R026173/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  XH has received funding from the Eu-  ropean Union’s Horizon 2020 research and innovation pro-  gramme under grant agreement No 956123, and is also sup-  ported by the UK EPSRC under project [EP/T026995/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  FW is funded by the Faculty of Environment, Science and  Economy at the University of Exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  We would like to thank Haozhe Wang, Anjan Dutta, and  the anonymous reviewers for their helpful comments and  Linyi Li for sharing the pretrained models with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  References  Alaifari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Elhoseiny, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' El-Gaaly, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Elgammal,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='  Balunovi´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Baader, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Singh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Gehr, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Vechev,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Certifying Geometric Robustness of Neural Net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Rosenfeld, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Kolter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Certi-  fied Adversarial Robustness via Randomized Smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In  ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Croce, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Hein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Reliable evaluation of adver-  sarial robustness with an ensemble of diverse parameter-free  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Tsipras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Schmidt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Madry,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Exploring the Landscape of Spatial Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In  ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Fischer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Baader, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Vechev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Cer-  tified Defense to Image Transformations via Randomized  Smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Modifications of the DIRECT  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' North Carolina state university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Hao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ying, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Dong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' GSmooth: Certified  Robustness against Semantic Transformations via General-  ized Randomized Smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Kroening, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Sharp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='  Thamo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' and Yi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A survey of safety  and trustworthiness of deep neural networks: Verification,  testing, adversarial attack and defence, and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Computer Science Review, 37: 100270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Simonyan,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Zisserman,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  and  Kavukcuoglu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Spatial Transformer Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In  NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' The DIRECT al-  gorithm: 25 years Later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Journal of Global Optimization,  79(3): 521–566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' and Martins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The DIRECT  algorithm: 25 years Later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='  Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='  Journal of Optimization Theory and Applications, 79: 157–  181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' TSS: Transformation-  Specific Smoothing for Robustness Certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Strong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Barrett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content=' Algorithms for verifying  deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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+page_content='  Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Fieldsend, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  PRoA:  A Probabilistic Robustness Assessment against Functional  Perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In ECML/PKDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Ruan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Gener-  alizing universal adversarial attacks beyond additive pertur-  bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In ICDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Appendix  Algorithm pseudocode  Algorithm 1: GeoRobust  Input: An input example x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the objective function ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the  bound of the parameter space P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the number of  function evaluation Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the number of iterations T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  the maximum depth D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' the size of candidates set α  Output: ℓmin with the corresponding solution cmin and an  estimation of the ground truth minimum ℓ∗  min  1 Normalise the parameter space to a unit hypercube with  centre point c0  2 t ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' q ← 0  3 Initialise the index set of hyperrectangles H = {0}  4 Initialise the set of potential optimal space P = {0}  5 while (t ≤ T) ∩ (q < Q) ∩ (P ̸= ∅) do  6  Initialise X = {}  7  for each potential optimal hyperrectangle p in P do  8  if hyperrectangle size σp = 3−D then  9  Continue  10  else  11  for each dimension i with long edge of  hyperrectangle p do  12  Append(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' cp ± δ  cp  j ei)  13  Append(H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' {q + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' q + 2})  14  q += 2  /* Conduct function evaluation via a  single forward propagation  /  15  Y = ℓ(X)  /* Space division  /  16  for each potential optimal hyperrectangle p in P do  17  Subdivide hyperrectangle p based on query results  in Y  18  Recording the size σ and local slope ˆK for all new  generated subspaces  19  Update p’s size σp and local slope ˆKcp  /* Record current best evaluation  and corresponding solution  /  20  ℓmin = minq∈H ℓ(cq),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' and cmin = arg mincq ℓ(cq)  21  Estimate the ground truth ℓ∗  min via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (15)  /* Select potential optimal  subspaces  /  22  Reset P = {}  23  for d ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' , D − 1} do  24  Build candidates set Hα from hyperrectangles  with σ = 1/3d  25  for each hyperrectangle q in Hα do  26  if q satisfies condition (11) then  27  Append(P, q)  28  t = t+1  Proofs  Proof of Lemma 1  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Given an input image example x ∈ RH×W ×C  and the ranges of transformation factors, the first-order  derivative of geometric transformation with bilinear sam-  pling w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' each transformation factor is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The derivative of pixel value Vi w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' xi is given by  ∂Vi  ∂xi =  H  �  n  W  �  m  Unm max(0, 1 − |yi − n|)  ×  �  �  �  �  �  0  if |m − xi| ≥ 1,  1  if |m − xi| < 1 and m ≥ xi,  −1  if |m − xi| < 1 and m < xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (16)  There are only four neighbouring pixels satisfy |m−xi| < 1  and |yi − n| < 1, so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (16) can then be written as  ∂Vi  ∂xi = U  ¯n ¯  m · (1 − yi + ¯n) + U¯n ¯  m · (1 − ¯n + yi)  − U¯n ¯m · (1 − ¯n + yi) − U  ¯n ¯m · (1 − yi + ¯n)  = (1 − yi + ¯n)  �  U  ¯n ¯  m − U  ¯n ¯m  �  + (1 + yi − ¯n)  �  U¯n ¯  m − U¯n ¯m  �  ,  (17)  where (¯n, ¯m) = (⌈yi⌉, ⌈xi⌉) and (¯n, ¯m) = (⌊yi⌋, ⌊xi⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We  can see that  (1 − yi + ¯n) + (1 + yi − ¯n) = 1,  (18)  which means Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (17) is taking a weighted average of the  difference between two pairs of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Without loss of gen-  erality, suppose the eligible pixel value is in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' We have  sup  x∈x(∂Vi  ∂x ) = 1,  (19)  and following a similar deduction, the same result can be ob-  tained for ∂Vi  ∂y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Then, the derivatives of x and y w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' trans-  formation matrix Aθ are  ∂xi  ∂Aθ  =  �  ∂xi  ∂θ11  ∂xi  ∂θ12  ∂xi  ∂θ13  0  0  0  �  =  �  x′  i  y′  i  1  0  0  0  �  ,  (20)  and  ∂yi  ∂Aθ  =  �  0  0  0  ∂yi  ∂θ21  ∂yi  ∂θ22  ∂yi  ∂θ23  �  =  �  0  0  0  x′  i  y′  i  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (21)  For each θ, we have  sup  θ∈Aθ  (∂xi  ∂θ ) = W, and sup  θ∈Aθ  (∂yi  ∂θ ) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (22)  In the final step, let us take the scaling factor λ as an exam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Following the chain rule, the partial derivative is given  by  ∂Vi  ∂λ = ∂Vi  ∂xi  ∂xi  ∂θ11  ∂θ11  ∂λ + ∂Vi  ∂yi  ∂yi  ∂θ22  ∂θ22  ∂λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (23)  Let R be the set of all eligible γ, we can substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (19)  and (22) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (23) and bound the derivative as  ∂Vi  ∂λ ≤ sup  γ∈R  (cos γ) · (W + H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (24)  Because there are finite numbers of pixels, the overall  derivative has an upper bound as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Similarly, by spec-  ifying the range of each transformation factor, their deriva-  tives are upper bound correspondingly, and this completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  𝜎  Search space  Satisfy the  potential  condition in  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (9-11)  Satisfy the  potential  condition in  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (10,11)  Figure 4: A visualisation about potential optimal condi-  tion (7) and α candidate set (α = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A partition of the  search space is presented in the upper figure, and the rela-  tionship between the sizes and corresponding function val-  ues of all subspaces is plotted in the lower figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust  would select both cp and c′  p as PO subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Proof of Lemma 2  Lemma 2 ((Gablonsky 2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Given the index set H and  a positive tolerance τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let ℓmin denote the current best  query result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let Hp  1 = {q ∈ H : σq < σp}, Hp  2 = {q ∈ H :  σq > σp} and Hp  3 = {q ∈ H : σq = σp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A hyperrectangle  Hp is said to be potentially optimal if  ℓ(cp) ≤ ℓ(cq), ∀q ∈ Hp  3,  (9)  and there is a ˜K > 0 such that  max  q∈Hp  1  ℓ (cp) − ℓ (cq)  σp − σq  ≤ ˜K ≤ min  q∈Hp  2  ℓ (cq) − ℓ (cp)  σq − σp  ,  (10)  and  �  τ ≤ ℓmin−ℓ(cp)  |ℓmin|  +  σp  |ℓmin| minq∈Hp  2  ℓ(cq)−ℓ(cp)  σq−σp  ,  if ℓmin ̸= 0,  ℓ (cp) ≤ σp minq∈Hp  2  ℓ(cq)−ℓ(cp)  σq−σp  ,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' For a hyperrectangle p, we can group all hyperrect-  angles into Hp  1, Hp  2, and Hp  3, then inequation (7) can be  rewritten into three inequalities,  ˜K ≥ ℓ (cj) − ℓ (ci)  σj − σi  , ∀i ∈ Hj  1,  (25)  ˜K ≤ ℓ (ci) − ℓ (cj)  σi − σj  , ∀i ∈ Hj  2,  (26)  and inequation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Putting inequalities (25) and (26) to-  gether gives inequity (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' If a hyperrectangle satisfies in-  equalities (9) and (10) at the same time, then the PO con-  dition (7) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' While we do not know the true ˜K in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (8), it can be replaced by an upper bound given in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Substituting condition (26) into condition (8) gives us in-  equalities (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Explanation of Definition 1  We encourage GeoRobust to select more PO subspace via  remove inequation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' A visualisation of α candidate set  is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 4, where both cp and c′  p would be se-  lected and queried by GeoRobust, while DIRECT optimisa-  tion would only choose cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Explanation of Remark 1  In every iteration, there is a hyperrectangle p satisfies σp =  maxq∈H σq and ℓ(cp) = minq∈Hp  3 ℓ(cq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Please recall that  Hp  3 contains the indexes of hyperrectangles with the same  size as p and Hp  1 contains the indexes of hyperrectangles  that are smaller than p, while Hp  2 is empty because no hy-  perrectangle larger than p exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Because Hp  2 = ∅, PO con-  dition (10) only produces a lower bound on ˜K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=',  max  i∈Hj  1  ℓ (cj) − ℓ (ci)  σj − σi  ≤ ˜K,  (27)  which means we can always find a slope that is large enough  to satisfy PO conditions, making hyperrectangle p a PO  subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore, GeoRobust would identify and parti-  tion at least one PO hyperrectangle throughout each itera-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Furthermore, for any hyperrectangle q, there is only  a finite number of hyperrectangles in its Hq  2 and Hq  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Un-  der the worst situation, hyperrectangle q will be selected  as a PO space and get divided in the next iteration when  Hq  2 ∪ Hi  3 = {q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Proof of Theorem 1  To prove Theorem 1, we need a relationship between the  depth of the largest subspace and the number of queries,  which is given in the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2 from (Gablonsky 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Gablonsky 2001) Assuming that only one  hyperrectangle gets divided in every iteration, the number of  iterations T after which no hyperrectangle of depth d − 1 is  left is given by  T = 3n−1�3nd − 1  3n − 1  �  < 3nd − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (28)  We can now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Let C be the n-dimensional united search space  and ˜K be the Lipschitz constant of ℓ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The gap be-  tween current minima and global minima after T iterations  can be written as  ℓmin − min  c∈C ℓ(c) ≤ ε < ˜K · (T + 1)− 1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (14)  Therefore, to achieve any desired ε, we need up to  O  �  ( ˜K/ε)n�  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  H1H3H2Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As the global minima must be contained in one of  the subspaces, and the objective function is Lipschitz con-  tinuous in the search space, we have  ℓmin − min  c∈C ℓ(c) ≤ ∀q ∈ H, ℓ(cq) − min  c∈C ℓ(c)  (29)  ≤ ε ≤ ˜K · 3−d,  (30)  where d is the depth of the current largest subspace in the  unit search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (28), we have d ≥  log3(T +1)  n  , and substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (29) gives  ε ≤ ˜K · 3− log3(T +1)  n  = ˜K · (T + 1)− 1  n ,  (31)  which leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The relationship between any de-  sired ε and the number of iterations T is then given by  T ≤ ( ˜K/ε)n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  (32)  We can see that the number of iterations is bound by  O  �  ( ˜K/ε)n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Detailed related works  Numerous studies have been conducted to find the worst-  case adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' While several adversarial at-  tacks, such as the projected gradient descent attack (Madry  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018), and Auto Attack (Croce and Hein 2020), can  generate strong adversarial examples, they cannot ensure  finding the worst-case perturbation (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Some complete verification technologies can be used to find  the worst-case perturbation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021), where com-  pleteness means that a method is guaranteed to find adver-  sarial examples within a given norm ball unless no adversar-  ial example exists, but most of them are computationally in-  efficient and have specific requirements for their target mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' ExactReach (Xiang, Tran, and Johnson 2017) and Relu-  Val (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018), for example, perform layer-by-layer  propagation through target models with only linear or ReLU  activations, requiring their target models to be fully accessi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Therefore, these methods only work under the white-  box setting and are unsuitable for large-scale neural net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Apart from the limitation on scalability, the layer-by-  layer propagation operation needs a Lp norm based pixel-  level or element-level bounding box of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' As illus-  trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1, it is difficult to establish such a bounding box  for geometric transformations because even a small transfor-  mation could affect a huge number of pixels and drastically  alter their value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DeepGo (Ruan, Huang, and Kwiatkowska  2018) is a global optimisation based method that operates  under the grey-box environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=', requiring no knowl-  edge of the model’s parameters but a pre-estimation of the  model’s Lipschitz constant, which is difficult to get in real-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Due to space limitations, we cannot cover all complete  verification methods here and refer readers to a recent survey  on verification techniques (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  On the other hand, there are also some studies on the  geometric robustness of DNNs, and we summarise the dif-  ference between our method and related works in Table 4  on finding the worst-case geometric transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Jader-  berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2015) proposed a differential module called  spatial transformer network (STN) to enhance neural net-  works’ learning ability regarding geometric transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Although the parameter space of control factors for most ge-  ometric manipulations is continued, the image pixels’ coor-  dinates are discrete, bounded integers, which means the pos-  sible outcomes for a particular set of transformations are fi-  nite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2017) empirically evaluated DNNs’  resistance toward geometric transformations by enumerat-  ing all possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Similarly, Engstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Engstrom  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019) employed random pick and grid search to dis-  cover the adversarial translation and rotation to deceive tar-  get models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DeepG (Balunovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019) computes a con-  vex relaxation of the bounding box for a set of geometric  transformations and then certifies the robustness property  via existing robustness verifier (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Mopha-  patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020) introduced a small net-  work, called Semantify-NN, to simulate the geometric ma-  nipulation and adopted existing verifier (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018)  to examine the hybrid model composed of Semantify-NN  and a target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Because FastLin (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018)  and DeepPoly (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019) are incompleteness ver-  ifiers, these two works may certify whether a set of geo-  metric transformations can affect the predictions of a tar-  get classifier but are unable to determine the worst-case  transformations precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Besides, these two verifiers use  layer-by-layer propagation, which is computationally inef-  ficient and limited to small networks in the white-box set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DistSPT (Fischer, Baader, and Vechev 2020), TSS (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021), and GSmooth (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022) utilise ran-  dom smoothing techniques to verify the geometric robust-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' TSS is a black-box verification method that is based  on random smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' DistSPT combines random smooth-  ing and interval bound propagation together to conduct the  verification on tasks beyond Lp norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GSmooth also uses  an image-to-image network to simulate the geometric trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Parallel to geometric transformation, several works (Alai-  fari, Alberti, and Gauksson 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2018) inves-  tigated spatial transformation, which is a spatial distortion  of the coordinates of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Please note that spatial trans-  formations performed using vector fields remain pixel-level  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Thus, it is fundamentally distinct from geomet-  ric transformation and beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Experiments  Implementation details within Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1  To make the comparison, we use GeoRobust to verify the  same models used in (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021) on the same subsets of  MNIST, CIFAR-10, and ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 5, we present the  benign accuracy reported by (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021) and obtained on  our machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' It can be seen that the reproduced accuracy of  MNIST and CIFAR-10 models are basically consistent with  the reported accuracy, while the ImageNet models are much  less accurate than expected (corresponding code is provided  for reviewing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Since we failed to load the ImageNet mod-  els properly, the comparison was only done on MNIST and  CIFAR-10 datasets in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  GeoRobust is carried out with D = 5 and α = 2, and  its computational budget is up to 200 iterations and 2000  Table 4: Comparison of methods for finding the worst-case transformation and providing the lower bound for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Method  Approach  Requirement  Efficiency  Scalability  Guarantee  Architecture  Scale  Lower bound  Worst-case  Exhaustive search (Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2017)  Query  None  �  �  �  �  �  Random pick (Engstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019)  Query  None  �  �  �  �  �  DeepG (Balunovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2019)  Layer-by-layer  propagation  Specify transformation  access all parameters  �  �  �  �  �  Semantify-NN (Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2020)  Surrogate network  and layer-by-layer  propagation  Specify transformation  access all parameters  �  �  �  �  �  DistSPT (Fischer, Baader, and Vechev 2020)  Random smoothing  and Layer-by-layer  propagation  Specify transformation  access all parameters  �  �  �  �  �  TSS (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2021)  Random smoothing  Specify transformation  �  �  �  �  �  GSmooth (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2022)  Surrogate network  and random smoothing  Specify transformation  access all parameters  �  �  �  �  �  GeoRobust (ours)  Query  None  �  �  �  �  �  Table 5: Benign accuracy of models trained and verified  by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The small CNN used for MNIST clas-  sification has 4 convolutional layers and 3 fully connected  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Model  Dataset  Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Reported acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Reproduced acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  small CNN  MNIST  R(50◦)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4%  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4%  S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4%  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4%  ResNet101  CIFAR-10  R(10◦)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2%  84%  R(30◦)  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6%  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2%  S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3)  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8%  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8%  ResNet50  ImageNet  R(30◦)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8%  S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6%  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' In practice, GeoRobust only conducted 244 queries  on average to verify the 1-dimensional adversarial geomet-  ric transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The average runtime on MNIST and  CIFAR-10 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='18 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='74 seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The grid  search, here, is carried out with 2000 queries, which is suf-  ficient for exploring 1-dimension transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Additional experiments on all combinations of geomet-  ric transformation  In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 2, we compared GeoRobust  to random pick and grid search under four combinations  of geometric transformations, while the comparison on all  combinations of geometric transformations is summarised in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' GeoRobust is carried out with D = 6 and α = 2, and  the computational budget is up to 150 iterations and 2000  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' It can be seen from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 6 that GeoRobust is sig-  nificantly more efficient than random pick and grid search  under all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' While three methods perform similarly on  verifying 1-dimensional geometric transformations, GeoRo-  bust can achieve the same and sometimes even better perfor-  mance than grid search when verifying multiple transforma-  tions together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Detailed benchmark on ImageNet classifiers  We present  a completed version of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3, presenting the average num-  bers of queries and runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' Here GeoRobust can run up to  150 iterations and 3000 queries per example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The depth and  candidate set are set to be D = 6 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The geomet-  ric transformations are R(20◦), S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1), and T(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 7, we can see that GeoRobust can conduct super ef-  ficient analysis on all ImageNet classifiers, and it only takes  GeoRobust less than 11 seconds to analyse one example on  the large Vit with 3 × 108 parameters, which is the largest  model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Table 6: Verifying geometric robustness on ImageNet against all combination of three transformations:R(20◦), S(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1), and  T(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' The target model is ResNet50, which achieves 74% classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' To make a fair comparison on  efficiency, the random pick and grid search are also implemented on GPU, where the batch size is 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  Transformations  GeoRobust  Random pick  Grid search  Verified Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  #Queries  Runtime (s)  Verified Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  #Queries  Runtime (s)  Verified Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  #Queries  Runtime (s)  R  58%  667 ± 29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4  58%  2000  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8  58%  2000  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='7  S  59%  679 ± 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4  59%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='7  59%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='7  R + S  54%  1096 ± 151  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8  56%  4000  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6  56%  5000  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1  T  57%  1046 ± 99  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6  57%  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4  59%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9  R + T  46%  1187 ± 112  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1  51%  6000  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4  49%  1 × 104  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3  S + T  57%  1170 ± 111  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='0  60%  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1  57%  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6  R + S + T  46%  1295 ± 117  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5  49%  8000  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1  46%  1 × 105  251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3  Table 7: A completed version of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content=' 3: Benchmarking Geometric Robustness on ImageNet  Models  Clean  Attack  Verified  #Parameters  # Queries  Runtime (s)  Inception v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='60%  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='20%  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='20%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4×107  1405±205  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5  Inception v3adv  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='60%  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4×107  1398±192  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5  Inception v4  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='20%  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3×107  1444±256  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9  ResNet34  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='60%  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='2×107  1475±245  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='4  ResNet50  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='12%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6×107  805±110  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='6  Wide ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='60%  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='9×107  1283±125  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5  ResNet101  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='00%  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='20%  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='20%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5×107  1291±134  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='5  ResNet152  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
+page_content='40%  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFMT4oBgHgl3EQf5jEL/content/2301.12456v1.pdf'}
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diff --git a/jtE2T4oBgHgl3EQfdQd0/content/tmp_files/2301.03904v1.pdf.txt b/jtE2T4oBgHgl3EQfdQd0/content/tmp_files/2301.03904v1.pdf.txt
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+1
+RedMule: A Mixed-Precision Matrix-Matrix
+Operation Engine for Flexible and Energy-Efficient
+On-Chip Linear Algebra and TinyML Training
+Acceleration
+Yvan Tortorella∗, Luca Bertaccini†, Luca Benini∗†, Davide Rossi∗, and Francesco Conti∗
+∗Department of Electrical, Electronic and Information Engineering (DEI), University of Bologna, Italy
+†IIS Integrated Systems Laboratory, ETH Zurich, Switzerland
+Abstract—The increasing interest in TinyML, i.e., near-sensor
+machine learning on power budgets of a few tens of mW, is
+currently pushing toward enabling TinyML-class training as
+opposed to inference only. Current training algorithms, based
+on various forms of error and gradient backpropagation, rely
+on floating-point matrix operations to meet the precision and
+dynamic range requirements. So far, the energy and power cost
+of these operations has been considered too high for TinyML
+scenarios. This paper addresses the open challenge of near-
+sensor training on a few mW power budget and presents
+RedMulE - Reduced-Precision Matrix Multiplication Engine, a
+low-power specialized accelerator conceived for multi-precision
+floating-point General Matrix-Matrix Operations (GEMM-Ops)
+acceleration, supporting FP16, as well as hybrid FP8 formats,
+with {sign, exponent, mantissa} = ({1, 4, 3}, {1, 5, 2}). We inte-
+grate RedMule into a Parallel Ultra-Low-Power (PULP) cluster
+containing eight energy-efficient RISC-V cores sharing a tightly-
+coupled data memory and implement the resulting system in
+a 22 nm technology. At its best efficiency point (@ 470 MHz,
+0.65 V), the RedMulE-augmented PULP cluster achieves 755
+GFLOPS/W and 920 GFLOPS/W during regular General Matrix-
+Matrix Multiplication (GEMM), and up to 1.19 TFLOPS/W and
+1.67 TFLOPS/W when executing GEMM-Ops, respectively, for
+FP16 and FP8 input/output tensors. In its best performance point
+(@ 613 MHz, 0.8 V), RedMulE achieves up to 58.5 GFLOPS
+and 117 GFLOPS for FP16 and FP8, respectively, with 99.4%
+utilization of the array of Computing Elements and consuming
+less than 60 mW on average, thus enabling on-device training
+of deep learning models in TinyML application scenarios while
+retaining the flexibility to tackle other classes of common linear
+algebra problems efficiently.
+Index Terms—General Matrix-Matrix Multiplication, General
+Matrix-Matrix Operations, Hardware Accelerator, Embedded-
+Systems, Online-Learning, TinyML.
+I. INTRODUCTION
+In the last few years, the number of Internet of Things (IoT)
+devices connected and executing Machine Learning (ML) and,
+in particular, Deep Learning (DL) based algorithms such as
+Deep Neural Networks (DNNs) increased considerably. To re-
+duce the amount of data sent over the network, improve energy
+efficiency, and prevent network congestion, the computation has
+been moved increasingly from data centers to energy-efficient
+IoT end-nodes with low power budgets (a few mW average,
+a hundred mW peak) [1], giving rise to the Tiny-ML field of
+research and application.
+Extreme-edge applications like training and inference of
+Neural Networks (NNs), graph analysis and manipulation [2],
+[3], short-distance problems [4], and model-based control rely
+on General Matrix-Matrix Multiplications (GEMMs) or Gen-
+eral Matrix-Matrix Operations (GEMM-Ops) as the most sig-
+nificant kernel. GEMM-Ops are operations that share the same
+structure of a GEMM but replace the canonical multiply/add
+with other mapping and reduction operations [5]. Due to
+the similarity of these computational patterns, it has recently
+been proposed [6] to augment TensorCores with GEMM-Ops
+support, thereby extending their acceleration capabilities to
+a broader class of applications. There is not yet an equal
+contribution targeting ultra-low-power embedded systems.
+In desktop, mobile, and data center computing, single and
+double-precision Floating-Point (FP) operations are typically
+employed for DL and linear algebra applications, providing
+high accuracy at an acceptable area and energy cost. However,
+on embedded devices, power and area constraints are much
+tighter. Recently, a significant effort has gone into adapting
+linear algebra-based algorithms as well as online learning [7]
+to low-precision formats, such as FP16 [8], [9] and FP8 [10],
+[11], while incurring in little accuracy loss. These algorith-
+mic advancements enabled performance and energy efficiency
+gains [12], [13], opening the way for deploying continual learn-
+ing and adaptation of DL models on extreme-edge computing
+systems such as smart wearable devices. However, the compu-
+tational capabilities of microcontroller units (MCUs), typically
+used in these devices, are minimal, especially concerning the
+execution of FP operations.
+In this paper, we present RedMulE (Reduced-precision ma-
+trix Multiplication Engine), a TinyML-class fully parametric
+open-source hardware accelerator designed to support on-chip
+mixed FP precision (FP8, FP16) linear algebra within RISC-V-
+based Parallel Ultra-Low-Power (PULP) [14] clusters. Since
+GEMM is commonly known to be the key kernel behind
+DL and ML training algorithms, RedMulE enables the de-
+ployment of on-chip learning and adaptation capabilities while
+efficiently supporting GEMM-Ops, on ultra-low-power System
+arXiv:2301.03904v1  [cs.AR]  10 Jan 2023
+
+2
+on-Chips (SoCs) suitable for TinyML applications. We pro-
+totyped our design within an 8-core PULP cluster in 22 nm
+CMOS technology, instantiating a RedMulE instance with 48
+internal Computing Elements (CEs). RedMulE occupies only
+0.15 mm2, accounting for 24% of the entire cluster area. It
+achieves up to 15× speedup during regular FP16 GEMM
+and up to 62× during GEMM-Ops with respect to parallel
+execution on the RISC-V cores, reaching up to 58.5 GFLOPS
+(99.4% CEs utilization) at 613 MHZ and 0.8 V. In its best
+efficiency point, i.e. 470 MHz at 0.65 V, RedMulE achieves up
+to 772 GFLOPS/W and 1.19 TFLOPS/W energy efficiency
+for GEMM and GEMM-Ops respectively, while reaching
+44.8 GFLOPS. When used with FP8 input/output tensors rep-
+resentation, a 96 CEs RedMulE implementation reaches up to
+117 GFLOPS at 613 MHz, achieving up to 920 GFLOPS/W.
+II. RELATED WORK
+The strong interest in executing linear algebra-based al-
+gorithms like inference and training of NNs led to the de-
+velopment of various hardware platforms specialized in this
+task, spanning from data-centers computing systems to ultra-
+low-power embedded platforms [15]. NVIDIA’s recent Hopper
+H100 [16] Graphic Processing Unit (GPU) is the most repre-
+sentative example of data-center computing platform for DL
+tasks like inference and training of NNs. The H100 achieves
+1978 TFLOPS at 700 W power consumption and can be used
+to train huge NN models like transformers by using narrow
+FP8 formats.
+On the other hand, enabling the execution of DL-based
+algorithms on ultra-low-power TinyML SoCs for extreme-
+edge devices such as smart wearable systems is challenging
+due to the strict power, energy, and cost constraints imposed.
+Extreme-edge inference is achievable in practical cases since it
+can be performed employing low-precision integer arithmetic,
+which reduces the model’s memory footprint and increases
+the energy efficiency of the underlying architecture with a
+limited accuracy loss [17], [18]. On the contrary, extreme-edge
+NNs training faces large memory requirements and the need
+for FP calculations, which typically leads to power envelopes
+exceeding the TinyML constraints [18], [19]. In this section,
+we focus on embedded platforms emphasizing edge training at
+moderate power.
+A. Inference Accelerators
+Hardware accelerators specialized for low-power DL infer-
+ence provide attractive alternatives to software-based execu-
+tions [18], [20]. Diana [21], a low-power NN SoC, features
+a digital NN inference accelerator and an analog in-memory-
+computing core integrated within a shared memory subsystem
+working only with narrow integer formats. DNPU [22] is a
+fully-digital energy-efficient DL processor for convolutional
+and recursive NN inference acceleration designed in 65 nm
+technology and based on a heterogeneous architecture support-
+ing 16-bit fixed-point arithmetic. Gemmini [23] is a 16 × 16
+systolic accelerator designed for inference of deep NNs with
+8-bit multiply-accumulate units with runtime-programmable
+weight stationary and output stationary dataflows.
+B. On-Device Learning
+On-device learning is an emerging and open challenge
+concerning training DL models on ultra-low-power general-
+purpose microcontrollers. To reach this aim, many works inves-
+tigated algorithms like direct feedback alignment or equilibrium
+propagation. However, such methods have been demonstrated to
+be less effective than the classical backpropagation method due
+to severe convergence difficulties [24]. TinyOL [25] and [26]
+focus on training NNs using the low-budget Arduino Nano mi-
+crocontroller based on Cortex-M core. On the other hand, PULP
+Trainlib [27], [28], and [29] are all examples of approaches to
+enable on-device learning and adaptation on RISC-V multi-core
+PULP clusters like Vega [30], that provide mixed FP precision
+capabilities, spanning from IEEE 754 Standard FP32 and FP16
+to bfloat. However, the low speed and number of available
+floating point units typical of ultra-low-power microcontrollers
+limit the performance of these libraries.
+C. Training Accelerators
+To address the limited training performance achievable by
+software libraries running on low-power processors, several
+researchers turned to hardware acceleration [15].
+Cambricon-Q [31] is a training-oriented chip for high accu-
+racy and energy efficiency based on 8-bit fixed-point arith-
+metic. However, many common training algorithms require
+floating-point operations to ensure convergence [32]. Most
+training-oriented chips employing FP arithmetic are all char-
+acterized by power envelopes unsuitable for extreme-edge ap-
+plications. IBM proposes [33], [34], an AI computing platform
+featuring 8 × 8 mixed-precision engines supporting FP16 and
+hybrid FP8 training, while [35] support only FP16 and FP32.
+Similarly, LNPU [36] supports mixed 8-bit and 16-bit FP
+precision for on-chip training. While these chips consume
+significantly less power than data-center GPUs during NN
+training (i.e. a few Watts as opposed to hundreds of Watts),
+they still do not meet the tens of mW power constraints of
+TinyML devices.
+Recently, a few training-oriented SoCs that fit the power
+budget of extreme-edge applications have been presented. T-
+PIM [37] is a processing-in-memory accelerator in 28 nm tech-
+nology for on-device learning. It reaches up to 250 GOPS/W
+during training with 0% of sparsity and within a power envelope
+of 51.23 mW at 280 MHz operating frequency. However, T-
+PIM and all the recently proposed PIM approaches do not
+support FP computations and are not suitable for standard back-
+propagation. To support NNs training at reduced power budgets,
+many training-oriented chips extensively employ pruning to
+increase sparsity during training [38], lacking generality. For
+example, TSUNAMI [39] and Trainer [40] are both accelerators
+designed for extreme-edge NN inference and training, meeting
+the TinyML power constraints by employing pruning and zero
+skipping. Anders et al. [41] propose a reconfigurable accelera-
+tor for dense-sparse matrix multiplications for mixed-precision
+computations, suitable for training-oriented applications since it
+features FP16 multiplications and FP32 accumulations with low
+area occupation and high energy efficiency. However, such an
+
+3
+TABLE I: Set of General Matrix-Matrix Operations supported
+by RedMulE
+Z = (X ◦ W) ⋆ Y
+Group
+Kernel
+◦
+⋆
+Res
+Matmul
+×
++
+Z = (X × W) + Y
+Group 1
+Maximum
+Critical Path
++
+max
+Z = max[Y, (X + W)]
+All-Pairs
+Shortest Paths
++
+min
+Z = min[Y, (X + W)]
+Maximum
+Reliability Path
+×
+max
+Z = max[Y, (X × W)]
+Minimum
+Reliability Path
+×
+min
+Z = min[Y, (X × W)]
+Group 2
+Minimum
+Spanning Tree
+max
+min
+Z = min[Y, max(X, W)]
+Maximym
+Capacity Path
+min
+max
+Z = max[Y, min(X, W)]
+accelerator is not parametric, thus not allowing a fast scale-up
+at design time when higher performance is needed. In addition,
+its integration into a real system has not been evaluated, and it
+does not support compressed FP8 input/output tensors, which
+allows for training larger NN models on edge devices where
+the memory resources are limited.
+D. GEMM-Ops Chips
+All examples of training and inference-oriented chips men-
+tioned so far target only the most common DL operations (such
+as matrix multiplications and convolutions). However, a large
+set of kernels share the same computational structure as GEMM
+but do not rely on multiplication and addition as elementary
+operations, falling into the GEMM-Ops scope. Graph analytics,
+such as breadth-first search [2], [3], short-distance problems [4]
+that are commonly used for path planning optimization in
+embedded drones navigation [42], and minimum spanning tree,
+used for computer vision [43], are examples of applications that
+make use of GEMM-Ops. SIMD2 [6] addresses this issue by
+building functional units for GEMM-Op acceleration on top of
+NVIDIA Streaming Multiprocessor architecture, resembling the
+TensorCores structure and providing dedicated ISA extensions.
+The design is implemented in 45 nm technology. Adding all
+the SIMD2 extensions to the baseline matrix multiplication
+unit results in up to 15.8× speedup with respect to executing
+the same kernel on CUDA cores at the cost of 69% of area
+overhead.
+In this paper, we propose an extended version of Red-
+MulE [44] with the following unique combination of features:
+• An array of Floating-Point Units-based Computing Ele-
+ments (CEs) for efficient training and inference of general
+DL models on embedded SoCs with additional support
+for reduced bit-width FP computation. We tightly couple
+RedMulE with a parallel cluster of RISC-V processors
+to achieve maximum flexibility in implementing complex
+training algorithms;
+• Supports for GEMM-Ops with a low area overhead (16%)
+with respect to a GEMM-only implementation to address
+a wider spectrum of applications;
+X
+W
++
+x
+=
+Y
+Z
+=
++
+x
+N
+N
+N
+N
+L
+L
+H
+a)
+X
+W
+Y
+Z
+=
++
+x
+x
++
+=
+N
+N
+L
+H
+L
+H
+L
+H
+b)
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+L
+H
+xN
+Fig. 1: Execution of a GEMM through a) scalar dot product
+microkernel and b) block-dot product (or outer product) micro-
+kernel.
+• A fully-parametric design that allows the instantiation of
+a wide range of CEs arrays, internal buffers and memory
+interface configurations.
+III. BACKGROUND
+A. Generalized Matrix-Matrix Operations
+In this work, we define Generalized Matrix-Matrix Op-
+erations (GEMM-Ops) as all the operations of the kind
+f2(Y, f1(X, W)), in particular they can be expressed as:
+Z = (X ◦ W) ⋆ Y
+(1)
+where ◦ corresponds to f1() and ⋆ corresponds to f2().
+Table I shows some examples of GEMM-Ops, divided into
+two groups. Group 1 includes all the GEMM-Ops where the ◦
+operator can be of the +/× kind while ⋆ can be min/max.
+Group 2 contains the GEMM-Ops kernels where the ◦ operator
+also belongs to the min/max kind. X is a matrix of size
+M × N, W is a matrix of size N × K, while Z and Y have
+size M × K.
+The similarity of GEMMs and GEMM-Ops makes matrix
+computing units good candidates to be extended for supporting
+GEMM-Ops, extending their flexibility to accelerate general-
+ized parallel algebraic operators. This class of algorithms is also
+well-suited for ML applications since matrices are the baseline
+structure of all DL models. To this purpose, it is essential
+to note that the structure of Equation 1 is symmetric. As a
+consequence, for ML applications, there is no need to identify
+X or W as input or weight matrices because their role can be
+flexibly exchanged.
+B. Asymptotic Optimality of Linear Algebra Acceleration
+Strategies
+Memory load/store operations enlarge the gap between the-
+oretical and practical performance and efficiency. Therefore,
+
+4
+RedMulE
+RISCY
+0
+RISCY
+7
+iCACHE SHARED
+PERIPH INTERCO
+CLUSTER AXI BUS
+DMAC
+HW
+SYNC
+TCDM
+0
+TCDM
+1
+TCDM
+15
+TCDM
+14
+TCDM
+2
+TCDM
+13
+LOG INTC
+HWPE INTC
+HCI
+CTRL
+32b
+32b
+32b
+288b
+STREAMER
+128b
+DataMover
+CTRL
+STREAMER
+32b
+TCDM
+3
+Fig. 2: PULP cluster architecture with HWPEs integration.
+maximizing the number of operations performed per memory
+access, i. e. the arithmetic intensity, is the key to an efficient
+design. As analyzed by Pedram [45], scalar dot products
+and vector units do not guarantee the best trade-off between
+the number of operations performed per memory load/store
+access. As shown in Fig. 1a, a simple scalar dot product that
+operates on a N-dimensional array performs 2 × N operations
+(N multiplications + N additions). The memory operations
+performed in this kernel are N loads of X, N loads of W,
+one load of Y and one store of Z. The resultant arithmetic
+intensity is:
+Intensity 1D =
+OPs
+LD/ST =
+2N
+2N + 2 ∼ 1,
+(N → ∞).
+(2)
+2-Dimensional L×H arrays exploit block-dot products (outer
+product) microkernels to perform GEMMs. Let us consider an
+L × H 2D array that can operate on L × 1 and 1 × H vectors,
+each made of N elements, like those shown in Fig. 1b. The
+operations performed on the two vectors are 2×L×H, repeated
+N times. The resulting load/store operations are L × N loads
+of X, H ×N loads of W, L×H loads of Y and L×H stores
+of Z. With these changes, Equation 2 becomes:
+OPs
+LD/ST =
+2LHN
+(L + H)N + 2LH ∼ 2LH
+L + H ,
+(N → ∞).
+(3)
+Equation 3 shows that if L = H, the number of operations
+is quadratic with the size of the 2-D array, while the number
+of memory accesses remains linear. This demonstrates that 2-
+dimensional arrays are more efficient with respect to scalar or
+vector units. Thus, we will exploit the outer-product approach
+for the RedMulE design.
+IV. ARCHITECTURE
+In this section, we describe the PULP cluster, the hardware
+template we rely upon, and the RedMulE micro-architecture.
+A. PULP Cluster and RedMulE
+In Fig. 2, we show the architecture of a PULP cluster,
+a multi-core architecture that features a parametric number
+(2–16) of 32-bit RISC-V general-purpose cores featuring a
+partially shared, partially private instruction cache. In this
+specific work, we focus on a PULP cluster containing 8 RISC-V
+cores, equipped with 128 kB of Tightly-Coupled Data Memory
+(TCDM) split among 16 banks for word-level interleaving with
+a low level of contention. The PULP cluster also features an
+event unit for flexible internal synchronization and a dedicated
+Direct Memory Access Controller (DMAC) to efficiently move
+data between the TCDM and external memories. A peripheral
+interconnect allows the RISC-V cores to program the on-board
+peripherals (like the DMAC), and an AXI4 full cross-bar inter-
+connect allows communications with the external environment.
+The capabilities of the PULP cluster can be further enhanced
+by integrating application-specific hardware accelerators called
+Hardware Processing Engines (HWPEs). HWPEs are software
+programmed by the RISC-V cores through the peripheral
+interconnect and share the TCDM with the RISC-V cores
+and the DMAC. In this sense, the HWPEs are tightly-coupled
+with the cluster cores [46]. The cores, the DMAC, and the
+accelerators access the shared TCDM through a single-cycle
+latency Heterogeneous Cluster Interconnect (HCI) [47]. Such
+interconnect features a logarithmic branch that allows all-to-
+all single-cycle accesses from 32-bit master ports, like those
+of the cores or the DMAC, to each of the word-interleaved
+memory banks. Conflicts are managed by granting only one
+initiator per bank with a round-robin scheme. The other branch
+is the shallow branch. It features a single n-bit parametric
+port, routed to adjacent 32-bit memory banks treated like a
+single wider bank without arbitration. This branch allows for
+simple integration of tightly-coupled accelerators like HWPEs.
+The bitwidth of the shallow branch port can be tuned to the
+HWPE requirements through a parameter. The TCDM banks
+are connected to the two HCI branches through a set of
+multiplexers, which grant access to one branch or the other
+according to a configurable starvation-free rotation scheme,
+allocating a configurable maximum of K < N consecutive
+cycles to the HWPE over a period of N cycles.
+During the execution of NNs workloads, particularly during
+inference and training, on-the-fly data marshalling operations
+are known to reduce performance significantly. For this reason,
+our PULP cluster features a DataMover [47]. The DataMover is
+a tiny accelerator capable of transposing 3-dimensional tensors
+stored in the TCDM, with 33% less time than eight RISC-V
+cores and up to 50× increased energy efficiency (the lower
+the precision of chunks to transpose the more significant the
+advantages). The accelerator works with a configurable data
+element bitwidth, from 32-bit down to 1-bit.
+B. RedMulE
+1) Global Architecture: RedMulE is a domain-specific pro-
+cessor designed to accelerate GEMM-Ops. Its architecture is
+shown in Fig. 3a. The core of RedMulE is the Datapath, a 2-
+Dimensional array of CEs interconnected as shown in Fig. 3b.
+The CEs are organized in L rows, each made of H columns.
+Within each row, a number of H CEs are cascaded so that each
+CE computing an intermediate product will pass its result to the
+next CE. The partial product computed by each row’s last CE is
+fed back as accumulation input of the same row’s first CE. The
+
+5
+RedMulE Parameters
+L
+# of rows of CEs
+H
+# of columns of CEs
+(# of CEs per row)
+P
+# of pipeline stages per CE
+CE
+0,0
+X
+W
+0
+Z
+ROW 0
+FNCOMP
+y
+x
+w
+z
+Computing Element (CE)
+COLUMN 0 COLUMN 1
+c)
+a)
+L
+H
+clk_1 clk_2
+en CLOCK
+GATING
+FMA
+!FMA
+TCDM
+RDATA
+STREAMER
+INPUT
+REDMULE CAST
+HFP8         FP16
+STREAMER
+OUTPUT
+TCDM
+DATA
+SEL = (CAST) ? 1 : 0
+1
+0
+1
+0
+REDMULE CAST
+HFP8         FP16
+d)
+X FIFO
+W FIFO
+Z FIFO
+X-BUFFER
+DATAPATH
+W-BUFFER
+Y/Z-BUFFER
+CONTROLLER
+PERIPH INTERCO
+W-BUFFER
+256b
+256b
+X
+W
+X
+W
+X
+W
+CE
+0,1
+CE
+0,2
+CE
+0,3
+COLUMN 2 COLUMN 3
+CE
+1,0
+X
+0
+Z
+Y
+ROW 1
+X
+X
+X
+Y
+CE
+1,1
+CE
+1,2
+CE
+1,3
+CE
+11,0
+X
+0
+Z
+Y
+ROW 11
+X
+X
+X
+CE
+11,1
+CE
+11,2
+CE
+11,3
+Y FIFO
+256b
+256b
+RedMulE
+TCDM
+288b
+b)
+STREAMER
+SCHEDULER
+Datapath
+RedMulE Cast
+Pipe
+COMB
+Pipe
+Pipe
+FMA
+COMB
+Pipe
+COMB
+Pipe
+Pipe
+FNCOMP
+COMB
+REDMULE CAST
+288b
+CE
+11,0
+CE
+1,0
+CE
+11,1
+e)
+Fig. 3: a) RedMulE internal architecture; b) RedMulE Datapath microarchitecture; c) RedMulE CE microarchitecture; d)
+RedMulE Cast module; e) Table with RedMulE design-time available parameters.
+RedMulE Datapath features a design-time configurable number
+of internal CEs, pipeline registers (P) for each CE, and internal
+computing precision (FP bitwidth). All RedMulE’s parameters
+are tunable at design time and are resumed in Fig. 3e.
+To feed the Datapath with data, RedMulE includes the
+Streamer, following the HWPE design strategy 1. The Streamer
+is a specialized memory access unit that connects RedMulE
+to the HCI shallow branch through a single wide port of
+parametric size (multiple of 32-bit), used for load and store
+operations. The incoming stream from the HCI is propagated to
+a single input-multiple output dispatcher that forwards the valid
+only to the selected output channel; simultaneously, each output
+channel propagates the incoming stream from the HCI to the
+accelerator input ports. On the other hand, the streams produced
+by RedMulE are propagated to the HCI interface during write
+operations.
+The Streamer is connected to three internal buffers: an X-
+Buffer that changes all the L inputs of a column once every
+H ×(P +1) cycles; a W-Buffer made of H shift registers, each
+broadcasting a new W-element to all the L CEs of a column
+every cycle; a Z-Buffer that buffers the computed Z-elements.
+The same buffer is used to pre-load Y-elements and push them
+into the Datapath. This solution saves area and power in the
+accelerator since there is no need for a separated buffer to store
+Y bias.
+The control side of the accelerator is divided into two sub-
+modules, namely Scheduler and the Controller, that contain the
+register file, accessed by the cores to program the accelerator
+and cooperate to regulate the accelerator execution.
+2) Computing
+Element
+Microarchitecture:
+The
+micro-
+architecture of each CE is shown in Fig. 3c. The CE is divided
+into two stages. The first stage is dedicated to the ◦ operation
+selection and contains one Fused Multiply-Add (FMA) and one
+1https://hwpe-doc.rtfd.io
+Floating-Point Non-Computational Operations (FNCOMP) like
+comparisons, which implements FP MIN/MAX operations. We
+adapted the FMA and FNCOMP modules from the open-source
+FPnew trans-precision Floating-Point Unit (FPU) [48], so that
+their internal pipeline registers could support backpressure
+coming from memory stalls during RedMulE’s operation.
+X and W elements are propagated into both the FMA and
+the FNCOMP modules. Depending on the desired ◦ operation,
+a multiplexer selects the result of either the FMA or the
+FNCOMP, while the clock gating module shown in Fig. 3b
+freezes the input operands of one module or the other so that
+there is no switching activity in the unused module. The Y
+element is propagated to the FMA in the first stage and is
+also directly sent to the input of the second stage, containing
+a fully combinational FNCOMP module. This architectural
+solution guarantees the execution of all the operations listed in
+Table I with a compact architectural implementation, in which
+we duplicate just what is strictly needed.
+3) Casting Module: Hybrid FP8 precision formats can be
+used as an efficient compression scheme to enable DL inference
+and training on extreme-edge devices. Hybrid FP8 precision
+means that the {sign, exponent, mantissa} structure used
+to represent the tensors can be either {1, 5, 2} or {1, 4, 3}2.
+The former format is best suited for backward propagation of
+gradients, as it provides a larger dynamic range but a lower ac-
+curacy, while the latter is a better fit for forward propagation of
+activations thanks to the larger mantissa [10], [11]. While 8-bit
+representation works for data compression, it could severely
+impact accuracy due to reduced-precision accumulations. To
+support this use case, RedMulE works internally with fixed
+FP16 precision but still accepts compressed FP8 formats as
+inputs and is capable of generating FP8 compressed output
+tensors. To do this, RedMulE is augmented with a dedicated
+2Also called E4M3 and E5M2 by NVIDIA [https://tinyurl.com/mkhbxj3v]
+
+6
+0
+Store
+Row_0
+Col_0
+X0,0
+X0,4
+X0,8
+W0,0
+W0,1
+......
+W0,15
+W0,2
+W4,0
+X0,0*W0,0
+X0,0*W0,1
+X0,0*W0,2
+X0,0*W0,3
+X0,0*W0,14
+X0,0*W0,15
+Y0 + X0,4*W4,0
+Y1 + X0,4*W4,1
+X0,0*W0,4
+X0,0*W0,5
+X0,0*W0,6
+X0,0*W0,7
+X0,0*W0,8
+X0,0*W0,9
+X0,0*W0,10
+X0,0*W0,11
+X0,0*W0,12
+X0,0*W0,13
+X0,1*W1,0
+X0,1*W1,1
+........
+X0,1*W1,0
+X0,1*W1,1
+.......
+X0,1*W1,1
+Y0
+Y1
+.......
+........
+K
+N
+M
+N
+L
+Hx(P+1)
+Col_1
+X0,1
+X0,5
+X0,9
+4 x 0
+W1,0
+......
+W1,15
+W1,1
+W5,0
+Col_2
+CE 03
+Col_3
+8 x 0
+W2,0
+......
+W2,15
+W2,1
+W6,0
+12 x 0
+W3,0
+......
+W3,15
+W3,1
+W7,0
+X0,2
+X0,6
+X0,10
+X0,3
+X0,7
+X0,11
+Y0,0   X0,0   W0,0
+Y0,1   X0,0   W0,1
+Y0,2   X0,0   W0,2
+Y0,3   X0,0   W0,3
+Y0,14   X0,0   W0,14
+Y0,15   X0,0   W0,15
+Z'0   X0,4   W4,0
+Z'1   X0,4   W4,1
+Y0,4   X0,0   W0,4
+Y0,5   X0,0   W0,5
+Y0,6   X0,0   W0,6
+Y0,7   X0,0   W0,7
+Y0,8   X0,0   W0,8
+Y0,9   X0,0   W0,9
+Y0,10   X0,0   W0,10
+Y0,11   X0,0   W0,11
+Y0,12   X0,0   W0,12
+Y0,13   X0,0   W0,13
+X0,1   W1,0
+X0,1   W1,1
+........
+X0,2   W2,0
+X0,2   W2,1
+.......
+X0,3   W3,0
+X0,3   W3,1
+Z'0
+Z'1
+.......
+........
+CE 00
+CE 01
+CE 02
+CE 03
+Time
+Column Index
+0
+1
+2
+3
+Partial 
+Result 
+a)
+Z
+W
+X
+Rw
+Rx
+Vz
+b)
+c)
+d)
+Load/Store Memory Access Schedule
+Broadcasted
+Time
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+16b
+Broadcasted
+Broadcasted
+Broadcasted
+CE 02
+CE 01
+CE 00
+Y0,0
+. . .
+Y0,15
+16b
+Ry
+L
+Hx(P+1)
+Y
+M
+K
+Accumulate
+Accumulate = 1
+0
+1
+Fig. 4: a) GEMM-Op execution displayed on matrices; b) Row of CEs within RedMulE Datapath; c) Memory access schedule
+in load/store mode described in terms of R (Ready) and V (Valid) handshake signals; d) Pipeline evolution within a row of ces.
+casting module placed between the Streamer and the HCI
+interface, as shown in Fig. 3d. The cast module contains two
+FP cast units: the input cast unit is used to cast 8-bit FP
+incoming stream into 16-bit FP to feed the accelerator so that
+the CEs in the Datapath can operate on larger precision, guar-
+anteeing enough accuracy during intermediate accumulations.
+After the computation, the output cast unit can be used to
+convert the 16-bit FP results produced by RedMulE to 8-bit
+encoded outgoing stream before writing it to memory. The cast
+units can be excluded from the path if the input tensors are
+represented with 16-bit. For DL use cases only, RedMulE can
+also be instantiated at design time so that it can only load and
+store HFP8 operands. In this use-case, the input and output
+tensors represented with 8-bit formats allow to read or write
+from and to the memory twice the number of elements while
+keeping the same memory bandwidth. Consequently, this allows
+for doubling the number of CEs inside each row, doubling
+RedMulE’s performance with respect to the 16-bit inputs case.
+C. RedMulE Computational Model
+Fig. 4a shows how RedMulE performs a GEMM-Op visual-
+ising it on the computed matrices, while Fig. 4b and Fig. 4d
+show the detailed sequence of the operations within a row
+of CEs providing an example of GEMM execution. For this
+discussion, let us focus on a RedMulE implementation that
+features L = 12, H = 4, and P = 3. The RedMulE operation
+starts by pre-loading the Z-Buffer with L rows from the Y-
+matrix, each row made of H × (P + 1) = 16 FP16 elements
+(256-bit memory width/16-bit internal precision), namely y0,0
+- y0,15 for Row 0, y1,0 - y1,15 for Row 1, and so on.
+Afterwards, RedMulE pre-loads the X-Buffer as well, following
+the same pattern, and then loads a set of H × (P + 1) = 16
+W-elements (w0,0 - w0,15) inside the first shift register of the
+W-buffer. Each W-element is broadcasted to all the L CEs in
+the first Datpath column. While W-elements are broadcasted,
+the Z-Buffer pushes Y-elements in the CEs array cycle-by-
+cycle to perform the ⋆ operation during the execution of the ◦
+one.
+After P + 1 cycles, each of the L CEs in the first column
+forwards its computed partial result to the neighbour CE in
+the second column. The accelerator loads another set of H ×
+(P + 1) W-elements (w1,0 - w1,15) to broadcast them to all
+the CEs in the second column. Once all the H CEs of a row
+have completed their computations, calculating a subset of H ×
+(P + 1) row-column intermediate results, RedMulE activates
+its feedback (accumulate = 1) to provide the intermediate
+results to the accumulation input of the first CEs of the given
+row, then reiterating the computation. Immediately after, the
+Streamer reloads the next Y-submatrix in the Z-Buffer so that
+it will be ready for the next calculation. During the Z-Buffer
+reload operation, the X-Buffer provides a new X-operand to the
+first column of CEs, and a new set of H ×(P +1) W-elements
+is reloaded in the first W shift register. After (P + 1) cycles,
+all the L CEs of the first column produce a new partial product
+and provide it to the CEs in the second column. The X-Buffer
+provides a new X-operand at the input of the second column
+of CEs, and the W-Buffer loads a new set of H × (P + 1) W-
+elements in the second W shift register for broadcasting, and
+the computation continues. Fig 4d shows the detailed sequence
+of data within the pipeline of a row of CEs from the beginning
+of a GEMM operation until the moment of the reuse of the
+partial results (accumulate = 1).
+To guarantee a continuous data flow in the accelerator, the
+W-buffer accesses the memory once every (P + 1)-cycles to
+load a new set of H ×(P +1) W-elements. Once the X-Buffer
+and the Z-Buffer are empty, RedMulE reuses the Streamer port
+to load the X and Y-operands. Such operation is made by
+interleaving the memory accesses to X or Y matrices between
+
+7
+RedMulE
+Baseline (8 Cores)
+Ideal (32 MAC/cycle)
+a)
+b)
+c)
+Fig. 5: RedMulE benchmarking with comparison with software executed on 8 RISC-V cores: a) Synthetic GEMM execution; b)
+ResNet8 execution; c) GEMM-Ops execution.
+two adjacent W-matrix accesses until the complete fulfilment
+of the X and Z buffers. Fig 4c shows how the memory accesses
+to different matrices are interleaved, describing the memory
+accesses in terms of Ready (R) and Valid (V) handshake signals.
+The Streamer load and store units fully support backpressure
+through a mechanism based on R/V handshake signals. Such
+a mechanism fully decouples the memory access and data
+consumption/production from the Datapath. The V signals for
+loads and the R signals for stores are generated within the
+Streamer itself depending only on memory stalls, which can
+be amortized by the presence of FIFO elements, and not on
+the actual usage from the Datapath. On the other hand, the
+Datapath uses the R signal of loads and the V signal of
+stores, as shown in Fig. 4c, to control the order of memory
+accesses interleaving them so that a continuous dataflow can
+be maintained. This choice is made to maximize the memory
+port utilization since having a single memory port also helps
+reduce the overall streamer area.
+After the conclusion of an entire row-column operation, the
+Z-Buffer buffers the final sub-matrices. Afterwards, store oper-
+ations are interleaved between two adjacent W load accesses
+until the Z-Buffer is empty and can be reloaded with Y-
+elements. With this approach, RedMulE optimizes the band-
+width utilization using a single wide memory port and achieves
+up to 99.4% CEs utilization.
+V. IMPLEMENTATION AND MEASUREMENTS
+A. Experimental Setup
+We focus our experiments on a RedMulE12x4 instance with
+H
+= 4, L = 12, P
+= 3, resulting in 48 CEs and a
+288-bit wide HCI port, for 256-bit + 32-bit non-word-aligned
+accesses. We also address a RedMulE12x8 since, as described
+in Section V-B3, it uses the same memory interface with twice
+the number of CEs.
+Our experiments target GlobalFoundries 22 nm technology
+using Synopsys Design Compiler for synthesis (slow corner at
+ftarg = 250 MHz, VDD = 0.59 V, T = 125 °C) and Cadence
+Innovus for full-cluster Place&Route in the same operating
+point. RedMulE’s timing analysis and power extraction were
+made using Prime Time with 100% annotated switching activity
+from post-layout simulation in typical corner at 25 °C, targeting
+two operating points: 470 MHz at 0.65 V for high energy
+efficiency and 613 MHz at 0.8 V for high performance.
+B. Performance Evaluation
+1) GEMM Performance Evaluation: We use square and
+rectangular matrices as a synthetic benchmark to evaluate
+RedMulE’s computation latency in cycles against the SW
+execution on 8 parallel RISC-V cores sharing 4 FPUs. On
+the given benchmark, RedMulE reaches a peak throughput of
+more than 95.4 OP/cycle, where we count both ⋆ and ◦ as
+one ”OP”, e.g. for a regular GEMM we count 1 MAC = 2
+OPs. RedMulE achieves up to 99.4% of CEs utilization on
+96×96 FP16 matrices (55 kB memory occupation), leading to
+58.5 GFLOPS at 613 MHz with 0.80 V supply. Fig. 5a shows
+the number of computing cycles required to compute various
+matrices during parallel FP16 software executed on 8 RISC-V
+cores and compares them on RedMulE, showing that it reaches
+15× average speedup over the software on large matrices. This
+performance increase with respect to the software counterpart
+settles around 13× with larger matrices since also the software
+execution becomes more efficient in those cases. We also
+consider the acceleration of a small 8 × 8 × 8 case, as shown
+in Fig. 5a in which the accelerator is under-utilized, but it still
+introduces 3.5× speedup over the software parallel execution.
+2) FP16 Network Training:
+To further evaluate Red-
+MulE performance on a real-case NN training, our target
+is TinyMLPerf [49], and in particular, we focused on the
+ResNet [50] example. For the software infrastructure, we rely
+on the pulp-TrainLib [27], and we compared RedMulE with
+a software baseline executed on 8 RISC-V cores sharing 4
+FPUs. The library takes into consideration all the training
+steps for the calculation of the gradients and backpropagation.
+Fig. 5b shows the execution of a single step in the ResNet8
+network when using 8 RISC-V cores in parallel and when using
+RedMulE for the matrix multiplication execution. RedMulE
+accelerates the matrix multiplication execution of 14.6× with
+respect to the parallel RISC-V execution in SW, speeding up
+the entire single step of the ResNet8 of 3.1×. RedMulE keeps
+its utilization constant at 99.1% (47.6 MAC/cycle) with the
+only exceptions in the first and the last layers where it drops to
+93.2% (44.7 MAC/cycle) and 32.3% (15.5 MAC/cycle) due
+to leftovers that do not allow to exploit the full potential
+of the array. From Fig. 5b, it is also evident that the data
+reorganization during the Im2Col accounts for approximately
+3 Millions computing cycles. To solve this problem, we aug-
+ment RedMulE’s operation with the support of the DataMover
+
+GEMM-Ops Speedup Over SW Execution
+RedMulE Cycles
+sGroup 1Group 2
+62 x
+1000000
+59 x
+47 x
+100000
+COMPUTING CYCLES
+25 x
+10000
+1000
+100
+10
+1
+8x8x8
+12x12x12
+32x32x32
+64x64x64
+MATRIX SIZESResNet 8 FP16 - RISC-V Cores vs RedMulE
+Single Training Step
+RedMulE+Datamover
+RedMulE+Datamover (FP8)
+RISC-V
+-RedMulE
+28,5 x
+COMPUTING CYCLES
+12000000
+6
+14,6
+10000000
+↑
+8000000
+3
+6000000
+9
+15
+4000000
+2,1 x
+2000000
+2,1 x
+0
+DMA
+Im2Col
+Im2Col
+Other
+Matmul
+TRAINING
+Transfers
+(Core)
+(Padding) Marshalling
+STEP
+RESNET PHASEHW vs. sW Execution
+XS'ET
+1E+6
+S
+x9's
+1
+x8'21
+ycl
+ 1E+5
+1
+1E+4
+Computing.
+5
+1E+3
+1E+2
+1E+1
+1E+0
+24x48x64
+24x32x32
+96x96x96
+6
+9
+Computed MACs8
+b)
+a)
+c)
+Fig. 6: a) RedMulE area breakdown with a focus on the
+datapath, b) RedMulE power distribution, c) PULP power
+distribution.
+engine, halving the number of computing cycles required to
+perform the two Im2Col operations and thus speeding up the
+overall training step execution up to 4.9×. As all the devices
+included in the PULP cluster (RISC-V cores and accelerators)
+are designed for synergistic cooperation and share the memory,
+the heterogeneity of the architecture can be efficiently and fully
+exploited.
+3) HFP8 Network Training: For the same training example,
+we consider a RedMulE12x8 instance used to train the ResNet
+network encoded on 8-bit FP inputs only. For the RedMulE12x4
+we considered until now, the memory port of the Streamer
+is 288-bit wide, meaning a 256-bit memory port with non-
+word aligned memory accesses capability. In this configuration,
+RedMulE12x4 can load 16 × FP16 elements at a time that are
+used to fill the pipeline during the computation. Having H = 4
+columns, the pipeline stages within each row are calculated as
+H ×(P +1), where P = 3 in this implementation, resulting in
+16 pipeline stages. Considering a fixed 8-bit input encoding,
+with the same 288-bit memory port, RedMulE can access up to
+32×FP8 elements at a time, meaning that we can implement a
+RedMule12x8 instance maintaining a 288-bit memory interface
+and obtaining 32 pipeline stages. We show how the ResNet8
+training can benefit from this configuration in the green bar of
+Fig. 5b. Matrix multiplication execution can be accelerated up
+to 28.5×, resulting in 5.5× speed-up over the entire training
+step execution, with 97% utilization.
+4) GEMM-Ops Performance Evaluation: To evaluate the
+GEMM-Ops performance, in Fig. 5c we compare the RedMulE
+GEMM-Ops execution against parallel SW execution on the
+RISC-V cores. RedMulE always takes the same number of
+computing cycles to perform any of the supported GEMM-
+Ops, while the parallel execution on the general-purpose cores
+changes depending on the executed kernel. All the kernels
+belonging to Group 1 (see Table I), i.e. for which ◦ operation is
+Fig. 7: RedMulE area sweep with several sizes of H and L.
++/× and ⋆ is max/min, require the same number of computing
+cycles when executed on the cores, while up to 47× speedup
+can be achieved when leveraging RedMulE. When also ⋆ are
+of the max/min kind, i.e. Group 2, the execution overhead for
+the general-purpose cores is even higher, and RedMulE can
+accelerate such kernels up to 62×.
+C. RedMulE Area
+1) Area
+Breakdown
+analysis:
+RedMulE12x4
+occupies
+0.15 mm2, corresponding to 23.8% of the entire PULP
+cluster area (0.64 mm2). Fig. 6a shows the breakdown of the
+RedMulE area, where the cast units account for the 7% to the
+overall accelerator area, and the FMA units account for the
+72%. The support for GEMM-Ops, namely the introduction
+of the two FNCOMP modules and the operation selection
+logic, introduces an overhead of just the 16% over the entire
+accelerator area. The 13% of this overhead resides in the first
+stage FNCOMP and is dominated by the pipeline introduced to
+match the number of cycles required by an FMA module. The
+second stage FNCOMP is fully combinational and accounts
+only for 3% of the overhead.
+2) RedMulE Area Sweep: We studied the area overhead
+introduced when changing the number of CEs within RedMulE,
+fixing the CEs’ internal pipeline stages to P = 3. Fig. 7 shows
+that RedMulE’s area occupation becomes comparable to the
+area of the entire PULP cluster when it contains 256 CEs,
+corresponding to a RedMulE16x16 instance. On the other hand,
+the area of RedMulE32x32 is 4× larger than the entire PULP
+cluster. Fig. 7 shows that changing the shape of the Datapath
+also affects the size of the Streamer. In particular, for each
+CE that is added to a row of the Datapath (or equivalently,
+changing the H parameter), other P + 1 pipeline registers are
+added within each Datapath row. The consequence is that the
+number of elements needed to keep a high CEs utilization
+increases by P + 1 as well. Keeping P = 3 as an example,
+increasing the H parameter by 1 requires the Streamer to
+provide P +1(= 4) additional FP16 elements to the Datapath.
+The consequence is that the streamer port must be enlarged of
+64-bit (= 4 × 16-bit), limiting the integration of RedMulE in
+the PULP cluster.
+
+2%3%
+10%
+5%
+4%
+76%RedMulE Area Sweep
+Control
+I Cast
+Streamer
+Buffers
+Datapath
+10000
+9000
+8000
+7000
+[kGE]
+6000
+5000
+Area
+4000
+3000
+PULP CLUSTER
+2000
+1000
+0
+8x4
+12x4
+12x8
+16x16
+32x32
+RedMulE Instance (LxH)9
+Fig. 8: Energy efficiency of RedMulE compared with the SW
+baseline executed on 8 RISC-V cores with 4 shared FPUs.
+D. RedMulE Power
+At a cluster level, the power consumption in the efficiency
+point amounts to 59.3 mW during GEMM operation. The
+RedMulE contribution dominates the power envelope account-
+ing for 66.8% of the overall consumption, while the TCDM
+banks and the HCI interconnect contribution is 13.3%. In this
+operating point, we reach a cluster peak energy efficiency of
+755 GFLOPS/W during GEMM execution, corresponding to
+12.5× higher energy efficiency with respect to the software
+baseline. During the execution of the algorithms belonging to
+GEMM-Ops’ Group 1 on RedMulE, the cluster-level power
+dissipation reaches 53.2 mW, leading to 842 GFLOPS/W,
+which is 57.2× higher than SW execution. On the other hand,
+during the execution of the algorithms in GEMM-Ops’ Group 2,
+the power consumption is further reduced to 37.6 mW resulting
+in 1.19 TFLOPS/W, thus 81.2× more efficient than software
+execution. Figure 8 compares the energy efficiency of RedMulE
+with the software baseline executed on 8 RISC-V cores with
+4 shared FPUs on FP16 elements during the execution of
+GEMM, GEMM-Ops’ Group 1 and GEMM-Ops’ Group 2
+kernels. Figure 6b and Figure 6c show respectively the power
+breakdown for RedMulE, where most of the power is consumed
+by the Datapath, and the PULP cluster during a GEMM
+operation, where the majority of the power is consumed by
+RedMulE and by the TCDM banks.
+VI. COMPARISON WITH THE STATE-OF-THE-ART
+Table II resumes the comparison of our work with different
+State of the Art (SoA) architectures.
+We compare our work with GPU architectures, in particular
+with NVIDIA H100 containing TensorCores, that guarantee up
+to 989 TFLOPS of performance in FP16 and 1979 TFLOPS
+in FP8, meaning 17000× than our work, but at the cost of
+700 W power consumption and 814 mm2, 12000× more power-
+consuming and 1300× larger than our work – representing an
+unfeasible solution for an IoT end-node.
+While RedMulE targets primarily training, it is also usable
+for inference. For this reason, we include in our comparison
+some inference-oriented chips, like DNPU [22]. DNPU’s per-
+formance is just 1.9× higher than our cluster, although DNPU
+contains 16× the number of CEs. Moreover, DNPU features
+Technology
+GF22FDX
+Cluster Area
+0.64 mm2 (48 CEs)
+0.73 mm2 (96 CEs)
+Cluster SRAM
+128 kB
+RedMulE Area
+0.15 mm2 (48 CEs)
+0.24 mm2 (96 CEs)
+RedMulE Perf.
+117 GFLOPS (FP8)
+58.5 GFLOPS (FP16)
+Freq. Range
+470 - 613 MHz
+VDD
+0.65 - 0.8 V
+Power Cons.
+37.6 - 193 mW
+GEMM En. Eff.
+920 GFLOPS/W (FP8)
+775 GFLOPS/W (FP16)
+GEMM-Ops En. Eff.
+1.1 TFLOPS/W (Group 1)
+1.67 TFLOPS/W (Group 2)
+Fig. 9: Area breakdown of the PULP cluster, layout and resume
+table.
+2.7× higher efficiency than RedMulE but is designed to work
+with fixed-point precision only, which helps increase energy
+efficiency. We also compared our work with Diana [21] and
+Gemmini [23], being designed in the same technology node
+of RedMulE. The former achieves 44.5% less performance
+than RedMulE12x8 and 12% less performance than RedMulE12x4
+in the energy efficient mode. Diana’s power consumption in
+efficiency mode is much lower than our design, but if we
+scale down the frequency to 50 MHz as they do, our PULP
+cluster with RedMulE12x4 consumes just 7.65 mW. The sig-
+nificant increase in Diana’s energy efficiency is justified as it
+uses only 8-bit integer arithmetic, which helps reduce power
+consumption and increases energy efficiency. On the contrary,
+Gemmini features one order of magnitude less energy efficiency
+than RedMulE12x4 despite it features 5× the number of CEs and
+works with 8-bit integer format.
+We also compared our design with other platforms specif-
+ically designed for on-chip training. IBM [33] demonstrated
+a 4-core AI chip in 7 nm technology which is just 2.4×
+more energy-efficient, 33.2× larger, and 74× more power-
+consuming than our PULP cluster with RedMulE12x4, despite
+the technology scaling and the reduced operating voltage. IBM
+also proposes a chip [35], with more than 1 W of power con-
+sumption, which is not acceptable for extreme-edge training.
+On the other hand, LNPU [36] is an extreme-edge processor
+that features a 6.67× higher power envelope than RedMulE12x4.
+Vega is a valid candidate for on-chip embedded training, but
+RedMulE12x4 achieves 7.8× higher performance and 3.2×
+higher energy efficiency. Cambricon-Q [31] is designed in
+45 nm and is 2.9× more energy-efficient than our design
+but makes use of narrow 8-bit fixed-point arithmetic, while
+generally available learning algorithms based on backpropaga-
+tion strictly require FP range and precision. Cambricon-Q is
+also 17.7× more power-hungry than our design, therefore not
+suitable for TinyML applications. Similar considerations hold
+for T-PIM [37], a training chip designed in 28 nm technology
+that features an in-memory computing core for high energy
+efficiency but only works with 16-bit integer precision, not
+satisfying the precision requirements to enable on-chip training.
+
+GEMM-Ops Energy Efficiency Over SW Execution
+Group 1
+Group 2
+GEMM
+RedMulE GOPS/W
+1000
+ENERGY EFFICIENCY. [GFLOPS/W]
+100
+10
+1
+8x8x8
+12x12x12
+32x32x32
+64x64x64
+MATRIX SIZES3%
+24%
+7%
+19%
+22%
+19%:10
+TABLE II: State of the art comparison. First line = Best Efficiency; Second line = Peak Performance. 1 MAC = 2 OPs.
+Category
+Design
+Tech
+nm
+Area
+mm2
+Freq
+MHz
+Volt
+V
+Power
+mW
+Perf
+GOPS
+Energy Eff
+GOPS/W
+CEs
+Precision
+GPU
+NVIDIA H100 [16]
+4
+814
+1830
+-
+700000
+1978900
+989400
+2827
+1413
+528
+FP8
+FP16
+SIMD2 [6]
+45
+19.5
+-
+-
+4190
+-
+-
+-
+INT16
+Inference
+Chips
+DNPU [22]
+65
+16
+50
+200
+0.7
+1.1
+34.6
+279
+72.6
+279
+2100
+1000
+768
+INT16
+Diana [21]
+22
+8.91
+50
+280
+0.55
+0.9
+9.96
+129
+40
+224
+4040
+1740
+256
+INT8
+Gemmini [23]
+22
+16
+700
+900
+0.75
+0.91
+-
+-
+-
+-
+70
+50
+256
+INT8
+Training
+Chips
+4-core IBM [33]
+7
+19.6
+1000
+1600
+0.55
+0.75
+4400
+13000
+8000
+12800
+1800
+980
+4096
+FP16
+LNPU [36]
+65
+16
+200
+0.78
+1.1
+367
+600
+300
+1630
+817
+768
+FP8
+FP16
+Oh, IBM [35]
+14
+9.8
+1000
+1500
+0.54
+0.62
+1428
+2727
+2000
+3000
+1400
+1100
+128
+FP32
+FP16
+T-PIM [37]
+28
+5.04
+50
+280
+0.75
+1.05
+5.25
+51.2
+39.8
+43
+7590
+840
+-
+INT16
+TSUNAMI [39]
+65
+16
+50
+200
+0.78
+1.1
+45
+419
+612
+3420
+1480
+2048
+FP8
+306
+1710
+740
+1024
+FP16
+Trainer [40]
+28
+21
+40
+440
+0.56
+1
+23
+363
+900
+4280
+8192
+FP8
+450
+2140
+4096
+FP16
+Cambricon-Q [31]
+45
+888
+1000
+0.6
+1030
+2000
+2240
+1024
+INT8
+Vega [30]
+22
+12
+450
+0.5
+0.8
+-
+49.4
+3.3
+7.5
+250
+180
+4
+FP16
+Mat-Mul
+Anders [41]
+14
+0.024
+2.1
+1090
+0.26
+0.9
+0.023
+82.7
+0.068
+34
+2970
+420
+16
+FP16
+GEMM
+This Work
+RedMulE12x4
+22
+0.64
+470
+613
+0.65
+0.8
+59.3
+116
+44.8
+58.5
+775
+506
+48
+FP16
+Group 1
+GEMM-Ops
+53.2
+103
+842
+576
+Group 2
+GEMM-Ops
+37.6
+71.5
+1193
+819
+GEMM
+This Work
+RedMulE12x8
+22
+0.73
+470
+613
+0.65
+0.8
+97.5
+193
+89.7
+117
+920
+608
+96
+FP8
+Group 1
+GEMM-Ops
+85.2
+168
+1052
+694
+Group 2
+GEMM-Ops
+54
+104
+1666
+1123
+TSUNAMI [39] and Trainer [40] are conceived for energy-
+efficient embedded training and extensively use pruning and
+sparse matrices generation to increase energy efficiency and
+reduce the number of required MAC operations during training
+with zero-skipping. We compare with the results they pro-
+vide during dense calculations. In their best efficiency points,
+TSUNAMI and Trainer’s power consumption is comparable
+to RedMulE’s. However, those points correspond to 50 MHz
+and 40 MHz for TSUNAMI and Trainer, while RedMule is
+evaluated at 470 MHz. Therefore, RedMulE would consume
+approximately one order of magnitude less power at compa-
+rable frequencies. TSUNAMI and Trainer reach up to 5× and
+8× higher performance, respectively, since they feature 21×
+and 85× the number of CEs, but feature a much lower CEs’
+utilization than our RedMulE (75% TSUNAMI and only 12.5%
+Trainer). The systolic architecture of RedMulE enables, in
+principle, almost arbitrary architecture scaling. Assuming linear
+performance, area, and power ratio, scaling to 1024 or 4096
+CEs (21× and 85× larger than RedMulE12x4), our utilization
+would still be 99.4%, leading to higher overall performance
+(1.25 TFLOPS and 5 TFLOPS respectively) once accounting
+overheads.
+We compare RedMulE12x4 with Anders et al. [41], proposing
+a hardware accelerator for matrix multiplications in 14 nm tech-
+nology that targets TinyML learning and inference applications.
+It reaches a peak energy efficiency of 2.97 TFLOPS/W in
+FP16 precision, 3.83× higher than RedMulE12x4, but only when
+operating at near-threshold voltage (260 mV) and extremely
+reduced frequency (2.1 MHz). In that operating point, their
+design is 659× less performant than RedMulE. Anders’ peak
+performance is obtained at 0.9 V and 1.09 GHz, leading to
+420 GFLOPS/W and 99.4% MAC units utilization, similarly
+to RedMulE’s. In 22 nm technology, at 613 MHz frequency
+and 0.8 V, RedMulE12x4 reaches 58.5 GFLOPS, 1.72× better
+than Anders et al., with a 20.5% higher energy efficiency of
+506 GFLOPS/W on FP16 GEMM.
+We also compared RedMulE with SIMD2 [6], the only
+other design that features GEMM-Ops extensions. Even though
+SIMD2 works only with integer precision and cannot thus target
+on-chip training, it features 36.1× higher power consumption
+than RedMulE. In their case, the authors also claim that
+the area overhead to build GEMM-Ops extensions on top of
+
+11
+NVIDIA Streaming Multiprocessor accounted for 69%, while
+in RedMulE, the area overhead introduced by GEMM-Ops
+extension is just 16%.
+VII. CONCLUSION
+In this paper, we presented RedMulE - Reduced-Precision
+Matrix Multiplication Engine, a fully-parametric open-source
+cluster-coupled accelerator enabling TinyML training on ultra-
+low-power devices, i.e. near-sensor training on a few tens
+of mW of power budget. RedMulE is conceived for FP16
+GEMM-Ops computation, and supports compressed FP8 inputs
+while also efficiently accelerating a wide range of operations
+that share the same structure of a GEMM. RedMulE allows
+the instantiation of a wide range of Floating-Point Units-
+based Computing Elements (CEs), internal buffers, and memory
+interface configurations. We integrated an instance of RedMulE,
+containing a 12×4 array of CEs into an ultra-low-power cluster
+containing 8 RISC-V cores, and implemented the resulting
+system in a 22 nm technology. RedMulE achieves 99.4%
+CEs utilization and an average 15× speedup during simple
+GEMM execution with respect to a parallel software baseline
+running on the eight cores. It occupies 0.15 mm2 accounting
+for only 24% of the cluster area. During GEMM-Ops execution,
+the performance speedup introduced by RedMulE over the
+RISC-V cores reaches up to 62×. In its best performance point
+(at 613 MHz, 0.8 V), RedMulE achieves 506 GFLOPS/W @
+58.5 GFLOPS when executing FP GEMM kernels; while,
+in its best efficiency point (at 470 MHz, 0.65 V), it reaches
+775 GFLOPS/W @ 44.8 GFLOPS. On a real example of NN
+training, RedMulE accelerates the matrix multiplication by up
+to 14.6× and 28.5× when the input tensors are represented with
+16-bit and 8-bit respectively, accelerating the whole training
+step of ResNet8 by 4.9× and 5.2×.
+ACKNOWLEDGMENTS
+This work was supported in part by Thales Alenia Space,
+The European PILOT (EuroHPC JU, g.a. 101034126), and
+NeuroSoC (Horizon EU g.a. 101070634).
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+Yvan Tortorella received his Master’s Degree in
+Electronic Engineering in October 2021 from the
+University of Bologna. He is currently pursuing a
+Ph. D. in Digital Systems Design in the group of
+Professor Luca Benini at the Department of Electrical
+and Information Engineering (DEI) of the University
+of Bologna. His research interests include the design
+of PULP (Parallel Ultra-Low Power)-based hardware
+accelerators for ultra-low power Machine Learning
+and the design of RISC-V-based computer architec-
+tures for satellite applications.
+Luca Bertaccini received the M.Sc. degree in Elec-
+tronic Engineering from the University of Bologna
+in 2020. He is currently pursuing a Ph.D. degree
+at ETH Z¨urich in the Digital Circuits and Systems
+group led by Prof. Luca Benini. His research inter-
+ests include heterogeneous systems-on-chip, energy-
+efficient hardware accelerators, computer arithmetic,
+and transprecision computing. He received the 2021
+IEEE ASAP Best Paper Honorable Mention.
+Luca Benini holds the chair of digital Circuits
+and systems at ETHZ and is Full Professor at the
+Universit`a di Bologna. He received a PhD from
+Stanford University. Dr. Benini’s research interests
+are in energy-efficient parallel computing systems,
+smart sensing micro-systems and machine learning
+hardware. He has published more than 1000 peer-
+reviewed papers and five books. He is a Fellow of the
+IEEE, of the ACM and a member of the Academia
+Europaea. He received the IEEE Mac Van Valkenburg
+award in 2016 and the ACM/IEEE A. Richard Newton
+Award in 2020.
+Davide Rossi received the Ph.D. degree from the Uni-
+versity of Bologna, Bologna, Italy, in 2012. He has
+been a Post-Doctoral Researcher with the Department
+of Electrical, Electronic and Information Engineering
+“Guglielmo Marconi,” University of Bologna, since
+2015, where he is currently an Assistant Professor.
+His research interests focus on energy-efficient digital
+architectures. In this field, he has published more than
+100 papers in international peer-reviewed conferences
+and journals. He is recipient of Donald O. Pederson
+Best Paper Award 2018, 2020 IEEE TCAS Darlington
+Best Paper Award, 2020 IEEE TVLSI Prize Paper Award.
+
+13
+Francesco Conti received the Ph.D. degree in elec-
+tronic engineering from the University of Bologna,
+Italy, in 2016. He is currently an Assistant Professor
+in the DEI Department of the University of Bologna.
+From 2016 to 2020, he held a research grant in
+the DEI department of University of Bologna and a
+position as postdoctoral researcher at the Integrated
+Systems Laboratory of ETH Zurich in the Digital
+Systems group. His research focuses on the devel-
+opment of deep learning based intelligence on top of
+ultra-low power, ultra-energy efficient programmable
+Systems-on-Chip. His research work has resulted in more than 40 publications
+in international conferences and journals and has been awarded several times,
+including the 2020 IEEE TCAS-I Darlington Best Paper Award.
+
diff --git a/jtE3T4oBgHgl3EQfJQmU/content/tmp_files/2301.04342v1.pdf.txt b/jtE3T4oBgHgl3EQfJQmU/content/tmp_files/2301.04342v1.pdf.txt
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@@ -0,0 +1,3033 @@
+Prepared for submission to JHEP
+Massive vector particle tunneling from
+Kerr-Newman-de Sitter black hole under
+generalized uncertainty principle
+Yenshembam Priyobarta Singh,a Telem Ibungochouba Singha,1
+aDepartment of Mathematics,
+Manipur University, Canchipur, Imphal, Manipur, India
+E-mail: priyoyensh@gmail.com, ibungochouba@rediffmail.com
+Abstract: The quantum tunneling of charged massive vector boson particles across the
+event horizon of Kerr-Newman-de Sitter black hole is investigated under the influence of
+quantum gravity effects. The Hawking temperature and the heat capacity of Kerr-Newman-
+de Sitter black hole are derived using the generalized field equation for charged massive
+vector bosons. It is found that the quantum gravity effects modified the Hawking tempera-
+ture and heat capacity. Moreover, they depend on the mass and angular momentum of the
+emitted vector boson particles. We also discuss the remnant and graphical analysis of the
+modified Hawking temperatures and heat capacities of the black hole.
+Keywords: Quantum tunneling; quantum gravity effects; Kerr-Newman-de Sitter black
+hole; Hawking temperature and heat capacity .
+1Corresponding author.
+arXiv:2301.04342v1  [gr-qc]  11 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Generalized field equations for massive vector bosons
+3
+3
+Quantum tunneling from KNdS black hole
+4
+4
+Remnant of KNdS black hole
+15
+5
+Graphical Analysis
+16
+5.1
+Temperature Td with radius of event horizon rH for 3-dimensional KNdS
+black hole
+16
+5.2
+Heat capacity CH with radius of event horizon rH for 3-dimensional KNdS
+black hole
+19
+5.3
+Hawking temperature Td4 with radius of event horizon for 4-dimensional
+KNdS black hole
+21
+5.4
+Heat Capacity CH4 versus horizon radius rH for 4-dimensional KNdS black
+hole
+24
+6
+Conclusions
+26
+A Coefficients of eq. (3.48)
+27
+B The expressions of χ1 and χ2 given in eq. (3.49)
+28
+1
+Introduction
+In the early 1970s, Hawking proposed a black hole radiation called Hawking radiation using
+quantum field theory techniques on a curve space-time background [1, 2]. The discovery of
+Hawking radiation is one of the significant developments in gravitational physics. It shows
+that black holes have a thermodynamical features. Refs. [3, 4] determined that the horizon
+area and the black hole’s entropy are proportional. More precisely, the black hole’s entropy
+is equal to one-fourth of its horizon area. Since then, theoretical physicists have paid a
+lot of attention to Hawking radiation and many approaches have been used to determine
+Hawking radiation.
+Generally, there are two approaches to investigate the imaginary part of the action.
+They are the radial null geodesic method and the Hamilton-Jacobi method. The radial null
+geodesic method was put forth by Parikh and Wilczek [5–8]. In this method, the emitted
+particles act as the potential barrier and the semiclassical WKB approximation is used to
+determine the imaginary part of the radial action. Later, Zhang and Zhao [9–11] have made
+– 1 –
+
+significant progress by extending Parikh’s method to non-spherical symmetric stationary
+black holes and the radiation of charged massive particle. The second is the Hamilton-Jacobi
+method, used by Angheben et al. [12] as an extension of the complex path integral method
+introduced by Padmanabhan et al. [13–15]. In both methods, the WKB approximation
+is used to investigate the tunneling probability for classically forbidden trajectory from
+inside to outside the horizon and it is given by Γ = exp
+�
+− 2
+ℏIm I
+�
+, where I and ℏ denote
+the semiclassical action of the outgoing particle and Planck’s constant. Kerner and Mann
+[16, 17] investigated the tunneling of spin-1/2 fermion particles from rotating and non-
+rotating black holes using Dirac equation, Pauli sigma matrix and Feynman prescription.
+By the fermions tunneling method the Hawking radiation of Dirac particles via tunneling
+from the black ring is also derived [18]. Using their method, many interesting results have
+been obtained in [19–22]. Damour and Ruffini [23] used a new approach called tortoise
+coordinate transformation to study Hawking radiation. Later, Sannan [24] extended the
+work of Damour and Ruffini by deriving the probability distributions for boson and fermion
+emissions from a black hole.
+The existence of a minimum measurable length that can be identified with the order
+of the Planck scale is predicted by numerous quantum gravity theories, such as string
+theory, loop quantum gravity, non-commutative geometry and Gedanken experiments [25–
+31]. Constructing new theoretical models is one of the fields to study quantum gravity
+effects. The Generalized Uncertainty Principle (GUP) is a modified theory that realizes this
+minimum length. It is the generalization of the Heisenberg uncertainty principle (HUP).
+The modified fundamental commutation relation proposed by Kempf et al. [32] is of the
+form
+�
+xi, pj
+�
+= iℏδi,j
+�
+1 + βp2�
+,
+(1.1)
+where the position xi and momentum operators pj are defined by
+xi = x0i,
+pj = p0j
+�
+1 + βp2
+0
+�
+.
+(1.2)
+Then x0i and p0j satisfy the canonical commutation relations as
+�
+x0i, p0j
+�
+= iℏδij. The
+corresponding GUP takes the form
+∆x∆p ≥ ℏ
+2
+�
+1 + (∆p)2β
+�
+,
+(1.3)
+where β =
+β0l2
+p
+ℏ2 =
+β0
+M2pc2 . Here β0(≤ 1034), lp and Mp represent the dimensionless parameter
+of order unity, Planck length and Planck mass respectively.
+The implications of the aspects of GUP have been investigated in many contexts such as
+modifications of the quantum Hall effect, neutrino oscillations, Landau levels, Newton’s law,
+cosmology, and the weak equivalence principle (WEP) [33–38]. It should be noted that the
+GUP has also influenced the thermodynamics of black holes. As a result, the GUP concept
+has been applied to various black holes in order to study their thermodynamics properties.
+Adler et al. [39] studied the influence of GUP on the thermodynamics of the Schwarzschild
+black hole.
+Further, using the Parikh-Wilczek tunnelling approach, Nozari and Saghaf
+– 2 –
+
+[40] studied Hawking radiation for massless scalar particles in the background geometry of
+Schwarzschild black hole and retrieved the tunnelling rate as well as the corrected Hawking
+temperature by taking GUP into account. Applying the WKB approximation to the Dirac
+equation, the tunnelling of fermions from the Kerr and Kerr-Newman black holes are studied
+in [41, 42]. Yale [43] investigated the tunneling of arbitrary scalars, fermions and spin-
+1 bosons by ignoring back-reaction effects.
+Banerjee and Majhi [44, 45] discussed the
+tunneling of fermions and bosons at the event horizon of black holes beyond the semiclassical
+approximation. They derived the modified Hawking temperature and change in black hole
+entropy. Ibungochouba et al. [46] studied the tunneling of fermions from the BTZ black
+hole.
+Many studies have shown that, GUP avoid the black hole to evaporate completely. The
+black hole has a phase transition close to the Planck scale and reaches thermal equilibrium
+with its surroundings.
+As a result, the black hole’s evaporation stops, leaving behind
+a stable remnant [47–51].
+Li [52] studied the tunneling of massive spin-1 particle from
+Reissner-Nordstrom and Kerr black hole under the effect of quantum gravity. Ovgun et
+al. [53] investigated the charged massive bosons tunneling from noncommutative charged
+black holes.
+In this paper, the tunneling of massive vector boson particles across the event horizon
+of the KNdS black hole is investigated. This article’s layout is constructed in the following
+manner: In section 2, we revisit the generalized field equation for massive vector boson
+particles. The quantum tunneling of massive vector boson from KNdS black hole under
+quantum gravity effects is presented in section 3. Further, section 4 provides the analysis
+of the remnant of KNdS black hole under quantum gravity effects. The graphical analysis
+of quantum corrected Hawking temperatures and heat capacity are presented in section 5.
+Lastly, some concluding remarks are presented in section 6.
+2
+Generalized field equations for massive vector bosons
+The GUP-corrected Lagrangian of massive vector field Ψµ is given by [52]
+LGUP = − 1
+2
+�
+D+
+µ Ψ+
+ν − D+
+ν Ψ+
+µ
+� �
+D−µΨ−ν − D−νΨ−µ�
+− m2
+ℏ2 W +
+µ W −µ
+− i
+ℏeF µνW +
+µ W −
+ν ,
+(2.1)
+where D±
+0 =
+�
+1 + βℏ2g00D±2
+0
+�
+D±
+0 , D±
+i =
+�
+1 − βℏ2giiD±2
+i
+�
+D±
+i ,
+Fµν = �∇µAν − �∇νAµ with D±
+µ
+= ∇µ ± i
+ℏeAµ, �∇0 =
+�
+1 + βℏ2g00∇2
+0
+�
+∇0 and �∇i =
+�
+1 − βℏ2gii∇2
+i
+�
+∇i. Here e, m and Aµ denote the charge of W + boson, mass of the vector
+particle and electromagnetic potential of the black hole respectively.
+The generalized action of the massive vector boson particles takes the form
+SGUP =
+�
+dx4√−g LGUP
+�
+Ψ±
+µ , ∂µΨ±
+ν , ∂µ∂ρΨ±
+ν , ∂µ∂ρ∂λΨ±
+ν
+�
+.
+(2.2)
+Following from eq.
+(2.2), the modified wave equation for massive vector bosons is
+obtained as
+∂µ(√−gΨµν) − 3β∂0
+�√−gg00 �
+e2A2
+0 + iℏe∇0A0
+�
+Ψ0v�
+– 3 –
+
++ 3β ∂i
+�√−ggii �
+e2A2
+i + iℏe∇iAi
+�
+Ψiv�
++ 3β∂0∂0
+�√−gg00iℏeA0Ψ0v�
+− 3β∂i∂i
+�√−ggiiiℏeAiΨiv�
++ βℏ2∂0∂0∂0
+�√−gg00Ψ0v�
+− βℏ2∂i∂i∂i
+�√−ggiiΨiv�
++ √−g i
+ℏeAµΨµν − √−gm2
+ℏ2 Ψν
+− √−g i
+ℏeF µνΨµ + β√−gg00
+�
+iℏe∇0∇0A0 + 3e2A0∇0A0 − i
+ℏe3A3
+0
+�
+Ψ0ν
+− β√−ggii
+�
+iℏe∇i∇iAi + 3e2Ai∇iAi − i
+ℏe3A3
+i
+�
+Ψiν = 0,
+(2.3)
+where GUP modified anti-symmetric tensor Ψµν is given by Ψµν = DµΨν − DνΨµ.
+3
+Quantum tunneling from KNdS black hole
+The line element of KNdS black hole in well known Boyer-Lindquist coordinates (t, r, θ, ϕ)
+[54] is expressed as
+ds2 = −
+�∆ − ∆θ a2 sin2 θ
+Σ2ρ2
+�
+dt2 − 2a sin2 θ[(r2 + a2)∆θ − ∆]
+Σ2ρ2
+dtdϕ + ρ2
+∆ dr2
++ ρ2
+∆θ
+dθ2 + sin2 θ
+�
+(r2 + a2)2 ∆θ − ∆a2 sin2 θ
+�
+Σ2ρ2
+dϕ2,
+(3.1)
+where
+Σ = 1 + Λa2
+3 , ρ2 = r2 + a2 cos2 θ, ∆θ = 1 + Λa2 cos2 θ
+3
+,
+∆ =
+�
+1 − Λa2
+3
+� �
+r2 + a2�
+− 2Mr + Q2.
+The electromagnetic potential Aµ of KNdS black hole is given by
+Aµ = Qr
+�
+δt
+µ − a sin2 θδϕ
+µ
+�
+ρ2Σ
+.
+(3.2)
+Eq.
+(3.1) describes a charged rotating black hole with mass m, spin parameter a and
+charge Q with the positive cosmological constant Λ. The case Λ = 0 gives the solution
+of the Kerr-Newman black hole. For Λ < 0, eq. (3.1) represents Kerr-Newman-(anti-)de
+Sitter (KNadS) black hole.
+The horizons of the KNdS can be obtained from the event horizon equation as
+∆ =
+�
+1 − Λr2
+3
+� �
+r2 + a2�
+− 2Mr + Q2 = 0.
+(3.3)
+Eq. (3.3) gives four real roots whenever 1
+Λ ≫ M2 > Q2+a2. The four roots are rC, rH,
+r+ and r− (rC > rH > r+ > r−). Here rC denotes the cosmological horizon, rH represents
+the event horizon and r− corresponds to the Cauchy horizon. One reaches singularity r = 0,
+θ = π
+2 and on other side of r = 0, r = r− is considered as another cosmological horizon [55].
+– 4 –
+
+In this coordinate system, the event horizon and the infinite red-shift surface do not
+coincide due to rotation. Because of this, it is inconvenient to study the characteristic of
+tunneling radiation. So, we perform the dragging coordinate transformation [56]
+dϕ
+dt = Ω = −gtϕ
+gϕϕ
+.
+(3.4)
+The line element (3.1) is reduced to
+ds2 = −
+∆∆θρ2
+Σ2 �
+∆θ(r2 + a2)2 − ∆a2 sin2 θ
+�dt2 + ρ2
+∆ dr2 + ρ2
+∆θ
+dθ2.
+(3.5)
+The non-vanishing component of electromagnetic potential A0 is given by
+A0 =
+(r2 + a2)∆θQr
+Σ
+�
+∆θ(r2 + a2)2 − ∆a2 sin2 θ
+�.
+(3.6)
+The dragging coordinate transformation makes the geometrical optical limit a reliable
+approximation; hence, the WKB approximation can be applied.
+Near the event horizon r = rH, we use the approximation
+∆(r) =∆(rH) + (r − rH)∆,r (rH) + O
+�
+(r − rH)2�
+≈ (r − rH)∆,r (rH),
+(3.7)
+where rH is defined as
+rH = 1
+α1
+�
+1 + 4ΛM2
+3β2
+1α1
++ . . .
+� �
+M +
+�
+M2 − (a2 + Q2)α1
+�
+,
+(3.8)
+where β =
+�
+1 − Λa2
+3
+and α1 =
+��
+1 + Λa2
+3
+�2
++ 4ΛQ2
+3
+. Then eq. (3.5) takes the form
+ds2 = −(r − rH)∆,r (rH) ρ2(rH)
+Σ2 �
+r2
+H + a2�2
+dt2 +
+ρ2(rH)
+(r − rH)∆,r (rH)dr2 + ρ2(rH)
+∆θ
+dθ2.
+(3.9)
+The metric determinant and contravariant components of the line element (3.9) are as
+follows
+g = −
+ρ6(rH)
+Σ2 �
+r2
+H + a2�2 ∆θ
+,
+g00 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH),
+g11 = (r − rH)∆,r (rH)
+ρ2(rH)
+,
+g22 =
+∆θ
+ρ2(rH),
+g01 = g02 = g10 = g12 = g20 = g21 = 0.
+(3.10)
+The angular velocity at the event horizon is given as
+Ω =
+a
+r2
+h + a2 .
+(3.11)
+– 5 –
+
+The surface gravity near the event horizon of the KNdS black hole is obtained as
+κ = lim
+g00→0
+�
+−1
+2
+�
+−g11
+g00
+dg00
+dr
+�
+=
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+(3.12)
+The original Hawking temperature of the KNdS black hole is obtained from the relation
+To =
+κ
+2π as [57]
+To =
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+2π
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+.
+(3.13)
+By using the expression of M from eq. (3.3), we derive the heat capacity of the black
+hole by using the relation Co = ∂M
+∂To as [58, 59]
+Co =
+�∂M
+∂rh
+� �∂rh
+∂To
+�
+=
+2π
+�
+a2 + r2�2 �
+3 + a2Λ
+� �
+a2 �
+3 + r2Λ
+�
++ 3
+�
+Q2 − r2 + r4Λ
+��
+3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)].
+(3.14)
+Applying WKB approximation, Ψµ given in eq. (2.3) is taken as
+Ψµ = cµ(t, r, θ) exp
+� i
+ℏS(t, r, θ)
+�
+,
+(3.15)
+where S is defined as
+S(t, r, θ) = S0(t, r, θ) + ℏ S1(t, r, θ) + ℏ2 S2(t, r, θ) + · · · .
+(3.16)
+By substituting eqs. (3.15), (3.16) and (3.9) in eq. (2.3) and keeping only the lowest
+order in ℏ, we obtain the equations for the coefficients cµ as
+(r − rH)∆,r (rH)
+ρ2(rH)
+�
+c0 B2
+1 (∂rS0)2 − c1 B0B1 (∂rS0) (∂tS0 + eA0)
+�
++
+∆θ
+ρ2(rH)
+�
+c0 B2
+2(∂θS0)2 − c2B0B2(∂θS0)(∂tS0 + eA0)
+�
++ m2c0 = 0,
+(3.17)
+−
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+�
+c1 B2
+0(∂tS0 + eA0)2 − c0 B0B1 (∂rS0) (∂tS0 + eA0)
+�
++
+∆θ
+ρ2(rH)
+�
+c1 B2
+2(∂θS0)2 − c2B1B2(∂rS0)(∂θS0)
+�
++ m2c1 = 0,
+(3.18)
+– 6 –
+
+−
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+�
+c2 B2
+0(∂tS0 + eA0)2 − c0 B0B2 (∂θS0) (∂tS0 + eA0)
+�
++ (r − rH)∆,r (rH)
+ρ2(rH)
+�
+c2 B2
+1(∂rS0)2 − c1B1B2(∂rS0)(∂θS0)
+�
++ m2c2 = 0,
+(3.19)
+where the Bµ’s are defined as
+B0 = 1 + β Σ2 �
+r2
+H + a2�2 (∂tS0 + eA0)2
+(r − rH)∆,r (rH) ρ2(rH)
+,
+B1 = 1 + β (r − rH)∆,r (rH)(∂rS0)2
+ρ2(rH)
+,
+B2 = 1 + β∆θ(∂θS0)2
+ρ2(rH)
+.
+(3.20)
+Considering the symmetries of the metric (3.9), we carry on the separation of variables
+as follows
+S0 = −(E − jΩ) t + R(r) + Θ(θ) + U,
+(3.21)
+where E is the energy of the emitted vector particles, j is the angular momentum and U is
+a complex constant. On inserting eq. (3.21) in eqs. (3.17)-(3.19), we get a matrix equation
+as
+F(c0, c1, c2)T = 0,
+(3.22)
+where F is a 3 × 3 matrix and all the entries are as follows
+F11 =(r − rH)∆,r (rH)
+ρ2(rH)
+B2
+1R′2 +
+∆θ
+ρ2(rH)B2
+2J2
+θ + m2,
+F12 = − (r − rH)∆,r (rH)
+ρ2(rH)
+(−E + jΩ + eA0)B0B1R′,
+F13 = −
+∆θ
+ρ2(rH)(−E + jΩ + eA0)B0B2Jθ,
+F21 =
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)B0B1R′,
+F22 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)2B2
+0 +
+∆θ
+ρ2(rH)B2
+2J2
+θ + m2,
+F23 = −
+∆θ
+ρ2(rH)B1B2JθR′,
+K31 =
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)B0B2Jθ,
+K32 = − (r − rH)∆,r (rH)
+ρ2(rH)
+B1B2JθR′,
+K33 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)2B2 + (r − rH)∆,r (rH)
+ρ2(rH)
+B2
+1R′2
++ m2,
+(3.23)
+– 7 –
+
+where R′ = ∂rR and Jθ = ∂θΘ.
+To find a non-trivial solution of eq. (3.22), we put the determinant of the matrix F
+equals zero, which in turn gives
+O(β4)(∂rR)12 + O(β3)(∂rR)10 + O(β2)(∂rR)8 +
+�
+H6 + O(β2)
+�
+(∂rR)6
++
+�
+H4 + O(β2)
+�
+(∂rR)4 +
+�
+H2 + O(β2)
+�
+(∂rR)2 + H0 + O(β2) = 0,
+(3.24)
+where
+H6 =4β
+�(r − rH)∆,r (rH)
+ρ2(rH)
+�3
+,
+H4 =(r − rH)2∆2
+,r(rH)
+ρ4(rH)
+�
+1 + 4β
+�
+−(−E + jΩ + eA0)2 Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
++ ∆θJ2
+θ
+ρ2(rH) + m2
+��
+,
+H2 =2(r − rH)∆,r (rH)
+ρ2(rH)
+�
+−(−E + jΩ + eA0)2 Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
++ ∆θJ2
+θ
+ρ2(rH) + m2
+−2β
+�
+(−E + jΩ + eA0)4 Σ4 �
+r2
+H + a2�4
+(r − rH)2∆2,r (rH) ρ4(rH)
+− ∆2
+θJ4
+θ
+ρ2(rH)
+��
+,
+H0 =
+�
+−(−E + jΩ + eA0)2 Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
++ ∆θJ2
+θ
+ρ2(rH) + m2
+� � ∆θJ2
+θ
+ρ2(rH) + m2
++(−E + jΩ + eA0)2 Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+− 4β
+�
+(−E + jΩ + eA0)4 Σ4 �
+r2
+H + a2�4
+(r − rH)2∆2,r (rH) ρ4(rH)
+− ∆2
+θJ4
+θ
+ρ4(rH)
+��
+.
+By neglecting the higher order term of β, eq. (3.24) becomes
+4β(r − rH)2∆2,r (rH) (∂rR)4
+ρ4(rH)
++ (r − rH)∆,r (rH) (∂rR)2
+ρ2(rH)
++ ∆θJ2
+θ
+ρ2(rH)
++ 4β
+�
+∆2
+θJ4
+θ
+ρ4(rH) − (−E + jΩ + eA0)4Σ4 �
+r2
+H + a2�4
+(r − rH)2∆2,r (rH) ρ4(rH)
+�
++ m2
+− (−E + jΩ + eA0)2Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+= 0.
+(3.25)
+We obtain the solution to the derivative of the radial action by neglecting the higher
+power of β as
+∂rR = ±
+�
+�
+�
+�−
+�
+J2
+θ ∆θ + m2ρ2�
+(r − rH) ∆,r (rH) + (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)2 (∆,r (rH))2
+�
+1 + β χ1
+χ2
+�
+, (3.26)
+– 8 –
+
+where
+χ1 =2∆,r (rH)(r − rH)
+�
+2J4
+θ ∆2
+θ + 2J2
+θ m2∆θρ2 + m4ρ4�
+− 4Σ2(−E + jΩ + eA0)2 �
+r2
+H + a2�2 �
+J2
+θ ∆θ + m2ρ2�
+,
+χ2 =ρ2 �
+(r − rH)∆,r (rH)
+�
+J2
+θ ∆θ + m2ρ2�
+− Σ2(−E + jΩ + eA0)2 �
+r2
+H + a2�2�
+.
+(3.27)
+The solution of the radial action is obtained by integrating eq. (3.26) around the pole
+r = rH. The imaginary part of the radial action gives the particle’s rate of tunneling as
+Im R± = ± Im
+�
+dr
+�
+�
+�
+�−
+�
+J2
+θ ∆θ + m2ρ2�
+(r − rH) ∆,r (rH) + (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)2 (∆,r (rH))2
+×
+�
+1 + β χ1
+χ2
+�
+= ±
+π(E − jΩ − eA0)
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+2
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+(1 + β Π) ,
+(3.28)
+where Π = 4m2+ 4J2
+θ ∆θ
+ρ2
+. R+ and R− stand for the radial action of the outgoing and ingoing
+particles respectively. Based on WKB approximation, the probabilities of the vector boson
+particles tunneling across the event horizon rH are
+Poutgoing = exp
+�
+−2
+�
+Im(R+) + Im(U)
+��
+and
+Pingoing = exp
+�
+−2
+�
+Im(R−) + Im(U)
+��
+.
+(3.29)
+According to the semiclassical WKB approximation, the ingoing vector boson particles have
+a 100% chance of entering the black hole. It shows that Im(U) = −Im(R−). Thus the
+tunneling rate of the vector boson particles is given by
+Γ = Poutgoing
+Pingoing
+= exp [−4 Im R+]
+= exp
+�
+�
+−2π(E − jΩ − eA0)
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+(1 + β Π)
+�
+� .
+(3.30)
+The modified Hawking temperature of the KNdS black hole under the quantum gravity
+effect is obtained as
+Td =
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+2π
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+(1 − β Π)
+=To (1 − β Π) ,
+(3.31)
+– 9 –
+
+where To =
+�
+(rH−M)− Λ rH
+3
+(2r2
+H+a2)
+�
+2π(r2
+H+a2)
+�
+1+ Λa2
+3
+�
+is the original Hawking temperature. It is obvious from
+the point Π > 0 that the modified Hawking temperature is reduced due to the quantum
+gravity effects. Further, the modified Hawking temperature relies on the mass and angular
+momentum of the emitted vector boson particles. In the absence of the quantum gravity
+effects i.e. β = 0, the original Hawking temperature of KNdS black hole is recovered.
+The modified heat capacity is calculated as
+CH =
+�∂M
+∂rh
+� � ∂rh
+∂Td
+�
+=
+2π
+�
+a2 + r2�2 �
+3 + a2Λ
+� �
+a2 �
+3 + r2Λ
+�
++ 3
+�
+Q2 − r2 + r4Λ
+��
+3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)]
+× (1 + β Π) .
+(3.32)
+From eq. (3.32), it is observed that the modified heat capacity tends to the original heat
+capacity when β = 0. Thus, in the absence of the quantum gravity effects, the original heat
+capacity is recovered, and the heat capacity increases due to the quantum gravity effects.
+Now using another coordinate transformation φ = ϕ − Ωt [60] where
+Ω =
+a [(r2 + a2)∆θ − ∆]
+(r2 + a2)2 ∆θ − ∆a2 sin2 θ,
+(3.33)
+then the line element (3.1) reduces to
+ds2 = −
+∆∆θρ2
+Σ2 �
+∆θ(r2 + a2)2 − ∆a2 sin2 θ
+�dt2 + ρ2
+∆ dr2 + ρ2
+∆θ
+dθ2
++
+�
+∆θ(r2 + a2)2 − ∆a2 sin2 θ
+�
+sin2 θ
+ρ2Σ2
+dφ2.
+(3.34)
+The corresponding non-zero electromagnetic potentials are given by
+A0 =
+(r2 + a2)∆θ Q r
+Σ
+�
+∆θ(r2 + a2)2 − ∆a2 sin2 θ
+�,
+A3 =
+−Qra sin2 θ
+(r2 + a2 cos2 θ)Σ.
+(3.35)
+Using eq. (3.7) in eq. (3.34), we get
+ds2 = − (r − rH)∆,r (rH) ρ2(rH)
+Σ2 �
+r2
+H + a2�2
+dt2 +
+ρ2(rH)
+(r − rH)∆,r (rH)dr2 + ρ2(rH)
+∆θ
+dθ2
++ ∆θ
+�
+r2
+H + a2�2 sin2 θ
+ρ2(rH)Σ2
+dφ2.
+(3.36)
+The metric determinant and the nonzero contravariant components of the line element
+given above are as follows
+g = ρ4 sin2 θ
+Σ4
+,
+g00 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH),
+g11 = (r − rH)∆,r (rH)
+ρ2(rH)
+,
+g22 =
+∆θ
+ρ2(rH),
+– 10 –
+
+g33 =
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+.
+(3.37)
+According to WKB approximation, Ψµ from eq. (2.3) can be written as
+Ψµ = cµ(t, r, θ, φ) exp
+� i
+ℏS(t, r, θ, φ)
+�
+,
+(3.38)
+where S is defined as
+S(t, r, θ) = S0(t, r, θ, φ) + ℏ S1(t, r, θ, φ) + ℏ2 S2(t, r, θ, φ) + · · · .
+(3.39)
+By substituting eqs. (3.38) and (3.39) in eq. (3.36), and keeping only the lowest order
+in ℏ, we derive the equations for the coefficient cµ as
+(r − rH)∆,r (rH)
+ρ2(rH)
+�
+c0 H2
+1 (∂rS0)2 − c1 H0H1 (∂rS0) (∂tS0 + eA0)
+�
++
+∆θ
+ρ2(rH)
+�
+c0 H2
+2 (∂θS0)2 − c2H0H2 (∂θH)(∂tS0 + eA0)
+�
++
+ρ2(rH) Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+�
+c0H2
+3 (∂φS0 + eA3)2 − c3 H0H3 (∂tS0 + eA0) (∂φS0 + eA3)
+�
++ m2c0 = 0,
+(3.40)
+−
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+�
+c1 H2
+0(∂tS0 + eA0)2 − c0 H0H1 (∂rS0) (∂tS0 + eA0)
+�
++
+∆θ
+ρ2(rH)
+�
+c1 H2
+2(∂θS0)2 − c2H1H2(∂rS0)(∂θS0)
+�
++
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+�
+c1 H2
+3(∂φS0 + eA3)2 − c3 H1H3 (∂rS0) (∂φS0 + eA3)
+�
++ m2c1 = 0,
+(3.41)
+−
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+�
+c2 H2
+0(∂tS0 + eA0)2 − c0 H0H2 (∂θS0) (∂tS0 + eA0)
+�
++ (r − rH)∆,r (rH)
+ρ2(rH)
+�
+c2 H2
+1(∂rS0)2 − c1H1H2(∂rS0)(∂θS0)
+�
++
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+�
+c2 H2
+3(∂φS0 + eA3)2 − c3 H2H3 (∂θS0) (∂φS0 + eA3)
+�
++ m2c2 = 0,
+(3.42)
+−
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)
+�
+c3 H2
+0(∂tS0 + eA0)2 − c0 H0H3(∂tS0 + eA0)
+(∂φS0 + eA3)] + (r − rH)∆,r (rH)
+ρ2(rH)
+�
+c3 H2
+1(∂rS0)2 − c1H1H3(∂rS0)(∂φS0 + eA3)
+�
++
+∆θ
+ρ2(rH)
+�
+c3 H2
+2 (∂θS0)2 − c2H2H3 (∂θS0)(∂φS0 + eA3)
+�
++ m2c3 = 0,
+(3.43)
+where the Hµ’s are defined as
+H0 = 1 + β Σ2 �
+r2
+H + a2�2 (∂tS0 + eA0)2
+(r − rH)∆,r (rH) ρ2(rH)
+,
+H1 = 1 + β (r − rH)∆,r (rH)(∂rS0)2
+ρ2(rH)
+,
+– 11 –
+
+H2 = 1 + β ∆θ (∂θS0)2
+ρ2(rH)
+,
+H3 = 1 + β ρ2(rH)Σ2 (∂φS0 + eA3)2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+.
+(3.44)
+Considering the symmetry of the black hole, the corresponding action of the vector
+boson particles can be written as
+S0 = −(E − jΩ) t + W(r) + Θ(θ, φ) + U,
+(3.45)
+where E is the energy of the emitted vector particles, j is the angular momentum and U
+is a complex constant. On inserting eq. (3.45) in eqs. (3.41)-(3.44), a matrix equation is
+obtained as
+F (c0, c1, c2, c3)T = 0,
+(3.46)
+where F is a 4 × 4 matrix, whose elements are as follows
+F11 =(r − rH)∆,r (rH)
+ρ2(rH)
+H2
+1 W′2 +
+∆θ
+ρ2(rH)H2
+2 J2
+θ +
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+H2
+3 (Jφ + eA3) + m2,
+F12 = − (r − rH)∆,r (rH)
+ρ2(rH)
+(−E + jΩ + eA0)H0H1W′,
+F13 = −
+∆θ
+ρ2(rH)(−E + jΩ + eA0)H0H2 Jθ,
+F14 = −
+ρ2(rH) Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+(−E + jΩ + eA0) (Jφ + eA3) H0H3,
+F21 =
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)H0H1W′,
+F22 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)2H2
+0 +
+∆θ
+ρ2(rH)H2
+2J2
+θ
++
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+(Jφ + eA3)2 H2
+3 + m2,
+F23 = −
+∆θ
+ρ2(rH)H1H2 Jθ W′,
+F24 = −
+ρ2(rH) Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+(Jφ + eA3) W′ H1H3,
+F31 =
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)H0H2 Jθ,
+F32 = − (r − rH)∆,r (rH)
+ρ2(rH)
+H1H2 Jθ W′,
+F33 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)2 H2
+0 + (r − rH)∆,r (rH)
+ρ2(rH)
+H2
+1 W′2
++
+ρ2(rH)Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+(Jφ + eA3)2 H2
+3 + m2,
+– 12 –
+
+F34 = −
+ρ2(rH) Σ2
+∆θ
+�
+r2
+H + a2�2 sin2 θ
+(Jφ + eA3) Jθ H2H3,
+F41 =
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0) (Jφ + eA3) H0H3,
+F42 = − (r − rH)∆,r (rH)
+ρ2(rH)
+(Jφ + eA3) H1H2 W′,
+F43 = −
+∆θ
+ρ2(rH) (Jφ + eA3) Jθ H2H3,
+F44 = −
+Σ2 �
+r2
+H + a2�2
+(r − rH)∆,r (rH) ρ2(rH)(−E + jΩ + eA0)2 H2
+0 + (r − rH)∆,r (rH)
+ρ2(rH)
+H2
+1 W′2
++
+∆θ
+ρ2(rH)H2
+2J2
+θ + m2,
+(3.47)
+where W′ = ∂rW, Jθ = ∂θΘ and Jφ = ∂φΘ.
+Eq. (3.46) has a non-trivial solution if the determinant of the matrix F equals zero. If
+det(F) = 0, then eq. (3.46) gives
+O(β6)(∂rW)18 + O(β5)(∂rW)16 + O(β4)(∂rW)14 + O(β3)(∂rW)12 + O(β2)
+(∂rW)10 + O(β2)(∂rW)8 + O(β2)(∂rW)6 + O(β2)(∂rW)4 + O(β2)(∂rW)2 + O(β2)
++
+�
+A∗
+0(∂rW)2 + A∗
+1
+� �
+B0 + B2(∂rW)2 + B4(∂rW)4 + B6(∂rW)6�
+= 0.
+(3.48)
+(The expressions for Ai and Bi are given in Appendix A.)
+Solving eq. (3.48) by neglecting the higher order terms of β , we obtain the solution of
+the derivative of the radial action as
+∂rW =
+�
+−m2ρ2(rH) + J2
+θ ∆θ
+(r − rH)∆,r (rH) + (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)2∆2,r (rH)
+−
+(Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ
+(r − rH)
+�
+r2
+H + a2�
+∆θ ∆,r (rH)
+�1
+2 ×
+�
+1 + χ1
+χ2
+β
+�
+.
+(3.49)
+(The expressions of χ1 and χ2 are given in Appendix B.)
+The imaginary part of the radial action is obtained by integrating eq. (3.49) at the
+pole, r = rH as
+ImW± = ± Im
+�
+dr
+�
+−m2ρ2(rH) + J2
+θ ∆θ
+(r − rH)∆,r (rH) + (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)2∆2,r (rH)
+−
+(Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ
+(r − rH)
+�
+r2
+H + a2�
+∆θ ∆,r (rH)
+�1
+2 ×
+�
+1 + χ1
+χ2
+β
+�
+= ±
+π(E − jΩ − eA0)
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+2
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+(1 + β Ξ) ,
+(3.50)
+– 13 –
+
+where
+Ξ = − 4 (Jφ + eA3)2 ρ2(rH) Σ2 csc2 θ
+�
+r2
+H + a2�2 ∆θ
++
+1
+2 ρ4(rH) Σ2 (Jφ + eA3) (Jφ + eA3 − 1)
+�
+3m2 �
+r2
+H + a2�2 ∆θ sin2 θ
+�
+5J2
+θ ∆θ + 3m2 ρ2(rH)
+�
++ (Jφ + eA3)3 ρ2(rH) Σ2
+�
+−8J2
+θ ∆θ (Jφ + eA3 − 1) + m2 ρ2(rH)
+�
+8 + 7 (Jφ + eA3)
+���
+.
+(3.51)
+Here W+ and W− indicate the radial action of the outgoing and ingoing particles
+respectively. According to WKB approximation, the tunneling probabilities are given by
+Poutgoing = exp
+�
+−2
+�
+Im(R+) + Im(U)
+��
+and
+Pingoing = exp
+�
+−2
+�
+Im(R−) + Im(U)
+��
+.
+(3.52)
+In accordance with the semiclassical WKB approximation, there is a 100% probability
+of ingoing particle to enter the black hole. Thus, the tunneling rate of W + boson particles
+is given by
+Γrate = Poutgoing
+Pingoing
+= exp
+�
+−4{Im(R+)
+�
+= exp
+�−2 π(E − jΩ − eA0)
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+(1 + β Ξ)
+�
+.
+(3.53)
+The Boltzman factor gives the Hawking temperature [17]. Thus, the GUP modified Hawking
+temperature is derived as
+Td4 =
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+2 π
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+(1 − β Ξ)
+= To (1 − β Ξ) ,
+(3.54)
+where To =
+�
+(rH − M) − Λ rH
+3
+�
+2r2
+H + a2��
+2 π
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+, is the original Hawking temperature of KNdS
+black hole without any quantum gravity effects. The modified Hawking temperature Td4
+may be lower or greater than the original Hawking temperature To according to Ξ > 0 or
+Ξ < 0. The modified Hawking temperature depends on the quantum numbers (mass and
+angular momentum) of the emitted vector boson particles.
+The modified heat capacity is calculated as
+CH4 =
+�∂M
+∂rh
+� � ∂rh
+∂Td
+�
+– 14 –
+
+=
+2π
+�
+a2 + r2�2 �
+3 + a2Λ
+� �
+a2 �
+3 + r2Λ
+�
++ 3
+�
+Q2 − r2 + r4Λ
+��
+3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)]
+× (1 + β Ξ) .
+(3.55)
+From eq. (3.55), it is observed that the modified heat capacity reduces to the original heat
+capacity in the absence of the quantum gravity effects. The modified heat capacity CH4 is
+higher or lower than the original heat capacity Co, according to Ξ > 0 or Ξ < 0.
+4
+Remnant of KNdS black hole
+Many studies have shown that the GUP effect could give a black hole remnant [47–51,
+62, 63]. From this point, we will investigate the remnant of 3-dimensional KNdS black
+hole only. The tunneling particle’s mass is no longer considered in the following discussion
+since the tunneling particles at the event horizon are effectively massless. According to the
+uncertainty principle, the lower limit of the tunneling particle energy can be expressed as
+[64, 65]
+E ≥
+ℏ
+∆x.
+(4.1)
+Near the event horizon, one may take the uncertainty of the position as the radius of the
+black hole [64, 65] as
+∆x ≈ rBH = rH.
+(4.2)
+From eq. (3.31), we obtain
+Td =
+�
+rH − 1
+2
+�
+1 − Λr2
+H
+3
+� �
+rH + a2
+rH
+�
+− Q2
+2rH
+− Λ rH
+3
+�
+2r2
+H + a2��
+2π
+�
+r2
+H + a2� �
+1 + Λa2
+3
+�
+×
+�
+1 −
+4βJ2
+θ ∆θ
+r2
+H + a2 cos2 θ
+�
+.
+(4.3)
+From eq. (4.3), it is observed that when
+rH ≤
+�
+4βJ2
+θ
+�
+1 + Λa2 cos2 θ
+�
+− 3a2 cos2 θ
+3
+,
+(4.4)
+the modified Hawking temperature becomes negative. This violates the law of black hole
+thermodynamics and thus has no physical meaning. It is clear that the evaporation will stop
+under the effects of GUP. Thus the Hawking temperature becomes zero when rH reaches
+the minimum radius, rmin as
+rmin =
+�
+4βJ2
+θ
+�
+1 + Λa2 cos2 θ
+�
+− 3a2 cos2 θ
+3
+.
+(4.5)
+– 15 –
+
+Using eq. (3.8) in eq. (3.31), we obtain the expression of Td in terms of the mass of the
+black hole as
+Td ≈ ζ1
+ζ2
+�
+�
+�
+�
+�1 −
+4βJ2
+θ ∆θ
+1
+α2
+1
+�
+1 + 4ΛM2
+3β2
+1α1
+�2 �
+M +
+�
+M2 − (a2 + Q2)α1
+�2
++ a2 cos2 θ
+�
+�
+�
+�
+� ,
+(4.6)
+where
+ζ1 = 1
+α1
+�
+1 + 4ΛM2
+3β2
+1α1
+� �
+M +
+�
+M2 − (a2 + Q2)α1
+�
+�
+1 − Λa2
+3
+− 2Λ
+3 α2
+1
+�
+1 + 4ΛM2
+3β2
+1α1
+�2 �
+M +
+�
+M2 − (a2 + Q2)α1
+�2
+�
+− M,
+ζ2 =2π
+�
+1 + Λa2
+3
+� �
+a2 + 1
+α2
+1
+�
+1 + 4ΛM2
+3β2
+1α1
+�2 �
+M +
+�
+M2 − (a2 + Q2)α1
+�2
+�
+.
+(4.7)
+To make the Hawking temperature T ≥ 0 i.e. to ensure the GUP corrected temperature
+has a physical meaning, the mass of the black hole must hold the inequality
+M ≥
+β2
+1
+8(a2 + Q2)Λ − 6β2
+1
+�
+3
+�
+−α1(a2 + Q2) + 3
+2
+�
+−4α1(a2 + Q2)
++8α1ζ3
+�
+3β2
+1 − 4(a2 + Q2)Λ
+� �
+−a2 + Q2 + 4J2
+θ α1β∆θ − a2α1 cos2 θ
+�
+3β2
+1
+� 1
+2
+�
+.
+(4.8)
+It is noted that the mass of the black hole has a minimum value which is given by
+Mmin =
+β2
+1
+8(a2 + Q2)Λ − 6β2
+1
+�
+3
+�
+−α1(a2 + Q2) + 3
+2
+�
+−4α1(a2 + Q2)
++8α1ζ3
+�
+3β2
+1 − 4(a2 + Q2)Λ
+� �
+−a2 + Q2 + 4J2
+θ α1β∆θ − a2α1 cos2 θ
+�
+3β2
+1
+� 1
+2
+�
+. (4.9)
+5
+Graphical Analysis
+In this section, we will examine graphically the effects of parameters β, Λ and m on the
+modified Hawking temperatures and modified heat capacities with respect to event horizon
+rH.
+5.1
+Temperature Td with radius of event horizon rH for 3-dimensional KNdS
+black hole
+This subsection is devoted to analysing the behaviour of modified Hawking temperature.
+The parameters are taken as follows a = 0.3, Q = 1, θ = π
+2 and Jθ = 0.1.
+– 16 –
+
+2
+4
+6
+8
+10
+rH
+-0.4
+-0.2
+0.2
+0.4
+0.6
+0.8
+Td
+Hawking Temperature vs Radius of Event Horizon
+β=5
+β=10
+β=15
+β=20
+β=25
+2
+4
+6
+8
+rH
+0.5
+1.0
+1.5
+2.0
+2.5
+Td
+Hawking Temperature vs Radius of Event Horizon
+β=25
+β=30
+β=50
+β=70
+β=90
+Figure 1.
+Hawking temperature Td with respect to radius of event horizon rH for different values
+of β.
+2
+4
+6
+8
+rH
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Td
+Hawking Temperature vs Radius of Event Horizon
+Λ=0.3
+Λ=0.4
+Λ=0.6
+Λ=0.8
+Λ=1
+Figure 2.
+Hawking temperature Td with respect to the radius of event horizon rH for different
+values of cosmological constant Λ.
+1. Figure 1 indicates the variation of the modified Hawking temperature with rH > 0,
+for different values of β with fixed value of the cosmological constant Λ = 1 and
+m = 0.1. We observe that for β = 25, the temperature decreases and tends to zero.
+As the radius of horizon increases for β < 25, the modified Hawking temperature
+becomes negative.
+This negative temperature and divergent behaviour reveal the
+nonphysically unstable state of the black hole [66]. Moreover, for β > 25, as the
+horizon increases, the temperature decreases and once the minimum value is reached,
+the temperature increases. It is worth mentioning that, for β < 25, we observe the
+nonphysical behaviour with negative temperature and for β = 25, the temperature
+vanishes. Furthermore, the temperatures are positive when β > 25. The β effects
+decelerate the increase in Hawking temperature which is also shown numerically in
+table 1.
+2. Figure 2 shows the behaviour of modified Hawking temperature with rH > 0, for
+different values of positive cosmological constant Λ > 0. The parameter are taken as
+follows: m = 0.1 and β = 40. It is observed that the modified Hawking temperatures
+decrease and attain its minimum values, which is also calculated numerically in table
+2. Later, it keeps on increasing as rH increases and its behaviour is linear. The change
+– 17 –
+
+in Λ gives the diverging temperature Td as rH increases.
+0
+2
+4
+6
+8
+rH
+0
+10
+20
+30
+40
+50
+Td
+Hawking Temperature vs Radius of Event Horizon
+m=0.1
+m=0.3
+m=0.5
+m=0.7
+m=0.9
+Figure 3. Hawking temperature Td with respect to radius of event horizon rH for different values
+of m.
+3. Figure 3 indicates the behaviour of Hawking temperature for different values of m. At
+first, the Hawking temperature drops suddenly and after attaining its minimum point,
+it keeps increasing with increasing the horizon radius. The increase of the parameter
+m makes the Hawking temperatures increases as shown numerically in table 3.
+0
+1
+2
+3
+4
+5
+rH
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Td
+Hawking Temperature vs Radius of Event Horizon
+a=0
+a=0.5
+a=1
+a=1.5
+a=2
+Figure 4. Hawking temperature Td versus radius of event horizon rH for different values of a.
+4. Figure 4 provides the graphical analysis of Td via horizon radius rH for different
+values of spin parameter a. The pink line, which corresponds to a = 0 represents
+the Hawking temperature graph for Reissner-Nordstrom-de Sitter (RNdS) black hole.
+The graph shows that the Hawking temperature of the RNdS black hole is greater
+than that of the KNdS black hole. The effect of spin parameter a decelerates the
+increase in Hawking temperature. This graphical presentation is compatible with the
+numerical calculation of table 4.
+– 18 –
+
+a=0.3;
+m=0.1;
+Λ = 1
+β
+25
+30
+50
+70
+90
+rc
+H
+7.97272
+1.81347
+1.43917
+1.3935
+1.37473
+T c
+d
+0.00953
+0.06208
+0.16458
+0.26337
+0.36169
+Table 1.
+The tabulated values of the critical radius rC
+H and the critical temperature T C
+H for
+different values of β.
+From figures 1 to 4 show that the temperature cools down to the minimum T C
+d , at
+rH = rC
+H and keeps on increasing as rH > rC
+H.
+a=0.3;
+m=0.1;
+β = 40
+Λ
+0.3
+0.4
+0.6
+0.8
+1
+rc
+H
+1.4842
+1.4876
+1.49087
+1.49664
+1.5016
+T c
+d
+0.01115
+0.02619
+0.05600
+0.08596
+0.11457
+Table 2.
+The tabulated values of the critical radius rC
+H and the critical temperature T C
+H for
+different values of Λ.
+a=0.3;
+Λ = 1;
+β = 40
+m
+0.1
+0.3
+0.5
+0.7
+0.9
+rc
+H
+1.5016
+1.18474
+1.16561
+1.16027
+1.15807
+T c
+d
+0.11457
+1.07833
+2.97447
+5.81695
+9.60655
+Table 3.
+The tabulated values of the critical radius rC
+H and the critical temperature T C
+H for
+different values of emitted particle mass m.
+m=0.1;
+Λ = 1;
+β = 40
+a
+0.3
+0.4
+0.6
+0.8
+1
+rc
+H
+1.44254
+1.86004
+1.96988
+2.07671
+2.17653
+T c
+d
+0.11791
+0.05845
+0.04575
+0.03314
+0.02359
+Table 4.
+The tabulated values of the critical radius rC
+H and the critical temperature T C
+H for
+different values of a.
+5.2
+Heat capacity CH with radius of event horizon rH for 3-dimensional KNdS
+black hole
+This subsection focuses on analysing modified heat capacity in different domains of event
+horizon radius rH with fixed parameters: a = 0.3, Q = 1, θ = π
+2 and J = 0.1. The heat
+capacity is connected to the local thermal stability. If the black hole has a negative heat
+capacity, it is unstable to thermal radiation and if it has a positive heat capacity, it is stable
+to thermal radiation.
+– 19 –
+
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+rH
+-10000
+-5000
+5000
+10000
+CH
+Heat Capacity vs Radius of Event Horizon
+β=0
+β=10
+β=30
+β=50
+β=70
+Figure 5. Heat capacity CH versus radius of event horizon rH for different values of β.
+(i) For fix parameters a = 0.3, Λ = 1 and m = 0.5, the variation of heat capacity
+CH with the horizon radius rH for different values of β is shown in figure 5. The
+position of phase transition remains unchanged in 3-dimensional KNdS black hole
+for different values of β. Further, it is observed that the phase transition occurs at
+rH = r∗
+H = 1.15467, r∗
+H denotes the position of the horizon radius at which the phase
+transition takes place.
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+rH
+-10000
+-5000
+5000
+10000
+CH
+Heat Capacity vs Radius of Event Horizon
+Λ=0.3
+Λ=0.5
+Λ=0.7
+Λ=0.9
+Figure 6. Heat capacity CH versus radius of event horizon rH for different values of Λ.
+(ii) Figure 6 illustrates the behaviour of CH w.r.t rH for fixed values of a = 0.3, β = 40,
+m = 0.5 and for different values of Λ. For different values of cosmological constant Λ,
+there are different positions of phase transition for 3-dimensional KNdS black hole.
+Increasing the values of Λ, the positions of phase transition r∗
+H are shifted towards
+the origin, which ensures the black hole faster stability for larger value of Λ. Different
+positions of phase transition are shown in table 5.
+(iii) Figure 7 shows the variation of CH for different values of spin parameter a and for
+fixed parameters Λ = 1, β = 40, m = 0.5. Increasing the value of rotation parameter
+a, the phase transition occurs at larger value of horizon radius r∗
+H which is also shown
+in table 6. Moreover, larger the value of a delays the stability of the black hole.
+– 20 –
+
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+rH
+-60000
+-40000
+-20000
+20000
+40000
+60000
+CH
+Heat Capacity vs Radius of Event Horizon
+a=0.5
+a=1
+a=1.5
+a=2
+Figure 7. Heat capacity CH versus radius of event horizon rH for different values of a.
+Figures 5 to 7 show that there is a phase transition when rH = r∗
+H. The positions
+of phase transition are shown in tables 5 and 6. The black holes are unstable in the
+region 0 ≤ rH ≤ r∗
+H and stable in the region r∗
+H ≤ rH ≤ ∞ i.e. the smaller black
+holes are less stable than larger black holes and vice versa.
+a=0.3;
+m = 0.5;
+β = 40
+Λ
+0.3
+0.5
+0.7
+0.9
+r∗
+H
+1.42403
+1.31282
+1.23652
+1.17887
+Table 5.
+The tabulated values of the position of phase transition r∗
+H for different values of Λ.
+Λ = 0.1;
+m = 0.5;
+β = 40
+a
+0.5
+1
+1.5
+2
+r∗
+H
+1.17485
+1.23815
+1.29861
+1.35098
+Table 6.
+The tabulated values of the position of phase transition r∗
+H for different values of a.
+5.3
+Hawking temperature Td4 with radius of event horizon for 4-dimensional
+KNdS black hole
+This subsection is devoted to analysing the behaviour of modified Hawking temperatures
+of KNdS black hole. The parameters are taken as follows: Q = 0.1, θ = π
+2 , e = 1, Jθ = 0.1
+and Jφ = 0.2.
+(i) Figure 8 shows the behaviour of Td4 for different values of β and for fixed values of
+a = 0.2, Λ = 1 and m = 0.1. Td4 increases exponentially and reach its maximum
+height T max
+d4
+for different values of β. Further, the temperature keeps on decreasing
+with increasing the horizon radius. It is noteworthy to mention that the temperature
+increases with increasing the values of β. Hence, the β effects accelerate the increase
+in Td4. The proof of the above statement is also calculated in table 7.
+– 21 –
+
+0.0
+0.2
+0.4
+0.6
+0.8
+rH
+20
+40
+60
+80
+100
+120
+140
+Td4
+Hawking Temperature vs Radius of Event Horizon
+β=10
+β=30
+β=50
+β=70
+β=90
+Figure 8. Hawking temperature Td4 with respect to radius of event horizon rH for different values
+of m.
+0.0
+0.2
+0.4
+0.6
+0.8
+rH
+20
+40
+60
+80
+100
+Td4
+Hawking Temperature vs Radius of Event Horizon
+Λ=0.2
+Λ=0.4
+Λ=0.6
+Λ=0.8
+Λ=1
+Figure 9. Hawking temperature Td4 with respect to radius of event horizon rH for different values
+of β.
+(ii) The behaviour of Td4 for varying cosmological constant Λ is depicted in figure 9. The
+parameters are as follows: a = 0.2, β = 50 and m = 0.1. The temperature increases
+to a certain height and attains its peak point T max
+d4
+at rH = rc
+H, then Td4 decreases
+as rH increases. It is noted that the Λ effects decelerate the increase in Td4 which is
+also shown numerically in table 8.
+(iii) The behaviour of Td4 w.r.t rH for varying mass of the vector boson particle m is
+depicted in Figure 10. The temperature increases to a certain height T max
+d4
+and then
+decreases with increasing rH. The rate of increase of temperature Td4 is dependent
+on the increase of m. The validity of the above statements is calculated numerically
+in table 9.
+(iv) Figure 11 shows the behaviour of Td4 w.r.t rH for varying a and fixed parameters
+β = 50, Λ = 1 and m = 0.1. The temperatures increase exponentially upto T max
+d4
+and decrease with increasing the horizon radius. The temperature gradually increases
+– 22 –
+
+with decreasing the values of rotation parameter, a. The numerical calculation shown
+in table 10 supports the above statement.
+0.0
+0.2
+0.4
+0.6
+0.8
+rH
+100
+200
+300
+400
+500
+600
+700
+Td4
+Hawking Temperature vs Radius of Event Horizon
+m=0.1
+m=0.3
+m=0.5
+m=0.7
+m=0.9
+Figure 10. Hawking temperature Td4 with respect to radius of event horizon rH for different values
+of m.
+0.0
+0.2
+0.4
+0.6
+0.8
+rH
+50
+100
+150
+Td4
+Hawking Temperature vs Radius of Event Horizon
+a=0.1
+a=0.15
+a=0.2
+a=0.25
+a=0.3
+Figure 11. Hawking temperature Td4 with respect to radius of event horizon rH for different values
+of a.
+We present the following tables to see the effect of β, Λ, m and a on the Hawking temper-
+ature. Table 7 confirms that T max
+d4
+gradually increases on increasing β (for fixed values of
+Λ, m and a). Similarly table 9 also confirms that T max
+d4
+gradually increases on increasing
+m (for fixed values of β, Λ, and a). On the contrary, tables 8 and 10 confirm that T max
+d4
+gradually decreases with increasing cosmological constant Λ (for fixed values of β, m and
+a) and rotational parameter a (for fixed values of β, Λ, and m) respectively. From table 9,
+it is observed that T max
+d4
+is highly dependent on m compared to that of β, Λ and a.
+– 23 –
+
+a=0.2;
+Λ = 1;
+m = 0.1
+β
+10
+30
+50
+70
+90
+rc
+H
+0.34861
+0.34849
+0.34847
+0.34846
+0.34845
+T max
+d4
+14.2974
+42.7385
+71.1796
+99.6206
+128.062
+Table 7.
+The tabulated values of rC
+H and T max
+d4
+for different values of β.
+a=0.2;
+Λ = 1;
+β = 50
+Λ
+0.2
+0.4
+0.6
+0.8
+1
+rc
+H
+0.35376
+0.35244
+0.35117
+0.34979
+0.34847
+T max
+d4
+87.2216
+82.793
+78.4615
+75.4107
+71.1796
+Table 8.
+The tabulated values of rC
+H and T max
+d4
+for different values of Λ.
+a = 0.2;
+Λ = 1;
+β = 50
+m
+0.1
+0.3
+0.5
+0.7
+0.9
+rc
+H
+0.34847
+0.39069
+0.45936
+0.50492
+0.53291
+T max
+d4
+71.1796
+111.502
+211.269
+386.449
+652.623
+Table 9.
+The tabulated values of rC
+H and T max
+d4
+for different values of m.
+β = 20;
+Λ = 1;
+m = 0.1
+a
+0.1
+0.15
+0.2
+0.25
+0.3
+rc
+H
+0.2075
+0.27555
+0.34852
+0.42236
+0.49421
+T max
+d4
+176.342
+65.3181
+28.518
+13.7598
+6.89087
+Table 10.
+The tabulated values of rC
+H and T max
+d4
+for different values of a.
+5.4
+Heat Capacity CH4 versus horizon radius rH for 4-dimensional KNdS black
+hole
+This subsection studies the modified heat capacity for different values of β, Λ and a for
+fixed values of Q = 1, θ = π
+2 , e = 1, Jθ = 0.1 and Jφ = 0.2.
+(i) The variation of modified heat capacity for different values of β is shown in figure
+12. The parameters taken are a = 0.2, Λ = 0.5 and m = 0.1. It is observed that
+there is one position of phase transition in the absence of GUP, but there are two
+positions of phase transition under the influence of GUP. The first phase transition in
+figure 12 is due to the quantum gravity effects and it occurs at rH = rH1 = 0.99338.
+The position of the second phase transition is observed at rH = rH2 = 1.29667 with
+or without quantum gravity effects. The variation of β doesn’t affect the position of
+phase transition.
+– 24 –
+
+(ii) For a = 0.2, β = 15 and m = 0.1, the variation of CH4 w.r.t rH changing the values
+of Λ are illustrated in figure 13. Varying the values of cosmological constant Λ, the
+positions of phase transition are also varied. With increasing the values of Λ, the
+position of phase transition is shifted toward the origin, which implies a slower rate
+of becoming a stable black hole. Table 11 is constructed numerically to show the
+different positions of phase transitions.
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+rH
+-100
+-50
+50
+100
+CH4
+Heat Capacity vs Radius of Event Horizon
+β=0
+β=10
+β=30
+Figure 12. Heat Capacity CH4 with respect to radius of event horizon rH for different values of β.
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+rH
+-40
+-20
+0
+20
+40
+CH4
+Heat Capacity vs Radius of Event Horizon
+Λ=0.3
+Λ=0.5
+Λ=0.7
+Figure 13. Heat Capacity CH4 with respect to radius of event horizon rH for different values of
+Λ.
+(iii) Figure 14 represents the behaviour of CH4 versus rH for fixed parameters β = 15,
+Λ = 0.5 and m = 0.1. The position of phase transition is shifted to far away from the
+origin toward the positive direction of rH with increasing the value of spin parameter
+a. It shows that the black hole becomes stable faster with increasing the value of
+rotation parameter a which is also indicated numerically by table 11.
+– 25 –
+
+0.5
+1.0
+1.5
+2.0
+rH
+-15
+-10
+-5
+0
+5
+10
+15
+CH4
+Heat Capacity vs Radius of Event Horizon
+a=0.1
+a=0.15
+a=0.2
+Figure 14. Heat Capacity CH4 with respect to radius of event horizon rH for different values of a.
+a = 0.2;
+β = 15;
+m = 0.1
+Λ = 0.5;
+β = 15;
+m = 0.1
+Λ
+rH1
+rH2
+a
+rH1
+rH2
+0.3
+0.99602
+1.40094
+0.1
+0.49917
+1.2864
+0.5
+0.99338
+1.29667
+0.4
+0.7472
+1.29073
+0.7
+0.99075
+1.22479
+0.6
+0.99338
+1.29667
+Table 11.
+The tabulated values of rH1 and rH2 for different values of Λ and a.
+6
+Conclusions
+This work studies the GUP effects on tunneling of massive vector boson particles from KNdS
+black hole. The Hawking temperatures and heat capacities near the horizon of KNdS black
+hole are studied using GUP-corrected Lagrangian of massive vector field, Feynman pre-
+scription and WKB approximation. It is noted that the Hawking temperatures and heat
+capacities are modified due to quantum gravity effects. They depend not only upon the
+quantum gravity parameter β, spin parameter a, mass of the emitted particle m, cosmolog-
+ical constant Λ, charge of the black hole Q but also on angular coordinates θ, Jθ and Jφ.
+We also discuss the stable and unstable formations of KNdS black hole in quantum gravity
+effects.
+The remnant of KNdS black hole is also discussed in the presence of quantum
+gravity effects. We also illustrate the graphs of modified Hawking temperatures and heat
+capacities and explore the effects of β, Λ, a and m. The modified Hawking temperature of
+a 3-dimensional KNdS black hole w.r.t rH tends to decrease for β < 25, but for β > 25, the
+temperature cools down till it reaches its minimum point and then increases, which leads
+to the formation of stable black hole. For a 4-dimensional KNdS black hole with the above
+fixed parameters, the modified Hawking temperature increases w.r.t rH and after attaining
+maximum height, the temperature eventually goes down. It is worth noting that there
+are one phase transition and two phase transitions for a non-zero horizon of 3-dimensional
+KNdS black hole and 4-dimensional KNdS black hole respectively. Different positions of
+phase transitions are due to the quantum gravity effects. It is worth mentioning that for
+– 26 –
+
+different values of dimensionless parameter β, the position of phase transitions remain the
+same in 3-dimensional and 4-dimensional KNdS black hole under the influence of quantum
+gravity effects. The modified Hawking temperatures and heat capacities are reduced to the
+original Hawking temperature and heat capacity in quantum gravity effects. Hence quan-
+tum gravity effects modified the Hawking temperature and heat capacity of KNdS black
+hole.
+A
+Coefficients of eq. (3.48)
+A∗
+0 =(r − rH)∆,r (rH)
+ρ2(rH)
+,
+(A.1)
+A∗
+1 =m2 + J2
+θ ∆θ
+ρ2(rH) − (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)∆,r (rH)ρ2(rH)
++ (Jφ + eA3)2 ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
+,
+(A.2)
+B6 =6m2β (r − rH)3 ∆2,r (rH)
+ρ6(rH)
+,
+(A.3)
+B4 =m2 (r − rH)2 ∆2,r (rH)
+ρ4(rH)
++ 2β (r − rH)2 ∆2,r (rH)
+ρ4(rH)
+�
+−
+�
+1
+(r − rH) ∆,r (rH) ρ2(rH)
+�
+(−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+�
+− 2 (Jφ + eA3 − 1) (Jφ + eA3) ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
++ 3m2
+���
++ m2
+�
+3m2 + 3J2
+θ ∆θ
+ρ2(rH) + (Jφ + eA3) (2 + Jφ + eA3) ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
+��
+,
+(A.4)
+B2 =(r − rH) ∆,r (rH)
+ρ2(rH)
+�
+(−E + jΩ + eA0)2 (Jφ + eA3) (Jφ + eA3 − 1) Σ4 csc2 θ
+(r − rH) ∆2,r (rH)
++ 2m4 + 2 J2
+θ m2∆θ
+ρ2(rH)
++ (Jφ + eA3) (1 + Jφ + eA3) m2ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
+− 2m2(−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH) ∆,r (rH) ρ2(rH)
++ 2β
+�
+1
+(r − rH)2 ∆2,r (rH) ρ4(rH)
+�
+(−E + jΩ + eA0)4 �
+r2
+H + a2�4 Σ4
+�(Jφ + eA3 − 1) (Jφ + eA3) ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
+− 3m2
+��
++ (−E + jΩ + eA0)2 (Jφ + eA3)3 (Jφ + eA3 − 1) ρ2(rH)Σ6 csc4 θ
+(r − rH) (r2
+H + a2)2 ∆2
+θ ∆,r (rH)
++ m2
+� 3J4
+θ ∆2
+θ
+ρ4(rH) + (Jφ + eA3)3 {1 + 2 (Jφ + eA3)} ρ4(rH)Σ4 csc4 θ
+(r2
+H + a2)4 ∆2
+θ
+���
+,
+(A.5)
+B0 = −
+�
+J2
+θ ∆θ
+ρ2(rH) − (−E + jΩ + eA0)2 �
+r2
+H + a2�2 Σ2
+(r − rH)∆,r (rH)ρ2(rH)
++ (Jφ + eA3)2 ρ2(rH)Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
++ m2
+�
++
+2 β
+(r − rH)3 (r2
+H + a2)6 ∆3
+θ ∆3,r (rH) ρ6(rH)
+�
+3(−E + jΩ + eA0)6∆6
+θΣ6
+�
+r2
+H + a2�10 �
+m2 �
+r2
+H + a2�2 ∆θ − (Jφ + eA3) (Jφ + eA3 − 1) ρ2(rH)Σ2 csc2 θ
+�
+– 27 –
+
+− (−E + jΩ + eA0)4(r − rH)
+�
+r2
+H + a2�8 ∆2
+θ ∆,r (rH)Σ4
+�
+3m2 �
+r2
+H + a2�2 ∆θ
+�
+J2
+θ ∆θ + m2ρ2(rH)
+�
+− (Jφ + eA3)2 ρ2(rH)
+�
+J2
+θ (Jφ + eA3 − 1) ∆θ − 3m2ρ2(rH)
+�
+Σ2 csc2 θ
+�
++ m2(r − rH)3∆3,r (rH)
+�
+3J6
+θ
+�
+r2
+H + a2�6 ∆6
+θ + J2
+θ ∆2
+θ ρ8(rH) Σ4 csc4 θ
+(Jφ + eA3)3
+�
+1 + 2 (Jφ + eA3)
+�
++ (Jφ + eA3)3 ρ10(rH) Σ4 csc4 θ
+�
+m2∆θ
+�
+1 + 2 (Jφ + eA3)
+� �
+r2
+H + a2�2 + 3ρ2(rH)Σ2 csc2 θ (Jφ + eA3)2
+�
++ J4
+θ ∆4
+θ ρ2(rH)
+�
+r2
+H + a2�4 �
+3m2 �
+r2
+H + a2�2 ∆θ + (Jφ + eA3) (2 + Jφ + eA3) ρ2(rH) Σ2 csc2 θ
+��
+− (−E + jΩ + eA0)2 �
+r2
+H + a2�2 (r − rH)2∆2,r (rH)Σ2
+�
+(1 − Jφ − eA3) (Jφ + eA3)
+ρ2(rH) Σ2 csc2 θ
+�
+2J2
+θ
+�
+r2
+H + a2�4 ∆4
+θ + J2
+θ (Jφ + eA3)2 �
+r2
+H + a2�2 ∆2
+θρ4(rH) Σ2
+csc2 θ + 3 (Jφ + eA3)4 ρ8(rH) Σ4 csc4 θ
+�
++ m2 �
+r2
+H + a2�2 ∆θ
+�
+3J4
+θ ∆4
+θ
+�
+r2
+H + a2�4
++ (Jφ + eA3)3 (2 + Jφ + eA3) ρ8(rH) Σ4 csc4 θ
+���
+.
+(A.6)
+B
+The expressions of χ1 and χ2 given in eq. (3.49)
+χ1 = − 3(−E + jΩ + eA0)6m2 �
+r2
+H + a2�6 Σ6
+(r − rH)3∆3,r (rH) ρ6(rH)
++
+(−E + jΩ + eA0)4 Σ4
+(r − rH)2 ∆2
+θ ∆2,r (rH) ρ6(rH)
+�
+9m2 �
+r2
+H + a2�2 ∆θ − 4 (Jφ + eA3 − 1) (Jφ + eA3) ρ2(rH) Σ2 csc2 θ
+�
+�
+∆θ
+�
+r2
+H + a2�2 �
+J2
+θ ∆θ + m2 ρ2(rH)
+�
++ (Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ
+�
++ m2
+�
+3
+�
+J2
+θ ∆θ + m2 ρ2(rH)
+�3
+ρ6(rH)
++ (Jφ + eA3) Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ ρ2(rH)
+�
+2J2
+θ m2 ∆θ ρ2(rH)
+�
+2 + 7 (Jφ + eA3)
+�
++ J2
+θ ∆2
+θ
+�
+4 + 5 (Jφ + eA3)
+�
++ m4 ρ4(rH)
+�
+4 + 5 (Jφ + eA3)
+�
+�
++ ρ2(rH) Σ4 csc4 θ (Jφ + eA3)3 �
+4 + 5 (Jφ + eA3)
+� �
+J2
+θ ∆θ + m2 ρ2(rH)
+�
+(r2
+H + a2)4 ∆2
+θ
+− (Jφ + eA3 − 4) (Jφ + eA3)5 ρ6(rH)Σ6 csc6 θ
+(r2
+H + a2)6 ∆3
+θ
+�
+− (jΩ + eA0 − E)2 �
+r2
+H + a2�2 Σ2
+(r − rH) ∆,r (rH) ρ2(rH)
+�
+9m6 + m2
+� J2
+θ ∆θ
+ρ2(rH) + (Jφ + eA3)2 ρ2(rH) Σ2 csc2 θ
+(r2
+H + a2)2 ∆θ
+�
+� 9Jθ∆1
+ρ2(rH) + ρ2(rH) Σ2 csc2 θ (Jφ + eA3) (8 + Jφ + eA3)
+(r2
+H + a2)2 ∆θ
+�
+– 28 –
+
++ 6m4
+�3J2
+θ ∆φ
+ρ2(rH) + ρ2(rH) Σ2 csc2 θ (Jφ + eA3) [1 + 2(Jφ + eA3)]
+(r2
+H + a2)2 ∆θ
+�
+− 4 Σ2 csc2 θ (Jφ + eA3 − 1) (Jφ + eA3)
+(r2
+H + a2)6 ∆3
+θ ρ2(rH)
+�
+J4
+θ ∆4
+θ
+�
+r2
+H + a2�4 + J2
+θ ∆2
+θ ρ4(rH)
+Σ2 csc2 θ (Jφ + eA3)2 �
+r2
+H + a2�2 + ρ8(rH)Σ4 csc4 θ (Jφ + eA3)4
+��
+,
+(B.1)
+χ2 =ρ2(rH) Σ2 csc2 θ (1 − Jφ − eA3) (Jφ + eA3)
+(r2
+H + a2)2 ∆θ
+�
+m2 − Σ2(−E + jΩ + eA0)2 �
+r2
+H + a2�2
+(r − rH) ∆,r (rH) ρ2(rH)
+�
+�
+m2 + J2
+θ ∆θ
+ρ2(rH) − Σ2(−E + jΩ + eA0)2 �
+r2
+H + a2�2
+(r − rH) ∆,r (rH) ρ2(rH)
++ ρ2(rH) Σ2 csc2 θ (Jφ + eA3)2
+(r2
+H + a2)2 ∆θ
+�
+. (B.2)
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diff --git a/jtE3T4oBgHgl3EQfJQmU/content/tmp_files/load_file.txt b/jtE3T4oBgHgl3EQfJQmU/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0850c3a3e6238c77770b1796fb7c8c65007b9a74
--- /dev/null
+++ b/jtE3T4oBgHgl3EQfJQmU/content/tmp_files/load_file.txt
@@ -0,0 +1,1374 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf,len=1373
+page_content='Prepared for submission to JHEP  Massive vector particle tunneling from  Kerr-Newman-de Sitter black hole under  generalized uncertainty principle  Yenshembam Priyobarta Singh,a Telem Ibungochouba Singha,1  aDepartment of Mathematics,  Manipur University, Canchipur, Imphal, Manipur, India  E-mail: priyoyensh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='com, ibungochouba@rediffmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='com  Abstract: The quantum tunneling of charged massive vector boson particles across the  event horizon of Kerr-Newman-de Sitter black hole is investigated under the influence of  quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The Hawking temperature and the heat capacity of Kerr-Newman-  de Sitter black hole are derived using the generalized field equation for charged massive  vector bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is found that the quantum gravity effects modified the Hawking tempera-  ture and heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Moreover, they depend on the mass and angular momentum of the  emitted vector boson particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' We also discuss the remnant and graphical analysis of the  modified Hawking temperatures and heat capacities of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Keywords: Quantum tunneling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' quantum gravity effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Kerr-Newman-de Sitter black  hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature and heat capacity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='04342v1  [gr-qc]  11 Jan 2023  Contents  1  Introduction  1  2  Generalized field equations for massive vector bosons  3  3  Quantum tunneling from KNdS black hole  4  4  Remnant of KNdS black hole  15  5  Graphical Analysis  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  Temperature Td with radius of event horizon rH for 3-dimensional KNdS  black hole  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  Heat capacity CH with radius of event horizon rH for 3-dimensional KNdS  black hole  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Hawking temperature Td4 with radius of event horizon for 4-dimensional  KNdS black hole  21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  Heat Capacity CH4 versus horizon radius rH for 4-dimensional KNdS black  hole  24  6  Conclusions  26  A Coefficients of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='48)  27  B The expressions of χ1 and χ2 given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49)  28  1  Introduction  In the early 1970s, Hawking proposed a black hole radiation called Hawking radiation using  quantum field theory techniques on a curve space-time background [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The discovery of  Hawking radiation is one of the significant developments in gravitational physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It shows  that black holes have a thermodynamical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [3, 4] determined that the horizon  area and the black hole’s entropy are proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' More precisely, the black hole’s entropy  is equal to one-fourth of its horizon area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Since then, theoretical physicists have paid a  lot of attention to Hawking radiation and many approaches have been used to determine  Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Generally, there are two approaches to investigate the imaginary part of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  They are the radial null geodesic method and the Hamilton-Jacobi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The radial null  geodesic method was put forth by Parikh and Wilczek [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' In this method, the emitted  particles act as the potential barrier and the semiclassical WKB approximation is used to  determine the imaginary part of the radial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Later, Zhang and Zhao [9–11] have made  – 1 –  significant progress by extending Parikh’s method to non-spherical symmetric stationary  black holes and the radiation of charged massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The second is the Hamilton-Jacobi  method, used by Angheben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [12] as an extension of the complex path integral method  introduced by Padmanabhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' In both methods, the WKB approximation  is used to investigate the tunneling probability for classically forbidden trajectory from  inside to outside the horizon and it is given by Γ = exp  �  − 2  ℏIm I  �  , where I and ℏ denote  the semiclassical action of the outgoing particle and Planck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Kerner and Mann  [16, 17] investigated the tunneling of spin-1/2 fermion particles from rotating and non-  rotating black holes using Dirac equation, Pauli sigma matrix and Feynman prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  By the fermions tunneling method the Hawking radiation of Dirac particles via tunneling  from the black ring is also derived [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Using their method, many interesting results have  been obtained in [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Damour and Ruffini [23] used a new approach called tortoise  coordinate transformation to study Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Later, Sannan [24] extended the  work of Damour and Ruffini by deriving the probability distributions for boson and fermion  emissions from a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The existence of a minimum measurable length that can be identified with the order  of the Planck scale is predicted by numerous quantum gravity theories, such as string  theory, loop quantum gravity, non-commutative geometry and Gedanken experiments [25–  31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Constructing new theoretical models is one of the fields to study quantum gravity  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The Generalized Uncertainty Principle (GUP) is a modified theory that realizes this  minimum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is the generalization of the Heisenberg uncertainty principle (HUP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The modified fundamental commutation relation proposed by Kempf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [32] is of the  form  �  xi, pj  �  = iℏδi,j  �  1 + βp2�  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  where the position xi and momentum operators pj are defined by  xi = x0i,  pj = p0j  �  1 + βp2  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  Then x0i and p0j satisfy the canonical commutation relations as  �  x0i, p0j  �  = iℏδij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The  corresponding GUP takes the form  ∆x∆p ≥ ℏ  2  �  1 + (∆p)2β  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3)  where β =  β0l2  p  ℏ2 =  β0  M2pc2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Here β0(≤ 1034), lp and Mp represent the dimensionless parameter  of order unity, Planck length and Planck mass respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The implications of the aspects of GUP have been investigated in many contexts such as  modifications of the quantum Hall effect, neutrino oscillations, Landau levels, Newton’s law,  cosmology, and the weak equivalence principle (WEP) [33–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It should be noted that the  GUP has also influenced the thermodynamics of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' As a result, the GUP concept  has been applied to various black holes in order to study their thermodynamics properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Adler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [39] studied the influence of GUP on the thermodynamics of the Schwarzschild  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Further, using the Parikh-Wilczek tunnelling approach, Nozari and Saghaf  – 2 –  [40] studied Hawking radiation for massless scalar particles in the background geometry of  Schwarzschild black hole and retrieved the tunnelling rate as well as the corrected Hawking  temperature by taking GUP into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Applying the WKB approximation to the Dirac  equation, the tunnelling of fermions from the Kerr and Kerr-Newman black holes are studied  in [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Yale [43] investigated the tunneling of arbitrary scalars, fermions and spin-  1 bosons by ignoring back-reaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Banerjee and Majhi [44, 45] discussed the  tunneling of fermions and bosons at the event horizon of black holes beyond the semiclassical  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' They derived the modified Hawking temperature and change in black hole  entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Ibungochouba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [46] studied the tunneling of fermions from the BTZ black  hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Many studies have shown that, GUP avoid the black hole to evaporate completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The  black hole has a phase transition close to the Planck scale and reaches thermal equilibrium  with its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  As a result, the black hole’s evaporation stops, leaving behind  a stable remnant [47–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Li [52] studied the tunneling of massive spin-1 particle from  Reissner-Nordstrom and Kerr black hole under the effect of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Ovgun et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' [53] investigated the charged massive bosons tunneling from noncommutative charged  black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  In this paper, the tunneling of massive vector boson particles across the event horizon  of the KNdS black hole is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' This article’s layout is constructed in the following  manner: In section 2, we revisit the generalized field equation for massive vector boson  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The quantum tunneling of massive vector boson from KNdS black hole under  quantum gravity effects is presented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Further, section 4 provides the analysis  of the remnant of KNdS black hole under quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The graphical analysis  of quantum corrected Hawking temperatures and heat capacity are presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Lastly, some concluding remarks are presented in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  2  Generalized field equations for massive vector bosons  The GUP-corrected Lagrangian of massive vector field Ψµ is given by [52]  LGUP = − 1  2  �  D+  µ Ψ+  ν − D+  ν Ψ+  µ  � �  D−µΨ−ν − D−νΨ−µ�  − m2  ℏ2 W +  µ W −µ  − i  ℏeF µνW +  µ W −  ν ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  where D±  0 =  �  1 + βℏ2g00D±2  0  �  D±  0 , D±  i =  �  1 − βℏ2giiD±2  i  �  D±  i ,  Fµν = �∇µAν − �∇νAµ with D±  µ  = ∇µ ± i  ℏeAµ, �∇0 =  �  1 + βℏ2g00∇2  0  �  ∇0 and �∇i =  �  1 − βℏ2gii∇2  i  �  ∇i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Here e, m and Aµ denote the charge of W + boson, mass of the vector  particle and electromagnetic potential of the black hole respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The generalized action of the massive vector boson particles takes the form  SGUP =  �  dx4√−g LGUP  �  Ψ±  µ , ∂µΨ±  ν , ∂µ∂ρΨ±  ν , ∂µ∂ρ∂λΨ±  ν  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  Following from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2), the modified wave equation for massive vector bosons is  obtained as  ∂µ(√−gΨµν) − 3β∂0  �√−gg00 �  e2A2  0 + iℏe∇0A0  �  Ψ0v�  – 3 –  + 3β ∂i  �√−ggii �  e2A2  i + iℏe∇iAi  �  Ψiv�  + 3β∂0∂0  �√−gg00iℏeA0Ψ0v�  − 3β∂i∂i  �√−ggiiiℏeAiΨiv�  + βℏ2∂0∂0∂0  �√−gg00Ψ0v�  − βℏ2∂i∂i∂i  �√−ggiiΨiv�  + √−g i  ℏeAµΨµν − √−gm2  ℏ2 Ψν  − √−g i  ℏeF µνΨµ + β√−gg00  �  iℏe∇0∇0A0 + 3e2A0∇0A0 − i  ℏe3A3  0  �  Ψ0ν  − β√−ggii  �  iℏe∇i∇iAi + 3e2Ai∇iAi − i  ℏe3A3  i  �  Ψiν = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3)  where GUP modified anti-symmetric tensor Ψµν is given by Ψµν = DµΨν − DνΨµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  3  Quantum tunneling from KNdS black hole  The line element of KNdS black hole in well known Boyer-Lindquist coordinates (t, r, θ, ϕ)  [54] is expressed as  ds2 = −  �∆ − ∆θ a2 sin2 θ  Σ2ρ2  �  dt2 − 2a sin2 θ[(r2 + a2)∆θ − ∆]  Σ2ρ2  dtdϕ + ρ2  ∆ dr2  + ρ2  ∆θ  dθ2 + sin2 θ  �  (r2 + a2)2 ∆θ − ∆a2 sin2 θ  �  Σ2ρ2  dϕ2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  where  Σ = 1 + Λa2  3 , ρ2 = r2 + a2 cos2 θ, ∆θ = 1 + Λa2 cos2 θ  3  ,  ∆ =  �  1 − Λa2  3  � �  r2 + a2�  − 2Mr + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The electromagnetic potential Aµ of KNdS black hole is given by  Aµ = Qr  �  δt  µ − a sin2 θδϕ  µ  �  ρ2Σ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1) describes a charged rotating black hole with mass m, spin parameter a and  charge Q with the positive cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The case Λ = 0 gives the solution  of the Kerr-Newman black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' For Λ < 0, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1) represents Kerr-Newman-(anti-)de  Sitter (KNadS) black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The horizons of the KNdS can be obtained from the event horizon equation as  ∆ =  �  1 − Λr2  3  � �  r2 + a2�  − 2Mr + Q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3) gives four real roots whenever 1  Λ ≫ M2 > Q2+a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The four roots are rC, rH,  r+ and r− (rC > rH > r+ > r−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Here rC denotes the cosmological horizon, rH represents  the event horizon and r− corresponds to the Cauchy horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' One reaches singularity r = 0,  θ = π  2 and on other side of r = 0, r = r− is considered as another cosmological horizon [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 4 –  In this coordinate system, the event horizon and the infinite red-shift surface do not  coincide due to rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Because of this, it is inconvenient to study the characteristic of  tunneling radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' So, we perform the dragging coordinate transformation [56]  dϕ  dt = Ω = −gtϕ  gϕϕ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4)  The line element (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1) is reduced to  ds2 = −  ∆∆θρ2  Σ2 �  ∆θ(r2 + a2)2 − ∆a2 sin2 θ  �dt2 + ρ2  ∆ dr2 + ρ2  ∆θ  dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5)  The non-vanishing component of electromagnetic potential A0 is given by  A0 =  (r2 + a2)∆θQr  Σ  �  ∆θ(r2 + a2)2 − ∆a2 sin2 θ  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6)  The dragging coordinate transformation makes the geometrical optical limit a reliable  approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' hence, the WKB approximation can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Near the event horizon r = rH, we use the approximation  ∆(r) =∆(rH) + (r − rH)∆,r (rH) + O  �  (r − rH)2�  ≈ (r − rH)∆,r (rH),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7)  where rH is defined as  rH = 1  α1  �  1 + 4ΛM2  3β2  1α1  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  � �  M +  �  M2 − (a2 + Q2)α1  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8)  where β =  �  1 − Λa2  3  and α1 =  ��  1 + Λa2  3  �2  + 4ΛQ2  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5) takes the form  ds2 = −(r − rH)∆,r (rH) ρ2(rH)  Σ2 �  r2  H + a2�2  dt2 +  ρ2(rH)  (r − rH)∆,r (rH)dr2 + ρ2(rH)  ∆θ  dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9)  The metric determinant and contravariant components of the line element (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9) are as  follows  g = −  ρ6(rH)  Σ2 �  r2  H + a2�2 ∆θ  ,  g00 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH),  g11 = (r − rH)∆,r (rH)  ρ2(rH)  ,  g22 =  ∆θ  ρ2(rH),  g01 = g02 = g10 = g12 = g20 = g21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='10)  The angular velocity at the event horizon is given as  Ω =  a  r2  h + a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='11)  – 5 –  The surface gravity near the event horizon of the KNdS black hole is obtained as  κ = lim  g00→0  �  −1  2  �  −g11  g00  dg00  dr  �  =  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  �  r2  H + a2� �  1 + Λa2  3  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='12)  The original Hawking temperature of the KNdS black hole is obtained from the relation  To =  κ  2π as [57]  To =  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  2π  �  r2  H + a2� �  1 + Λa2  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='13)  By using the expression of M from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3), we derive the heat capacity of the black  hole by using the relation Co = ∂M  ∂To as [58, 59]  Co =  �∂M  ∂rh  � �∂rh  ∂To  �  =  2π  �  a2 + r2�2 �  3 + a2Λ  � �  a2 �  3 + r2Λ  �  + 3  �  Q2 − r2 + r4Λ  ��  3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='14)  Applying WKB approximation, Ψµ given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3) is taken as  Ψµ = cµ(t, r, θ) exp  � i  ℏS(t, r, θ)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15)  where S is defined as  S(t, r, θ) = S0(t, r, θ) + ℏ S1(t, r, θ) + ℏ2 S2(t, r, θ) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='16)  By substituting eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3) and keeping only the lowest  order in ℏ, we obtain the equations for the coefficients cµ as  (r − rH)∆,r (rH)  ρ2(rH)  �  c0 B2  1 (∂rS0)2 − c1 B0B1 (∂rS0) (∂tS0 + eA0)  �  +  ∆θ  ρ2(rH)  �  c0 B2  2(∂θS0)2 − c2B0B2(∂θS0)(∂tS0 + eA0)  �  + m2c0 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='17)  −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  �  c1 B2  0(∂tS0 + eA0)2 − c0 B0B1 (∂rS0) (∂tS0 + eA0)  �  +  ∆θ  ρ2(rH)  �  c1 B2  2(∂θS0)2 − c2B1B2(∂rS0)(∂θS0)  �  + m2c1 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='18)  – 6 –  −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  �  c2 B2  0(∂tS0 + eA0)2 − c0 B0B2 (∂θS0) (∂tS0 + eA0)  �  + (r − rH)∆,r (rH)  ρ2(rH)  �  c2 B2  1(∂rS0)2 − c1B1B2(∂rS0)(∂θS0)  �  + m2c2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='19)  where the Bµ’s are defined as  B0 = 1 + β Σ2 �  r2  H + a2�2 (∂tS0 + eA0)2  (r − rH)∆,r (rH) ρ2(rH)  ,  B1 = 1 + β (r − rH)∆,r (rH)(∂rS0)2  ρ2(rH)  ,  B2 = 1 + β∆θ(∂θS0)2  ρ2(rH)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='20)  Considering the symmetries of the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9), we carry on the separation of variables  as follows  S0 = −(E − jΩ) t + R(r) + Θ(θ) + U,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='21)  where E is the energy of the emitted vector particles, j is the angular momentum and U is  a complex constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' On inserting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='21) in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='17)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='19), we get a matrix equation  as  F(c0, c1, c2)T = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='22)  where F is a 3 × 3 matrix and all the entries are as follows  F11 =(r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  B2  1R′2 +  ∆θ  ρ2(rH)B2  2J2  θ + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F12 = − (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  (−E + jΩ + eA0)B0B1R′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F13 = −  ∆θ  ρ2(rH)(−E + jΩ + eA0)B0B2Jθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F21 =  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)B0B1R′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F22 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)2B2  0 +  ∆θ  ρ2(rH)B2  2J2  θ + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F23 = −  ∆θ  ρ2(rH)B1B2JθR′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  K31 =  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)B0B2Jθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  K32 = − (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  B1B2JθR′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  K33 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)2B2 + (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  B2  1R′2  + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='23)  – 7 –  where R′ = ∂rR and Jθ = ∂θΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  To find a non-trivial solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='22), we put the determinant of the matrix F  equals zero, which in turn gives  O(β4)(∂rR)12 + O(β3)(∂rR)10 + O(β2)(∂rR)8 +  �  H6 + O(β2)  �  (∂rR)6  +  �  H4 + O(β2)  �  (∂rR)4 +  �  H2 + O(β2)  �  (∂rR)2 + H0 + O(β2) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='24)  where  H6 =4β  �(r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  �3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  H4 =(r − rH)2∆2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r(rH)  ρ4(rH)  �  1 + 4β  �  −(−E + jΩ + eA0)2 Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  + ∆θJ2  θ  ρ2(rH) + m2  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  H2 =2(r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  �  −(−E + jΩ + eA0)2 Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  + ∆θJ2  θ  ρ2(rH) + m2  −2β  �  (−E + jΩ + eA0)4 Σ4 �  r2  H + a2�4  (r − rH)2∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ4(rH)  − ∆2  θJ4  θ  ρ2(rH)  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  H0 =  �  −(−E + jΩ + eA0)2 Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  + ∆θJ2  θ  ρ2(rH) + m2  � � ∆θJ2  θ  ρ2(rH) + m2  +(−E + jΩ + eA0)2 Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  − 4β  �  (−E + jΩ + eA0)4 Σ4 �  r2  H + a2�4  (r − rH)2∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ4(rH)  − ∆2  θJ4  θ  ρ4(rH)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  By neglecting the higher order term of β, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='24) becomes  4β(r − rH)2∆2,r (rH) (∂rR)4  ρ4(rH)  + (r − rH)∆,r (rH) (∂rR)2  ρ2(rH)  + ∆θJ2  θ  ρ2(rH)  + 4β  �  ∆2  θJ4  θ  ρ4(rH) − (−E + jΩ + eA0)4Σ4 �  r2  H + a2�4  (r − rH)2∆2,r (rH) ρ4(rH)  �  + m2  − (−E + jΩ + eA0)2Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='25)  We obtain the solution to the derivative of the radial action by neglecting the higher  power of β as  ∂rR = ±  �  �  �  �−  �  J2  θ ∆θ + m2ρ2�  (r − rH) ∆,r (rH) + (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)2 (∆,r (rH))2  �  1 + β χ1  χ2  �  , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='26)  – 8 –  where  χ1 =2∆,r (rH)(r − rH)  �  2J4  θ ∆2  θ + 2J2  θ m2∆θρ2 + m4ρ4�  − 4Σ2(−E + jΩ + eA0)2 �  r2  H + a2�2 �  J2  θ ∆θ + m2ρ2�  ,  χ2 =ρ2 �  (r − rH)∆,r (rH)  �  J2  θ ∆θ + m2ρ2�  − Σ2(−E + jΩ + eA0)2 �  r2  H + a2�2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='27)  The solution of the radial action is obtained by integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='26) around the pole  r = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The imaginary part of the radial action gives the particle’s rate of tunneling as  Im R± = ± Im  �  dr  �  �  �  �−  �  J2  θ ∆θ + m2ρ2�  (r − rH) ∆,r (rH) + (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)2 (∆,r (rH))2  ×  �  1 + β χ1  χ2  �  = ±  π(E − jΩ − eA0)  �  r2  H + a2� �  1 + Λa2  3  �  2  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  (1 + β Π) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='28)  where Π = 4m2+ 4J2  θ ∆θ  ρ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' R+ and R− stand for the radial action of the outgoing and ingoing  particles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Based on WKB approximation, the probabilities of the vector boson  particles tunneling across the event horizon rH are  Poutgoing = exp  �  −2  �  Im(R+) + Im(U)  ��  and  Pingoing = exp  �  −2  �  Im(R−) + Im(U)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29)  According to the semiclassical WKB approximation, the ingoing vector boson particles have  a 100% chance of entering the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It shows that Im(U) = −Im(R−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Thus the  tunneling rate of the vector boson particles is given by  Γ = Poutgoing  Pingoing  = exp [−4 Im R+]  = exp  �  �  −2π(E − jΩ − eA0)  �  r2  H + a2� �  1 + Λa2  3  �  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  (1 + β Π)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='30)  The modified Hawking temperature of the KNdS black hole under the quantum gravity  effect is obtained as  Td =  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  2π  �  r2  H + a2� �  1 + Λa2  3  �  (1 − β Π)  =To (1 − β Π) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='31)  – 9 –  where To =  �  (rH−M)− Λ rH  3  (2r2  H+a2)  �  2π(r2  H+a2)  �  1+ Λa2  3  �  is the original Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is obvious from  the point Π > 0 that the modified Hawking temperature is reduced due to the quantum  gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Further, the modified Hawking temperature relies on the mass and angular  momentum of the emitted vector boson particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' In the absence of the quantum gravity  effects i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' β = 0, the original Hawking temperature of KNdS black hole is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The modified heat capacity is calculated as  CH =  �∂M  ∂rh  � � ∂rh  ∂Td  �  =  2π  �  a2 + r2�2 �  3 + a2Λ  � �  a2 �  3 + r2Λ  �  + 3  �  Q2 − r2 + r4Λ  ��  3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)]  × (1 + β Π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='32)  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='32), it is observed that the modified heat capacity tends to the original heat  capacity when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Thus, in the absence of the quantum gravity effects, the original heat  capacity is recovered, and the heat capacity increases due to the quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Now using another coordinate transformation φ = ϕ − Ωt [60] where  Ω =  a [(r2 + a2)∆θ − ∆]  (r2 + a2)2 ∆θ − ∆a2 sin2 θ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='33)  then the line element (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1) reduces to  ds2 = −  ∆∆θρ2  Σ2 �  ∆θ(r2 + a2)2 − ∆a2 sin2 θ  �dt2 + ρ2  ∆ dr2 + ρ2  ∆θ  dθ2  +  �  ∆θ(r2 + a2)2 − ∆a2 sin2 θ  �  sin2 θ  ρ2Σ2  dφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34)  The corresponding non-zero electromagnetic potentials are given by  A0 =  (r2 + a2)∆θ Q r  Σ  �  ∆θ(r2 + a2)2 − ∆a2 sin2 θ  �,  A3 =  −Qra sin2 θ  (r2 + a2 cos2 θ)Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='35)  Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34), we get  ds2 = − (r − rH)∆,r (rH) ρ2(rH)  Σ2 �  r2  H + a2�2  dt2 +  ρ2(rH)  (r − rH)∆,r (rH)dr2 + ρ2(rH)  ∆θ  dθ2  + ∆θ  �  r2  H + a2�2 sin2 θ  ρ2(rH)Σ2  dφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='36)  The metric determinant and the nonzero contravariant components of the line element  given above are as follows  g = ρ4 sin2 θ  Σ4  ,  g00 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH),  g11 = (r − rH)∆,r (rH)  ρ2(rH)  ,  g22 =  ∆θ  ρ2(rH),  – 10 –  g33 =  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='37)  According to WKB approximation, Ψµ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3) can be written as  Ψµ = cµ(t, r, θ, φ) exp  � i  ℏS(t, r, θ, φ)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='38)  where S is defined as  S(t, r, θ) = S0(t, r, θ, φ) + ℏ S1(t, r, θ, φ) + ℏ2 S2(t, r, θ, φ) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='39)  By substituting eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='38) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='39) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='36), and keeping only the lowest order  in ℏ, we derive the equations for the coefficient cµ as  (r − rH)∆,r (rH)  ρ2(rH)  �  c0 H2  1 (∂rS0)2 − c1 H0H1 (∂rS0) (∂tS0 + eA0)  �  +  ∆θ  ρ2(rH)  �  c0 H2  2 (∂θS0)2 − c2H0H2 (∂θH)(∂tS0 + eA0)  �  +  ρ2(rH) Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  �  c0H2  3 (∂φS0 + eA3)2 − c3 H0H3 (∂tS0 + eA0) (∂φS0 + eA3)  �  + m2c0 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='40)  −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  �  c1 H2  0(∂tS0 + eA0)2 − c0 H0H1 (∂rS0) (∂tS0 + eA0)  �  +  ∆θ  ρ2(rH)  �  c1 H2  2(∂θS0)2 − c2H1H2(∂rS0)(∂θS0)  �  +  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  �  c1 H2  3(∂φS0 + eA3)2 − c3 H1H3 (∂rS0) (∂φS0 + eA3)  �  + m2c1 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='41)  −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  �  c2 H2  0(∂tS0 + eA0)2 − c0 H0H2 (∂θS0) (∂tS0 + eA0)  �  + (r − rH)∆,r (rH)  ρ2(rH)  �  c2 H2  1(∂rS0)2 − c1H1H2(∂rS0)(∂θS0)  �  +  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  �  c2 H2  3(∂φS0 + eA3)2 − c3 H2H3 (∂θS0) (∂φS0 + eA3)  �  + m2c2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='42)  −  Σ2 �  r2  H + a2�2  (r − rH)∆,r (rH) ρ2(rH)  �  c3 H2  0(∂tS0 + eA0)2 − c0 H0H3(∂tS0 + eA0)  (∂φS0 + eA3)] + (r − rH)∆,r (rH)  ρ2(rH)  �  c3 H2  1(∂rS0)2 − c1H1H3(∂rS0)(∂φS0 + eA3)  �  +  ∆θ  ρ2(rH)  �  c3 H2  2 (∂θS0)2 − c2H2H3 (∂θS0)(∂φS0 + eA3)  �  + m2c3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='43)  where the Hµ’s are defined as  H0 = 1 + β Σ2 �  r2  H + a2�2 (∂tS0 + eA0)2  (r − rH)∆,r (rH) ρ2(rH)  ,  H1 = 1 + β (r − rH)∆,r (rH)(∂rS0)2  ρ2(rH)  ,  – 11 –  H2 = 1 + β ∆θ (∂θS0)2  ρ2(rH)  ,  H3 = 1 + β ρ2(rH)Σ2 (∂φS0 + eA3)2  ∆θ  �  r2  H + a2�2 sin2 θ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='44)  Considering the symmetry of the black hole, the corresponding action of the vector  boson particles can be written as  S0 = −(E − jΩ) t + W(r) + Θ(θ, φ) + U,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='45)  where E is the energy of the emitted vector particles, j is the angular momentum and U  is a complex constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' On inserting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='45) in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='41)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='44), a matrix equation is  obtained as  F (c0, c1, c2, c3)T = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='46)  where F is a 4 × 4 matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' whose elements are as follows  F11 =(r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  H2  1 W′2 +  ∆θ  ρ2(rH)H2  2 J2  θ +  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  H2  3 (Jφ + eA3) + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F12 = − (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  (−E + jΩ + eA0)H0H1W′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F13 = −  ∆θ  ρ2(rH)(−E + jΩ + eA0)H0H2 Jθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F14 = −  ρ2(rH) Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  (−E + jΩ + eA0) (Jφ + eA3) H0H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F21 =  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)H0H1W′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F22 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)2H2  0 +  ∆θ  ρ2(rH)H2  2J2  θ  +  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  (Jφ + eA3)2 H2  3 + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F23 = −  ∆θ  ρ2(rH)H1H2 Jθ W′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F24 = −  ρ2(rH) Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  (Jφ + eA3) W′ H1H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F31 =  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)H0H2 Jθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F32 = − (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  H1H2 Jθ W′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F33 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)2 H2  0 + (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  H2  1 W′2  +  ρ2(rH)Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  (Jφ + eA3)2 H2  3 + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 12 –  F34 = −  ρ2(rH) Σ2  ∆θ  �  r2  H + a2�2 sin2 θ  (Jφ + eA3) Jθ H2H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F41 =  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0) (Jφ + eA3) H0H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F42 = − (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  (Jφ + eA3) H1H2 W′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F43 = −  ∆θ  ρ2(rH) (Jφ + eA3) Jθ H2H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  F44 = −  Σ2 �  r2  H + a2�2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)(−E + jΩ + eA0)2 H2  0 + (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  H2  1 W′2  +  ∆θ  ρ2(rH)H2  2J2  θ + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='47)  where W′ = ∂rW, Jθ = ∂θΘ and Jφ = ∂φΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='46) has a non-trivial solution if the determinant of the matrix F equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' If  det(F) = 0, then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='46) gives  O(β6)(∂rW)18 + O(β5)(∂rW)16 + O(β4)(∂rW)14 + O(β3)(∂rW)12 + O(β2)  (∂rW)10 + O(β2)(∂rW)8 + O(β2)(∂rW)6 + O(β2)(∂rW)4 + O(β2)(∂rW)2 + O(β2)  +  �  A∗  0(∂rW)2 + A∗  1  � �  B0 + B2(∂rW)2 + B4(∂rW)4 + B6(∂rW)6�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='48)  (The expressions for Ai and Bi are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=')  Solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='48) by neglecting the higher order terms of β , we obtain the solution of  the derivative of the radial action as  ∂rW =  �  −m2ρ2(rH) + J2  θ ∆θ  (r − rH)∆,r (rH) + (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)2∆2,r (rH)  −  (Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ  (r − rH)  �  r2  H + a2�  ∆θ ∆,r (rH)  �1  2 ×  �  1 + χ1  χ2  β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49)  (The expressions of χ1 and χ2 are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=')  The imaginary part of the radial action is obtained by integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49) at the  pole, r = rH as  ImW± = ± Im  �  dr  �  −m2ρ2(rH) + J2  θ ∆θ  (r − rH)∆,r (rH) + (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)2∆2,r (rH)  −  (Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ  (r − rH)  �  r2  H + a2�  ∆θ ∆,r (rH)  �1  2 ×  �  1 + χ1  χ2  β  �  = ±  π(E − jΩ − eA0)  �  r2  H + a2� �  1 + Λa2  3  �  2  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  (1 + β Ξ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='50)  – 13 –  where  Ξ = − 4 (Jφ + eA3)2 ρ2(rH) Σ2 csc2 θ  �  r2  H + a2�2 ∆θ  +  1  2 ρ4(rH) Σ2 (Jφ + eA3) (Jφ + eA3 − 1)  �  3m2 �  r2  H + a2�2 ∆θ sin2 θ  �  5J2  θ ∆θ + 3m2 ρ2(rH)  �  + (Jφ + eA3)3 ρ2(rH) Σ2  �  −8J2  θ ∆θ (Jφ + eA3 − 1) + m2 ρ2(rH)  �  8 + 7 (Jφ + eA3)  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='51)  Here W+ and W− indicate the radial action of the outgoing and ingoing particles  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' According to WKB approximation, the tunneling probabilities are given by  Poutgoing = exp  �  −2  �  Im(R+) + Im(U)  ��  and  Pingoing = exp  �  −2  �  Im(R−) + Im(U)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='52)  In accordance with the semiclassical WKB approximation, there is a 100% probability  of ingoing particle to enter the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Thus, the tunneling rate of W + boson particles  is given by  Γrate = Poutgoing  Pingoing  = exp  �  −4{Im(R+)  �  = exp  �−2 π(E − jΩ − eA0)  �  r2  H + a2� �  1 + Λa2  3  �  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  (1 + β Ξ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='53)  The Boltzman factor gives the Hawking temperature [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Thus, the GUP modified Hawking  temperature is derived as  Td4 =  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  2 π  �  r2  H + a2� �  1 + Λa2  3  �  (1 − β Ξ)  = To (1 − β Ξ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='54)  where To =  �  (rH − M) − Λ rH  3  �  2r2  H + a2��  2 π  �  r2  H + a2� �  1 + Λa2  3  �  , is the original Hawking temperature of KNdS  black hole without any quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The modified Hawking temperature Td4  may be lower or greater than the original Hawking temperature To according to Ξ > 0 or  Ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The modified Hawking temperature depends on the quantum numbers (mass and  angular momentum) of the emitted vector boson particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The modified heat capacity is calculated as  CH4 =  �∂M  ∂rh  � � ∂rh  ∂Td  �  – 14 –  =  2π  �  a2 + r2�2 �  3 + a2Λ  � �  a2 �  3 + r2Λ  �  + 3  �  Q2 − r2 + r4Λ  ��  3 [a4 (r2Λ − 3) + 3r2 (r2 + r4Λ − 3Q2) + a2 (8r4Λ − 3Q2 − 12r2)]  × (1 + β Ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='55)  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='55), it is observed that the modified heat capacity reduces to the original heat  capacity in the absence of the quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The modified heat capacity CH4 is  higher or lower than the original heat capacity Co, according to Ξ > 0 or Ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  4  Remnant of KNdS black hole  Many studies have shown that the GUP effect could give a black hole remnant [47–51,  62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' From this point, we will investigate the remnant of 3-dimensional KNdS black  hole only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The tunneling particle’s mass is no longer considered in the following discussion  since the tunneling particles at the event horizon are effectively massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' According to the  uncertainty principle, the lower limit of the tunneling particle energy can be expressed as  [64, 65]  E ≥  ℏ  ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  Near the event horizon, one may take the uncertainty of the position as the radius of the  black hole [64, 65] as  ∆x ≈ rBH = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='31), we obtain  Td =  �  rH − 1  2  �  1 − Λr2  H  3  � �  rH + a2  rH  �  − Q2  2rH  − Λ rH  3  �  2r2  H + a2��  2π  �  r2  H + a2� �  1 + Λa2  3  �  ×  �  1 −  4βJ2  θ ∆θ  r2  H + a2 cos2 θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3)  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3), it is observed that when  rH ≤  �  4βJ2  θ  �  1 + Λa2 cos2 θ  �  − 3a2 cos2 θ  3  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4)  the modified Hawking temperature becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' This violates the law of black hole  thermodynamics and thus has no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is clear that the evaporation will stop  under the effects of GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Thus the Hawking temperature becomes zero when rH reaches  the minimum radius, rmin as  rmin =  �  4βJ2  θ  �  1 + Λa2 cos2 θ  �  − 3a2 cos2 θ  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5)  – 15 –  Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='31), we obtain the expression of Td in terms of the mass of the  black hole as  Td ≈ ζ1  ζ2  �  �  �  �  �1 −  4βJ2  θ ∆θ  1  α2  1  �  1 + 4ΛM2  3β2  1α1  �2 �  M +  �  M2 − (a2 + Q2)α1  �2  + a2 cos2 θ  �  �  �  �  � ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6)  where  ζ1 = 1  α1  �  1 + 4ΛM2  3β2  1α1  � �  M +  �  M2 − (a2 + Q2)α1  �  �  1 − Λa2  3  − 2Λ  3 α2  1  �  1 + 4ΛM2  3β2  1α1  �2 �  M +  �  M2 − (a2 + Q2)α1  �2  �  − M,  ζ2 =2π  �  1 + Λa2  3  � �  a2 + 1  α2  1  �  1 + 4ΛM2  3β2  1α1  �2 �  M +  �  M2 − (a2 + Q2)α1  �2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7)  To make the Hawking temperature T ≥ 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' to ensure the GUP corrected temperature  has a physical meaning, the mass of the black hole must hold the inequality  M ≥  β2  1  8(a2 + Q2)Λ − 6β2  1  �  3  �  −α1(a2 + Q2) + 3  2  �  −4α1(a2 + Q2)  +8α1ζ3  �  3β2  1 − 4(a2 + Q2)Λ  � �  −a2 + Q2 + 4J2  θ α1β∆θ − a2α1 cos2 θ  �  3β2  1  � 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8)  It is noted that the mass of the black hole has a minimum value which is given by  Mmin =  β2  1  8(a2 + Q2)Λ − 6β2  1  �  3  �  −α1(a2 + Q2) + 3  2  �  −4α1(a2 + Q2)  +8α1ζ3  �  3β2  1 − 4(a2 + Q2)Λ  � �  −a2 + Q2 + 4J2  θ α1β∆θ − a2α1 cos2 θ  �  3β2  1  � 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9)  5  Graphical Analysis  In this section, we will examine graphically the effects of parameters β, Λ and m on the  modified Hawking temperatures and modified heat capacities with respect to event horizon  rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  Temperature Td with radius of event horizon rH for 3-dimensional KNdS  black hole  This subsection is devoted to analysing the behaviour of modified Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The parameters are taken as follows a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3, Q = 1, θ = π  2 and Jθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 16 –  2  4  6  8  10  rH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  Td  Hawking Temperature vs Radius of Event Horizon  β=5  β=10  β=15  β=20  β=25  2  4  6  8  rH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  Td  Hawking Temperature vs Radius of Event Horizon  β=25  β=30  β=50  β=70  β=90  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Hawking temperature Td with respect to radius of event horizon rH for different values  of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  2  4  6  8  rH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  Td  Hawking Temperature vs Radius of Event Horizon  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  Λ=1  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Hawking temperature Td with respect to the radius of event horizon rH for different  values of cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Figure 1 indicates the variation of the modified Hawking temperature with rH > 0,  for different values of β with fixed value of the cosmological constant Λ = 1 and  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' We observe that for β = 25, the temperature decreases and tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  As the radius of horizon increases for β < 25, the modified Hawking temperature  becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  This negative temperature and divergent behaviour reveal the  nonphysically unstable state of the black hole [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Moreover, for β > 25, as the  horizon increases, the temperature decreases and once the minimum value is reached,  the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is worth mentioning that, for β < 25, we observe the  nonphysical behaviour with negative temperature and for β = 25, the temperature  vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Furthermore, the temperatures are positive when β > 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The β effects  decelerate the increase in Hawking temperature which is also shown numerically in  table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Figure 2 shows the behaviour of modified Hawking temperature with rH > 0, for  different values of positive cosmological constant Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The parameter are taken as  follows: m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1 and β = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is observed that the modified Hawking temperatures  decrease and attain its minimum values, which is also calculated numerically in table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Later, it keeps on increasing as rH increases and its behaviour is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The change  – 17 –  in Λ gives the diverging temperature Td as rH increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0  2  4  6  8  rH  0  10  20  30  40  50  Td  Hawking Temperature vs Radius of Event Horizon  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td with respect to radius of event horizon rH for different values  of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Figure 3 indicates the behaviour of Hawking temperature for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' At  first, the Hawking temperature drops suddenly and after attaining its minimum point,  it keeps increasing with increasing the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The increase of the parameter  m makes the Hawking temperatures increases as shown numerically in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0  1  2  3  4  5  rH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='30  Td  Hawking Temperature vs Radius of Event Horizon  a=0  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  a=1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  a=2  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td versus radius of event horizon rH for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Figure 4 provides the graphical analysis of Td via horizon radius rH for different  values of spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The pink line, which corresponds to a = 0 represents  the Hawking temperature graph for Reissner-Nordstrom-de Sitter (RNdS) black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The graph shows that the Hawking temperature of the RNdS black hole is greater  than that of the KNdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The effect of spin parameter a decelerates the  increase in Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' This graphical presentation is compatible with the  numerical calculation of table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 18 –  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1  β  25  30  50  70  90  rc  H  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='97272  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='81347  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='43917  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3935  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='37473  T c  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='00953  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='06208  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='16458  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='26337  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='36169  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the critical radius rC  H and the critical temperature T C  H for  different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  From figures 1 to 4 show that the temperature cools down to the minimum T C  d , at  rH = rC  H and keeps on increasing as rH > rC  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 40  Λ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  1  rc  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4842  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4876  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49087  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49664  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5016  T c  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='01115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='02619  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='05600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='08596  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='11457  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the critical radius rC  H and the critical temperature T C  H for  different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 40  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  rc  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5016  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='18474  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='16561  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='16027  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15807  T c  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='11457  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='07833  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='97447  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='81695  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='60655  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the critical radius rC  H and the critical temperature T C  H for  different values of emitted particle mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 40  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  1  rc  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='44254  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='86004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='96988  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='07671  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='17653  T c  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='11791  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='05845  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='04575  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='03314  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='02359  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the critical radius rC  H and the critical temperature T C  H for  different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  Heat capacity CH with radius of event horizon rH for 3-dimensional KNdS  black hole  This subsection focuses on analysing modified heat capacity in different domains of event  horizon radius rH with fixed parameters: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3, Q = 1, θ = π  2 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The heat  capacity is connected to the local thermal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' If the black hole has a negative heat  capacity, it is unstable to thermal radiation and if it has a positive heat capacity, it is stable  to thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 19 –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='30  rH  10000  5000  5000  10000  CH  Heat Capacity vs Radius of Event Horizon  β=0  β=10  β=30  β=50  β=70  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat capacity CH versus radius of event horizon rH for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (i) For fix parameters a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3, Λ = 1 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5, the variation of heat capacity  CH with the horizon radius rH for different values of β is shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The  position of phase transition remains unchanged in 3-dimensional KNdS black hole  for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Further, it is observed that the phase transition occurs at  rH = r∗  H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15467, r∗  H denotes the position of the horizon radius at which the phase  transition takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  rH  10000  5000  5000  10000  CH  Heat Capacity vs Radius of Event Horizon  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat capacity CH versus radius of event horizon rH for different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (ii) Figure 6 illustrates the behaviour of CH w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH for fixed values of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3, β = 40,  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5 and for different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' For different values of cosmological constant Λ,  there are different positions of phase transition for 3-dimensional KNdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Increasing the values of Λ, the positions of phase transition r∗  H are shifted towards  the origin, which ensures the black hole faster stability for larger value of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Different  positions of phase transition are shown in table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (iii) Figure 7 shows the variation of CH for different values of spin parameter a and for  fixed parameters Λ = 1, β = 40, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Increasing the value of rotation parameter  a, the phase transition occurs at larger value of horizon radius r∗  H which is also shown  in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Moreover, larger the value of a delays the stability of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 20 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  rH  60000  40000  20000  20000  40000  60000  CH  Heat Capacity vs Radius of Event Horizon  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  a=1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  a=2  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat capacity CH versus radius of event horizon rH for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Figures 5 to 7 show that there is a phase transition when rH = r∗  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The positions  of phase transition are shown in tables 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The black holes are unstable in the  region 0 ≤ rH ≤ r∗  H and stable in the region r∗  H ≤ rH ≤ ∞ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' the smaller black  holes are less stable than larger black holes and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 40  Λ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  r∗  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='42403  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='31282  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='23652  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='17887  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the position of phase transition r∗  H for different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 40  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  2  r∗  H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='17485  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='23815  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29861  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='35098  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of the position of phase transition r∗  H for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Hawking temperature Td4 with radius of event horizon for 4-dimensional  KNdS black hole  This subsection is devoted to analysing the behaviour of modified Hawking temperatures  of KNdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The parameters are taken as follows: Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1, θ = π  2 , e = 1, Jθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  and Jφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (i) Figure 8 shows the behaviour of Td4 for different values of β and for fixed values of  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2, Λ = 1 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Td4 increases exponentially and reach its maximum  height T max  d4  for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Further, the temperature keeps on decreasing  with increasing the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is noteworthy to mention that the temperature  increases with increasing the values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hence, the β effects accelerate the increase  in Td4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The proof of the above statement is also calculated in table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 21 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  rH  20  40  60  80  100  120  140  Td4  Hawking Temperature vs Radius of Event Horizon  β=10  β=30  β=50  β=70  β=90  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td4 with respect to radius of event horizon rH for different values  of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  rH  20  40  60  80  100  Td4  Hawking Temperature vs Radius of Event Horizon  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  Λ=1  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td4 with respect to radius of event horizon rH for different values  of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (ii) The behaviour of Td4 for varying cosmological constant Λ is depicted in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The  parameters are as follows: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2, β = 50 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The temperature increases  to a certain height and attains its peak point T max  d4  at rH = rc  H, then Td4 decreases  as rH increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is noted that the Λ effects decelerate the increase in Td4 which is  also shown numerically in table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (iii) The behaviour of Td4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH for varying mass of the vector boson particle m is  depicted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The temperature increases to a certain height T max  d4  and then  decreases with increasing rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The rate of increase of temperature Td4 is dependent  on the increase of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The validity of the above statements is calculated numerically  in table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (iv) Figure 11 shows the behaviour of Td4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH for varying a and fixed parameters  β = 50, Λ = 1 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The temperatures increase exponentially upto T max  d4  and decrease with increasing the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The temperature gradually increases  – 22 –  with decreasing the values of rotation parameter, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The numerical calculation shown  in table 10 supports the above statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  rH  100  200  300  400  500  600  700  Td4  Hawking Temperature vs Radius of Event Horizon  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td4 with respect to radius of event horizon rH for different values  of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  rH  50  100  150  Td4  Hawking Temperature vs Radius of Event Horizon  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='25  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking temperature Td4 with respect to radius of event horizon rH for different values  of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  We present the following tables to see the effect of β, Λ, m and a on the Hawking temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Table 7 confirms that T max  d4  gradually increases on increasing β (for fixed values of  Λ, m and a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Similarly table 9 also confirms that T max  d4  gradually increases on increasing  m (for fixed values of β, Λ, and a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' On the contrary, tables 8 and 10 confirm that T max  d4  gradually decreases with increasing cosmological constant Λ (for fixed values of β, m and  a) and rotational parameter a (for fixed values of β, Λ, and m) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' From table 9,  it is observed that T max  d4  is highly dependent on m compared to that of β, Λ and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 23 –  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  β  10  30  50  70  90  rc  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34861  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34849  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34847  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34846  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34845  T max  d4  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2974  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7385  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1796  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6206  128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='062  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of rC  H and T max  d4  for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 50  Λ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  1  rc  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='35376  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='35244  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='35117  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34847  T max  d4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2216  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='793  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4615  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4107  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1796  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of rC  H and T max  d4  for different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 50  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  rc  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34847  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='39069  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='45936  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='50492  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='53291  T max  d4  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1796  111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='502  211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='269  386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='449  652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='623  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of rC  H and T max  d4  for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  rc  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='27555  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='34852  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='42236  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49421  T max  d4  176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='342  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3181  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='518  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7598  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='89087  Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of rC  H and T max  d4  for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  Heat Capacity CH4 versus horizon radius rH for 4-dimensional KNdS black  hole  This subsection studies the modified heat capacity for different values of β, Λ and a for  fixed values of Q = 1, θ = π  2 , e = 1, Jθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1 and Jφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (i) The variation of modified heat capacity for different values of β is shown in figure  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The parameters taken are a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2, Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is observed that  there is one position of phase transition in the absence of GUP, but there are two  positions of phase transition under the influence of GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The first phase transition in  figure 12 is due to the quantum gravity effects and it occurs at rH = rH1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='99338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The position of the second phase transition is observed at rH = rH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29667 with  or without quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The variation of β doesn’t affect the position of  phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 24 –  (ii) For a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2, β = 15 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1, the variation of CH4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH changing the values  of Λ are illustrated in figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Varying the values of cosmological constant Λ, the  positions of phase transition are also varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' With increasing the values of Λ, the  position of phase transition is shifted toward the origin, which implies a slower rate  of becoming a stable black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Table 11 is constructed numerically to show the  different positions of phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  rH  100  50  50  100  CH4  Heat Capacity vs Radius of Event Horizon  β=0  β=10  β=30  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat Capacity CH4 with respect to radius of event horizon rH for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  rH  40  20  0  20  40  CH4  Heat Capacity vs Radius of Event Horizon  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  Λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat Capacity CH4 with respect to radius of event horizon rH for different values of  Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (iii) Figure 14 represents the behaviour of CH4 versus rH for fixed parameters β = 15,  Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The position of phase transition is shifted to far away from the  origin toward the positive direction of rH with increasing the value of spin parameter  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It shows that the black hole becomes stable faster with increasing the value of  rotation parameter a which is also indicated numerically by table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  – 25 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='0  rH  15  10  5  0  5  10  15  CH4  Heat Capacity vs Radius of Event Horizon  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='15  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Heat Capacity CH4 with respect to radius of event horizon rH for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  β = 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  Λ  rH1  rH2  a  rH1  rH2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='99602  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='40094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49917  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2864  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='99338  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7472  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='99075  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='22479  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='99338  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='29667  Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The tabulated values of rH1 and rH2 for different values of Λ and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  6  Conclusions  This work studies the GUP effects on tunneling of massive vector boson particles from KNdS  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The Hawking temperatures and heat capacities near the horizon of KNdS black  hole are studied using GUP-corrected Lagrangian of massive vector field, Feynman pre-  scription and WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is noted that the Hawking temperatures and heat  capacities are modified due to quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' They depend not only upon the  quantum gravity parameter β, spin parameter a, mass of the emitted particle m, cosmolog-  ical constant Λ, charge of the black hole Q but also on angular coordinates θ, Jθ and Jφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  We also discuss the stable and unstable formations of KNdS black hole in quantum gravity  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  The remnant of KNdS black hole is also discussed in the presence of quantum  gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' We also illustrate the graphs of modified Hawking temperatures and heat  capacities and explore the effects of β, Λ, a and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The modified Hawking temperature of  a 3-dimensional KNdS black hole w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH tends to decrease for β < 25, but for β > 25, the  temperature cools down till it reaches its minimum point and then increases, which leads  to the formation of stable black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' For a 4-dimensional KNdS black hole with the above  fixed parameters, the modified Hawking temperature increases w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='t rH and after attaining  maximum height, the temperature eventually goes down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is worth noting that there  are one phase transition and two phase transitions for a non-zero horizon of 3-dimensional  KNdS black hole and 4-dimensional KNdS black hole respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Different positions of  phase transitions are due to the quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' It is worth mentioning that for  – 26 –  different values of dimensionless parameter β, the position of phase transitions remain the  same in 3-dimensional and 4-dimensional KNdS black hole under the influence of quantum  gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' The modified Hawking temperatures and heat capacities are reduced to the  original Hawking temperature and heat capacity in quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hence quan-  tum gravity effects modified the Hawking temperature and heat capacity of KNdS black  hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  A  Coefficients of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='48)  A∗  0 =(r − rH)∆,r (rH)  ρ2(rH)  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  A∗  1 =m2 + J2  θ ∆θ  ρ2(rH) − (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)∆,r (rH)ρ2(rH)  + (Jφ + eA3)2 ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  B6 =6m2β (r − rH)3 ∆2,r (rH)  ρ6(rH)  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3)  B4 =m2 (r − rH)2 ∆2,r (rH)  ρ4(rH)  + 2β (r − rH)2 ∆2,r (rH)  ρ4(rH)  �  −  �  1  (r − rH) ∆,r (rH) ρ2(rH)  �  (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  �  − 2 (Jφ + eA3 − 1) (Jφ + eA3) ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  + 3m2  ���  + m2  �  3m2 + 3J2  θ ∆θ  ρ2(rH) + (Jφ + eA3) (2 + Jφ + eA3) ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  ��  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4)  B2 =(r − rH) ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  ρ2(rH)  �  (−E + jΩ + eA0)2 (Jφ + eA3) (Jφ + eA3 − 1) Σ4 csc2 θ  (r − rH) ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  + 2m4 + 2 J2  θ m2∆θ  ρ2(rH)  + (Jφ + eA3) (1 + Jφ + eA3) m2ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  − 2m2(−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH) ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  + 2β  �  1  (r − rH)2 ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ4(rH)  �  (−E + jΩ + eA0)4 �  r2  H + a2�4 Σ4  �(Jφ + eA3 − 1) (Jφ + eA3) ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  − 3m2  ��  + (−E + jΩ + eA0)2 (Jφ + eA3)3 (Jφ + eA3 − 1) ρ2(rH)Σ6 csc4 θ  (r − rH) (r2  H + a2)2 ∆2  θ ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  + m2  � 3J4  θ ∆2  θ  ρ4(rH) + (Jφ + eA3)3 {1 + 2 (Jφ + eA3)} ρ4(rH)Σ4 csc4 θ  (r2  H + a2)4 ∆2  θ  ���  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='5)  B0 = −  �  J2  θ ∆θ  ρ2(rH) − (−E + jΩ + eA0)2 �  r2  H + a2�2 Σ2  (r − rH)∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)ρ2(rH)  + (Jφ + eA3)2 ρ2(rH)Σ2 csc2 θ  (r2  H + a2)2 ∆θ  + m2  �  +  2 β  (r − rH)3 (r2  H + a2)6 ∆3  θ ∆3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ6(rH)  �  3(−E + jΩ + eA0)6∆6  θΣ6  �  r2  H + a2�10 �  m2 �  r2  H + a2�2 ∆θ − (Jφ + eA3) (Jφ + eA3 − 1) ρ2(rH)Σ2 csc2 θ  �  – 27 –  − (−E + jΩ + eA0)4(r − rH)  �  r2  H + a2�8 ∆2  θ ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)Σ4  �  3m2 �  r2  H + a2�2 ∆θ  �  J2  θ ∆θ + m2ρ2(rH)  �  − (Jφ + eA3)2 ρ2(rH)  �  J2  θ (Jφ + eA3 − 1) ∆θ − 3m2ρ2(rH)  �  Σ2 csc2 θ  �  + m2(r − rH)3∆3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)  �  3J6  θ  �  r2  H + a2�6 ∆6  θ + J2  θ ∆2  θ ρ8(rH) Σ4 csc4 θ  (Jφ + eA3)3  �  1 + 2 (Jφ + eA3)  �  + (Jφ + eA3)3 ρ10(rH) Σ4 csc4 θ  �  m2∆θ  �  1 + 2 (Jφ + eA3)  � �  r2  H + a2�2 + 3ρ2(rH)Σ2 csc2 θ (Jφ + eA3)2  �  + J4  θ ∆4  θ ρ2(rH)  �  r2  H + a2�4 �  3m2 �  r2  H + a2�2 ∆θ + (Jφ + eA3) (2 + Jφ + eA3) ρ2(rH) Σ2 csc2 θ  ��  − (−E + jΩ + eA0)2 �  r2  H + a2�2 (r − rH)2∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH)Σ2  �  (1 − Jφ − eA3) (Jφ + eA3)  ρ2(rH) Σ2 csc2 θ  �  2J2  θ  �  r2  H + a2�4 ∆4  θ + J2  θ (Jφ + eA3)2 �  r2  H + a2�2 ∆2  θρ4(rH) Σ2  csc2 θ + 3 (Jφ + eA3)4 ρ8(rH) Σ4 csc4 θ  �  + m2 �  r2  H + a2�2 ∆θ  �  3J4  θ ∆4  θ  �  r2  H + a2�4  + (Jφ + eA3)3 (2 + Jφ + eA3) ρ8(rH) Σ4 csc4 θ  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='6)  B  The expressions of χ1 and χ2 given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='49)  χ1 = − 3(−E + jΩ + eA0)6m2 �  r2  H + a2�6 Σ6  (r − rH)3∆3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ6(rH)  +  (−E + jΩ + eA0)4 Σ4  (r − rH)2 ∆2  θ ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ6(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='∆θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='θ ∆θ + m2 ρ2(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='+ (Jφ + eA3)2 ρ4(rH) Σ2 csc2 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='+ m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='+ (Jφ + eA3) Σ2 csc2 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='(r2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='H + a2)2 ∆θ ρ2(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2J2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='θ m2 ∆θ ρ2(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2 + 7 (Jφ + eA3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='+ J2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='θ ∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4 + 5 (Jφ + eA3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='+ m4 ρ4(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4 + 5 (Jφ + eA3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='+ ρ2(rH) Σ4 csc4 θ (Jφ + eA3)3 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='4 + 5 (Jφ + eA3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='J2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='θ ∆θ + m2 ρ2(rH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='(r2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='H + a2)4 ∆2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='− (Jφ + eA3 − 4) (Jφ + eA3)5 ρ6(rH)Σ6 csc6 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='(r2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='H + a2)6 ∆3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='− (jΩ + eA0 − E)2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='H + a2�2 Σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='(r − rH) ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='r (rH) ρ2(rH)  �  9m6 + m2  � J2  θ ∆θ  ρ2(rH) + (Jφ + eA3)2 ρ2(rH) Σ2 csc2 θ  (r2  H + a2)2 ∆θ  �  � 9Jθ∆1  ρ2(rH) + ρ2(rH) Σ2 csc2 θ (Jφ + eA3) (8 + Jφ + eA3)  (r2  H + a2)2 ∆θ  �  – 28 –  + 6m4  �3J2  θ ∆φ  ρ2(rH) + ρ2(rH) Σ2 csc2 θ (Jφ + eA3) [1 + 2(Jφ + eA3)]  (r2  H + a2)2 ∆θ  �  − 4 Σ2 csc2 θ (Jφ + eA3 − 1) (Jφ + eA3)  (r2  H + a2)6 ∆3  θ ρ2(rH)  �  J4  θ ∆4  θ  �  r2  H + a2�4 + J2  θ ∆2  θ ρ4(rH)  Σ2 csc2 θ (Jφ + eA3)2 �  r2  H + a2�2 + ρ8(rH)Σ4 csc4 θ (Jφ + eA3)4  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='1)  χ2 =ρ2(rH) Σ2 csc2 θ (1 − Jφ − eA3) (Jφ + eA3)  (r2  H + a2)2 ∆θ  �  m2 − Σ2(−E + jΩ + eA0)2 �  r2  H + a2�2  (r − rH) ∆,r (rH) ρ2(rH)  �  �  m2 + J2  θ ∆θ  ρ2(rH) − Σ2(−E + jΩ + eA0)2 �  r2  H + a2�2  (r − rH) ∆,r (rH) ρ2(rH)  + ρ2(rH) Σ2 csc2 θ (Jφ + eA3)2  (r2  H + a2)2 ∆θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='2)  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking, Black hole explosions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=', Nature 248 (1974) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Hawking, Particle creation by black holes, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Bekenstein, Black Holes and Entropy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Wilczek, Some applications of a simple stationary line element for the  Schwarzschild geometry, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Wilczek, Self-interaction correction to black hole radiance, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Kraus and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Wilczek, Effect of self-interaction on charged black hole radiance, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' B 437 (1995) 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Parikh and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Wilczek, Hawking radiation as tunneling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Adler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Chen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Amelino-Camelia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Arzano and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' D 70 (2004) 107501 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content='  [66] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Javed and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' Babar, Fermions Tunneling and Quantum Corrections for Quintessential  Kerr-Newman-AdS Black Hole, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
+page_content=' 2019 (2019) 2759641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQfJQmU/content/2301.04342v1.pdf'}
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+Optical Telecommunications-Band Clock based on Neutral Titanium Atoms
+Scott Eustice,1, 2 Dmytro Filin,3 Jackson Schrott,1, 2 Sergey Porsev,3 Charles
+Cheung,3 Diego Novoa,1, 2 Dan M. Stamper-Kurn,1, 2, 4 and Marianna S. Safronova3, 5
+1Department of Physics, University of California, Berkeley, CA 94720
+2Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720
+3Department of Physics and Astronomy, University of Delaware, Newark, DE 19716
+4Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
+5Joint Quantum Institute, National Institute of Standards and Technology
+and the University of Maryland, College Park, Maryland 20742
+(Dated: February 1, 2023)
+We propose an optical clock based on narrow, spin-forbidden M1 and E2 transitions in laser-
+cooled neutral titanium. These transitions exhibit much smaller black body radiation shifts than
+those in alkaline earth atoms, small quadratic Zeeman shifts, and have wavelengths in the S, C,
+and L-bands of fiber-optic telecommunication standards, allowing for integration with robust laser
+technology. We calculate lifetimes; transition matrix elements; dynamic scalar, vector, and tensor
+polarizabilities; and black body radiation shifts of the clock transitions using a high-precision rel-
+ativistic hybrid method that combines a configuration interaction and coupled cluster approaches.
+We also calculate the line strengths and branching ratios of the transitions used for laser cooling. To
+identify magic trapping wavelengths, we have completed the largest-to-date direct dynamical polar-
+izability calculations. Finally, we identify new challenges that arise in precision measurements due
+to magnetic dipole-dipole interactions and describe an approach to overcome them. Direct access
+to a telecommunications-band atomic frequency standard will aid the deployment of optical clock
+networks and clock comparisons over long distances.
+Optical atomic clocks have taken a giant leap in re-
+cent years, with several experiments reaching uncertain-
+ties at the 10−18 level [1–3]. The comparison of clocks
+based on different atomic standards [4] or placed in sepa-
+rate locations [5] enables important applications such as
+relativistic geodesy [6], tests of fundamental physics [7],
+and dark matter searches [8]. These applications moti-
+vate the development of synchronized clock networks and
+transportable clocks that operate in extreme and distant
+environments [9].
+The leading neutral-atom optical clocks operate on
+wavelengths of 698 nm (Sr) [10] and 578 nm (Yb) [11].
+Light at these wavelengths is strongly attenuated in opti-
+cal fibers, posing a challenge to long-distance time trans-
+fer.
+These wavelengths are also inconvenient for con-
+structing the ultrastable lasers that are an essential com-
+ponent of optical clocks.
+By comparison, an optical atomic clock operating in
+the telecommunication wavelength band would have clear
+advantages. The S-, C- and L-bands, ranging altogether
+between about 1460 and 1625 nm, feature low losses
+in standard optical fibers. Stable light sources and ro-
+bust optical amplifiers are also available across these
+ranges [12]. These features would support the develop-
+ment of fiber-linked terrestrial clock networks over con-
+tinental distances.
+We propose the use of ultra-narrow optical tran-
+sitions in atomic titanium (Ti) as the basis of a
+telecommunications-band atomic clock. It has recently
+been pointed out that numerous transition-metal ele-
+ments, including Ti, can be laser-cooled on near-cycling
+optical transitions [13], allowing for the adoption of op-
+tical lattice or tweezer trapping techniques [14] used in
+J
+b)
+a)
+E [103cm−1]
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+20
+25
+cooling
+498 nm
+E2
+1573 nm
+M1
+1549 nm
+3d24s2 a3F
+|g⟩
+3d3(4F)4s a5F
+|e⟩
+3d3(4F)4p y 5G
+◦
+6
+lattice lasers
+clock laser
+x
+y
+z
+⃗B
+FIG. 1. (a) Relevant atomic structure in Ti for an optical
+clock.
+The a F
+3
+and a F
+5
+terms serve as the basis for the
+optical clock, while the excited y G
+5
+◦
+6 level serves as the excited
+state for laser cooling of Ti. The two optical clock transitions
+highlighted in the text are shown as maroon arrows; the laser
+cooling transition is shown in cyan.
+(b) A diagram of the
+proposed experimental system.
+Polarizations are indicated
+on a given beam by a small arrow of the same color as the
+beam itself.
+today’s leading neutral-atom clocks.
+We identify sev-
+eral transitions between the 3d24s2 a F
+3
+and 3d3( F
+4
+)4s
+a F
+5
+fine structure manifolds in Ti with transition wave-
+lengths between 1483 and 1610 nm (see Fig. 1 and Ta-
+ble I) that can serve as optical clock references for ultra-
+stable telecommunication-band light sources.
+From a numerical calculation of the Ti level structure,
+we identify several key features that make Ti an attrac-
+tive atom for clock applications: the extreme narrowness
+of the candidate clock transitions, a weak clock sensi-
+tivity to blackbody radiation shifts, and the existence
+arXiv:2301.13363v1  [physics.atom-ph]  31 Jan 2023
+
+2
+of several magic wavelengths for optical trapping. While
+we identify challenges posed by the non-zero angular mo-
+mentum of the clock states in Ti, we show that a proper
+magic-wavelength condition for optical trapping, which
+imposes a significant differential tensor ac Stark shift,
+mitigates their effects.
+Our analysis relies on high precision atomic structure
+calculations, by which we characterize 85 levels of neu-
+tral Ti. For this, we employ a hybrid method that com-
+bines the configuration interaction (CI) and linearized
+coupled-cluster (CC) approaches (referred to as CI +
+all order method [15, 16]).
+In this method, the corre-
+lations between four valence electrons are included via a
+large-scale CI computation using a highly parallel mes-
+sage passing interface (MPI) CI code [16, 17]. Several
+computations with increased number of configurations
+were carried out to ensure convergence. The core-core
+and core-valence correlations are included using an effec-
+tive Hamiltonian formalism [15]. We construct the effec-
+tive Hamiltonian using second-order many-body pertur-
+bation theory (MBPT) and more accurate CC methods.
+The difference between these results gives the size of the
+higher-order corrections, which we use to estimate un-
+certainties on all theory values [16]. The results are used
+to calculate transition rates, dynamical polarizabilities,
+and systematic shifts in the clock transitions. Further
+details of the computational methods are given in the
+Supplemental Material [18].
+Several clock transitions are identified in Table I. Tran-
+sitions between the a F
+3
+and a F
+5
+manifolds occur via spin
+forbidden electric quadrupole (E2) and magnetic dipole
+(M1) transitions. Calculated reduced matrix elements for
+these transitions are tabulated. The calculated natural
+linewidths account for both the decay of the upper state
+to the lower manifold on the listed E2 and M1 transitions
+and the M1 decays within each fine-structure manifold.
+The transitions are all exceptionally narrow, allowing for
+optical atomic clocks with long coherence times.
+In this letter, we focus on the a F
+3
+4 → a F
+5
+5 transi-
+tion at 1549 nm unless otherwise noted. An advantage
+of this transition is that the a F
+5
+5 state is the lower level
+of the near-cycling 498 nm transition, which is suited for
+laser cooling. Our calculations predict that the cooling
+transition has low branching ratios to other even parity
+states (∼ 10−6), enabling single-laser state preparation
+and readout for atoms in the upper clock state. For de-
+tails on calculations relevant to the laser cooling transi-
+tion, see the Supplemental Material [18]. An additional
+benefit is that light at the 1549 nm clock wavelength can
+be generated by narrow-linewidth, high-power Er-doped
+fiber lasers, simplifying the required optical setup.
+We consider the three titanium isotopes which have
+zero nuclear spin, and therefore no hyperfine structure
+(46,48,50Ti). To make the clock insensitive to first-order
+differential Zeeman shifts from stray magnetic fields, we
+drive the |mJ = 0⟩ → |m′
+J = 0⟩ transition, with mJ be-
+ing the magnetic quantum number and the primed sym-
+bols and numbers referring to the upper a F
+5
+state. Be-
+J
+J′
+λ (nm)
+Tele.
+DM1
+DE2
+Γ
+Band
+(10−3µB)
+(a.u.)
+(10−6s−1)
+4
+5
+1548.926
+C
+1.0(5)
+0.140(4)
+242(5)
+4
+4
+1573.346
+L
+0.36(18)
+0.134(8)
+239(5)
+4
+3
+1593.846
+L
+1.02(12)
+0.0015(3)
+227(5)
+4
+2
+1609.816
+L
+N/A
+0.0314(27)
+214(5)
+3
+5
+1498.615
+S
+N/A
+0.0472(7)
+162.2(2.6)
+3
+4
+1521.463
+S
+0.4(4)
+0.027(10)
+159.1(2.5)
+3
+3
+1540.625
+C
+0.2(2)
+0.124(4)
+147.2(2.6)
+3
+2
+1555.541
+C
+0.3(4)
+0.0204(22)
+134.3(2.6)
+3
+1
+1565.754
+L
+N/A
+0.0463(23)
+129.2(2.5)
+2
+4
+1483.073
+S
+N/A
+0.0196(26)
+32.75(29)
+2
+3
+1501.275
+S
+0.40(16)
+0.024(7)
+20.83(36)
+2
+2
+1515.435
+S
+0.1(1)
+0.1006(24)
+7.93(38)
+2
+1
+1525.127
+S
+0.23(2)
+0.0643(11)
+2.85(11)
+TABLE I. List of proposed optical clock transitions in Ti. All
+transitions are between the lower a F
+3
+and upper a F
+5
+terms.
+The lower (upper) states are indexed by J (J′). Transition
+wavelengths λ are taken from Ref. [19]. The telecomm band
+is indicated, with S (short), C (conventional) and L (long)
+bands noted.
+M1, E2 reduced matrix elements DM1, DE2
+and transition linewidths Γ are calculated.
+The two clock
+transitions highlighted in the text are in bold.
+cause the E2 matrix element for this transition is zero,
+only the M1 matrix element contributes to a direct one-
+photon drive of the clock transition. Choosing quantiza-
+tion, clock-laser polarization, and clock-laser propagation
+axes as shown in Fig. 1, we calculate that for a driving in-
+tensity of 0.1 W/mm2, we achieve a clock Rabi frequency
+of 91(46) Hz.
+To compare the strength of this M1 transition to that
+of an E2 transition in the same set of transitions, we also
+consider driving the |a F
+3
+4, mJ = 0⟩ → |a F
+5
+4, m′
+J = 0⟩
+transition at a wavelength of 1573 nm. For this transi-
+tion, the M1 matrix element vanishes while the E2 matrix
+element does not. With the same intensity and polariza-
+tion as in Fig. 1, but propagating along the z axis, the
+Rabi frequency for such an E2 transition is 214(13) Hz.
+For a detailed derivation of these Rabi frequencies, see
+the Supplemental Material [18].
+Neutral-atom optical clocks often use optical lattice
+potentials to confine atoms, allowing for a long interro-
+gation time. In order to avoid imposing large differential
+ac Stark shifts between the upper and lower states of
+the clock transition, it is necessary to use lattice light
+which is at a “magic wavelength”, at which the dynamic
+polarizabilities of the lower and upper clock states are
+identical [20].
+In Fig. 2 and Table II, we report several magic wave-
+lengths for the |a F
+3
+4, mJ = 0⟩ → |a F
+5
+5, m′
+J = 0⟩ clock
+transition. As with most states in Ti, the clock states ex-
+perience significant vector and tensor ac Stark shifts [13],
+owing to their non-zero angular momentum and Ti’s com-
+plex spectrum. To account for these shifts, we consider
+the specific lattice configuration shown in Fig. 1. Here, a
+magnetic field applied in the z direction imposes a linear
+
+3
+λmagic
+α
+αS
+a F
+3
+4
+αS
+a F
+5
+5
+αV
+a F
+3
+4
+αV
+a F
+5
+5
+αT
+a F
+3
+4
+αT
+a F
+5
+5
+1036.6+0.4
+−0.4
+116(10)
+115(3)
+66(12)
+-2(2)
+-470(30)
+4(4)
+154(8)
+887+4
+−4
+122(5)
+121(4)
+159.3(2.6)
+-3(2)
+-104(5)
+5(4)
+-111(3)
+789+5
+−2.2
+129(5)
+127(4)
+127.1(1.4)
+-4(3)
+127(4)
+5(4)
+6.5(1.6)
+781+3
+−7
+130(5)
+128(4)
+126.3(1.3)
+-4(3)
+138(3)
+6(4)
+11.5(1.5)
+TABLE II. Data for the magic wavelengths for the a F
+3
+4 to a F
+5
+5 clock transition.
+Wavelengths are given in units of nm,
+polarizabilities are given in atomic units.
+1100
+1050
+1000
+950
+900
+850
+800
+750
+ [nm]
+0
+200
+400
+S [a.u.]
+a5F5
+a3F4
+FIG. 2. The scalar dynamic polarizability of the mJ = 0 sub-
+levels of the a F
+3
+4 (red) and a F
+5
+5 (blue) states in Ti from 1100
+nm to 750 nm as calculated by the sum-over-states method.
+The angle between the polarization and the B field direction
+is set to 90◦. The locations of magic wavelengths considered
+in the rest of this work are circled.
+Zeeman shift and defines the quantization direction. All
+lattice light is linearly polarized in the transverse x − y
+plane. In this configuration, the clock transition is shifted
+only by the differential scalar and tensor ac Stark effects
+(the vector shift is zero on the mJ = m′
+J = 0 sublevels).
+The sum of the scalar and tensor dynamic polarizabilities
+(αS
+i and αT
+i respectively) on the transition is then given
+by
+∆α = αS
+a F
+5
+5 − αS
+a F
+3
+4 + 1
+2
+�2
+3αT
+a F
+5
+5 − 5
+7αT
+a F
+3
+4
+�
+(1)
+At the identified magic wavelengths, the net transition
+ac Stark shift is zero. For a more detailed description of
+the ac Stark shifts, see the Supplemental Material [18].
+Calculations of the polarizabilities were performed by
+two methods.
+First, the sum-over-states method was
+used to roughly calculate polarizabilities over a wide
+range of frequencies. The 76 transitions with the largest
+contributions to dc polarizability were used in the case of
+the a F
+3
+4 states, while 51 transitions were used in the case
+of the a F
+5
+5 states. Once promising candidates for magic
+wavelengths were found, we performed direct dynamical
+polarizability calculations to identify the location of the
+magic wavelengths more precisely. Direct computations
+for two of the magic wavelengths allow us to predict the
+remaining values accurately. Previously, the direct com-
+putation method was only used for divalent systems such
+as Sr [21, 22], Mg [23], Yb [24, 25], Cd [26], or Tm [27].
+For more complex atoms, the rapidly increasing num-
+ber of relevant configurations makes such a direct compu-
+tation intractable. Here, we apply instead a truncation
+approximation: we order the configurations by weight
+to select the most important ones and then start re-
+moving configurations while checking the accuracy of the
+energies and relevant matrix elements. This procedure
+drastically reduces the number of Slater determinants
+required to maintain numerical accuracy.
+Further de-
+tails on our method are found in the Supplemental Mate-
+rial [18]. We emphasize that our approach is not specific
+to Ti; it should allow for the computation of polarizabil-
+ities, magic wavelengths, and other atomic properties for
+other atoms with a complex electronic structure.
+Using the lattice configuration and magic wavelength
+described above not only eliminates the differential light
+shift, but also protects against the effects of dipole-
+dipole interactions between Ti atoms.
+These effects
+are not present in lattice clocks of Sr, Yb, or Hg as
+those clocks operate on transitions between non-magnetic
+J = 0 states. In contrast, the magnetic moments of the
+proposed Ti clock states are both large, with µa F
+3
+4 =
+5.00 µB and µa F
+5
+5 = 7.05 µB.
+There are three processes associated with the dipole-
+dipole interaction that we consider: dipolar relaxation,
+elastic spin-spin energy shifts, and inelastic spin-spin
+mixing [28]. Dipolar relaxation is the process by which
+Zeeman energy is converted to kinetic energy, depleting
+atoms from the clock states. Such relaxation can be sup-
+pressed for atoms trapped in a deep 3D optical lattice by
+ensuring the bandgap far exceeds the Zeeman energy [29].
+The band energy scale in a lattice is set by the lattice re-
+coil energy Er = h2/(8ma2), where a is the lattice spac-
+ing.
+For a 48Ti atom in the magic-wavelength lattice
+described above, the recoil energy is Er = h × 6.8 kHz.
+3D optical lattice clocks typically use deep lattices to
+suppress tunneling and atom-atom contact interactions.
+As of 2019, the fermionic Sr 3D lattice clock at JILA
+operated at a lattice depth of V0 = 80Er [30]. In deep
+lattices, the gap above the ground band is Eg ≈ 2√V0Er.
+In the case of Ti, a comparable lattice operating at the
+magic wavelength near 781 nm could be achieved by in-
+tersecting six 3.5 W beams with waists of 0.1 mm. This
+would give a lattice depth of V0 ≈ 79Er = h × 540 kHz
+and band gap of Eg ≈ 18Er = h × 120 kHz. Setting the
+Zeeman energy below this band gap requires the ambient
+magnetic field be well below B ∼ Eg/µB = 60 mG.
+The second two processes associated with the dipole-
+dipole interaction are captured in the so-called secular
+Hamiltonian, which is obtained by time-averaging the
+
+4
+dipole-dipole interaction over Larmor precession:
+Hdd = µ0µ2
+B
+8π
+�
+⟨i,j⟩
+gJigJj
+r3
+ij
+�
+1 − 3 cos2 θij
+�
+×
+�
+Jz
+i Jz
+j − 1
+4
+�
+J+
+i J−
+j + J−
+i J+
+j
+��
+,
+(2)
+Here, i and j label two atoms held at different sites of a
+lattice, separated by a distance vector of length rij and
+polar angle θij with respect to the quantization axis. gJi
+is the Land´e g-factor of the atom at lattice site i.
+The elastic spin-spin energy shift corresponds to the
+Jz
+i Jz
+j term in the secular Hamiltonian. In theory, this
+term generates shifts to the transition frequency between
+atomic states with non-zero angular momenta. However,
+for a clock transition between mJ = m′
+J = 0 magnetic
+sublevels, the shift is zero and can be ignored.
+The final process is the spin-mixing interaction, which
+corresponds to the J+
+i J−
+j + J−
+i J+
+j
+term in the Hamil-
+tonian.
+This term couples atoms in an initial two-
+body state |m(1)
+J
+= 0, m(2)
+J
+= 0⟩ to final states |m(1)
+J
+=
+±n, m(2)
+J
+= ∓n⟩, n ∈ {1, . . . , J}. If not controlled, this
+would lead to rapid loss of population from the mJ = 0
+clock states.
+The maximal strength of the coupling is
+ℏΩSM = µ0µ2
+Bg2
+JJ(J+1)
+√
+2/16π(λ/2)3. In a λ = 781 nm
+optical lattice, this gives spin mixing strengths of h × 2.4
+Hz (4.6 Hz) within the lower (upper) clock state manifold.
+Spin mixing between atoms in the upper and lower clock
+states is energetically suppressed because of the signifi-
+cant differential Zeeman splitting. For a 30 mG magnetic
+field, the splitting between the |m(1)
+J
+= 0, m′(2)
+J
+= 0⟩ and
+|m(1)
+J
+= ±1, m′(2)
+J
+= ∓1⟩ states is h × 6.7 kHz.
+In the case where both atoms occupy either the up-
+per or lower clock state, spin mixing is suppressed by the
+tensor ac Stark shift imparted by the optical lattice light.
+The tensor light shift creates an energy splitting between
+the |m(1)
+J
+= 0, m(2)
+J
+= 0⟩ and |m(1)
+J
+= ±n, m(2)
+J
+= ∓n⟩
+two-atom states.
+Using the same optical lattice con-
+figuration described above, the splitting between the
+|m(1)
+J
+= 0, m(2)
+J
+= 0⟩ and |m(1)
+J
+= ±1, m(2)
+J
+= ∓1⟩ states
+is ∆Etens = h × 4(2) kHz (h × 4.8(6) kHz) within the
+lower (upper) clock state manifold. Since the differen-
+tial Zeeman splitting and ∆Etens are much larger than
+ℏΩSM, spin mixing is highly suppressed.
+In this regime, spin mixing enters as a second-order
+perturbative effect. The |m(1)
+J
+= 0, m(2)
+J
+= 0⟩ two-atom
+states in both the lower and upper clock manifolds are
+weakly coupled to the corresponding |m(1)
+J
+= ±1, m(2)
+J
+=
+∓1⟩ states by ΩSM. Both clock states experience an en-
+ergy shift on the order of ∼ Ω2
+SM/∆Etens. The difference
+between the shifts leads to a shift of the clock frequency,
+while the sum of the shifts leads to decoherence between
+the clock states. For two atoms, the shift is ∼ 3 mHz
+and the rate of decoherence is ∼ 6 mHz. For more dis-
+cussion of the dipole-dipole interaction, see the Supple-
+mental Materials [18].
+One complication in our scheme of using tensor light
+shifts to combat magnetic dipole-dipole interactions is
+that deviations from the lattice-light polarization shown
+in Fig. 1 will introduce clock frequency shifts. Consid-
+ering the example parameters from above, a 0.5◦ tilt of
+the linear polarization away from the desired orientation
+would introduce a ∼ 4 Hz overall shift in the clock transi-
+tion frequency, and a much smaller differential shift spa-
+tially across the lattice owing to variation in the light in-
+tensity of the Gaussian-focused beams. Standard meth-
+ods for reducing and calibrating this residual shift, in-
+cluding measuring the variation of the clock frequency
+with lattice-light intensity, should allow the systematic
+uncertainty to be reduced to an acceptable level [30, 31].
+Additional terms in the light shift, such as the hyper-
+polarizability and the M1 and E2 polarizabilities would
+also need to be taken into account, but their effects are
+small (below 10−18 levels in Sr [32–34]), and their con-
+sideration is beyond the scope of this paper.
+Another significant systematic uncertainty in optical
+clocks is the blackbody radiation (BBR) shift, which has
+been the subject of significant past investigation [21, 35].
+We model the BBR shift for the Ti clock line as:
+∆BBR = −κ
+�
+α0
+a F
+5
+5 − α0
+a F
+3
+4
+� � T
+300
+�4
+(1 + η)
+(3)
+where κ = 1
+2(831.9[V/m])2 is a constant of proportional-
+ity, α0
+i is the dc scalar polarizability of the i state of Ti,
+T is the thermal background temperature measured in
+K, and η is a small dynamical correction omitted in the
+present work. The same CI+all-order approach is used to
+compute dc and dynamic polarizabilities. In this case, we
+find that α0
+a F
+5
+5 = 128.53 a.u. and α0
+a F
+3
+4 = 100.39 a.u.,
+which leads to ∆BBR = −0.24 Hz at T = 300 K. This
+value is approximately an order of magnitude lower than
+that in Sr, where the BBR shift is known to be -2.2789
+Hz [36].
+The final systematic uncertainty that we consider is the
+quadratic Zeeman shift (QZS). For the 46,48,50Ti isotopes,
+the effect will be small since it will arise only from the
+mixing of neighboring fine structure states, whereas in
+atoms with nonzero nuclear spin, a stronger QZS arises
+from mixing of hyperfine states. For the states in the
+Ti clock, the QZS of the mJ = 0 sublevels are ∆(a F
+3
+4)
+QZS
+=
+0.129[Hz/G2]B2 and ∆(a F
+5
+5)
+QZS
+= 0.434[Hz/G2]B2, and the
+QZS on the transition is thus ∆QZS = 0.305[Hz/G2]B2.
+Given that a Ti clock must operate at a magnetic field
+well below 60 mG to suppress dipolar relaxation, the QZS
+of the clock transition will be below 1 mHz. This is ap-
+proximately an order of magnitude lower than the QZS
+that is present in Sr optical lattice clocks, of almost 10
+mHz [30, 31].
+Altogether, we have shown that laser-cooled Ti is an
+attractive choice for realizing a telecommunications-band
+optical atomic clock.
+Operating Ti clocks on several
+of the available telecommunications-band optical tran-
+sitions would allow for clock comparisons as a powerful
+
+5
+method for identifying and reducing systematic correc-
+tions. We have advanced atomic structure calculations
+to determine critical properties of such clocks, including
+identifying magic wavelengths for optical trapping, esti-
+mating clock transition widths and line strengths, and
+determining that the BBR shift for Ti clock transitions
+is an order of magnitude smaller than the shift that dom-
+inates current Sr-based clock systematics
+[30, 31]. We
+also describe potential effects of, and mitigation mea-
+sures against, magnetic dipole-dipole interactions. These
+measures are relevant to other potential applications of
+dipole-interacting atoms and molecules for precision mea-
+surement.
+ACKNOWLEDGMENTS
+We thank Mikhail Kozlov, Andrey Bondarev, and Ilya
+Tupitsyn for helpful discussions of polarizability com-
+putations.
+This work is supported by a collaboration
+between the US DOE and other Agencies. This mate-
+rial is based upon work supported by the U.S. Depart-
+ment of Energy, Office of Science, National Quantum In-
+formation Science Research Centers, Quantum Systems
+Accelerator.
+Additional support is acknowledged from
+the ONR (Grant Nos. N00014-20-1-2513 and N00014-
+22-1-2280), NSF (PHY-2012068 and the QLCI program
+through Grant No. OMA-2016245), and European Re-
+search Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation program (Grant No.
+856415). This research was supported in part through
+the use of University of Delaware HPC Caviness and
+DARWIN computing systems: DARWIN - A Resource
+for Computational and Data-intensive Research at the
+University of Delaware and in the Delaware Region,
+Rudolf Eigenmann, Benjamin E. Bagozzi, Arthi Jayara-
+man, William Totten, and Cathy H. Wu, University of
+Delaware, 2021 [37].
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+
+Supplemental Material for
+“Optical Telecommunications-Band Clock based on Neutral Titanium Atoms”
+Scott Eustice,1, 2 Dmytro Filin,3 Jackson Schrott,1, 2 Sergey Porsev,3 Charles
+Cheung,3 Diego Novoa,1, 2 Dan M. Stamper-Kurn,1, 2, 4 and Marianna S. Safronova3, 5
+1Department of Physics, University of California, Berkeley, CA 94720
+2Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720
+3Department of Physics and Astronomy, University of Delaware, Newark, DE 19716
+4Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
+5Joint Quantum Institute, National Institute of Standards and Technology
+and the University of Maryland, College Park, Maryland 20742
+(Dated: February 1, 2023)
+S1.
+CALCULATION OF E1, M1, AND E2
+TRANSITION RATES IN TI I
+A.
+Theoretical framework: CI+all-order method
+We use the CI+all-order method [S1] that combines
+linearized coupled cluster and configuration interaction
+(CI) approaches. In this method, the electrons are sepa-
+rated into the 1s22s22p63s23p6 core and four remaining
+valence electrons. First, the coupled cluster method is
+used to construct the effective Hamiltonian Heff that ac-
+counts for core and core-valence correlations and can be
+constructed using second-order many-body perturbation
+theory in the CI + MBPT method or the coupled clus-
+ter approach (CI + all-order method). The CI method
+is used to correlate the remaining four valence electrons
+using this effective Hamiltonian rather than the usual
+bare Hamiltonian. This procedure effectively includes all
+types of correlation effects in the core and valence spaces.
+The CI wave function is constructed as a linear com-
+bination of all distinct states of a specified angular mo-
+mentum J and parity,
+ψJ =
+�
+i
+ciΦi,
+(S1)
+where {Φi} is the set of Slater determinants generated
+by exciting electrons from the reference configuration to
+higher orbitals. The many-electron Schr¨odinger equation
+can be written as
+HeffΨ = EΨ,
+(S2)
+where the effective Hamiltonian has the form
+Heff = HCI + Σ.
+(S3)
+Here, HCI is the CI Hamiltonian described by the equa-
+tion
+HCI = Ecore +
+�
+i>Ncore
+hi,CI +
+�
+j>i>Mcore
+Vij,
+(S4)
+where Ecore is the energy of the frozen core, Ncore is
+the number of core electrons, hi,CI represents the kinetic
+energy of the valence electrons and their interaction with
+the central field and Vij accounts for the valence–valence
+correlations.
+The core-valence correlation potential,
+Σ = Σ1 + Σ2,
+is obtained from the all-order method. Here, Σ1 and Σ2
+are the one- and two-electron parts of the core–valence
+correlation potential, respectively.
+After Eq. (S2) is
+solved using the CI technique and the wave functions are
+obtained, they are used to calculate matrix elements of
+the electric-dipole, magnetic-dipole, electric-quadrupole,
+and other one-electron operators.
+B.
+Energy level calculation
+When applying the CI+all-order method to atomic Ti,
+we used a a V N−4 potential of the 1s22s22p63s23p6 frozen
+core. We solve Dirac-Hartree-Fock equations in this po-
+tential to generate 3d, 4s, 4p, 5s, 4d, 5p, and 4f orbitals.
+All other orbitals are constructed in a spherical cavity of
+40 a.u. using B-splines.
+The set of CI configurations has to be constructed
+separately for even and odd states. We carry out sev-
+eral calculations with increasing number of configura-
+tions to ensure convergence of the CI with the num-
+ber of included configurations. For even states, we find
+it sufficient to make all possible single and double ex-
+citations to a 20spd18f16g basis starting from 4s23d2,
+4s3d25s, 4s23d4d, 3d24p2, 3d25s2, 4s3d3, 4s3d24d, 3d35s,
+and 3d25s4d configurations. We verified that a subset of
+triple excitations give a negligible contribution. For odd
+states, the CI configuration space is sufficiently saturated
+by the single and double excitations to the same large ba-
+sis from the 4s3d24p, 3d34p, 4s3d25p, 3d24p5s, 4s3d4p4d,
+3d25s5p, and 3d24p4d configurations.
+We have carried out calculation of energies for 85 levels
+and compared them with experiment. Most of the the-
+oretical energies differ from experimental values by only
+0.1-2.5%. We present results only for the lines and lev-
+els that can be used to pump titanium optically into the
+metastable state to perform laser cooling and to drive the
+arXiv:2301.13363v1  [physics.atom-ph]  31 Jan 2023
+
+2
+TABLE S1. Comparison of theoretical even energy levels (in
+cm−1) with experiment [S2].
+Configuration
+Term
+Expt
+Theory
+Diff
+Diff %
+3d24s2
+a F
+3
+2
+0
+0
+0
+a F
+3
+3
+170
+177
+7
+4.1%
+a F
+3
+4
+387
+396
+9
+2.3%
+3d3(4F)4s
+a F
+5
+1
+6557
+6374
+-183
+-2.8%
+a F
+5
+2
+6599
+6416
+-183
+-2.8%
+a F
+5
+3
+6661
+6477
+-184
+-2.8%
+a F
+5
+4
+6743
+6557
+-186
+-2.8%
+a F
+5
+5
+6843
+6652
+-191
+-2.8%
+3d3(2G)4s
+a G
+3
+5
+15220
+15497
+276
+1.8%
+3d3(2H)4s
+a H
+3
+5
+18141
+18450
+308
+1.7%
+a H
+3
+6
+18193
+18498
+306
+1.7%
+3d3(2H)4s
+a H
+1
+5
+20796
+21171
+375
+1.8%
+TABLE S2.
+Comparison of theoretical odd energy levels (in
+cm−1) with experiment [S2].
+Configuration
+Term
+Expt
+Theory
+Diff
+Diff %
+3d2(3F)4s4p(3P o)
+z G
+5
+o
+6
+16459
+16454
+-5
+0.0%
+3d2(3F)4s4p(3P o)
+z S
+1
+o
+0
+24174
+3d2(3P)4s4p(3P o)
+D
+5
+o
+4
+25927
+26081
+154
+0.6%
+3d3(4F)4p
+y G
+5
+o
+6
+26911
+26982
+71
+0.3%
+relevant clock transitions. The selected energy levels in
+cm−1 are listed in Tables S1 and S2.
+We compute the expectation values ⟨L2⟩ and ⟨S2⟩,
+where L and S are the total electron orbital and spin an-
+gular momentum operators, to obtain approximate quan-
+tum numbers L and S, where ⟨L2⟩ = L(L + 1) and
+⟨S2⟩ = S(S+1), which allowed us to unambiguously iden-
+tify all terms in Tables S1 and S2. As a result, we iden-
+tify a level, 3d24s4p 1So
+0, not listed in the NIST database.
+This level is included in Table S2.
+C.
+Optical clock transitions
+We study forbidden transitions between the a F
+3
+and
+a F
+5
+terms to identify the most suitable clock transition.
+The main text summarizes the properties of the clock
+transitions. The total clock transition linewidth accounts
+not only for spontaneous decay on the clock transition it-
+self, but also for decay of the upper and lower levels of the
+clock transition to other states, leading to an overall clock
+transition linewidth that is larger than the spontaneous
+decay rate on the clock transition alone. In Table S3,
+we list the contributions that determine the linewidth of
+the a3F4 → a5F5 clock transition.
+We note that the
+clock transition linewidth is dominated by M1 decays
+of both the lower and upper states to other fine struc-
+ture states within their respective manifolds. The same
+calculation was performed to calculate the linewidth of
+all other clock transitions. The random-phase approx-
+imation (RPA) corrections are included to the effective
+electric quadupole and magnetic dipole operators, see,
+for example Ref. [S3]. Such effective operators account
+for the core-valence correlations in analogy with the effec-
+tive Hamiltonian Heff discussed above. The uncertainties
+in the values of matrix elements were estimated as dif-
+ference between values obtained using CI+all-order and
+CI+MBPT methods.
+D.
+Laser cooling transitions
+Ref. [S4] identified two candidate transitions on which
+Ti may be laser cooled. To support ongoing experimen-
+tal efforts to realize laser cooling of Ti, we character-
+ized these two transitions theoretically using our atomic-
+structure calculations described herein. Specifically, we
+calculated the strengths of the two electric-dipole laser
+cooling transitions, and also, critically to experimental ef-
+forts, determined the small leakage rate (branching ratio)
+out of the laser cooling transitions. Results for two of the
+cooling transitions, 3d3( F
+4
+)4p y G
+5
+◦
+6 - 3d3( F
+4
+)4s a F
+5
+5 at
+λ = 498 nm and 3d3( F
+3
+)4s4p( P
+3
+o) z G
+5
+o
+6 - 3d3( F
+4
+)4s
+a F
+5
+5 at λ = 1040 nm are listed in Table S4).
+In Ref. [S4] it was noted that while there do exist other
+even parity states to which a Ti atom in the excited state
+of the laser cooling transition can decay, such scattering
+would be strongly suppressed because the transitions are
+spin forbidden. Our calculations confirm this expecta-
+tion.
+Indeed, we find the branching to those states is
+exceptionally low, at the 10−6 level for the 498 nm tran-
+sition and at the 3·10−7 level for the 1040 nm transition.
+These are low enough branching ratios to enable the typ-
+ical tools of modern ultracold atomic physics, including
+both Doppler and sub-Doppler cooling techniques and
+single-atom fluorescenence detection in quantum-gas mi-
+croscopes or optical tweezers, without the need for addi-
+tional repumping lasers.
+To obtain transition rates and branching ratios for the
+cooling transitions (see Table S4) we use the electric-
+dipole reduced matrix elements calculated with the ef-
+fective electric-dipole operator in the random-phase ap-
+proximation. We also considered other correction to the
+E1 operator beyond RPA: the core-Brueckner (σ), struc-
+tural radiation (SR), two-particle (2P), and normaliza-
+tion (Norm) corrections [S5–S7]. As has been noted for
+the case of Sr [S3], these corrections cannot be omitted
+at the 1% level of accuracy.
+In Table S4 we include transition matrix elements,
+Dtot, obtained taking into account all the corrections
+mentioned above. The uncertainties were estimated by
+taking the difference between the values obtained using
+CI+all-order and CI+MBPT methods. For very small
+matrix elements (marked by the tilde symbol), we pro-
+
+3
+TABLE S3. Results of calculations for the a F
+3
+4 → a F
+5
+5 clock transitions of Ti. Wavelength, λ, (in units of nm), E2 and
+M1 reduced matrix elements (ME) (in a.u. for E2 transitions, in µB for M1 transitions), transition rates (in ×10−7s−1), and
+branch ratio.
+Upper
+Term
+Level
+Lower
+Term
+Level
+λ
+ME
+Tr. rate
+Branching ratio
+3d3(4F)4s
+a F
+5
+5
+6843
+3d24s2
+a F
+3
+3
+170
+1498.6
+E2
+0.0472(7)
+3.00(9)
+0.00838(25)
+a F
+3
+4
+387
+1548.9
+M1
+0.0010(5)
+7(6)
+0.020(17)
+E2
+0.140(4)
+22.4(1.3)
+0.063(4)
+3d3(4F)4s
+a F
+5
+4
+6743
+99794.4
+M1
+3.632(10)
+325.43(18)
+0.909(16)
+E2
+1.748(18)
+3.14(8) × 10−6
+8.78(22) × 10−9
+3d24s2
+a F
+3
+4
+387
+3d24s2
+a F
+3
+3
+170
+46138.
+M1
+2.597(0.026)
+2059(53)
+1
+TABLE S4. Wavelengths (in nm), electric-dipole reduced matrix elements Dtot (in a.u.), transition rates (in s−1), and branching
+ratios for cooling transitions of Ti I. The Dtot values (in a.u.) are calculated with CI+all-order method and include the random-
+phase approximation (RPA), the core-Brueckner (σ), structural radiation (SR), two-particle (2P), and normalization (Norm)
+corrections. For approximate values (indicated by a ∼ symbol), the precise value of the matrix element is highly uncertain,
+and the reported value should be interpreted as correct only within an order of magnitude.
+Upper
+Term
+Level
+Lower
+Term
+Level
+λ
+Dtot
+Tr. rate
+Branch ratio
+3d3(4F)4p
+y G
+5
+o
+6
+26911
+3d3(4F)4s
+a F
+5
+5
+6843
+498
+E1
+7.337(12)
+6.780(23)×107
+0.9999989(5)
+3d3(2G)4s
+a G
+3
+5
+15220
+855
+E1
+∼ 0.007
+∼ 11
+∼ 2 × 10−7
+3d3(2H)4s
+a H
+3
+5
+18141
+1140
+E1
+0.0055(14)
+3.2(1.4)
+4.7(2.0)×10−8
+3d3(2H)4s
+a H
+3
+6
+18193
+1147
+E1
+0.025(4)
+62(19)
+9(3)×10−7
+3d3(2H)4s
+a H
+1
+5
+20796
+1635
+E1
+∼ 0.0004
+∼ 0.005
+∼ 7 × 10−11
+3d2(3F)4s4p(3P o)
+z G
+5
+o
+6
+16459
+3d3(4F)4s
+a F
+5
+5
+6843
+1040
+E1
+0.86(3)
+1.03(8)×105
+0.99999969(2)
+3d3(2G)4s
+a G
+3
+5
+15220
+8076
+E1
+0.01040(10)
+0.0318(25)
+3.1(2)×10−7
+TABLE S5.
+Wavelengths (in nm), calculated and observed transition rates (in s−1×106), and branching ratios for optical
+pumping of Ti. The upper state for all transitions is the 3d2(3P)4s4p(3P o)
+D
+5
+o
+4 level, at 25967 cm−1.
+Lower
+Term
+Level
+λ
+Tr. rate, theory
+Tr. rate, lit
+Branch ratio
+3d24s2
+a F
+3
+3
+170
+388.2
+0.13(6)
+0.28(6)
+0.049
+a F
+3
+4
+387
+391.5
+1.8(8)
+2.11(28)
+0.68
+3d3(4F)4s
+a F
+5
+3
+6661
+519.1
+0.00036(12)
+0.00014
+a F
+5
+4
+6743
+521.3
+0.08(8)
+0.31(16)
+0.030
+a F
+5
+5
+6843
+524.0
+0.4(4)
+0.17
+b F
+3
+4
+11777
+706.7
+0.06(3)
+0.02
+3d3(4P)4s
+a P
+5
+3
+14106
+845.9
+0.129(4)
+0.049
+
+4
+vide approximate values without explicit evaluation of
+errors. For these, the error should be assumed to be at
+the same order of magnitude as the quantity itself.
+E.
+Optical pumping transitions
+Gas phase Ti atoms may be produced with a ther-
+mal source that operates between 1200◦C to 1800◦C. At
+these temperatures, a low population of the Ti atoms
+produced would be in the a F
+5
+5 laser-coolable state. It
+would therefore be necessary to transfer atoms to this
+state from the a F
+3
+ground state manifold via optical
+pumping. Through our calculations, we examined multi-
+ple potential excited states that could be used to trans-
+fer population efficiently to the a F
+5
+5 state. We found the
+3d2( P
+3
+)4s4p( P
+3
+o) y D
+5
+o
+4 state has a significant branching
+ratio to both the a F
+5
+5 state and a F
+3
+states, allowing for
+optical pumping of atoms from the ground states to the
+laser cooling state. The transition wavelengths are also
+amenable to current laser technology.
+S2.
+OPTICAL CLOCK RABI FREQUENCIES
+The optical clock transitions of Ti can be driven by
+either the M1 or E2 multipoles of the electromagnetic
+field, both of which must be accounted for when calculat-
+ing the overall Rabi frequency. In each of the transitions
+we highlighted in the main text (mJ = 0 → m′
+J = 0,
+J = 4 → J′ = 4, 5) only one of the two multipoles is
+allowed.
+Given a reduced M1 matrix element, DM1, for a tran-
+sitions between lower and upper levels J and J′, and
+a magnetic field B of the clock laser, the M1 Rabi fre-
+quency between a lower and upper sublevel mJ and m′
+J
+is given by
+ΩM1 = −DM1
+ℏ
+(−1)J′−m′
+J ˆe∗
+m′
+J−mJ · B
+×
+�
+J
+1
+J′
+mJ m′
+J − mJ −m′
+J
+�
+.
+(S5)
+For an electric quadrupole transition with reduced ma-
+trix element DE2 driven by a plane-wave clock laser field
+E with polarization ε and propagation wavevector k, the
+Rabi frequency is
+ΩE2 = iDE2
+2ℏ (−1)J′−m′
+J
+�
+J
+2
+J′
+mJ m′
+J − mJ −m′
+J
+�
+×
+�
+i,j
+Mij(m′
+J − mJ)kiεj.
+(S6)
+Here, Mij(q) is a geometric factor given by
+Mij(q) = (−1)q√
+5
+�
+q1,q2
+(ˆei · ˆe∗
+q1)(ˆej · ˆe∗
+q1)
+�
+1
+1
+2
+q1 q2 −q
+�
+,
+(S7)
+where ˆei and ˆeq are the cartesian and spherical basis
+vectors respectively.
+S3.
+DYNAMICAL POLARIZABILITY OF TI I
+In order to find the magic wavelengths of the clock
+transitions, it is necessary to calculate the polarizabil-
+ity of Ti in both the a F
+3
+4 and a F
+5
+5 states over a wide
+range of frequencies. Performing this calculation directly
+remains a computational challenge for complex atoms
+like Ti. Direct calculation requires the inversion of huge
+matrices — 350,000 × 350,000 in our case — which is
+computationally intractable. Instead, we use an iterative
+approach included in the pCI Code Package [S8] imple-
+menting the CI+all-order technique to calculate the po-
+larizability of the considered states of Ti. This approach
+allows us to get accurate results using the inversion of
+smaller matrices (15,000 × 15,000), making the task fea-
+sible. Unfortunately, this method does not work at all
+frequencies, owing to the potential divergence of the it-
+erative process. We successfully obtained static polariz-
+abilities for both 3d24s2 a F
+3
+4 and 3d3(4F)4s a F
+5
+5 levels,
+but for dynamic polarizabilities the calculation diverged
+for wavelengths shorter than 750 nm (900 nm) for the
+a F
+3
+4 (a F
+5
+5) level. To overcome this, we used a combi-
+nation of the CI+all-order technique and the sum-over-
+states method to calculate the scalar, vector and tensor
+polarizability from 1100 nm to 400 nm.
+The sum-over-states method involves using only bound
+states of an atom, and there is always some inaccuracy
+due to missing contributions from continuum states and
+bound states not included in the calculation.
+For the
+best accuracy, it is advisable to use as many states as
+possible in the sum.
+However, there are always limits
+on the accuracy of calculations of highly excited states.
+To balance these trade-offs, we use in the sum-over-state
+method for the lower lying states that contribute the ma-
+jority of the dc polarizability but still can be properly
+calculated. For this purpose, we used 73 states to obtain
+the polarizability of the 3d24s2 a F
+3
+4 level and 49 states
+for the polarizability of the 3d3(4F)4s a F
+5
+5 level. The
+most important contributions to both polarizabilities are
+shown in Tables S6, S7. The polarizability is generally
+divided into three terms: αS - scalar, αV -vector, and
+αT - tensor polarizabilities. They are represented for an
+arbitrary state i as follows [S9, S10]:
+αS
+i (ω) =
+2
+3(2Ji + 1)
+�
+n
+(En − Ei)|⟨n||D||i⟩|2
+(En − Ei)2 − ω2
+(S8)
+αV
+i (ω) = C1
+�
+n
+(−1)Jn+Ji
+�
+1
+1
+1
+Ji Ji Jn
+�
+ω|⟨n||D||i⟩|2
+(En − Ei)2 − ω2
+(S9)
+
+5
+TABLE S6. Contributions to the static electric-dipole polarizability α0 with the appropriate reduced matrix elements D (in
+a.u.) of the 3d24s2 a F
+5
+5 level. For comparison, the experimental and theoretical energy levels (in cm−1) are shown.
+Configuration
+Term
+Exp. Level
+Theor. Level
+Diff %
+D
+Contrib. to α0
+3d2(3F)4s4p(3P o)
+z G
+5
+o
+6
+16459
+16454
+0.0%
+0.86(3)
+1.00(8)
+z F
+5
+o
+5
+17215
+17185
+-0.2%
+2.48(6)
+7.7(4)
+z D
+5
+o
+4
+18695
+18769
+0.4%
+2.08(3)
+4.77(16)
+3d3(4F)4p
+y G
+5
+o
+5
+26773
+26850
+0.3%
+2.299(5)
+3.48(2)
+y G
+5
+o
+6
+26911
+26982
+0.3%
+7.837(12)
+40.16(12)
+y F
+5
+o
+4
+28788
+29061
+1.0%
+2.410(7)
+3.45(2)
+y F
+5
+o
+5
+28896
+29164
+0.9%
+7.251(6)
+31.07(5)
+x D
+5
+o
+4
+30060
+30474
+1.4%
+6.723(12)
+25.24(12)
+Other
+11.7
+Total
+128.5(2.0)
+TABLE S7. Contributions to the static electric-dipole polarizability α0 with the appropriate reduced matrix elements D (in
+a.u.) of the 3d24s2 a F
+3
+4 level. For comparison, the experimental and theoretical energy levels (in cm−1) are shown.
+Configuration
+Term
+Exp. Level
+Theor. Level
+Diff %
+D
+Contrib. to α0
+3d2(3F)4s4p(3P o)
+z F
+3
+o
+4
+19574
+19632
+0.3%
+1.682(13)
+2.39(4)
+z D
+3
+o
+3
+20126
+20115
+-0.1%
+1.467(9)
+1.774(22)
+z G
+3
+o
+5
+21740
+22025
+1.3%
+1.297(4)
+1.263(8)
+z G
+1
+o
+4
+24695
+25163
+1.9%
+1.1(4)
+0.8(6)
+3d2(3F)4s4p(1P o)
+y F
+3
+o
+3
+25227
+25351
+0.5%
+1.479(13)
+1.423(25)
+y F
+3
+o
+4
+25388
+25531
+0.6%
+3.9(3)
+9.8(1.8)
+3d3(4F)4p
+y D
+3
+o
+3
+25644
+25591
+-0.2%
+3.12(17)
+6.3(7)
+3d2(1D)4s4p(3P o)
+x F
+3
+o
+4
+27026
+27325
+1.1%
+3.36(11)
+6.8(5)
+3d2(3F)4s4p(1P o)
+y G
+3
+o
+5
+27750
+27846
+0.35%
+4.65(25)
+12.7(1.4)
+w D
+3
+o
+3
+29912
+30145
+0.8%
+2.983(17)
+4.86(5)
+3d2(1G)4s4p(3P o)
+x G
+3
+o
+5
+30039
+30349
+1.0%
+4.416(10)
+10.53(5)
+3d3(4F)4p
+w G
+3
+o
+5
+31629
+32053
+1.3%
+4.61(20)
+10.9(1.0)
+3d2(1G)4s4p(3P o)
+v F
+3
+o
+4
+34205
+34742
+1.6%
+3.60(14)
+6.1(5)
+3d3(2D2)4p
+u F
+3
+o
+3
+37744
+38871
+3.0%
+2.74(13)
+3.2(3)
+3d2(3P)4s4p(1P o)
+u D
+3
+o
+3
+38159
+38909
+2.0%
+1.74(5)
+1.28(7)
+3d3(2G)4p
+t F
+3
+o
+4
+38671
+39560
+2.3%
+2.6(4)
+2.9(8)
+3d3(2D2)4p
+s D
+3
+o
+3
+39715
+40515
+2.0%
+1.5(4)
+0.9(5)
+3d24s(4F)5p
+o D
+3
+o
+3
+44234
+45270
+2.3%
+2.11(8)
+1.61(12)
+Other
+14.9
+Total
+100.4(1.8)
+
+6
+αT
+i (ω) = C2
+�
+n
+(−1)Jn+Ji
+�
+1
+1
+2
+Ji Ji Jn
+�
+(En − Ei)|⟨n||D||i⟩|2
+(En − Ei)2 − ω2
+,
+(S10)
+where
+C1 = −2
+�
+6Ji
+(Ji + 1)(2Ji + 1)
+and
+C2 = 4
+�
+5Ji(2Ji − 1)
+6(Ji + 1)(2Ji + 1)(2Ji + 3).
+The index n refers to the states in the sum-over-states
+that contribute to the polarizability of the state i; En,i is
+the energy and Jn,i is the total angular momentum of the
+state; ⟨n||D||i⟩ is the reduced matrix element between the
+two states, and ω is the frequency of the external electric
+field. The total dynamic polarizability can be expressed
+as follows:
+αi(ω) = αS
+i (ω) + ε cos(θk)mJi
+2Ji
+αV
+i (ω)+
++
+�3 cos2 θp − 1
+2
+� 3m2
+Ji − Ji(Ji + 1)
+Ji(2Ji − 1)
+αT
+i (ω),
+(S11)
+where ε is the ellipticity of the polarization, θk is the
+angle between the direction of propagation of the light
+and the quantization axis, θp is the angle between the
+polarization of the light and the quantization axis, and
+mJi is a magnetic quantum number.
+To obtain an accurate value of dynamic polarizabil-
+ity with the sum-over-states method, one has to estimate
+the residual contribution to the polarizability from states
+that are not included in summation. We made this esti-
+mate using the fact that |ω/(En − Ei)| ≪ 1 for all such
+residual states in the range of wavelengths that we are
+considering. Indeed, using this ratio as a small parame-
+ter and expanding to the lowest non-zero order in, Eq. S8
+can be simplified to
+αS
+i (ω) = αSN
+i
+(ω) + αSres
+i
+(ω)
+(S12)
+where
+αSN
+i
+(ω) =
+2
+3(2Ji + 1)
+N
+�
+n=1
+(En − Ei)|⟨n|D|i⟩|2
+(En − Ei)2 − ω2
+(S13)
+and where N is the number of states used in direct sum-
+mation. Moreover, we can write
+αSres
+i
+(ω) = A0 + B0 ω2
+(S14)
+where A0 and B0 are expansion constants for the long
+wavelength scalar polarizability.
+The same simplifica-
+tion works for αV
+i (ω) and αT
+i (ω), by writing αVN
+i
+(ω) and
+αTN
+i
+(ω) as a summation as in S13. This yields the ex-
+pansion of the long wavelength residuals:
+αVres
+i
+(ω) = B1ω
+αTres
+i
+(ω) = A2 + B2 ω2
+(S15)
+where B1, A2, and B2 are additional expansion constants.
+Using the results of the sum-over-states method, we
+identified the four candidate magic wavelengths discussed
+in the main text. The atoms were assumed to have mag-
+netic quantum numbers mJ = 0 and the optical field
+parameters were ε = 0 and θp = θk = 90◦. However, the
+more accurate iterative approach used by the pCI code
+package was found to not converge at all of the candidate
+wavelengths. Using the fact that convergence in the iter-
+ative polarizability can be achieved at ω = 0.043989 a.u.
+(λ = 1035.8 nm) and for all smaller values of ω down
+to ω = 0, we subtracted the values of αi(0.043989) and
+αi(0) obtained from the sum-over-states method from
+the corresponding results computed with the pCI code.
+In this way, we found the residuals αSres
+i
+(0), αVres
+i
+(0),
+αTres
+i
+(0) and αSres
+i
+(0.043989 a.u.), αVres
+i
+(0.043989 a.u.),
+αTres
+i
+(0.043989 a.u.). This allowed us to determine the
+parameters A0,2 and B0,1,2 in Eqs. S14, S15 and thus
+extend the accuracy of the direct iterative polarizability
+calculation to frequencies where the iterative calculation
+fails to converge.
+The uncertainty on the polarizabilities were obtained
+by comparing the calculations of the more accurate
+CI+all-order technique with CI+MBPT at the points
+where the direct calculation converged. The difference
+between the values of the appropriate reduced matrix el-
+ements |⟨n||D||i⟩|CI+all−order and |⟨n||D||i⟩|CI+MBP T is
+an additional possible inaccuracy of the method and thus
+is used to estimate the uncertainties of the polarizabili-
+ties. Furthermore, possible errors in the parameters A0,2
+and B0,1,2 were included in the final uncertainties as the
+difference between the correspondent parameters calcu-
+lated with the CI+all-order and CI+MBPT methods.
+S4.
+DIPOLE-DIPOLE INTERACTIONS
+As discussed in the main text, dipole-dipole interac-
+tions underlie several effects that impact the operation
+of a Ti atomic clock. One can distinguish between inter-
+actions that are inelastic or elastic in the motional de-
+grees of freedom. Inelastic interactions can be described
+as spin relaxation, a dipole-mediated conversion of inter-
+nal Zeeman energy into external motional energy. For
+example, in an optical lattice spin relaxation may couple
+atoms from lower to higher bands of the lattice. As dis-
+cussed in the main text, inelastic motional interactions
+can be suppressed by trapping atoms in a lattice with a
+sufficiently large band gap [S11].
+One may then focus on elastic motional interactions.
+We describe these interactions purely in the spin sec-
+tor using the secular Hamiltonian, which accounts for
+
+7
+cycle-averaging over the Larmor precession of the atomic
+spins [S12]. This secular Hamiltonian, whose form is pre-
+sented also in the main text, is given for two atoms as
+Hdd = µ0µ2
+B
+4π
+gJ1gJ2
+r3
+12
+�
+1 − 3 cos2 θ12
+�
+×
+�
+Jz
+1 Jz
+2 − 1
+4
+�
+J+
+1 J−
+2 + J−
+1 J+
+2
+��
+(S16)
+with 1 and 2 labelling the two atoms, and r12 and θ12
+describing their position difference vector and the angle
+it makes with the quantization axis.
+One can consider separately two effects of this secu-
+lar Hamiltonian. The first is a spin-elastic interaction,
+which is diagonal in the separable basis of magnetic sub-
+level states. This shift is zero on the mJ = 0 magnetic
+sublevels and therefore does not affect a clock based on
+an mJ = m′
+J = 0 transition.
+The second effect is the spin mixing interaction, which
+is off-diagonal in the separable magnetic sublevel basis.
+This interaction conserves the total magnetic quantum
+number of the two atoms, m(tot)
+J
+= m(1)
+J
++ m(2)
+J , but
+changes the magnetic quantum numbers of the individual
+atoms. In the absence of dipolar interactions, the entire
+manifold of two-atom states with identical m(tot)
+J
+is de-
+generate for atoms with equal Zeeman splittings. Dipo-
+lar interactions can then generate significant mixing and
+energy shifts within this manifold, leading to imprecise
+measurement of the clock transition.
+While several methods have been developed in NMR to
+control dipolar spin mixing, e.g. multiple pulse sequences
+[S13] and magic angle spinning [S14], these techniques are
+not needed to suppress spin mixing in a Ti clock system.
+As described in the main text, spin mixing within the
+clock states is mitigated by the tensor light shift imposed
+by the optical lattice beams.
+From Eq. S11, the ten-
+sor light shift includes a part proportional to m2
+J, which
+splits the degeneracy of the aforementioned states and
+energetically suppresses the spin mixing process.
+To study the suppression of spin mixing, we simulate
+the secular dipole-dipole interaction of two nearest neigh-
+bor atoms in an optical lattice with and without account-
+ing for the lattice-induced ac Stark shift. Figure S1 shows
+a schematic of the system under consideration. We no-
+tate the single atom states by |ξ(i)
+mJ⟩ where ξ ∈ {g, e}
+refers to the upper or lower clock states, the superscript
+denotes the ithatom, and the mJ subscript is the mag-
+netic sublevel. We use g to refer to the lower a3F4 clock
+state, and e to refer to the upper a5F5 state. The two
+atom system consists of [(2Jg + 1) + (2Je + 1)]2 = 400
+states.
+We simulate a simple Ramsey interferometry sequence
+which includes the dynamics of Hdd, the Zeeman Hamil-
+tonian HZ, and, optionally, the ac Stark Hamiltonian
+Hac. We initialize two atoms in the |g(1)
+0 ; g(2)
+0 ⟩ state, ap-
+ply a π/2 pulse to each atom on the g0 → e0 clock tran-
+sition, allow the two-atom state to evolve for a time T
+-1 0 1
+-1 0 1
+-1 0 1
+-1 0 1
+M1
+M1
+Hdd
+Hdd
+J = 5
+J = 4
+|
+(1)
+mJ
+|
+(2)
+mJ
+e
+g
+FIG. S1. The secular dipole-dipole interaction of two atoms in
+the a3F4 (g) and a5F5 (e) clock states. The g and e states have
+angular momentum quantum numbers J = 4 and J = 5, giv-
+ing rise to 2J +1 magnetic sublevels each. For brevity, the fig-
+ure shows only three of these mJ sublevels in each state, split
+by Zeeman shifts. The clock drive ΩM1 couples the |gmJ =0⟩
+and |emJ =0⟩ states, and Hdd couples the subspaces of the two
+atoms.
+0.0
+0.1
+0.2
+T [s]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+C(T)
+Hac on
+Hac off
+40
+80
+2 /T
+0
+2 /T
+ [s
+1]
+0.00
+0.25
+0.50
+0.75
+1.00
+Pe(T = 0.2)
+0.25
+FIG. S2. (left) Ramsey contrast of the clock transition as a
+function of the free evolution time T. The contrast, shown in
+violet (green), dies quickly (slowly) when the ac Stark shift,
+Hac, is excluded (included) from the simulation.
+Note the
+split in the T axis and change in scale at 0.25 s. (right) The
+resulting Ramsey fringes for T = 0.2 s (indicated by circles
+in the left plot). We plot the probability an atom is found
+in state |e0⟩ as a function of the detuning δ from the clock
+transition..
+under HZ, Hdd, and (optionally) Hac; apply a second π/2
+pulse to each atom; and then determine the probability
+of an atom being in the e0 state. We vary the detun-
+ing of the clock drive frequency, δ, to generate a Ramsey
+interferometry fringe.
+From this fringe, we obtain the
+contrast C(T) = (max Pe(T, δ) − min Pe(T, δ)).
+Figure S2 shows the results of the simulation.
+The
+left panel demonstrates the decay of the Ramsey contrast
+with and without Hac included. We break the axis at 0.25
+s to illustrate that the tensor ac Stark shift extends the
+decay time of the contrast significantly. Because we only
+simulate unitary dynamics of two neighboring atoms, re-
+
+8
+S(|g(1)
+mJ ; g(2)
+mJ )
+S(|g(1)
+0 ; e(2)
+0
+)
+S(|e(1)
+mJ ; e(2)
+mJ )
+Hac off
+Hac on
+|g(1)
+0 ; g(2)
+0
+S(|g(1)
+0 ; e(2)
+0
+)
+|e(1)
+0 ; e(2)
+0
+S(|g(1)
+1 ; g(2)
+1 )
+S(|e(1)
+1 ; e(2)
+1 )
+g
+SM
+e
+SM
+FIG. S3.
+(left) The 2-atom level structure with and with-
+out the ac Stark shifts applied (Hac).
+Only states that
+are symmetric under particle exchange are shown, as these
+are the only states involved in the dynamics.
+The states
+S(|g(1)
+mJ ̸=0; e(2)
+−mJ ⟩) are omitted because they are split from the
+S(|g(1)
+0 ; e(2)
+0 ⟩) state by the differential Zeeman shift. The dot-
+ted lines show that the degeneracy of the mtot
+J
+= 0 sublevels
+is lifted when Hac is turned on. The five boxed levels include
+the clock states and the states most strongly coupled to them
+by Hdd. (right) The simplified picture, including only the lev-
+els boxed in the left panel. Hdd couples the states |ξ(1)
+0 ; ξ(2)
+0 ⟩
+and S(|ξ(1)
+mJ ; ξ(2)
+−mJ ⟩) with rate Ωξ
+SM.
+vivals of the contrast are observed in the simulation be-
+yond the times plotted in Figure S2, but these would not
+occur in a true many-body situation as the coherence
+would spread between many particles and be lost. The
+right panel shows the resulting Ramsey fringes taken at
+the time highlighted on the left side of the figure. When
+no optical lattice is applied, the coherence on the clock
+transition quickly vanishes, leading to a loss of the Ram-
+sey signal. The fringe survives for ∼ 85 s when the lattice
+beams are on.
+Figure S3 gives a simplified picture of the level struc-
+ture at play.
+S(|ψ⟩) is defined as the function that
+symmetrizes a multiparticle state by adding states with
+swapped mJ and, if necessary, states in which the excited
+atom is switched.
+The initial state of the Ramsey se-
+quence (|g(1)
+0 ; g(2)
+0 ⟩) is symmetric under particle exchange
+and the Hamiltonian commutes with the exchange oper-
+ator, so all states involved in the dynamics must remain
+symmetric. The left panel of Figure S3 illustrates that in
+the absence of a tensor light field or dipole-dipole shifts,
+the |ξ(1)
+0 ; ξ(2)
+0 ⟩ states are degenerate with all the sym-
+metrized states S(|ξ(1)
+mJ; ξ(2)
+−mJ⟩) = 1/
+√
+2(|ξ(1)
+mJ; ξ(2)
+−mJ⟩ +
+|ξ(1)
+−mJ; ξ(2)
+mJ⟩). On the other hand, the figure shows the
+singly excited state S(|g(1)
+0 ; e(2)
+0 ⟩) = 1/
+√
+2(|g(1)
+0 ; e(2)
+0 ⟩ +
+|e(1)
+0 ; g(2)
+0 ⟩) is not degenerate with states of mixed angu-
+lar momentum (e.g. S(|g(1)
+1 ; e(2)
+−1⟩)) because the g and e
+states have different Zeeman splittings.
+The dotted lines show the lifting of the degeneracy of
+the spin-mixed states by the ac Stark shift (Hac). The
+splitting leads to the simplified level structure shown
+in the right panel of Figure S3.
+After the splitting,
+only the S(|ξ(1)
+1 ; ξ(2)
+−1⟩) states remain energetically nearby
+the |ξ(1)
+0 ; ξ(2)
+0 ⟩ clock states.
+The splitting between the
+|ξ(1)
+0 ; ξ(2)
+0 ⟩ and S(|ξ(1)
+1 ; ξ(2)
+−1⟩) states by the tensor ac light
+shift is denoted ∆Eξ
+tens.
+With the parameters previ-
+ously used to describe the optical lattice, this splitting
+is h×4(2) kHz for gg atoms, h×4.8(6) kHz for ee atoms.
+The spin mixing term in Hdd couples the nearby states
+with rates Ωξ
+SM (h × 2.4 Hz for gg atoms, h × 4.6 Hz for
+ee atoms). The coupling produces first-order level mixing
+at the Ωξ
+SM/∆Eξ
+tens ∼ 10−3 level between these states —
+small enough to not directly impact the coherence of the
+clock.
+However at second-order, both of the clock states will
+experience an energy shift of ∆Eξξ ∼ −(Ωξ
+SM)2/∆Eξ
+tens.
+It is clear that the difference between these two shifts
+leads to a shift of the clock frequency, which from the
+simplified treatment is ∼ 3 mHz for two atoms. The sum
+of the shifts to the clock states also leads to decoherence
+through entanglement generation.
+To understand this
+effect, we consider the state of the two atom system after
+it has evolved for a time t during the Ramsey sequence:
+|ψ(t)⟩ = 1
+2
+�
+e−i∆Eggt/ℏ|g(1)
+0 ; g(2)
+0 ⟩ + e−i∆Eeet/ℏ|e(1)
+0 ; e(2)
+0 ⟩
++ |g(1)
+0 ; e(2)
+0 ⟩ + |e(1)
+0 ; g(2)
+0 ⟩
+�
+(S17)
+where we’ve included only the non-trivial phases accu-
+mulated from the second-order dipole-dipole interaction.
+If we trace over the second atom state space, we find the
+single atom reduced density matrix is:
+ρ(1)(t) =|g(1)
+0 ⟩⟨g(1)
+0 | + |e(1)
+0 ⟩⟨e(1)
+0 |
++
+�
+(1 + e−i(∆Egg+∆Eee)t/ℏ)|g(1)
+0 ⟩⟨e(1)
+0 | + h.c.
+�
+(S18)
+From this, we can see the coherence is fully lost when
+(∆Egg +∆Eee)t/ℏ = ±π. Given the second-order dipole-
+dipole shifts calculated above, this corresponds to a full
+loss of coherence at t ∼ 86 s. This is in agreement with
+the lifetime seen in the simulation from Figure S2.
+[S1] M. S. Safronova, M. G. Kozlov, W. R. Johnson, and
+D. Jiang, Development of a configuration-interaction
+plus all-order method for atomic calculations, Physical
+
+9
+Review A 80, 012516 (2009).
+[S2] A. Kramida and Y. Ralchenko, NIST Atomic Spec-
+tra Database, NIST Standard Reference Database 78
+(1999), type: dataset.
+[S3] M. S. Safronova, S. G. Porsev, U. I. Safronova, M. G.
+Kozlov, and C. W. Clark, Blackbody-radiation shift in
+the Sr optical atomic clock, Physical Review A 87,
+012509 (2013).
+[S4] S. Eustice, K. Cassella, and D. Stamper-Kurn, Laser
+cooling of transition-metal atoms, Physical Review A
+102, 053327 (2020).
+[S5] V. A. Dzuba, V. V. Flambaum, M. G. Kozlov, and S. G.
+Porsev, Using effective operators in calculating the hy-
+perfine structure of atoms, Journal of Experimental and
+Theoretical Physics 87, 885 (1998).
+[S6] S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov,
+Electric-dipole amplitudes, lifetimes, and polarizabili-
+ties of the low-lying levels of atomic ytterbium, Physical
+Review A 60, 2781 (1999).
+[S7] S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov, Cal-
+culation of hyperfine structure constants for ytterbium,
+Journal of Physics B: Atomic, Molecular and Optical
+Physics 32, 1113 (1999).
+[S8] C. Cheung, M. Safronova, and S. Porsev, Scalable Codes
+for Precision Calculations of Properties of Complex
+Atomic Systems, Symmetry 13, 621 (2021).
+[S9] F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel,
+Dynamical polarizability of atoms in arbitrary light
+fields: general theory and application to cesium, The
+European Physical Journal D 67, 10.1140/epjd/e2013-
+30729-x (2013).
+[S10] J. Bai, X. Wang, X. Hou, W. Liu, and J. Wang, Angle-
+Dependent Magic Optical Trap for the 6S1/2-nP3/2 Ry-
+dberg Transition of Cesium Atoms, Photonics 9, 303
+(2022).
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+O. Gorceix, and B. Laburthe-Tolra, Resonant demag-
+netization of a dipolar Bose-Einstein condensate in a
+three-dimensional optical lattice, Physical Review A 87,
+051609 (2013).
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+Tolra, B. L. Lev, and T. Pfau, Dipolar physics:
+A
+review of experiments with magnetic quantum gases
+(2022), number:
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+[cond-mat, physics:physics, physics:quant-ph].
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+neering of Many-Body Spin Systems, Physical Review
+X 10, 031002 (2020).
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+
diff --git a/jtFQT4oBgHgl3EQflzZ1/content/tmp_files/load_file.txt b/jtFQT4oBgHgl3EQflzZ1/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1199 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf,len=1198
+page_content='Optical Telecommunications-Band Clock based on Neutral Titanium Atoms  Scott Eustice,1, 2 Dmytro Filin,3 Jackson Schrott,1, 2 Sergey Porsev,3 Charles  Cheung,3 Diego Novoa,1, 2 Dan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Stamper-Kurn,1, 2, 4 and Marianna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 5  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  2Challenge Institute for Quantum Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of Delaware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Newark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' DE 19716  4Materials Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  5Joint Quantum Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' National Institute of Standards and Technology  and the University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Maryland 20742  (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 2023)  We propose an optical clock based on narrow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' spin-forbidden M1 and E2 transitions in laser-  cooled neutral titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' These transitions exhibit much smaller black body radiation shifts than  those in alkaline earth atoms, small quadratic Zeeman shifts, and have wavelengths in the S, C,  and L-bands of fiber-optic telecommunication standards, allowing for integration with robust laser  technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We calculate lifetimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' transition matrix elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' dynamic scalar, vector, and tensor  polarizabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' and black body radiation shifts of the clock transitions using a high-precision rel-  ativistic hybrid method that combines a configuration interaction and coupled cluster approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We also calculate the line strengths and branching ratios of the transitions used for laser cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' To  identify magic trapping wavelengths, we have completed the largest-to-date direct dynamical polar-  izability calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Finally, we identify new challenges that arise in precision measurements due  to magnetic dipole-dipole interactions and describe an approach to overcome them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Direct access  to a telecommunications-band atomic frequency standard will aid the deployment of optical clock  networks and clock comparisons over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Optical atomic clocks have taken a giant leap in re-  cent years, with several experiments reaching uncertain-  ties at the 10−18 level [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The comparison of clocks  based on different atomic standards [4] or placed in sepa-  rate locations [5] enables important applications such as  relativistic geodesy [6], tests of fundamental physics [7],  and dark matter searches [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' These applications moti-  vate the development of synchronized clock networks and  transportable clocks that operate in extreme and distant  environments [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The leading neutral-atom optical clocks operate on  wavelengths of 698 nm (Sr) [10] and 578 nm (Yb) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Light at these wavelengths is strongly attenuated in opti-  cal fibers, posing a challenge to long-distance time trans-  fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  These wavelengths are also inconvenient for con-  structing the ultrastable lasers that are an essential com-  ponent of optical clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  By comparison, an optical atomic clock operating in  the telecommunication wavelength band would have clear  advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The S-, C- and L-bands, ranging altogether  between about 1460 and 1625 nm, feature low losses  in standard optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Stable light sources and ro-  bust optical amplifiers are also available across these  ranges [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' These features would support the develop-  ment of fiber-linked terrestrial clock networks over con-  tinental distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We propose the use of ultra-narrow optical tran-  sitions in atomic titanium (Ti) as the basis of a  telecommunications-band atomic clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' It has recently  been pointed out that numerous transition-metal ele-  ments, including Ti, can be laser-cooled on near-cycling  optical transitions [13], allowing for the adoption of op-  tical lattice or tweezer trapping techniques [14] used in  J  b)  a)  E [103cm−1]  1  2  3  4  5  6  0  5  10  15  20  25  cooling  498 nm  E2  1573 nm  M1  1549 nm  3d24s2 a3F  |g⟩  3d3(4F)4s a5F  |e⟩  3d3(4F)4p y 5G 6  lattice lasers  clock laser  x  y  z  ⃗B  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' (a) Relevant atomic structure in Ti for an optical  clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The a F  3  and a F  5  terms serve as the basis for the  optical clock, while the excited y G  5 6 level serves as the excited  state for laser cooling of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The two optical clock transitions  highlighted in the text are shown as maroon arrows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' the laser  cooling transition is shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (b) A diagram of the  proposed experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Polarizations are indicated  on a given beam by a small arrow of the same color as the  beam itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  today’s leading neutral-atom clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We identify sev-  eral transitions between the 3d24s2 a F  3  and 3d3( F  4  )4s  a F  5  fine structure manifolds in Ti with transition wave-  lengths between 1483 and 1610 nm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1 and Ta-  ble I) that can serve as optical clock references for ultra-  stable telecommunication-band light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  From a numerical calculation of the Ti level structure,  we identify several key features that make Ti an attrac-  tive atom for clock applications: the extreme narrowness  of the candidate clock transitions, a weak clock sensi-  tivity to blackbody radiation shifts, and the existence  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='13363v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='atom-ph]  31 Jan 2023  2  of several magic wavelengths for optical trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' While  we identify challenges posed by the non-zero angular mo-  mentum of the clock states in Ti, we show that a proper  magic-wavelength condition for optical trapping, which  imposes a significant differential tensor ac Stark shift,  mitigates their effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Our analysis relies on high precision atomic structure  calculations, by which we characterize 85 levels of neu-  tral Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For this, we employ a hybrid method that com-  bines the configuration interaction (CI) and linearized  coupled-cluster (CC) approaches (referred to as CI +  all order method [15, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In this method, the corre-  lations between four valence electrons are included via a  large-scale CI computation using a highly parallel mes-  sage passing interface (MPI) CI code [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Several  computations with increased number of configurations  were carried out to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The core-core  and core-valence correlations are included using an effec-  tive Hamiltonian formalism [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We construct the effec-  tive Hamiltonian using second-order many-body pertur-  bation theory (MBPT) and more accurate CC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The difference between these results gives the size of the  higher-order corrections, which we use to estimate un-  certainties on all theory values [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The results are used  to calculate transition rates, dynamical polarizabilities,  and systematic shifts in the clock transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Further  details of the computational methods are given in the  Supplemental Material [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Several clock transitions are identified in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Tran-  sitions between the a F  3  and a F  5  manifolds occur via spin  forbidden electric quadrupole (E2) and magnetic dipole  (M1) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Calculated reduced matrix elements for  these transitions are tabulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The calculated natural  linewidths account for both the decay of the upper state  to the lower manifold on the listed E2 and M1 transitions  and the M1 decays within each fine-structure manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The transitions are all exceptionally narrow, allowing for  optical atomic clocks with long coherence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In this letter, we focus on the a F  3  4 → a F  5  5 transi-  tion at 1549 nm unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' An advantage  of this transition is that the a F  5  5 state is the lower level  of the near-cycling 498 nm transition, which is suited for  laser cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Our calculations predict that the cooling  transition has low branching ratios to other even parity  states (∼ 10−6), enabling single-laser state preparation  and readout for atoms in the upper clock state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For de-  tails on calculations relevant to the laser cooling transi-  tion, see the Supplemental Material [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' An additional  benefit is that light at the 1549 nm clock wavelength can  be generated by narrow-linewidth, high-power Er-doped  fiber lasers, simplifying the required optical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We consider the three titanium isotopes which have  zero nuclear spin, and therefore no hyperfine structure  (46,48,50Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' To make the clock insensitive to first-order  differential Zeeman shifts from stray magnetic fields, we  drive the |mJ = 0⟩ → |m′  J = 0⟩ transition, with mJ be-  ing the magnetic quantum number and the primed sym-  bols and numbers referring to the upper a F  5  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Be-  J  J′  λ (nm)  Tele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content=')  (10−6s−1)  4  5  1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='926  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content='93(38)  2  1  1525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content='85(11)  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' List of proposed optical clock transitions in Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' All  transitions are between the lower a F  3  and upper a F  5  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The lower (upper) states are indexed by J (J′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Transition  wavelengths λ are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The telecomm band  is indicated, with S (short), C (conventional) and L (long)  bands noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  M1, E2 reduced matrix elements DM1, DE2  and transition linewidths Γ are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The two clock  transitions highlighted in the text are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  cause the E2 matrix element for this transition is zero,  only the M1 matrix element contributes to a direct one-  photon drive of the clock transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Choosing quantiza-  tion, clock-laser polarization, and clock-laser propagation  axes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1, we calculate that for a driving in-  tensity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1 W/mm2, we achieve a clock Rabi frequency  of 91(46) Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To compare the strength of this M1 transition to that  of an E2 transition in the same set of transitions, we also  consider driving the |a F  3  4, mJ = 0⟩ → |a F  5  4, m′  J = 0⟩  transition at a wavelength of 1573 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For this transi-  tion, the M1 matrix element vanishes while the E2 matrix  element does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' With the same intensity and polariza-  tion as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1, but propagating along the z axis, the  Rabi frequency for such an E2 transition is 214(13) Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  For a detailed derivation of these Rabi frequencies, see  the Supplemental Material [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Neutral-atom optical clocks often use optical lattice  potentials to confine atoms, allowing for a long interro-  gation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In order to avoid imposing large differential  ac Stark shifts between the upper and lower states of  the clock transition, it is necessary to use lattice light  which is at a “magic wavelength”, at which the dynamic  polarizabilities of the lower and upper clock states are  identical [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 2 and Table II, we report several magic wave-  lengths for the |a F  3  4, mJ = 0⟩ → |a F  5  5, m′  J = 0⟩ clock  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' As with most states in Ti, the clock states ex-  perience significant vector and tensor ac Stark shifts [13],  owing to their non-zero angular momentum and Ti’s com-  plex spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' To account for these shifts, we consider  the specific lattice configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Here, a  magnetic field applied in the z direction imposes a linear  3  λmagic  α  αS  a F  3  4  αS  a F  5  5  αV  a F  3  4  αV  a F  5  5  αT  a F  3  4  αT  a F  5  5  1036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4  116(10)  115(3)  66(12)  2(2)  470(30)  4(4)  154(8)  887+4  −4  122(5)  121(4)  159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6)  3(2)  104(5)  5(4)  111(3)  789+5  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2  129(5)  127(4)  127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4)  4(3)  127(4)  5(4)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6)  781+3  −7  130(5)  128(4)  126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3)  4(3)  138(3)  6(4)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5)  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Data for the magic wavelengths for the a F  3  4 to a F  5  5 clock transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Wavelengths are given in units of nm,  polarizabilities are given in atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  1100  1050  1000  950  900  850  800  750  [nm]  0  200  400  S [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=']  a5F5  a3F4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The scalar dynamic polarizability of the mJ = 0 sub-  levels of the a F  3  4 (red) and a F  5  5 (blue) states in Ti from 1100  nm to 750 nm as calculated by the sum-over-states method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The angle between the polarization and the B field direction  is set to 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The locations of magic wavelengths considered  in the rest of this work are circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Zeeman shift and defines the quantization direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' All  lattice light is linearly polarized in the transverse x − y  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In this configuration, the clock transition is shifted  only by the differential scalar and tensor ac Stark effects  (the vector shift is zero on the mJ = m′  J = 0 sublevels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The sum of the scalar and tensor dynamic polarizabilities  (αS  i and αT  i respectively) on the transition is then given  by  ∆α = αS  a F  5  5 − αS  a F  3  4 + 1  2  �2  3αT  a F  5  5 − 5  7αT  a F  3  4  �  (1)  At the identified magic wavelengths, the net transition  ac Stark shift is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For a more detailed description of  the ac Stark shifts, see the Supplemental Material [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Calculations of the polarizabilities were performed by  two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  First, the sum-over-states method was  used to roughly calculate polarizabilities over a wide  range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The 76 transitions with the largest  contributions to dc polarizability were used in the case of  the a F  3  4 states, while 51 transitions were used in the case  of the a F  5  5 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Once promising candidates for magic  wavelengths were found, we performed direct dynamical  polarizability calculations to identify the location of the  magic wavelengths more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Direct computations  for two of the magic wavelengths allow us to predict the  remaining values accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Previously, the direct com-  putation method was only used for divalent systems such  as Sr [21, 22], Mg [23], Yb [24, 25], Cd [26], or Tm [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  For more complex atoms, the rapidly increasing num-  ber of relevant configurations makes such a direct compu-  tation intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Here, we apply instead a truncation  approximation: we order the configurations by weight  to select the most important ones and then start re-  moving configurations while checking the accuracy of the  energies and relevant matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This procedure  drastically reduces the number of Slater determinants  required to maintain numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Further de-  tails on our method are found in the Supplemental Mate-  rial [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We emphasize that our approach is not specific  to Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' it should allow for the computation of polarizabil-  ities, magic wavelengths, and other atomic properties for  other atoms with a complex electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Using the lattice configuration and magic wavelength  described above not only eliminates the differential light  shift, but also protects against the effects of dipole-  dipole interactions between Ti atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  These effects  are not present in lattice clocks of Sr, Yb, or Hg as  those clocks operate on transitions between non-magnetic  J = 0 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In contrast, the magnetic moments of the  proposed Ti clock states are both large, with µa F  3  4 =  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00 µB and µa F  5  5 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='05 µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  There are three processes associated with the dipole-  dipole interaction that we consider: dipolar relaxation,  elastic spin-spin energy shifts, and inelastic spin-spin  mixing [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Dipolar relaxation is the process by which  Zeeman energy is converted to kinetic energy, depleting  atoms from the clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Such relaxation can be sup-  pressed for atoms trapped in a deep 3D optical lattice by  ensuring the bandgap far exceeds the Zeeman energy [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The band energy scale in a lattice is set by the lattice re-  coil energy Er = h2/(8ma2), where a is the lattice spac-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  For a 48Ti atom in the magic-wavelength lattice  described above, the recoil energy is Er = h × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  3D optical lattice clocks typically use deep lattices to  suppress tunneling and atom-atom contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  As of 2019, the fermionic Sr 3D lattice clock at JILA  operated at a lattice depth of V0 = 80Er [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In deep  lattices, the gap above the ground band is Eg ≈ 2√V0Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In the case of Ti, a comparable lattice operating at the  magic wavelength near 781 nm could be achieved by in-  tersecting six 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5 W beams with waists of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This  would give a lattice depth of V0 ≈ 79Er = h × 540 kHz  and band gap of Eg ≈ 18Er = h × 120 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Setting the  Zeeman energy below this band gap requires the ambient  magnetic field be well below B ∼ Eg/µB = 60 mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The second two processes associated with the dipole-  dipole interaction are captured in the so-called secular  Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' which is obtained by time-averaging the  4  dipole-dipole interaction over Larmor precession:  Hdd = µ0µ2  B  8π  �  ⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='j⟩  gJigJj  r3  ij  �  1 − 3 cos2 θij  �  ×  �  Jz  i Jz  j − 1  4  �  J+  i J−  j + J−  i J+  j  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (2)  Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' i and j label two atoms held at different sites of a  lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' separated by a distance vector of length rij and  polar angle θij with respect to the quantization axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' gJi  is the Land´e g-factor of the atom at lattice site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The elastic spin-spin energy shift corresponds to the  Jz  i Jz  j term in the secular Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In theory, this  term generates shifts to the transition frequency between  atomic states with non-zero angular momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' However,  for a clock transition between mJ = m′  J = 0 magnetic  sublevels, the shift is zero and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The final process is the spin-mixing interaction, which  corresponds to the J+  i J−  j + J−  i J+  j  term in the Hamil-  tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  This term couples atoms in an initial two-  body state |m(1)  J  = 0, m(2)  J  = 0⟩ to final states |m(1)  J  =  ±n, m(2)  J  = ∓n⟩, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' If not controlled, this  would lead to rapid loss of population from the mJ = 0  clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The maximal strength of the coupling is  ℏΩSM = µ0µ2  Bg2  JJ(J+1)  √  2/16π(λ/2)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In a λ = 781 nm  optical lattice, this gives spin mixing strengths of h × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4  Hz (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6 Hz) within the lower (upper) clock state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Spin mixing between atoms in the upper and lower clock  states is energetically suppressed because of the signifi-  cant differential Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For a 30 mG magnetic  field, the splitting between the |m(1)  J  = 0, m′(2)  J  = 0⟩ and  |m(1)  J  = ±1, m′(2)  J  = ∓1⟩ states is h × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In the case where both atoms occupy either the up-  per or lower clock state, spin mixing is suppressed by the  tensor ac Stark shift imparted by the optical lattice light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The tensor light shift creates an energy splitting between  the |m(1)  J  = 0, m(2)  J  = 0⟩ and |m(1)  J  = ±n, m(2)  J  = ∓n⟩  two-atom states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Using the same optical lattice con-  figuration described above, the splitting between the  |m(1)  J  = 0, m(2)  J  = 0⟩ and |m(1)  J  = ±1, m(2)  J  = ∓1⟩ states  is ∆Etens = h × 4(2) kHz (h × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8(6) kHz) within the  lower (upper) clock state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Since the differen-  tial Zeeman splitting and ∆Etens are much larger than  ℏΩSM, spin mixing is highly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In this regime, spin mixing enters as a second-order  perturbative effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The |m(1)  J  = 0, m(2)  J  = 0⟩ two-atom  states in both the lower and upper clock manifolds are  weakly coupled to the corresponding |m(1)  J  = ±1, m(2)  J  =  ∓1⟩ states by ΩSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Both clock states experience an en-  ergy shift on the order of ∼ Ω2  SM/∆Etens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The difference  between the shifts leads to a shift of the clock frequency,  while the sum of the shifts leads to decoherence between  the clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For two atoms, the shift is ∼ 3 mHz  and the rate of decoherence is ∼ 6 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For more dis-  cussion of the dipole-dipole interaction, see the Supple-  mental Materials [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  One complication in our scheme of using tensor light  shifts to combat magnetic dipole-dipole interactions is  that deviations from the lattice-light polarization shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 1 will introduce clock frequency shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Consid-  ering the example parameters from above, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5◦ tilt of  the linear polarization away from the desired orientation  would introduce a ∼ 4 Hz overall shift in the clock transi-  tion frequency, and a much smaller differential shift spa-  tially across the lattice owing to variation in the light in-  tensity of the Gaussian-focused beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Standard meth-  ods for reducing and calibrating this residual shift, in-  cluding measuring the variation of the clock frequency  with lattice-light intensity, should allow the systematic  uncertainty to be reduced to an acceptable level [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Additional terms in the light shift, such as the hyper-  polarizability and the M1 and E2 polarizabilities would  also need to be taken into account, but their effects are  small (below 10−18 levels in Sr [32–34]), and their con-  sideration is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Another significant systematic uncertainty in optical  clocks is the blackbody radiation (BBR) shift, which has  been the subject of significant past investigation [21, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We model the BBR shift for the Ti clock line as:  ∆BBR = −κ  �  α0  a F  5  5 − α0  a F  3  4  � � T  300  �4  (1 + η)  (3)  where κ = 1  2(831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='9[V/m])2 is a constant of proportional-  ity, α0  i is the dc scalar polarizability of the i state of Ti,  T is the thermal background temperature measured in  K, and η is a small dynamical correction omitted in the  present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The same CI+all-order approach is used to  compute dc and dynamic polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In this case, we  find that α0  a F  5  5 = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='53 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' and α0  a F  3  4 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='39 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=',  which leads to ∆BBR = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='24 Hz at T = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This  value is approximately an order of magnitude lower than  that in Sr, where the BBR shift is known to be -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2789  Hz [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The final systematic uncertainty that we consider is the  quadratic Zeeman shift (QZS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For the 46,48,50Ti isotopes,  the effect will be small since it will arise only from the  mixing of neighboring fine structure states, whereas in  atoms with nonzero nuclear spin, a stronger QZS arises  from mixing of hyperfine states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For the states in the  Ti clock, the QZS of the mJ = 0 sublevels are ∆(a F  3  4)  QZS  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='129[Hz/G2]B2 and ∆(a F  5  5)  QZS  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='434[Hz/G2]B2, and the  QZS on the transition is thus ∆QZS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='305[Hz/G2]B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Given that a Ti clock must operate at a magnetic field  well below 60 mG to suppress dipolar relaxation, the QZS  of the clock transition will be below 1 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This is ap-  proximately an order of magnitude lower than the QZS  that is present in Sr optical lattice clocks, of almost 10  mHz [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Altogether, we have shown that laser-cooled Ti is an  attractive choice for realizing a telecommunications-band  optical atomic clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Operating Ti clocks on several  of the available telecommunications-band optical tran-  sitions would allow for clock comparisons as a powerful  5  method for identifying and reducing systematic correc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We have advanced atomic structure calculations  to determine critical properties of such clocks, including  identifying magic wavelengths for optical trapping, esti-  mating clock transition widths and line strengths, and  determining that the BBR shift for Ti clock transitions  is an order of magnitude smaller than the shift that dom-  inates current Sr-based clock systematics  [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We  also describe potential effects of, and mitigation mea-  sures against, magnetic dipole-dipole interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' These  measures are relevant to other potential applications of  dipole-interacting atoms and molecules for precision mea-  surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank Mikhail Kozlov, Andrey Bondarev, and Ilya  Tupitsyn for helpful discussions of polarizability com-  putations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  This work is supported by a collaboration  between the US DOE and other Agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This mate-  rial is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Depart-  ment of Energy, Office of Science, National Quantum In-  formation Science Research Centers, Quantum Systems  Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Additional support is acknowledged from  the ONR (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' N00014-20-1-2513 and N00014-  22-1-2280), NSF (PHY-2012068 and the QLCI program  through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' OMA-2016245), and European Re-  search Council (ERC) under the European Union’s Hori-  zon 2020 research and innovation program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  856415).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This research was supported in part through  the use of University of Delaware HPC Caviness and  DARWIN computing systems: DARWIN - A Resource  for Computational and Data-intensive Research at the  University of Delaware and in the Delaware Region,  Rudolf Eigenmann, Benjamin E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Bagozzi, Arthi Jayara-  man, William Totten, and Cathy H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content='  McNally,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Zhang,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Barrett,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Strouse, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Tew, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Ye, Systematic  evaluation of an atomic clock at 2 × 10-18 total uncer-  tainty, Nature Communications 6, 6896 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content=' Porsev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Kozlov,  Multipolar Polarizabilities and Hyperpolarizabilities in  the Sr Optical Lattice Clock, Physical Review Letters  120, 063204 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content=' Takamoto, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Katori, Operational  Magic Intensity for Sr Optical Lattice Clocks, Physical  Review Letters 121, 263202 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content=' Kestler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Filin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Cheung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Schneeweiss,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Hoinkes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Volz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Rauschen-  beutel, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Barreiro, State-Insensitive Trapping  of Alkaline-Earth Atoms in a Nanofiber-Based Optical  Dipole Trap (2022), publisher: arXiv Version Number:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Porsev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Derevianko, Multipolar theory of  blackbody radiation shift of atomic energy levels and its  implications for optical lattice clocks, Physical Review A  74, 020502 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Lisdat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' D¨orscher, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Nosske, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Sterr, Black-  body radiation shift in strontium lattice clocks revisited,  Physical Review Research 3, L042036 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  [37] https://udspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='udel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='edu/handle/19716/29071  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Supplemental Material for  “Optical Telecommunications-Band Clock based on Neutral Titanium Atoms”  Scott Eustice,1, 2 Dmytro Filin,3 Jackson Schrott,1, 2 Sergey Porsev,3 Charles  Cheung,3 Diego Novoa,1, 2 Dan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Stamper-Kurn,1, 2, 4 and Marianna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Safronova3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 5  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  2Challenge Institute for Quantum Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' University of Delaware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Newark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' DE 19716  4Materials Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' CA 94720  5Joint Quantum Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' National Institute of Standards and Technology  and the University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Maryland 20742  (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' 2023)  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  CALCULATION OF E1, M1, AND E2  TRANSITION RATES IN TI I  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Theoretical framework: CI+all-order method  We use the CI+all-order method [S1] that combines  linearized coupled cluster and configuration interaction  (CI) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In this method, the electrons are sepa-  rated into the 1s22s22p63s23p6 core and four remaining  valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' First, the coupled cluster method is  used to construct the effective Hamiltonian Heff that ac-  counts for core and core-valence correlations and can be  constructed using second-order many-body perturbation  theory in the CI + MBPT method or the coupled clus-  ter approach (CI + all-order method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The CI method  is used to correlate the remaining four valence electrons  using this effective Hamiltonian rather than the usual  bare Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This procedure effectively includes all  types of correlation effects in the core and valence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The CI wave function is constructed as a linear com-  bination of all distinct states of a specified angular mo-  mentum J and parity,  ψJ =  �  i  ciΦi,  (S1)  where {Φi} is the set of Slater determinants generated  by exciting electrons from the reference configuration to  higher orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The many-electron Schr¨odinger equation  can be written as  HeffΨ = EΨ,  (S2)  where the effective Hamiltonian has the form  Heff = HCI + Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (S3)  Here, HCI is the CI Hamiltonian described by the equa-  tion  HCI = Ecore +  �  i>Ncore  hi,CI +  �  j>i>Mcore  Vij,  (S4)  where Ecore is the energy of the frozen core, Ncore is  the number of core electrons, hi,CI represents the kinetic  energy of the valence electrons and their interaction with  the central field and Vij accounts for the valence–valence  correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The core-valence correlation potential,  Σ = Σ1 + Σ2,  is obtained from the all-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Here, Σ1 and Σ2  are the one- and two-electron parts of the core–valence  correlation potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  After Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' (S2) is  solved using the CI technique and the wave functions are  obtained, they are used to calculate matrix elements of  the electric-dipole, magnetic-dipole, electric-quadrupole,  and other one-electron operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Energy level calculation  When applying the CI+all-order method to atomic Ti,  we used a a V N−4 potential of the 1s22s22p63s23p6 frozen  core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We solve Dirac-Hartree-Fock equations in this po-  tential to generate 3d, 4s, 4p, 5s, 4d, 5p, and 4f orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  All other orbitals are constructed in a spherical cavity of  40 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' using B-splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The set of CI configurations has to be constructed  separately for even and odd states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We carry out sev-  eral calculations with increasing number of configura-  tions to ensure convergence of the CI with the num-  ber of included configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For even states, we find  it sufficient to make all possible single and double ex-  citations to a 20spd18f16g basis starting from 4s23d2,  4s3d25s, 4s23d4d, 3d24p2, 3d25s2, 4s3d3, 4s3d24d, 3d35s,  and 3d25s4d configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We verified that a subset of  triple excitations give a negligible contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For odd  states, the CI configuration space is sufficiently saturated  by the single and double excitations to the same large ba-  sis from the 4s3d24p, 3d34p, 4s3d25p, 3d24p5s, 4s3d4p4d,  3d25s5p, and 3d24p4d configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We have carried out calculation of energies for 85 levels  and compared them with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Most of the the-  oretical energies differ from experimental values by only  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We present results only for the lines and lev-  els that can be used to pump titanium optically into the  metastable state to perform laser cooling and to drive the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='13363v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='atom-ph]  31 Jan 2023  2  TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Comparison of theoretical even energy levels (in  cm−1) with experiment [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Configuration  Term  Expt  Theory  Diff  Diff %  3d24s2  a F  3  2  0  0  0  a F  3  3  170  177  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1%  a F  3  4  387  396  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3%  3d3(4F)4s  a F  5  1  6557  6374  183  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  a F  5  2  6599  6416  183  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  a F  5  3  6661  6477  184  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  a F  5  4  6743  6557  186  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  a F  5  5  6843  6652  191  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  3d3(2G)4s  a G  3  5  15220  15497  276  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  3d3(2H)4s  a H  3  5  18141  18450  308  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7%  a H  3  6  18193  18498  306  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7%  3d3(2H)4s  a H  1  5  20796  21171  375  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8%  TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Comparison of theoretical odd energy levels (in  cm−1) with experiment [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Configuration  Term  Expt  Theory  Diff  Diff %  3d2(3F)4s4p(3P o)  z G  5  o  6  16459  16454  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0%  3d2(3F)4s4p(3P o)  z S  1  o  0  24174  3d2(3P)4s4p(3P o)  D  5  o  4  25927  26081  154  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6%  3d3(4F)4p  y G  5  o  6  26911  26982  71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3%  relevant clock transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The selected energy levels in  cm−1 are listed in Tables S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We compute the expectation values ⟨L2⟩ and ⟨S2⟩,  where L and S are the total electron orbital and spin an-  gular momentum operators, to obtain approximate quan-  tum numbers L and S, where ⟨L2⟩ = L(L + 1) and  ⟨S2⟩ = S(S+1), which allowed us to unambiguously iden-  tify all terms in Tables S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' As a result, we iden-  tify a level, 3d24s4p 1So  0, not listed in the NIST database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  This level is included in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Optical clock transitions  We study forbidden transitions between the a F  3  and  a F  5  terms to identify the most suitable clock transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The main text summarizes the properties of the clock  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The total clock transition linewidth accounts  not only for spontaneous decay on the clock transition it-  self, but also for decay of the upper and lower levels of the  clock transition to other states, leading to an overall clock  transition linewidth that is larger than the spontaneous  decay rate on the clock transition alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In Table S3,  we list the contributions that determine the linewidth of  the a3F4 → a5F5 clock transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We note that the  clock transition linewidth is dominated by M1 decays  of both the lower and upper states to other fine struc-  ture states within their respective manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The same  calculation was performed to calculate the linewidth of  all other clock transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The random-phase approx-  imation (RPA) corrections are included to the effective  electric quadupole and magnetic dipole operators, see,  for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' [S3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Such effective operators account  for the core-valence correlations in analogy with the effec-  tive Hamiltonian Heff discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The uncertainties  in the values of matrix elements were estimated as dif-  ference between values obtained using CI+all-order and  CI+MBPT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Laser cooling transitions  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' [S4] identified two candidate transitions on which  Ti may be laser cooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' To support ongoing experimen-  tal efforts to realize laser cooling of Ti, we character-  ized these two transitions theoretically using our atomic-  structure calculations described herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Specifically, we  calculated the strengths of the two electric-dipole laser  cooling transitions, and also, critically to experimental ef-  forts, determined the small leakage rate (branching ratio)  out of the laser cooling transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Results for two of the  cooling transitions, 3d3( F  4  )4p y G  5 6 - 3d3( F  4  )4s a F  5  5 at  λ = 498 nm and 3d3( F  3  )4s4p( P  3  o) z G  5  o  6 - 3d3( F  4  )4s  a F  5  5 at λ = 1040 nm are listed in Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' [S4] it was noted that while there do exist other  even parity states to which a Ti atom in the excited state  of the laser cooling transition can decay, such scattering  would be strongly suppressed because the transitions are  spin forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Our calculations confirm this expecta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Indeed, we find the branching to those states is  exceptionally low, at the 10−6 level for the 498 nm tran-  sition and at the 3·10−7 level for the 1040 nm transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  These are low enough branching ratios to enable the typ-  ical tools of modern ultracold atomic physics, including  both Doppler and sub-Doppler cooling techniques and  single-atom fluorescenence detection in quantum-gas mi-  croscopes or optical tweezers, without the need for addi-  tional repumping lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To obtain transition rates and branching ratios for the  cooling transitions (see Table S4) we use the electric-  dipole reduced matrix elements calculated with the ef-  fective electric-dipole operator in the random-phase ap-  proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We also considered other correction to the  E1 operator beyond RPA: the core-Brueckner (σ), struc-  tural radiation (SR), two-particle (2P), and normaliza-  tion (Norm) corrections [S5–S7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' As has been noted for  the case of Sr [S3], these corrections cannot be omitted  at the 1% level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In Table S4 we include transition matrix elements,  Dtot, obtained taking into account all the corrections  mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The uncertainties were estimated by  taking the difference between the values obtained using  CI+all-order and CI+MBPT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For very small  matrix elements (marked by the tilde symbol), we pro-  3  TABLE S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Results of calculations for the a F  3  4 → a F  5  5 clock transitions of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Wavelength, λ, (in units of nm), E2 and  M1 reduced matrix elements (ME) (in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' for E2 transitions, in µB for M1 transitions), transition rates (in ×10−7s−1), and  branch ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Upper  Term  Level  Lower  Term  Level  λ  ME  Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' rate  Branching ratio  3d3(4F)4s  a F  5  5  6843  3d24s2  a F  3  3  170  1498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6  E2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0472(7)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00(9)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00838(25)  a F  3  4  387  1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='9  M1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0010(5)  7(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='020(17)  E2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='140(4)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='063(4)  3d3(4F)4s  a F  5  4  6743  99794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4  M1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='632(10)  325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='43(18)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='909(16)  E2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='748(18)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='14(8) × 10−6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='78(22) × 10−9  3d24s2  a F  3  4  387  3d24s2  a F  3  3  170  46138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  M1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='597(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='026)  2059(53)  1  TABLE S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Wavelengths (in nm), electric-dipole reduced matrix elements Dtot (in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ), transition rates (in s−1), and branching  ratios for cooling transitions of Ti I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The Dtot values (in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=') are calculated with CI+all-order method and include the random-  phase approximation (RPA), the core-Brueckner (σ), structural radiation (SR), two-particle (2P), and normalization (Norm)  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For approximate values (indicated by a ∼ symbol), the precise value of the matrix element is highly uncertain,  and the reported value should be interpreted as correct only within an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Upper  Term  Level  Lower  Term  Level  λ  Dtot  Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' rate  Branch ratio  3d3(4F)4p  y G  5  o  6  26911  3d3(4F)4s  a F  5  5  6843  498  E1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='337(12)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='780(23)×107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='9999989(5)  3d3(2G)4s  a G  3  5  15220  855  E1  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='007  ∼ 11  ∼ 2 × 10−7  3d3(2H)4s  a H  3  5  18141  1140  E1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0055(14)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0)×10−8  3d3(2H)4s  a H  3  6  18193  1147  E1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='025(4)  62(19)  9(3)×10−7  3d3(2H)4s  a H  1  5  20796  1635  E1  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0004  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='005  ∼ 7 × 10−11  3d2(3F)4s4p(3P o)  z G  5  o  6  16459  3d3(4F)4s  a F  5  5  6843  1040  E1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='86(3)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='03(8)×105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='99999969(2)  3d3(2G)4s  a G  3  5  15220  8076  E1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='01040(10)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0318(25)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1(2)×10−7  TABLE S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Wavelengths (in nm), calculated and observed transition rates (in s−1×106), and branching ratios for optical  pumping of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The upper state for all transitions is the 3d2(3P)4s4p(3P o)  D  5  o  4 level, at 25967 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Lower  Term  Level  λ  Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' rate, theory  Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' rate, lit  Branch ratio  3d24s2  a F  3  3  170  388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='13(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='28(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='049  a F  3  4  387  391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8(8)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='11(28)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='68  3d3(4F)4s  a F  5  3  6661  519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00036(12)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00014  a F  5  4  6743  521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='08(8)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='31(16)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='030  a F  5  5  6843  524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='17  b F  3  4  11777  706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='06(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='02  3d3(4P)4s  a P  5  3  14106  845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='129(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='049  4  vide approximate values without explicit evaluation of  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For these, the error should be assumed to be at  the same order of magnitude as the quantity itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Optical pumping transitions  Gas phase Ti atoms may be produced with a ther-  mal source that operates between 1200◦C to 1800◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' At  these temperatures, a low population of the Ti atoms  produced would be in the a F  5  5 laser-coolable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' It  would therefore be necessary to transfer atoms to this  state from the a F  3  ground state manifold via optical  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Through our calculations, we examined multi-  ple potential excited states that could be used to trans-  fer population efficiently to the a F  5  5 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We found the  3d2( P  3  )4s4p( P  3  o) y D  5  o  4 state has a significant branching  ratio to both the a F  5  5 state and a F  3  states, allowing for  optical pumping of atoms from the ground states to the  laser cooling state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The transition wavelengths are also  amenable to current laser technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  OPTICAL CLOCK RABI FREQUENCIES  The optical clock transitions of Ti can be driven by  either the M1 or E2 multipoles of the electromagnetic  field, both of which must be accounted for when calculat-  ing the overall Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In each of the transitions  we highlighted in the main text (mJ = 0 → m′  J = 0,  J = 4 → J′ = 4, 5) only one of the two multipoles is  allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Given a reduced M1 matrix element, DM1, for a tran-  sitions between lower and upper levels J and J′, and  a magnetic field B of the clock laser, the M1 Rabi fre-  quency between a lower and upper sublevel mJ and m′  J  is given by  ΩM1 = −DM1  ℏ  (−1)J′−m′  J ˆe∗  m′  J−mJ · B  ×  �  J  1  J′  mJ m′  J − mJ −m′  J  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (S5)  For an electric quadrupole transition with reduced ma-  trix element DE2 driven by a plane-wave clock laser field  E with polarization ε and propagation wavevector k, the  Rabi frequency is  ΩE2 = iDE2  2ℏ (−1)J′−m′  J  �  J  2  J′  mJ m′  J − mJ −m′  J  �  ×  �  i,j  Mij(m′  J − mJ)kiεj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (S6)  Here, Mij(q) is a geometric factor given by  Mij(q) = (−1)q√  5  �  q1,q2  (ˆei · ˆe∗  q1)(ˆej · ˆe∗  q1)  �  1  1  2  q1 q2 −q  �  ,  (S7)  where ˆei and ˆeq are the cartesian and spherical basis  vectors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  DYNAMICAL POLARIZABILITY OF TI I  In order to find the magic wavelengths of the clock  transitions, it is necessary to calculate the polarizabil-  ity of Ti in both the a F  3  4 and a F  5  5 states over a wide  range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Performing this calculation directly  remains a computational challenge for complex atoms  like Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Direct calculation requires the inversion of huge  matrices — 350,000 × 350,000 in our case — which is  computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Instead, we use an iterative  approach included in the pCI Code Package [S8] imple-  menting the CI+all-order technique to calculate the po-  larizability of the considered states of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This approach  allows us to get accurate results using the inversion of  smaller matrices (15,000 × 15,000), making the task fea-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Unfortunately, this method does not work at all  frequencies, owing to the potential divergence of the it-  erative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We successfully obtained static polariz-  abilities for both 3d24s2 a F  3  4 and 3d3(4F)4s a F  5  5 levels,  but for dynamic polarizabilities the calculation diverged  for wavelengths shorter than 750 nm (900 nm) for the  a F  3  4 (a F  5  5) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' To overcome this, we used a combi-  nation of the CI+all-order technique and the sum-over-  states method to calculate the scalar, vector and tensor  polarizability from 1100 nm to 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The sum-over-states method involves using only bound  states of an atom, and there is always some inaccuracy  due to missing contributions from continuum states and  bound states not included in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  For the  best accuracy, it is advisable to use as many states as  possible in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  However, there are always limits  on the accuracy of calculations of highly excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To balance these trade-offs, we use in the sum-over-state  method for the lower lying states that contribute the ma-  jority of the dc polarizability but still can be properly  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For this purpose, we used 73 states to obtain  the polarizability of the 3d24s2 a F  3  4 level and 49 states  for the polarizability of the 3d3(4F)4s a F  5  5 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The  most important contributions to both polarizabilities are  shown in Tables S6, S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The polarizability is generally  divided into three terms: αS - scalar, αV -vector, and  αT - tensor polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' They are represented for an  arbitrary state i as follows [S9, S10]:  αS  i (ω) =  2  3(2Ji + 1)  �  n  (En − Ei)|⟨n||D||i⟩|2  (En − Ei)2 − ω2  (S8)  αV  i (ω) = C1  �  n  (−1)Jn+Ji  �  1  1  1  Ji Ji Jn  �  ω|⟨n||D||i⟩|2  (En − Ei)2 − ω2  (S9)  5  TABLE S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Contributions to the static electric-dipole polarizability α0 with the appropriate reduced matrix elements D (in  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=') of the 3d24s2 a F  5  5 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For comparison, the experimental and theoretical energy levels (in cm−1) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Configuration  Term  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Level  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Level  Diff %  D  Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' to α0  3d2(3F)4s4p(3P o)  z G  5  o  6  16459  16454  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='86(3)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00(8)  z F  5  o  5  17215  17185  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='48(6)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7(4)  z D  5  o  4  18695  18769  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='08(3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='77(16)  3d3(4F)4p  y G  5  o  5  26773  26850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='299(5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='48(2)  y G  5  o  6  26911  26982  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='3%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='837(12)  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='16(12)  y F  5  o  4  28788  29061  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='410(7)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='45(2)  y F  5  o  5  28896  29164  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='9%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='251(6)  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='07(5)  x D  5  o  4  30060  30474  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='723(12)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='24(12)  Other  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='7  Total  128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='5(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0)  TABLE S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Contributions to the static electric-dipole polarizability α0 with the appropriate reduced matrix elements D (in  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=') of the 3d24s2 a F  3  4 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For comparison, the experimental and theoretical energy levels (in cm−1) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Configuration  Term  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Level  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content='9  Total  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content='8)  6  αT  i (ω) = C2  �  n  (−1)Jn+Ji  �  1  1  2  Ji Ji Jn  �  (En − Ei)|⟨n||D||i⟩|2  (En − Ei)2 − ω2  ,  (S10)  where  C1 = −2  �  6Ji  (Ji + 1)(2Ji + 1)  and  C2 = 4  �  5Ji(2Ji − 1)  6(Ji + 1)(2Ji + 1)(2Ji + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The index n refers to the states in the sum-over-states  that contribute to the polarizability of the state i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' En,i is  the energy and Jn,i is the total angular momentum of the  state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ⟨n||D||i⟩ is the reduced matrix element between the  two states, and ω is the frequency of the external electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The total dynamic polarizability can be expressed  as follows:  αi(ω) = αS  i (ω) + ε cos(θk)mJi  2Ji  αV  i (ω)+  +  �3 cos2 θp − 1  2  � 3m2  Ji − Ji(Ji + 1)  Ji(2Ji − 1)  αT  i (ω),  (S11)  where ε is the ellipticity of the polarization, θk is the  angle between the direction of propagation of the light  and the quantization axis, θp is the angle between the  polarization of the light and the quantization axis, and  mJi is a magnetic quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To obtain an accurate value of dynamic polarizabil-  ity with the sum-over-states method, one has to estimate  the residual contribution to the polarizability from states  that are not included in summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We made this esti-  mate using the fact that |ω/(En − Ei)| ≪ 1 for all such  residual states in the range of wavelengths that we are  considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Indeed, using this ratio as a small parame-  ter and expanding to the lowest non-zero order in, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S8  can be simplified to  αS  i (ω) = αSN  i  (ω) + αSres  i  (ω)  (S12)  where  αSN  i  (ω) =  2  3(2Ji + 1)  N  �  n=1  (En − Ei)|⟨n|D|i⟩|2  (En − Ei)2 − ω2  (S13)  and where N is the number of states used in direct sum-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Moreover, we can write  αSres  i  (ω) = A0 + B0 ω2  (S14)  where A0 and B0 are expansion constants for the long  wavelength scalar polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The same simplifica-  tion works for αV  i (ω) and αT  i (ω), by writing αVN  i  (ω) and  αTN  i  (ω) as a summation as in S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This yields the ex-  pansion of the long wavelength residuals:  αVres  i  (ω) = B1ω  αTres  i  (ω) = A2 + B2 ω2  (S15)  where B1, A2, and B2 are additional expansion constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Using the results of the sum-over-states method, we  identified the four candidate magic wavelengths discussed  in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The atoms were assumed to have mag-  netic quantum numbers mJ = 0 and the optical field  parameters were ε = 0 and θp = θk = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' However, the  more accurate iterative approach used by the pCI code  package was found to not converge at all of the candidate  wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Using the fact that convergence in the iter-  ative polarizability can be achieved at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='043989 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (λ = 1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8 nm) and for all smaller values of ω down  to ω = 0, we subtracted the values of αi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='043989) and  αi(0) obtained from the sum-over-states method from  the corresponding results computed with the pCI code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  In this way, we found the residuals αSres  i  (0), αVres  i  (0),  αTres  i  (0) and αSres  i  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='043989 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ), αVres  i  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='043989 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ),  αTres  i  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='043989 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This allowed us to determine the  parameters A0,2 and B0,1,2 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S14, S15 and thus  extend the accuracy of the direct iterative polarizability  calculation to frequencies where the iterative calculation  fails to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The uncertainty on the polarizabilities were obtained  by comparing the calculations of the more accurate  CI+all-order technique with CI+MBPT at the points  where the direct calculation converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The difference  between the values of the appropriate reduced matrix el-  ements |⟨n||D||i⟩|CI+all−order and |⟨n||D||i⟩|CI+MBP T is  an additional possible inaccuracy of the method and thus  is used to estimate the uncertainties of the polarizabili-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Furthermore, possible errors in the parameters A0,2  and B0,1,2 were included in the final uncertainties as the  difference between the correspondent parameters calcu-  lated with the CI+all-order and CI+MBPT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  DIPOLE-DIPOLE INTERACTIONS  As discussed in the main text, dipole-dipole interac-  tions underlie several effects that impact the operation  of a Ti atomic clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' One can distinguish between inter-  actions that are inelastic or elastic in the motional de-  grees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Inelastic interactions can be described  as spin relaxation, a dipole-mediated conversion of inter-  nal Zeeman energy into external motional energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For  example, in an optical lattice spin relaxation may couple  atoms from lower to higher bands of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' As dis-  cussed in the main text, inelastic motional interactions  can be suppressed by trapping atoms in a lattice with a  sufficiently large band gap [S11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  One may then focus on elastic motional interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We describe these interactions purely in the spin sec-  tor using the secular Hamiltonian, which accounts for  7  cycle-averaging over the Larmor precession of the atomic  spins [S12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This secular Hamiltonian, whose form is pre-  sented also in the main text, is given for two atoms as  Hdd = µ0µ2  B  4π  gJ1gJ2  r3  12  �  1 − 3 cos2 θ12  �  ×  �  Jz  1 Jz  2 − 1  4  �  J+  1 J−  2 + J−  1 J+  2  ��  (S16)  with 1 and 2 labelling the two atoms, and r12 and θ12  describing their position difference vector and the angle  it makes with the quantization axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  One can consider separately two effects of this secu-  lar Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The first is a spin-elastic interaction,  which is diagonal in the separable basis of magnetic sub-  level states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This shift is zero on the mJ = 0 magnetic  sublevels and therefore does not affect a clock based on  an mJ = m′  J = 0 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The second effect is the spin mixing interaction, which  is off-diagonal in the separable magnetic sublevel basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  This interaction conserves the total magnetic quantum  number of the two atoms, m(tot)  J  = m(1)  J  + m(2)  J , but  changes the magnetic quantum numbers of the individual  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' In the absence of dipolar interactions, the entire  manifold of two-atom states with identical m(tot)  J  is de-  generate for atoms with equal Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Dipo-  lar interactions can then generate significant mixing and  energy shifts within this manifold, leading to imprecise  measurement of the clock transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  While several methods have been developed in NMR to  control dipolar spin mixing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' multiple pulse sequences  [S13] and magic angle spinning [S14], these techniques are  not needed to suppress spin mixing in a Ti clock system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  As described in the main text, spin mixing within the  clock states is mitigated by the tensor light shift imposed  by the optical lattice beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S11, the ten-  sor light shift includes a part proportional to m2  J, which  splits the degeneracy of the aforementioned states and  energetically suppresses the spin mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To study the suppression of spin mixing, we simulate  the secular dipole-dipole interaction of two nearest neigh-  bor atoms in an optical lattice with and without account-  ing for the lattice-induced ac Stark shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Figure S1 shows  a schematic of the system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We no-  tate the single atom states by |ξ(i)  mJ⟩ where ξ ∈ {g, e}  refers to the upper or lower clock states, the superscript  denotes the ithatom, and the mJ subscript is the mag-  netic sublevel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We use g to refer to the lower a3F4 clock  state, and e to refer to the upper a5F5 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The two  atom system consists of [(2Jg + 1) + (2Je + 1)]2 = 400  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  We simulate a simple Ramsey interferometry sequence  which includes the dynamics of Hdd, the Zeeman Hamil-  tonian HZ, and, optionally, the ac Stark Hamiltonian  Hac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We initialize two atoms in the |g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0 ⟩ state, ap-  ply a π/2 pulse to each atom on the g0 → e0 clock tran-  sition, allow the two-atom state to evolve for a time T  1 0 1  1 0 1  1 0 1  1 0 1  M1  M1  Hdd  Hdd  J = 5  J = 4  |  (1)  mJ  |  (2)  mJ  e  g  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The secular dipole-dipole interaction of two atoms in  the a3F4 (g) and a5F5 (e) clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The g and e states have  angular momentum quantum numbers J = 4 and J = 5, giv-  ing rise to 2J +1 magnetic sublevels each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' For brevity, the fig-  ure shows only three of these mJ sublevels in each state, split  by Zeeman shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The clock drive ΩM1 couples the |gmJ =0⟩  and |emJ =0⟩ states, and Hdd couples the subspaces of the two  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2  T [s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='0  C(T)  Hac on  Hac off  40  80  2 /T  0  2 /T  [s  1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='00  Pe(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='25  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' (left) Ramsey contrast of the clock transition as a  function of the free evolution time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The contrast, shown in  violet (green), dies quickly (slowly) when the ac Stark shift,  Hac, is excluded (included) from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Note the  split in the T axis and change in scale at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='25 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' (right) The  resulting Ramsey fringes for T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='2 s (indicated by circles  in the left plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We plot the probability an atom is found  in state |e0⟩ as a function of the detuning δ from the clock  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='.  under HZ, Hdd, and (optionally) Hac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' apply a second π/2  pulse to each atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' and then determine the probability  of an atom being in the e0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We vary the detun-  ing of the clock drive frequency, δ, to generate a Ramsey  interferometry fringe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  From this fringe, we obtain the  contrast C(T) = (max Pe(T, δ) − min Pe(T, δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Figure S2 shows the results of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The  left panel demonstrates the decay of the Ramsey contrast  with and without Hac included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' We break the axis at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='25  s to illustrate that the tensor ac Stark shift extends the  decay time of the contrast significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Because we only  simulate unitary dynamics of two neighboring atoms, re-  8  S(|g(1)  mJ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  mJ )  S(|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0  )  S(|e(1)  mJ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  mJ )  Hac off  Hac on  |g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0  S(|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0  )  |e(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0  S(|g(1)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  1 )  S(|e(1)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  1 )  g  SM  e  SM  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  (left) The 2-atom level structure with and with-  out the ac Stark shifts applied (Hac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Only states that  are symmetric under particle exchange are shown, as these  are the only states involved in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The states  S(|g(1)  mJ ̸=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  −mJ ⟩) are omitted because they are split from the  S(|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0 ⟩) state by the differential Zeeman shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The dot-  ted lines show that the degeneracy of the mtot  J  = 0 sublevels  is lifted when Hac is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The five boxed levels include  the clock states and the states most strongly coupled to them  by Hdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' (right) The simplified picture, including only the lev-  els boxed in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Hdd couples the states |ξ(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  0 ⟩  and S(|ξ(1)  mJ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  −mJ ⟩) with rate Ωξ  SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  vivals of the contrast are observed in the simulation be-  yond the times plotted in Figure S2, but these would not  occur in a true many-body situation as the coherence  would spread between many particles and be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The  right panel shows the resulting Ramsey fringes taken at  the time highlighted on the left side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' When  no optical lattice is applied, the coherence on the clock  transition quickly vanishes, leading to a loss of the Ram-  sey signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The fringe survives for ∼ 85 s when the lattice  beams are on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  Figure S3 gives a simplified picture of the level struc-  ture at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  S(|ψ⟩) is defined as the function that  symmetrizes a multiparticle state by adding states with  swapped mJ and, if necessary, states in which the excited  atom is switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The initial state of the Ramsey se-  quence (|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0 ⟩) is symmetric under particle exchange  and the Hamiltonian commutes with the exchange oper-  ator, so all states involved in the dynamics must remain  symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The left panel of Figure S3 illustrates that in  the absence of a tensor light field or dipole-dipole shifts,  the |ξ(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  0 ⟩ states are degenerate with all the sym-  metrized states S(|ξ(1)  mJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  −mJ⟩) = 1/  √  2(|ξ(1)  mJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  −mJ⟩ +  |ξ(1)  −mJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  mJ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' On the other hand, the figure shows the  singly excited state S(|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0 ⟩) = 1/  √  2(|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0 ⟩ +  |e(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0 ⟩) is not degenerate with states of mixed angu-  lar momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' S(|g(1)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  −1⟩)) because the g and e  states have different Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The dotted lines show the lifting of the degeneracy of  the spin-mixed states by the ac Stark shift (Hac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The  splitting leads to the simplified level structure shown  in the right panel of Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  After the splitting,  only the S(|ξ(1)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  −1⟩) states remain energetically nearby  the |ξ(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  0 ⟩ clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The splitting between the  |ξ(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  0 ⟩ and S(|ξ(1)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' ξ(2)  −1⟩) states by the tensor ac light  shift is denoted ∆Eξ  tens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  With the parameters previ-  ously used to describe the optical lattice, this splitting  is h×4(2) kHz for gg atoms, h×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='8(6) kHz for ee atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  The spin mixing term in Hdd couples the nearby states  with rates Ωξ  SM (h × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='4 Hz for gg atoms, h × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='6 Hz for  ee atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The coupling produces first-order level mixing  at the Ωξ  SM/∆Eξ  tens ∼ 10−3 level between these states —  small enough to not directly impact the coherence of the  clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  However at second-order, both of the clock states will  experience an energy shift of ∆Eξξ ∼ −(Ωξ  SM)2/∆Eξ  tens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  It is clear that the difference between these two shifts  leads to a shift of the clock frequency, which from the  simplified treatment is ∼ 3 mHz for two atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' The sum  of the shifts to the clock states also leads to decoherence  through entanglement generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  To understand this  effect, we consider the state of the two atom system after  it has evolved for a time t during the Ramsey sequence:  |ψ(t)⟩ = 1  2  �  e−i∆Eggt/ℏ|g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0 ⟩ + e−i∆Eeet/ℏ|e(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0 ⟩  + |g(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' e(2)  0 ⟩ + |e(1)  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' g(2)  0 ⟩  �  (S17)  where we’ve included only the non-trivial phases accu-  mulated from the second-order dipole-dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  If we trace over the second atom state space, we find the  single atom reduced density matrix is:  ρ(1)(t) =|g(1)  0 ⟩⟨g(1)  0 | + |e(1)  0 ⟩⟨e(1)  0 |  +  �  (1 + e−i(∆Egg+∆Eee)t/ℏ)|g(1)  0 ⟩⟨e(1)  0 | + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content='  �  (S18)  From this, we can see the coherence is fully lost when  (∆Egg +∆Eee)t/ℏ = ±π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' Given the second-order dipole-  dipole shifts calculated above, this corresponds to a full  loss of coherence at t ∼ 86 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
+page_content=' This is in agreement with  the lifetime seen in the simulation from Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFQT4oBgHgl3EQflzZ1/content/2301.13363v1.pdf'}
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+Robust empirical risk minimization via Newton’s method
+Eirini Ioannou
+University of Edinburgh
+ei250@cantab.ac.uk
+Muni Sreenivas Pydi
+LAMSADE, Universit´e Paris Dauphine-PSL
+muni.pydi@lamsade.dauphine.fr
+Po-Ling Loh
+University of Cambridge
+pll28@cam.ac.uk
+Abstract
+We study a variant of Newton’s method for empirical risk minimization, where at each
+iteration of the optimization algorithm, we replace the gradient and Hessian of the objective
+function by robust estimators taken from existing literature on robust mean estimation for
+multivariate data. After proving a general theorem about the convergence of successive iterates
+to a small ball around the population-level minimizer, we study consequences of our theory in
+generalized linear models, when data are generated from Huber’s epsilon-contamination model
+and/or heavy-tailed distributions. We also propose an algorithm for obtaining robust Newton
+directions based on the conjugate gradient method, which may be more appropriate for high-
+dimensional settings, and provide conjectures about the convergence of the resulting algorithm.
+Compared to the robust gradient descent algorithm proposed by Prasad et al. (2020), our
+algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-
+order algorithms for convex problems, i.e., quadratic convergence in a neighborhood of the
+optimum, with a stepsize that may be chosen adaptively via backtracking linesearch.
+1
+Introduction
+Statistical estimation via classical procedures often depends on strong model assumptions, which
+only hold in the absence of outliers and other deviations. However, many real-life datasets do not
+typically follow these model assumptions, necessitating the use of robust statistical methods [14,
+29, 22], which remain reasonably accurate even under deviations from the model assumptions. In
+this paper, we focus on situations where data are sampled from a small ball around a parametric
+distribution, according to Huber’s ϵ-contamination model. In other words, we have samples of the
+form zi ∼ (1 − ϵ)Pθ∗ + ϵQ, where Q is an arbitrary distribution and the goal is to estimate the
+unknown parameter θ∗ based on an observed data set {zi}n
+i=1. We also analyze the behavior of the
+same algorithms in situations where data are generated from a heavy-tailed distribution. Although
+the parameter corresponds to the true data-generating distribution, “outliers” are observed in
+the data set due to random sampling, and the goal is to obtain an estimator with similar high-
+probability guarantees as in the case of standard parameter estimation techniques for lighter-tailed
+distributions.
+Classical robust statistics [14] suggests the use of M-estimators, which involve optimizing an ap-
+propriate loss function over the space of parameters. More specifically, suppose we wish to estimate
+the parameter θ∗ = arg minθ∈Θ R(θ), where the risk R(θ) = E[L(θ, (x, y))] is the expectation of a
+loss function. In practice, one uses an empirical risk minimizer �θ ∈ arg minθ∈Θ 1
+n
+�n
+i=1 L(θ, (xi, yi)).
+1
+arXiv:2301.13192v1  [stat.ML]  30 Jan 2023
+
+Standard theory of parametric statistics shows that the optimal choice of L corresponds to the log-
+likelihood function when data are not contaminated. However, taking into account ϵ-contamination
+leads to the use of other losses such as the Huber loss, which can be shown to be optimal in a min-
+imax sense when the uncontaminated data are normally distributed [14].
+Similarly, while the
+least-squares loss corresponds to maximum likelihood for Gaussian errors, minimizing a different
+loss function may be advantageous in the case of heavy-tailed data.
+In this paper, we adopt an alternative approach inspired by optimization methods [2]. Rather
+than seeking to design a robust loss, we introduce robustness into the estimation algorithm by
+implementing robust updates in an iterative second-order optimization procedure. Our work is di-
+rectly inspired by the work of Prasad et al. [28], who proposed and analyzed a first-order version of
+this method. Our algorithm, which we call “robust Newton’s method,” utilizes the AgnosticMean
+algorithm from Lai et al. [15] in the Huber contamination setting to obtain robust gradient and
+Hessian estimates on each iterate of our algorithm. Moreover, given appropriate assumptions, we
+prove that the rate of convergence of this algorithm is faster than that of robust gradient descent,
+and successive iterates converge quadratically to a small ball around θ∗. Furthermore, a suitable
+stepsize may be determined adaptively using a robust variant of backtracking linesearch. Our anal-
+ysis of the Newton iterates is fairly general, and can be used to derive convergence guarantees when
+alternative procedures are employed for gradient/Hessian estimation. We consequently propose a
+method based on the conjugate gradient method [30] for obtaining approximate Newton directions
+which may be useful in higher dimensions, and discuss some conjectures about the corresponding
+convergence rate on ϵ-contaminated data.
+1.1
+Related Work
+Here, we discuss several other general approaches to robust empirical risk minimization which have
+appeared in the literature. A variety of algorithms have been proposed based on median-of-means
+estimators, which give a robust alternative to a mean estimator (a more detailed description is
+provided in Section 2.4 below). Median-of-mean tournaments [21, 20, 19] provide a method for
+comparing pairs of candidate regression functions based on the number of blocks in which the em-
+pirical mean of the loss function is smaller for one function than the other. The final estimator is
+a function which “wins” the most pairwise matches among other candidate functions. Another use
+of median-of-means estimators derives an estimator by considering a “minimaximization” problem
+formed by increments of the objective function, where a median-of-means estimate is used in place
+of the expectation appearing in the population-level version of the problem [16, 17, 7]. Finally, and
+more similar in spirit to the approach taken in our paper, we mention a method which involves
+modifying gradient descent by computing a gradient with respect to a median block on each itera-
+tion [18]. The median block is defined as the block with the smallest empirical mean (with respect
+to the objective function value) on the current iteration. Excess risk bounds are then derived for a
+class of binary classification problems, where a certain fraction of the data are arbitrarily generated
+outliers and the rest are drawn i.i.d. from the uncontaminated model.
+The SEVER algorithm [9] also operates via an appropriate modification of an iterative opti-
+mization procedure. It uses any “approximate learner” algorithm, which can find an approximate
+critical point of an empirical risk minimization problem, as a subroutine (e.g., gradient descent,
+stochastic gradient descent, or Newton’s method). On successive iterations, the SEVER algorithm
+filters out data points by applying the approximate learner to the currently remaining set of data
+points and then filtering out any points with outlying gradients computed at the parameter chosen
+by the approximate learner. Statistical error bounds are derived for the output of the SEVER algo-
+rithm on classification and regression problems, where data are drawn from a possibly heavy-tailed
+2
+
+model and then corrupted by a small fraction of adversarial outliers.
+Finally, our work is most closely related to the work of Prasad et al. [28], which may be seen as
+a first-order version of our second-order algorithm. In that paper, the authors propose to perform
+parameter estimation by running a variant of gradient descent on the empirical risk objective, where
+successive gradients are computed by treating each gradient computation as an approximation of
+a population-level mean, and then applying a robust mean estimation procedure for multivariate
+data. As in our work, they use the mean estimation algorithm by Lai et al. [15] for their multivariate
+estimation procedure in the case of Huber’s ϵ-contamination model. They also derive statistical
+error bounds for successive iterates, which hold with high probability. The main difference with
+our work is that we are able to derive faster rates of convergence due to the use of second-order
+algorithms, while enjoying the broad applicability of their approach.
+1.2
+Outline
+The remainder of our paper is organized as follows: In Section 2, we discuss the setup of the
+problem we are aiming to solve. In Section 3, we introduce our novel robust Newton’s method
+and present two theorems concerning its convergence. In Section 4, we discuss applications of our
+general theory to generalized linear models. In Section 5, we provide some illustrative numerical
+results and comparisons. In Section 6, we present a version of robust Newton’s method based on
+the conjugate gradient method and provide some conjectures. Finally, we conclude our paper with
+a discussion of open directions in Section 7.
+1.3
+Notation
+For a matrix A ∈ Rp×p, we use ∥A∥2 to denote the spectral norm, λmin(A) to denote the minimum
+eigenvalue, and tr(A) to denote the trace. We use c, C, c1, C1, c2, C2, . . . to denote universal positive
+constants whose specific values may change from line to line. For functions f(n) and g(n), we write
+f(n) = O(g(n)) to mean that f(n) ≤ Cg(n) for some constant C > 0, and also write f(n) ≾ g(n)
+and g(n) ≿ f(n). We write f(n) ≍ g(n) when both inequalities hold simultaneously.
+2
+Background
+We consider a parametric estimation problem, wherein the data {zi}n
+i=1 ⊆ Z sampled from a true
+distribution P are to be fit to a model with parameter θ ∈ Θ. A loss function L : Θ × Z → R
+measures the goodness of fit of the model. The optimal parameter θ∗ ∈ Θ minimizes the population
+risk of the model, which is the expected loss incurred by the model over the true data distribution:
+θ∗ = argmin
+θ∈Θ
+R(θ) := Ez∼P [L(θ, z)].
+(1)
+Given n i.i.d. data points {zi}n
+i=1 sampled from the true distribution P, the goal in empirical
+risk minimization (ERM) is to estimate the parameter �θn that minimizes the empirical risk of the
+model, which is the average loss incurred by the model over the n data points:
+�θn = argmin
+θ∈Θ
+�Rn(θ) := 1
+n
+n
+�
+i=1
+L(θ, zi).
+(2)
+3
+
+2.1
+Examples
+Linear regression:
+In linear regression, data z ∈ Z are of the form z = (x, y) ∈ Rp × R, where
+the covariate x and response y are related via
+y = xT θ∗ + w,
+where w ∈ R is noise that is sampled independently from x and y. The loss function we use for
+this model is the squared loss function,
+L(θ, (x, y)) = 1
+2(y − xT θ)2.
+Generalized linear models:
+In a generalized linear model (GLM), data z = (z, y) ∈ Rp × R
+are sampled from a true distribution P that satisfies the following relation on the conditional
+probability of x given y:
+P(y|x) ∝ exp
+�yxT θ∗ − Φ(xT θ∗)
+c(σ)
+�
+,
+(3)
+where c(σ) is the scale parameter and Φ : R → R is a convex link function. The loss function we
+use for a GLM is the negative log-likelihood,
+L(θ, (x, y)) = −yxT θ + Φ(xT θ).
+(4)
+2.2
+Optimization Algorithms
+In practice, we seek efficient algorithms for solving the ERM problem (1). A popular algorithm is
+gradient descent [1]. Given an initial guess for the parameter θ0 ∈ Θ and a stepsize η, the gradient
+descent algorithm generates a sequence of iterates {θt}∞
+t=1, as follows:
+θt+1 = θt − η∇R(θt).
+Another popular algorithm is Newton’s method [30, 2], whose iterates are given by the following
+update equation:
+θt+1 = θt − (∇2R(θt))−1∇R(θt).
+(5)
+Whereas gradient descent uses only gradient information at the current iterate θt, Newton’s
+method uses both gradient and Hessian information at the current iterate.
+2.3
+Huber’s ϵ-Contamination Model
+In Huber’s ϵ-contamination model, samples are drawn from a mixture distribution of the form
+Pϵ = (1 − ϵ)P + ϵQ,
+(6)
+where P is the true data distribution and Q is an arbitrary noise distribution. The goal is to
+estimate a parameter θ∗ ∈ Θ corresponding to the uncontaminated component P, given n i.i.d.
+samples drawn from the corrupted distribution Pϵ.
+Huber’s contamination model is a classical model studied in robust statistics [14, 29, 22], with
+many exciting theoretical breakthroughs in estimation and inference.
+More recently, as robust
+4
+
+statistics received renewed attention in the theoretical computer science community, additional
+questions were raised, particularly concerning computational tractability for optimal robust esti-
+mators in high dimensions. The contemporaneous work of Lai et al. [15] and Diakonikolas et al. [8]
+studied computationally-tractable mean estimation in multivariate Gaussian settings, where the
+former paper studied contamination with respect to Huber’s model and the latter paper studied
+a stronger form of “adversarial” contamination. The subroutine which we call Algorithm 1 comes
+from Lai et al. [15]—we state it in the slightly adapted version studied in Prasad et al. [28].
+Algorithm 1 Huber Estimator
+Require: Samples S = {si}n
+i=1, Corruption level ϵ, Dimension p, Failure probability δ
+1: function HuberEstimator(S = {si}n
+i=1, ϵ, p, δ)
+2:
+Set �S = HuberOutlierTruncation(S, ϵ, p, δ)
+3:
+if p = 1 then
+4:
+return mean(�S)
+5:
+else
+6:
+Compute Σ�S, the covariance matrix of �S
+7:
+Compute V , the span of the top p/2 principal components of Σ�S and W, its complement
+8:
+Set S1 := PV (�S), where PV is the projection operation onto V
+9:
+Set �µV := HuberEstimator(S1, ϵ, p/2, δ)
+10:
+Set �µW := mean(PW �S)
+11:
+Set �µ ∈ Rp such that PV (�µ) = �µV and PW (�µ) = �µW
+return �µ.
+12:
+end if
+13: end function
+14:
+15: function HuberOutlierGradientTruncation(S, ϵ, p, δ)
+16:
+if p = 1 then
+17:
+Let [a, b] be the smallest interval containing
+�
+1 − ϵ − C
+�
+log(|S|/δ)
+|S|
+�
+(1 − ϵ) fraction of
+points
+18:
+�S ← S ∩ [a, b]
+19:
+return �S
+20:
+else
+21:
+Let [S]i be the samples with the ith coordinates only, [S]i = {⟨x, ei⟩ |x ∈ S}
+22:
+for i = 1 to p do
+23:
+a[i] = HuberEstimator([S]i, ϵ, 1, δ/p)
+24:
+end for
+25:
+Let B(r, a) be the ball of smallest radius centered at a containing
+(1 − ϵ −
+Cp
+��
+p
+|S| log
+�
+|S|
+pδ
+��
+(1 − ϵ) fraction of points in S
+26:
+�S ← S ∩ B(r, a)
+27:
+return �S
+28:
+end if
+29: end function
+5
+
+2.4
+Heavy-Tailed Model
+In the heavy-tailed model, we assume that data are drawn i.i.d. from a distribution with some
+number of finite moments. Note that the heavy-tailed model does not involve a contaminating
+distribution Q. However, the i.i.d. data may still appear to have “outlier” points due to random
+sampling.
+A popular approach for heavy-tailed mean estimation in the probably approximately correct
+(PAC) framework—obtaining high-probability deviation bounds which are as tight as possible under
+minimal distributional assumptions—is to use a median-of-means (MOM) estimator.
+Roughly
+speaking, data are randomly partitioned into blocks, the mean of each block is computed, and the
+median of all of the block means is returned as the estimator. In multiple dimensions, different
+notions of medians exist, leading to different flavors of MOM estimators.
+For a more detailed
+overview, see the survey [19] and the references cited therein. The MOM algorithm is summarized
+in Algorithm 2. In particular, we will employ a version of the algorithm from Minsker [24], which
+combines the mean estimates using the geometric median, i.e., the point which minimizes the sum
+of ℓ2-distances to the block means.
+Algorithm 2 Heavy-Tailed Estimator
+Require: Samples S = {si}n
+i=1, Failure probability δ
+1: function HeavyTailedEstimator(S = {si}n
+i=1, δ)
+2:
+Set b = 1 + ⌊3.5 log 1/δ⌋, the number of buckets
+3:
+Partition S into b blocks B1, . . . , Bb, each of size ⌊n/b⌋
+4:
+for i = 1 . . . n do
+5:
+�µi =
+1
+|Bi|
+�
+s∈Bi
+s
+6:
+end for
+7:
+Set �µ = argmin
+µ
+b
+�
+i=1
+∥µ − �µi∥2
+return �µ
+8: end function
+3
+Robust Newton’s Method
+We now present our variant of robust Newton’s method. At each iterate, we will use gradient and
+Hessian estimates (g(θ), H(θ)) in place of (∇R(θ), ∇2R(θ)) in the update equation (5). We assume
+that these estimates satisfy the conditions described in the following definitions:
+Definition 1. (Prasad et al. [28]) A function g(θ) is a robust gradient estimator for a data set
+S = {zi}n
+i=1, if for functions αg and βg, with probability at least 1 − δ, at any fixed θ ∈ Θ, the
+estimator satisfies the following inequality:
+∥g(θ) − ∇R(θ)∥2 ≤ αg(n, δ)∥θ − θ∗∥2 + βg(n, δ).
+(7)
+Definition 2. A function H(θ) is a robust Hessian estimator for a data set S = {zi}n
+i=1, if for
+functions αh and βh, with probability at least 1 − δ, at any fixed θ ∈ Θ, the estimator satisfies the
+following inequality:
+∥H(θ) − ∇2R(θ)∥2 ≤ αh(n, δ)∥θ − θ∗∥2 + βh(n, δ).
+(8)
+6
+
+Successive iterates then take the form
+θt+1 = θt − αtH(θt)−1g(θt),
+where αt is chosen via a version of backtracking linesearch [2]. The exit condition of backtracking
+linesearch differs from its non-robust version in that function evaluations are replaced by robust
+estimates (cf. Lemmas 7 and 8 below) and an extra tolerance parameter ζ is included. The full
+algorithm is provided in Algorithm 3.
+Algorithm 3 Robust Newton’s Method
+Require: Data samples S = {zi}n
+i=1, Number of iterations T, Initial guess θ0 ∈ Θ, Backtracking
+linesearch parameters κ1 ∈ (0, 0.5), κ2 ∈ (0, 1), and ζ
+1: function RobustNewton(S, ξ, θ0, , κ1, κ2, ζ)
+2:
+for t = 0 to T − 1 do
+3:
+Compute losses {L(θt, zi)}n
+i=1 and gradients {∇L(θt, zi)}n
+i=1
+4:
+Compute gradient estimate g(θt) = RobustGradientEstimate(S, θt)
+5:
+Compute Hessian estimate H(θt) = RobustHessianEstimate(S, θt)
+6:
+Compute Newton step ∆θnt = −H(θt)−1g(θt)
+7:
+Compute stepsize α = BacktrackingLineSearch(S, θt, ∆θnt, g(θt), κ1, κ2, ζ)
+8:
+Update θt+1 = θt + α∆θnt
+9:
+end for
+return θT
+10: end function
+11:
+12: function BacktrackingLineSearch((S, θ, ∆θnt), g(θ), κ1, κ2, ζ)
+13:
+Set α = 1
+14:
+while RobustEstimate({L(θ + α∆θnt, zi)}n
+i=1) > RobustEstimate({L(θ, zi)}n
+i=1) +
+κ1αg(θ)∆θnt + ζ do
+15:
+Update α = κ2α
+16:
+end while
+return α
+17: end function
+3.1
+General Analysis for Robust Newton’s Method
+For the results of this section, we assume that f(θ) := R(θ) is twice-differentiable and satisfies
+the Lipschitz condition ∥∇2f(θ1) − ∇2f(θ2)∥2 ≤ L∥θ1 − θ2∥2, for all θ1, θ2. We also assume that
+f satisfies the strong convexity and smoothness conditions mI ⪯ ∇2f(θ) ⪯ MI, for all θ close
+enough to the initialization θ0. (We will verify these conditions for GLMs in Propositions 1 and 2
+below.) Finally, we will assume that at each iterate, the gradient and Hessian estimates g(θ) and
+H(θ) satisfy inequalities (7) and (8), respectively. As demonstrated in Theorems 3 and 4 later, the
+last condition can typically be justified w.h.p. via a union bound. Observe that in this setting, the
+unique global minimum of f is the true parameter θ∗.
+The first result shows that if ∥∇f(θ0)∥2 is sufficiently small, the backtracking linesearch pro-
+cedure will always choose stepsize 1. (This is known as the “pure Newton” phase.) Furthermore,
+successive iterates converge at a geometric rate to a small ball around θ∗. Recall that the parameters
+(κ1, κ2) of backtracking linesearch are defined as in Algorithm 3.
+7
+
+Theorem 1. Suppose ∥∇f(θ0)∥2 < η, where
+η := m2
+8L · min {3(1 − 2κ1), 2} .
+(9)
+Suppose the gradient and Hessian errors satisfy the bounds
+γg := 2ηαg
+m
++ βg ≤ η,
+and
+γh := 2ηαh
+m
++ βh ≤ m
+2 .
+(10)
+Also suppose the robust estimates satisfy
+|RobustEstimate({L(θt + α∆θt, zi)}n
+i=1) − f(θt + α∆θt)| ≤ ζ
+4,
+(11)
+for each evaluation of backtracking linesearch, where we set the linesearch parameter to be
+ζ ≥ 8γgη
+m
++ 16γhη2
+m2
+.
+(12)
+Then backtracking linesearch chooses unit steps on all successive iterates, and we have ∥∇f(θt)∥2 <
+η and
+∥θt − θ∗∥2 ≤ m
+L
+�1
+2
+�2t
++ 6c2
+m
+(13)
+for all t ≥ 1, where
+c2 = η
+�4γgL
+m2 + 2γh
+m
+�
++ 2Lγ2
+g
+m2
++ γg + 2γgγh
+m
+,
+and we further assume that (γg, γh) are small enough so that
+c2 ≤ min
+�η
+2, m2
+24L
+�
+.
+(14)
+Proof. Our first step is to show that backtracking linesearch chooses unit steps whenever the gra-
+dient is small, i.e., ∥∇f(θt)∥2 < η. In other words, we want to prove that
+˜f(θ + ∆θnt) ≤ ˜f(θ) − κ1˜λ(θ)2 + ζ,
+where θ = θt denotes the iterate, ˜f denotes the robust estimate of f, and we have defined the noisy
+Newton decrement
+˜λ(θ) :=
+�
+g(θ)T H−1(θ)g(θ)
+�1/2 .
+(15)
+Recall that ∆θnt = −H(θt)−1g(θt). Note that since ∥∇f(θt)∥2 < η, we have [2, Equation (9.11)]
+∥θt − θ∗∥2 ≤ 2
+m∥∇f(θt)∥2 < 2η
+m := γ0.
+(16)
+In particular, this implies a bound of γg := αgγ0 + βg on the error of the gradient, and a bound of
+γh := αhγ0 + βh on the error of the Hessian, according to Definitions 1 and 2. We will show that
+f(θ + ∆θnt) ≤ f(θ) − κ1˜λ(θ)2 + ζ
+2,
+(17)
+from which the desired result clearly follows by the accuracy bound (11) on the robust estimates
+and the triangle inequality.
+8
+
+Note that ˜λ(θ)2 = ∆θT
+ntH(θ)∆θnt, implying that
+˜λ(θ)2 ≥ (m − γh)∥∆θnt∥2
+2 > m
+2 ∥∆θnt∥2
+2
+(18)
+(where we assume γh ≤ m
+2 ). Furthermore, by the Lipschitz condition, for u ≥ 0, we have
+∥∇2f(θ + u∆θnt) − ∇2f(θ)∥2 ≤ uL∥∆θnt∥2,
+so
+��∆θT
+nt
+�
+∇2f(θ + u∆θnt) − ∇2f(θ)
+�
+∆θnt
+�� ≤ uL∥∆θnt∥3
+2.
+(19)
+Defining ¯f(u) := f(θ + u∆θnt), we have ¯f′′(u) = ∆θT
+nt∇2f(θ + u∆θnt)∆θnt, so we can rewrite
+inequality (19) as
+| ¯f′′(u) − ¯f′′(0)| ≤ uL∥∆θnt∥3
+2,
+implying that
+¯f′′(u) ≤ ¯f′′(0) + uL∥∆θnt∥3
+2 ≤ ¯f′′(0) + uL
+� 2
+m
+�3/2
+˜λ(θ)3,
+using inequality (18). Integrating with respect to u gives
+¯f′(u) ≤ ¯f′(0) + u ¯f′′(0) + u2L
+2
+� 2
+m
+�3/2
+˜λ(θ)3,
+and a second integration gives
+¯f(u) ≤ ¯f(0) + u ¯f′(0) + u2
+2
+¯f′′(0) + u3L
+6
+� 2
+m
+�3/2
+˜λ(θ)3.
+(20)
+Now note that
+¯f′(0) = ∇f(θ)T ∆θnt
+= −∇f(θ)T H−1(θ)g(θ)
+= −˜λ(θ)2 + (g(θ) − ∇f(θ))H−1(θ)g(θ)
+≤ −˜λ(θ)2 + γg
+1
+m − γh
+(∥∇f(θ)∥2 + γg)
+≤ −˜λ(θ)2 + γg
+1
+m − γh
+(η + γg) ,
+(21)
+whereas
+¯f′′(0) = ∆θT
+nt∇2f(θ)∆θnt
+= ˜λ(θ)2 + ∆θT
+nt
+�
+∇2f(θ) − H(θ)
+�
+∆θnt
+≤ ˜λ(θ)2 + γh∥∆θnt∥2
+2
+≤ ˜λ(θ)2
+�
+1 + 2γh
+m
+�
+,
+(22)
+using the bound (18) in the last inequality. Plugging inequalities (21) and (22) into inequality (20)
+(with u = 1) then gives
+f(θ + ∆θnt) ≤ f(θ) +
+�
+−˜λ(θ)2 + γg(η + γg)
+m − γh
+�
++
+˜λ(θ)2
+2
+�
+1 + 2γh
+m
+�
++ L
+6
+� 2
+m
+�3/2
+˜λ(θ)3.
+9
+
+Finally, note that
+˜λ(θ) ≤ ∥H−1/2(θ)∥2 · ∥g(θ)∥2 ≤ ∥∇f(θ)∥2 + γg
+√m − γh
+≤
+η + γg
+√m − γh
+.
+Since
+ζ ≥ 2
+�
+�γg · 2η
+m/2 + γh
+m
+�
+2η
+�
+m/2
+�2�
+�
+and using the assumptions γh ≤ m
+2 and γg ≤ η, we then have
+f(θ + ∆θnt) ≤ f(θ) − ˜λ(θ)2
+�
+1
+2 − L
+6
+� 2
+m
+�3/2
+˜λ(θ)
+�
++ ζ
+2.
+In particular, if ˜λ(θ) ≤
+3−6κ1
+L(2/m)3/2 , which is guaranteed if η is chosen sufficiently small so
+η + γg
+√m − γh
+≤
+3 − 6κ1
+L(2/m)3/2 ,
+then inequality (17) is indeed satisfied.
+We can guarantee this last inequality by taking η ≤
+3m2(1−2κ1)
+8L
+, assuming γh ≤ m
+2 and γg ≤ η.
+To derive the geometric convergence rate (13), we will use induction. We first establish an
+inequality of the form
+∥∇f(θ + ∆θnt)∥2 ≤ c1∥∇f(θ)∥2
+2 + c2,
+(23)
+assuming ∥∇f(θ)∥2 < η. Note that
+∥∇f(θ + ∆θnt)∥2 = ∥∇f(θ + ∆θnt) − g(θ) − H(θ)∆θnt∥2
+≤ ∥∇f(θ + ∆θnt) − ∇f(θ) − ∇2f(θ)∆θnt∥2 + γg + γh∥∆θnt∥2
+=
+����
+� 1
+0
+�
+∇2f(θ + u∆θnt) − ∇2f(θ)
+�
+∆θntdu
+����
+2
++ γg + γh∥∆θnt∥2
+≤ L
+2 ∥∆θnt∥2
+2 + γg + γh∥∆θnt∥2,
+(24)
+using the Lipschitz condition in the second inequality. Next, we use the bound
+∥∆θnt∥2 = ∥H−1(θ)g(θ)∥2 ≤
+1
+m − γh
+(∥∇f(θ)∥2 + γg) ≤ 2
+m (∥∇f(θ)∥2 + γg) ,
+assuming γh ≤ m
+2 . Plugging back into inequality (24) gives
+∥∇f(θ + ∆θnt)∥2 ≤ L
+2
+�2 (∥∇f(θ)∥2 + γg)
+m
+�2
++ γg + γh
+�2 (∥∇f(θ)∥2 + γg)
+m
+�
+= 2L
+m2 ∥∇f(θ)∥2
+2 + ∥∇f(θ)∥2
+�4γgL
+m2 + 2γh
+m
+�
++
+�
+2Lγ2
+g
+m2
++ γg + 2γgγh
+m
+�
+≤ 2L
+m2 ∥∇f(θ)∥2
+2 + η
+�4γgL
+m2 + 2γh
+m
+�
++ 2Lγ2
+g
+m2
++ γg + 2γgγh
+m
+,
+(25)
+10
+
+giving inequality (23) with c1 = 2L
+m2 and c2 = η
+�
+4γgL
+m2 + 2γh
+m
+�
++
+2Lγ2
+g
+m2 + γg + 2γgγh
+m . In particular, c2
+can be made small if we choose γg and γh small enough, and we will assume that c2 ≤ η
+2. We will
+also assume that c1c2 ≤ 1
+12.
+We are now ready for our induction. Using the notation yt := c1∥∇f(θt)∥2, we will prove that
+yt < c1η and yt ≤ y2t
+0 + c1c2 for all t ≥ 1. For the base case t = 1, note that
+y1 ≤ y2
+0 + c1c2 = y21
+0 + c1c2,
+using inequality (23). Furthermore, since y0 < c1η < 1
+2 and c2 ≤ η
+2 by assumption, we have
+y1 ≤ y0
+2 + c1η
+2
+< c1η
+2 + c1η
+2
+= c1η.
+For the inductive step, suppose t ≥ 1, and we have ys < c1η and ys ≤ y2s
+0 + 3c1c2 for all s ≤ t.
+Then by inequality (23), we have
+yt+1 ≤ y2
+t + c1c2.
+(26)
+Furthermore, using the assumption η ≤ m2
+4L , we have yt < 1
+2, so if c2 ≤ η
+2, this implies that
+yt+1 ≤ yt
+2 + c1η
+2
+< c1η
+2 + c1η
+2
+≤ c1η.
+By inequality (26) and the induction hypothesis, we now write
+yt+1 ≤ y2
+t + c1c2
+≤
+�
+y2t
+0 + 3c1c2
+�2
++ c1c2
+= y2t+1
+0
++ 6c1c2y2t
+0 + 9c2
+1c2
+2 + c1c2
+≤ y2t+1
+0
++ 3
+2c1c2 + 3
+4c1c2 + c1c2
+≤ y2t+1
+0
++ 3c1c2,
+using the assumption c1c2 ≤ 1
+12. This completes the induction.
+Thus,
+c1∥∇f(θt)∥2 ≤
+�1
+2
+�2t
++ 3c1c2.
+Applying inequality (16) then gives the convergence rate
+∥θt+1 − θ∗∥2 ≤ 2
+m · m2
+2L
+��1
+2
+�2t+1
++ 6L
+m2 c2
+�
+,
+completing the proof.
+Next, we show that after a finite number of steps, the iterates will indeed satisfy ∥∇f(θt)∥2 < η,
+for an appropriate η.
+Theorem 2. Suppose the parameters η and ζ are as defined in Theorem 1. Define
+γ := κ1 · κ2
+m
+M
+�1
+2 − κ1
+�
+·
+η2
+4
+√
+2M
+.
+11
+
+Suppose (γg, γh) are chosen small enough such that ζ ≤ γ
+2 and conditions (10) and (11) are satisfied.
+Also suppose
+2αg
+m
+�
+2M (f(θ0) − f(θ∗)) + βg ≤ min
+�
+η
+2,
+�m
+2 · 1
+2
+�
+η2
+4
+√
+2M
+�
+,
+(27)
+and
+2αh
+m
+�
+2M (f(θ0) − f(θ∗)) + βh ≤ M.
+(28)
+If ∥∇f(θt)∥2 ≥ η for an iterate t ≥ 0, then f(θt) ≤ f(θ0) and f(θt+1) − f(θt) < − γ
+2.
+Proof. First, we show that we have an upper bound γ′
+0 :=
+2
+m
+�
+2M (f(θ0) − f(θ∗)) on ∥θt − θ∗∥2.
+We can then translate this into upper bounds γ′
+g := αgγ′
+0 + βg and γ′
+h := αhγ′
+0 + βh on the gradient
+and Hessian deviations, respectively. By the result of Theorem 1, we must have ∥∇f(θs)∥2 ≥ η for
+all 0 ≤ s ≤ t. We now show by induction that:
+1. f(θs) ≤ f(θ0), and
+2. ∥θs − θ∗∥2 ≤ γ′
+0,
+for all 0 ≤ s ≤ t. For the base case s = 0, note that claim (1) is obvious. We can establish claim
+(2) by noting that
+∥θs − θ∗∥2 ≤ 2
+m∥∇f(θs)∥2 ≤ 2
+m
+�
+2M(f(θs) − f(θ∗)) ≤ 2
+m
+�
+2M(f(θ0) − f(θ∗)) = γ′
+0,
+(29)
+using inequality (16), inequality (9.14) of Boyd and Vandenberghe [2], and claim (1).
+Turning to the inductive step, suppose claims (1) and (2) hold for all s ≤ s′, where 0 ≤ s′ < t.
+We wish to establish the claims for s = s′ + 1. Note that if we prove claim (1), then claim (2)
+follows by the same chain of inequalities (29). Thus, it remains to establish claim (1).
+Assuming γ′
+g ≤
+η
+2, we have ∥g(θs)∥2 ≥
+η
+2 for all 0 ≤ s ≤ s′ by claim (2), the fact that
+∥∇f(θs)∥2 ≥ η, and the triangle inequality. Using the same notation for the Newton decrement (15),
+we note that
+˜λ(θs)2 ≥
+1
+�
+M + γ′
+h
+∥g(θs)∥2
+2 ≥
+η2
+4
+√
+2M
+,
+(30)
+where we assume γ′
+h ≤ M.
+First, we will prove that the exit condition of BacktrackingLineSearch function will be satis-
+fied, i.e., we want to prove that
+˜f(θs′ + α∆θnt) ≤ ˜f(θ) − κ1α˜λ(θs′)2 + ζ
+(31)
+holds for small enough α, where ˜f is the robust estimate of f. For convenience, we use the notation
+θ = θs′ in what follows. In fact, we will show that
+f(θ + α∆θnt) ≤ f(θ) − κ1α˜λ(θ)2
+(32)
+for small enough α, which clearly then implies inequality (31) by the triangle inequality and the
+condition (11).
+12
+
+Consider the following:
+f(θ + α∆θnt) ≤ f(θ) + α∇f(θ)T ∆θnt + M
+2 ∥∆θnt∥2
+2α2
+≤ f(θ) + αg(θ)∆θnt + αγ′
+g∥∆θnt∥2 + M
+2
+˜λ(θ)2 2
+mα2
+= f(θ) − α˜λ(θ)2 + αγ′
+g∥∆θnt∥2 + M
+m
+˜λ(θ)2α2
+≤ f(θ) − α˜λ(θ)2 + αγ′
+g˜λ(θ)
+�
+2
+m + M
+m
+˜λ(θ)2α2,
+where we use the relation −˜λ(θ)2 = g(θ)T ∆θnt and inequality (18). Assuming γ′
+g
+�
+2
+m ≤ 1
+2
+�
+η2
+4
+√
+2M
+and using inequality (30), the last expression is upper-bounded as
+f(θ + α∆θnt) ≤ f(θ) − α
+2
+˜λ(θ)2 + M
+m
+˜λ(θ)2α2
+= f(θ) − ˜λ(θ)2α
+�1
+2 − M
+m α
+�
+.
+(33)
+Hence, the condition (32) is indeed satisfied for sufficiently small α, i.e., α ≤ m
+M
+� 1
+2 − κ1
+�
+, and in
+particular, the linesearch procedure must return a stepsize satisfying α ≥ κ2 m
+M
+� 1
+2 − κ1
+�
+. Plugging
+such a stepsize into inequality (31), we have
+f(θs′+1) ≤ f(θs′) − κ1 · κ2
+m
+M
+�1
+2 − κ1
+�
+·
+η2
+4
+√
+2M
++ ζ ≤ f(θ0) − γ + ζ ≤ f(θ0) − γ
+2,
+(34)
+using inequality (30) and the induction hypothesis. This implies that claim (1) is true, completing
+the induction.
+Finally, note that the inequality f(θt+1)−f(θt) < − γ
+2 follows by the same argument in inequal-
+ity (34) with θ = θt, completing the proof.
+The preceding theorem directly implies that after a finite number of steps (known as the
+“damped Newton” phase), all successive iterates of the algorithm satisfy ∥∇f(θt)∥2 < η.
+3.2
+Robust Estimation of Gradients and Hessians
+In this subsection, we explain how robust estimators for gradients and Hessians can be obtained
+under two models of contamination, namely the Huber ϵ-contamination model and the heavy-tailed
+model.
+3.2.1
+Robust Gradient Estimation
+For the ϵ-contamination model, we obtain a robust gradient estimate by applying Algorithm 1 to
+the gradients computed on each of the n sampled data points. Similarly, for the heavy-tailed model,
+we use Algorithm 2 to obtain a robust gradient estimate. For completeness, we summarize this
+procedure in Algorithm 4.
+13
+
+Algorithm 4 Robust Gradient Estimator
+Require: Samples S = {zi}n
+i=1, Parameter θ, Contamination type Type
+Require: (If Type = Huber) Corruption Level ϵ, Dimension p, Failure probability δ
+Require: (If Type = Heavy-tail) Failure probability δ
+1: function RobustGradientEstimator(S, θ, Type, ϵ, p, δ)
+2:
+Compute {∇L(θ, zi)}n
+i=1, the gradient of the loss at each data point in S
+3:
+if Type = Huber then
+return HuberEstimator({∇L(θ, zi)}n
+i=1, ϵ, p, δ)
+4:
+end if
+5:
+if Type = Heavy-tail then
+return HeavyTailedEstimator({∇L(θ, zi)}n
+i=1, δ)
+6:
+end if
+7: end function
+The following lemmas, borrowed from Prasad et al. [28], show that Algorithm 4 returns a robust
+gradient estimator that satisfies Definition 1.
+Lemma 1 (Lemma 1 of Prasad et al. [28]). Let {zi}n
+i=1 be n i.i.d. samples drawn from a Huber
+ϵ-contaminated distribution (6). Let the true distribution of gradients ∇L(θ, z), with z drawn from
+P, have bounded fourth moments. Then Algorithm 4 with S = {zi}n
+i=1, Type = Huber, and any
+θ ∈ Θ returns a gradient estimate g(θ) that satisfies
+∥g(θ) − E[∇L(θ, z)]∥2 ≤ C1(√ϵ + γ(n, p, δ, ϵ))
+�
+∥ Cov(∇L(θ, z))∥2 log p,
+(35)
+with probability at least 1 − δ, where C1 > 0 is a constant and γ is given by
+γ(n, p, δ, ϵ) =
+�p log(p) log(n/(pδ))
+n
+�3/8
++
+�ϵp2 log(p) log(p log(p)/δ)
+n
+�1/4
+.
+(36)
+Lemma 2 (Lemma 2 of Prasad et al. [28]). Let {zi}n
+i=1 be n i.i.d. samples drawn from a heavy-tailed
+distribution P such that the true distribution of gradients ∇L(θ, z) has bounded second moments.
+Then Algorithm 4 with S = {zi}n
+i=1, Type = Heavy-tail, and any θ ∈ Θ returns a gradient estimate
+g(θ) that satisfies
+∥g(θ) − E[∇L(θ, z)]∥2 ≤ 11
+�
+tr(Cov(∇L(θ, z))) log(1.4/δ)
+n
+,
+(37)
+with probability at least 1 − δ.
+3.2.2
+Robust Hessian Estimation: The Vectorizing Approach
+The procedure for obtaining a robust Hessian, summarized in Algorithm 5, is similar to that
+of Algorithm 4, except that the appropriate multivariate estimation procedure is applied to a
+vectorized version of the Hessian matrix (where we use flatten(A) to denote a vectorized version of
+the matrix A, and use unflatten() to denote the inverse function).
+14
+
+Algorithm 5 Robust Hessian Estimator
+Require: Samples S = {zi}n
+i=1, Parameter θ, Contamination type Type
+Require: (If Type = Huber) Corruption Level ϵ, Dimension p, Failure probability δ
+Require: (If Type = Heavy-tail) Failure probability δ
+1: function RobustHessianEstimator(S, θ, Type, ϵ, p, δ)
+2:
+Compute {∇2L(θ, zi)}n
+i=1, the Hessian of the loss at each data point in S
+3:
+if Type = Huber then
+return unflatten(HuberEstimator({flatten(∇2L(θ, zi))}n
+i=1, ϵ, p, δ))
+4:
+end if
+5:
+if Type = Heavy-tail then
+return unflatten(HeavyTailedEstimator({flatten(∇2L(θ, zi))}n
+i=1, δ))
+6:
+end if
+7: end function
+The next two lemmas follow immediately from the arguments used to derive Lemmas 1 and 2:
+Lemma 3. Let {zi}n
+i=1 be n i.i.d. samples drawn from a Huber ϵ-contaminated distribution (6).
+Suppose Cov(flatten(∇2L(θ, z))) is finite and flatten(∇2L(θ, z))) has bounded fourth moments.
+Then Algorithm 4 with S = {zi}n
+i=1, Type = Huber, and any θ ∈ Θ returns a Hessian estimate
+H(θ) that satisfies
+∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C2(√ϵ + γ(n, p, δ, ϵ))
+�
+∥ Cov(flatten(∇2L(θ, z)))∥2 log p,
+(38)
+with probability at least 1 − δ, where C2 > 0 is a constant and γ is given by equation (36).
+Lemma 4. Let {zi}n
+i=1 be n i.i.d. samples drawn from a heavy-tailed distribution P.
+Suppose
+Cov(flatten(∇2L(θ, z))) is finite. Then Algorithm 4 with S = {zi}n
+i=1, Type = Heavy-tail, and any
+θ ∈ Θ returns a Hessian estimate H(θ) that satisfies
+∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C3
+�
+tr(Cov(flatten(∇2L(θ, z)))) log(1.4/δ)
+n
+,
+(39)
+with probability at least 1 − δ, where C3 > 0 is a constant.
+4
+Application to GLMs
+In this section, we apply the robust Newton method to parametric estimation in GLMs. We consider
+the Huber ϵ-contamination model in Section 4.1, and we consider the heavy-tailed contamination
+model in Section 4.2.
+Throughout this section, we will assume that the uncontaminated model is a GLM of the
+form (3). Consider the loss function in equation (4). We assume that the link function Φ of the
+GLM satisfies the following bounds:
+E
+�
+|Φ′(xT
+i θ) − Φ′(xT
+i θ∗)|2k�
+≤ LΦ,2k∥θ − θ∗∥2
+2 + BΦ,2k,
+∀θ ∈ Θ,
+(40)
+E
+�
+|Φ(t)(xT
+i θ∗)|k�
+≤ MΦ,t,k,
+(41)
+and
+∥Φ(t)∥∞ ≤ MΦ,t,
+(42)
+15
+
+for pairs (k, t) to be specified in the sequel, where Φ(t) is the tth derivative of Φ.
+We also make assumptions on the boundedness of moments of xi ∈ Rp. We say that xi has
+bounded 2kth moments if there is a constant �C2k > 0 such that for every unit vector v ∈ Rd, we
+have E[(xT
+i v)2k] ≤ �C2k
+�
+E[(xT
+i v)2]
+�k. We assume that the covariate distribution has bounded eighth
+moments and a finite covariance matrix Σx.
+From Lemmas 1 and 2, we see that the term Cov(∇L(θ, z)) plays a crucial role in proving
+that our gradient estimates are robust. Likewise, Lemmas 3 and 4 show the importance of the
+term Cov(flatten(∇2L(θ, z))) in proving that the Hessian estimates are robust. The following two
+lemmas provide upper bounds on these two terms for the specific case of GLMs:
+Lemma 5 (Lemma 4 in Prasad et al. [28]). Let {zi}n
+i=1 be n i.i.d. samples drawn from a distribution
+that satisfies the GLM model (3) with the assumptions above.
+Let the link function Φ satisfy
+inequalities (40) and (41) for k ∈ {1, 2} and t ∈ {2, 4}. Then the true distribution of gradients
+∇L(θ, z) has bounded fourth moments. Moreover,
+∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2
+��
+LΦ,4 + LΦ,2
+�
+∥θ − θ∗∥2
+2
++ C2∥Σx∥2
+�
+BΦ,2 +
+�
+BΦ,4 + c(σ)
+�
+MΦ,2,2 +
+�
+c(σ)3MΦ,4,1
+�
+,
+(43)
+where C1, C2 > 0 are constants.
+Lemma 6. Let {zi}n
+i=1 be n i.i.d. samples drawn from a distribution that satisfies the GLM
+model (3) with the assumptions above. Let the link function Φ satisfy inequality (42) for t = 2. Then
+the distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth moments. Moreover,
+we have
+tr(Cov(flatten(∇2L(θ, z)))) ≤ M
+2
+Φ,2 �C4p2∥Σx∥2
+2.
+(44)
+Proof. From the definition of the loss function (4), we have E[∇2L(θ, z)] = E[Φ′′(xT
+i θ)xixT
+i ]. By
+our assumptions on the boundedness of Φ′′ and bounded eighth moments of xi, we see that the
+distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth moments. We then
+write
+tr(Cov(flatten(∇2L(θ, z)))) = tr(Cov(flatten(Φ′′(xT
+i θ)xixT
+i )))
+≤ tr(E[flatten(Φ′′(xT
+i θ)xixT
+i ) flatten(Φ′′(xT
+i θ)xixT
+i )T ])
+≤ M
+2
+Φ,2 tr(E[flatten(xixT
+i ) flatten(xixT
+i )T ])
+= M
+2
+Φ,2
+p
+�
+j,k=1
+E[x2
+ijx2
+ik]
+≤ M
+2
+Φ,2E
+�
+�
+�
+�
+p
+�
+j=1
+xij
+�
+�
+4�
+�
+≤ M
+2
+Φ,2E
+��
+xT
+i 1
+�4�
+,
+where 1 denotes the all-ones vector. Finally, note that
+E
+��
+xT
+i 1
+�4�
+≤ �C4E
+��
+xT
+i 1
+�2�2
+≤ �C4p2∥Σx∥2
+2,
+implying the desired result.
+16
+
+Remark 1. Note that under additional assumptions (e.g., 4-wise independence of the components
+of the xi’s), we can prove that
+∥ Cov(flatten(∇2L(θ, z)))∥2 ≤ C �C4∥Σx∥2
+2,
+for some constant C > 0, which avoids an extra dimension-dependent factor in comparison to
+inequality (44) (cf. Proposition 4.2 in Lai et al. [15]) for the Huber contamination setting. Indeed,
+only the spectral norm of the covariance of the flattened Hessian appears in the deviation bound
+of Lemma 3 (Huber’s ϵ-contamination model); the trace of the covariance appears in Lemma 4
+(heavy-tailed model).
+In order to apply Theorems 1 and 2 to GLMs, we need the Hessian ∇2R(θ) to be Lipschitz
+smooth and satisfy mI ⪯ ∇2R(θ) ⪯ MI for all θ ∈ Θ close enough to the initialization θ0. We now
+verify these assumptions.
+Proposition 1. Let the link function Φ satisfy inequality (42) for t = 3. Then the Hessian ∇2R(θ)
+is L-Lipschitz and satisfies ∇2R(θ) ⪯ MI with L :=
+�
+�C4MΦ,3∥Σx∥2 and M := MΦ,2
+�
+�C4∥Σx∥2.
+Proof. For the Lipschitz condition, note that for any θ1, θ2 ∈ Rp, we have
+∥∇2R(θ1) − ∇2R(θ2)∥2 = ∥∇2L(θ1, z) − ∇2L(θ2, z)∥2
+=
+��E
+�
+xixT
+i
+�
+Φ′′(xT
+i θ1) − Φ′′(xT
+i θ2)
+����
+2
+=
+sup
+u∈Sp−1 uT E
+�
+xixT
+i
+�
+Φ′′(xT
+i θ1) − Φ′′(xT
+i θ2)
+��
+u
+=
+sup
+u∈Sp−1 E
+�
+(uT xi)2 �
+Φ′′(xT
+i θ1) − Φ′′(xT
+i θ2)
+��
+≤
+sup
+u∈Sp−1 E
+�
+(uT xi)4� 1
+2 E
+��
+Φ′′(xT
+i θ1) − Φ′′(xT
+i θ2)
+�2� 1
+2
+≤
+sup
+u∈Sp−1
+�
+�C4E
+�
+(uT xi)2� �
+MΦ,3∥θ1 − θ2∥2
+≤
+�
+�C4MΦ,3∥Σx∥2∥θ1 − θ2∥2,
+where we use the mean value theorem to upper-bound the expectation in the second-to-last in-
+equality.
+For any θ ∈ Rp, we have
+∥∇2R(θ)∥2 = ∥∇2L(θ, z)∥2
+=
+��E
+�
+xixT
+i
+�
+Φ′′(xT
+i θ)
+����
+2
+=
+sup
+u∈Sp−1 uT E
+�
+xixT
+i
+�
+Φ′′(xT
+i θ)
+��
+u
+=
+sup
+u∈Sp−1 E
+�
+(uT xi)2 �
+Φ′′(xT
+i θ)
+��
+≤
+sup
+u∈Sp−1 E
+�
+(uT xi)4� 1
+2 E
+��
+Φ′′(xT
+i θ)
+�2� 1
+2
+≤
+sup
+u∈Sp−1
+�
+�C4E
+�
+(uT xi)2�
+MΦ,2
+≤ MΦ,2
+�
+�C4∥Σx∥2.
+17
+
+Proposition 2. Suppose there exist constants B, τ > 0 such that for any θ ∈ Rp such that ∥θ∥2 ≤
+B, we have
+�C4∥Σx∥2
+2 · P(|xT
+i θ| > τ) ≤ 1
+4λ2
+min(Σx).
+(45)
+Define bτ := inf|u|≤τ Φ′′(u). Then bτ
+2 λmin(Σx)I ⪯ ∇2R(θ) for all θ ∈ Rp such that ∥θ∥2 ≤ B.
+Proof. Suppose v ∈ Rp is a unit vector. We write
+vT ∇2R(θ)v = E
+�
+(vT xi)2 · Φ′′(xT
+i θ)
+�
+≥ E
+�
+(vT xi)2 · bτ1{|xT
+i θ| ≤ τ}
+�
+= bτ
+�
+E
+�
+(vT xi)2�
+− E
+�
+(vT xi)2 · 1{|xT
+i θ| > τ}
+��
+≥ bτ
+�
+λmin(Σx) −
+�
+E [(vT xi)4] · P
+�
+|xT
+i θ| > τ
+��
+≥ bτ
+�
+λmin(Σx) −
+�
+�C4∥Σx∥2
+2 · P
+�
+|xT
+i θ| > τ
+��
+≥ bτ
+2 λmin(Σx),
+where we have used the fact that Φ′′ is always nonnegative in the first inequality, applied Cauchy-
+Schwarz in the second inequality, and used the assumption (45) in the last inequality.
+Remark 2. Note that when the covariates are sub-Gaussian, we can certainly guarantee that the
+tail condition (45) is satisfied for sufficiently large τ, since xT
+i θ is sub-Gaussian with parameter
+scaling with B and the sub-Gaussian parameter σ2
+x of xi. Thus, we have
+P(|xT
+i θ| > τ) ≤ c1 exp
+�
+− c2τ 2
+B2σ2x
+�
+,
+and it suffices to take τ = c3Bσx log1/2 � c4∥Σx∥2
+2
+λ2
+min(Σx)
+�
+. Furthermore, in the proofs of Theorems 1 and 2
+(cf. inequalities (16) and (29), respectively), we show that ∥θt − θ∗∥2 remains bounded (where the
+bound depends on θ0 and the problem parameters).
+In the case of logistic regression, we have Φ′′(u) =
+eu
+(1+eu)2 , and it is easy to see that bτ > 0 for
+any value of τ.
+For applying Theorems 1 and 2, we also need the robust estimate of the losses to be close to the
+population risk, as in inequality (11). In the following two lemmas, we show that this assumption
+holds with high probability for the robust estimates obtained by applying Algorithms 1 and 2 on
+the losses. Further note that the following lemmas require boundedness of higher-order moments
+of L(θ, z), which can be justified in our scenario if θ is bounded. As mentioned in Remark 2, we
+can indeed assume that the iterates {θt}, to which Lemmas 7 and 8 are applied in the sequel, are
+bounded.
+Lemma 7. Let {zi}n
+i=1 be n i.i.d. samples drawn from a Huber ϵ-contaminated distribution (6),
+where the true distribution satisfies the GLM model (3) with the assumptions above. Let L(θ, z)
+have bounded fourth moments. Then with probability at least 1 − δ, the robust estimate returned by
+Algorithm 1 satisfies
+|HuberEstimate({L(θ, zi)}n
+i=1) − R(θ)|
+≤ C1
+�
+ϵ +
+�
+log(n/δ)
+n
+� 3
+4
++ C2
+�
+ϵ +
+�
+log(n/δ)
+n
+� 1
+2 log(1/δ)
+n
+,
+18
+
+where C1, C2 > 0 are constants.
+Proof. The desired result follows from an application of Lemma 14 in Prasad et al. [28].
+Lemma 8. Let {zi}n
+i=1 be n i.i.d. samples drawn from a heavy-tailed distribution P that satisfies
+the GLM model (3) with the assumptions above. Let L(θ, z) have bounded second moments. Then
+with probability at least 1 − δ, the robust estimate returned by Algorithm 2 satisfies
+|HeavyTailedEstimate({L(θ, zi)}n
+i=1) − R(θ)| ≤ C
+�
+log
+� 1.4
+δ
+�
+n
+,
+with probability at least 1 − δ, where C > 0 is a constant.
+Proof. This result follows from similar arguments to those in Lemma 2.
+The following lemma shows how small the parameters (αg, βg, αh, βh) in the robust gradient
+and Hessian estimates need to be in order to satisfy the assumptions of Theorems 1 and 2.
+Lemma 9. Define
+�αg := min
+�
+m
+64,
+ηm
+8
+�
+2M(f(θ0) − f(θ∗))
+,
+�
+η2m
+8
+√
+2M
+.
+m
+8
+�
+2M(f(θ0) − f(θ∗))
+�
+,
+�βg := min
+�
+η
+32,
+ηm
+8
+�
+2M(f(θ0) − f(θ∗))
+, 1
+4
+�
+η2m
+8
+√
+2M
+�
+,
+�αh := min
+�
+m2
+256η,
+mM
+4
+�
+2M(f(θ0) − f(θ∗))
+�
+,
+�βh := min
+� m
+128, M
+2
+�
+.
+Suppose αg ≤ �αg, βg ≤ �βg, αh ≤ �αh, and βh ≤ �βh. Then the bounds (10) and (14) of Theorem 1,
+as well as the bounds (27) and (28) of Theorem 2, are satisfied.
+Proof. Under the assumptions, we have
+γg = 2ηαg
+m
++ βg ≤ η
+32 + η
+32 = η
+16,
+γh = 2ηαh
+m
++ βh ≤ m
+128 + m
+128 = m
+64,
+2αg
+m
+�
+2M (f(θ0) − f(θ∗)) + βg ≤ min
+�
+η
+2,
+�m
+2 · 1
+2
+�
+η2
+4
+√
+2M
+�
+,
+2αh
+m
+�
+2M (f(θ0) − f(θ∗)) + βh ≤ M.
+Hence, inequalities (10), (27), and (28) are satisfied. Using the fact that η := m2
+8L ·min {3(1 − 2κ1), 2} ≤
+19
+
+m2
+4L , we have
+L
+m2 ≤
+1
+4η. Then
+c2 = η
+�4γgL
+m2 + 2γh
+m
+�
++ 2Lγ2
+g
+m2
++ γg + 2γgγh
+m
+≤ η
+�4η
+16 · 1
+4η + 2
+m · m
+64
+�
++ 2η2
+256 · 1
+4η + η
+16 + 2
+m · η
+16 · m
+64
+< η
+6
+≤ m2
+24L.
+Hence, inequality (14) is satisfied.
+In Propositions 3 and 4, we will derive expressions for (αg, βg, αh, βh) for the Huber contami-
+nation and heavy-tailed models, which will then allow us to translate the conditions of Lemma 9
+into assumptions involving the contamination level and/or minimum sample size required for our
+theoretical results to hold.
+4.1
+Huber Contamination
+In this subsection, we assume that the link function Φ satisfies inequality (40) for k ∈ {1, 2, 4},
+inequality (41) for t ∈ {2, 4}, and inequality (42) for t ∈ {2, 3}. We assume that the population
+risk R(θ) is m-strongly convex. Define L and M as in Proposition 1.
+Proposition 3. Under the assumptions above, the gradient and Hessian estimates with Type =
+Huber returned by Algorithms 4 and 5, respectively, satisfy the conditions of Definitions 1 and 2
+with the following parameters:
+αg = c1(√ϵ + γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p,
+βg = c2(√ϵ + γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p,
+αh = 0,
+βh = c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p
+�
+log p,
+with probability at least 1 − δ.
+Proof. By Lemma 5, the true distribution of the gradients ∇L(θ, z) has bounded fourth moments.
+Moreover,
+∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2
+��
+LΦ,4 + LΦ,2
+�
+∥θ − θ∗∥2
+2
++ C2∥Σx∥2
+�
+BΦ,2 +
+�
+BΦ,4 + c(σ)
+�
+MΦ,2,2 +
+�
+c(σ)3MΦ,4,1
+�
+.
+Plugging the above bound into inequality (35) of Lemma 1, we obtain
+∥g(θ) − E[∇L(θ, z)]∥2 ≤ C′
+1(√ϵ + γ(n, p, δ, ϵ))
+�
+∥ Cov(∇L(θ, z))∥2 log p
+≤ c1(√ϵ + γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p · ∥θ − θ∗∥2
++ c2(√ϵ + γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p.
+20
+
+Hence, the gradient estimate returned by Algorithm 4 satisfies Definition 1 with αg = c1(√ϵ +
+γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p and βg = c2(√ϵ + γ(n, p, δ, ϵ))
+�
+∥Σx∥2 log p.
+By Lemma 6, the true distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded
+fourth moments. Moreover, combining Lemma 6 with Lemma 3, we obtain
+∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C′
+2(√ϵ + γ(n, p, δ, ϵ))
+�
+∥ Cov(flatten(∇2L(θ, z))))∥2 log p
+≤ c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p
+�
+log p.
+Hence, the Hessian estimate returned by Algorithm 5 satisfies Definition 2 with αh = 0 and βh =
+c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p√log p.
+We then have the following result:
+Theorem 3. Let {zi}n
+i=1 be n i.i.d. samples drawn from a Huber ϵ-contaminated distribution (6),
+where the true distribution satisfies the GLM model (3) and the aforementioned assumptions are
+satisfied. Define (η, c2, γg, γh) as in Theorem 1 and γ as in Theorem 2. Suppose
+ζ = C max
+�
+�
+�
+γgη
+m + γhη2
+m2 ,
+�
+ϵ +
+�
+log(n/δ)
+n
+� 3
+4
++
+�
+ϵ +
+�
+log(n/δ)
+n
+� 1
+2 log(1/δ)
+n
+�
+�
+� ≤ γ
+2.
+(46)
+Let δ > 0. Define
+�γ := min
+�
+�αg
+c1
+�
+∥Σx∥2 log p
+,
+�βg
+c2
+�
+∥Σx∥2 log p
+,
+�βh
+c3∥Σx∥2p√log p
+�
+,
+(47)
+where (�αg, �βh, �βg) are as defined in Lemma 9. Suppose n and ϵ are such that
+√ϵ + γ (n, p, δ, ϵ) < �γ.
+(48)
+Then applying Algorithm 3 on {zi}n
+i=1 with initialization θ0 ∈ Θ and number of iterations
+T ≥ R(θ0) − R(θ∗)
+γ/2
++ log2 log2
+�6c2L
+m2
+�
+returns an output such that
+∥θT − θ∗∥2 ≤ 12c2
+m
+= O
+�
+p2�
+ϵ log p
+�
+,
+with probability at least 1 − Tδ
+�
+5 +
+� log( m
+M ( 1
+2 −κ1))
+log κ2
+��
+.
+Proof. We will apply Theorems 1 and 2 to show that Algorithm 5 returns �θT such that ∥θT −θ∗∥2 ≤
+12c2
+m = O
+�
+ϵ log(p) +
+�
+ϵ log(p)
+�
+.
+Under the assumption that √ϵ + γ (n, p, δ, ϵ) < �γ, and using Proposition 3, the assumptions
+of Lemma 9 are satisfied. Note that the assumptions of Lemma 7 are likewise satisfied by the
+condition (46). Applying Theorem 2, the risk R(θt) is reduced by at least γ
+2 in each step of the
+damped Newton phase of the algorithm. Hence, the number of such iterations cannot exceed
+Tdamp := R(θ0) − R(θ∗)
+γ/2
+.
+(49)
+21
+
+Define
+Tpure := log2 log2
+�6c2L
+m2
+�
+.
+(50)
+Applying Theorem 1, we observe that after Tpure iterations in the pure Newton phase, we have
+m
+L
+� 1
+2
+�2t
+< 6c2
+m . Therefore, from inequality (13), we have ∥�θ − θ∗∥2 ≤ 12c2
+m . Combining inequali-
+ties (49) and (50), we obtain the bound on the total number of iterations T.
+From the preceding analysis on the robust gradient and Hessian estimators, observe that η, L,
+and m are independent of ϵ and p, while γg is O(√ϵ log p) and γh is O(p√ϵ log p). Hence, from
+inequality (13), we have ∥θT − θ∗∥2 = O
+�
+p√ϵ log p
+�
+.
+We now compute the error probability of the algorithm via a union bound. For each of the
+T gradient and Hessian calculations, we have a possible error of δ.
+Furthermore, each call of
+backtracking linesearch incurs a possible error from the robust estimates, by Lemma 7; once at θt
+and once for each value of α used in the linesearch. This is a total of 2Tpure evaluations for the
+pure Newton steps, and a maximum of Tdamp
+�
+1 +
+� log( m
+M ( 1
+2 −κ1))
+log κ2
+��
+evaluations for the damped
+Newton steps. Thus, the overall probability of error is at most
+2Tδ + 2Tpureδ + Tdamp
+�
+1 +
+�
+log
+� m
+M
+� 1
+2 − κ1
+��
+log κ2
+��
+δ ≤ Tδ
+�
+5 +
+�
+log
+� m
+M
+� 1
+2 − κ1
+��
+log κ2
+��
+.
+Remark 3. Examining the condition (48), we see that (assuming (m, M, L, η, ∥Σx∥2) are all con-
+stants) we have a required upper bound of �γ2 ≍
+1
+log p on the contamination proportion ϵ. Further-
+more, from the expression (36), we have (ignoring log factors) n ≿ max
+�
+p, ϵp2�
+. The condition (46)
+likewise gives a minimum sample size requirement on n in terms of δ.
+Remark 4. It is instructive to compare the result of Theorem 3 to Theorem 4 in Prasad et al. [28],
+which gives a convergence statement of the form
+∥θt − θ∗∥2 ≤ κt∥θ0 − θ∗∥2 + C∥Σx∥1/2
+2
+√log p
+1 − κ
+�√ϵ + γ(n, p, δ, ϵ)
+�
+for iterates {θt}t≥0 of robust gradient descent. For sufficiently large t, the second term dominates,
+leaving an error term of O(√ϵ log p). Our theorem has a dominant factor of O(p√ϵ log p), which
+can be reduced to O(√ϵ log p) if we assume 4-wise independence of the coordinates of the covariate
+distribution (cf. Remark 1 above). In terms of the convergence rate of the optimization procedure,
+however, we just need T ≍ log log 1
+ϵ, compared to T ≍ log 1
+ϵ in the case of robust gradient descent.
+Linear regression is of course a special case of GLMs, for which Theorem 3 readily applies.
+On the other hand, note that a much more direct way to obtain a robust estimator for linear
+regression would be to directly robustify the estimator �θOLS =
+�
+XT X
+n
+�−1 �
+XT y
+n
+�
+, where we apply
+Algorithm 1 to obtain robust estimates of E[yixi] and E[xixT
+i ] (the latter matrix being vectorized
+before applying the agnostic mean algorithm). Indeed, in the non-robust case, applying Newton’s
+method to the ordinary least squares objective converges in a single step. A careful analysis of this
+so-called “robust plug-in estimator” would also give an error of O(√ϵ) in the robust case, but a
+direct analysis would provide an error bound which depends on ∥θ∗∥2, since Cov(xi, yi) would scale
+with ∥θ∗∥2 (cf. Corollary 3 in Prasad et al. [28]). On the other hand, the guarantee of Theorem 3
+for the full robust Newton’s method does not involve ∥θ∗∥2.
+22
+
+Remark 5. A natural question is whether the estimation error upper bounds in Theorem 3 are tight:
+For i.i.d. samples from a GLM with Huber ϵ-contamination, is it possible to derive estimators with
+error smaller than C√ϵ? For the case of linear regression, this problem has been studied quite
+carefully, and it has been established that when the uncontaminated data are Gaussian with an
+isotropic covariance, the rate should be Θ(ϵ) [3, 11, 27]. In the case when the covariates only follow
+a bounded fourth moment assumption, the rate improves to Θ(√ϵ) [5, 6]. We are not aware of
+existing lower bounds in the literature for more general GLMs.
+4.2
+Heavy-Tailed Distributions
+In this subsection, we assume that the link function Φ satisfies inequality (40) for k ∈ {1, 2},
+inequality (41) for t ∈ {2, 4}, and inequality (42) for t ∈ {2, 3}. As in the case of Huber contam-
+ination, we assume that the population risk R(θ) is m-strongly convex. Define L and M as in
+Proposition 1.
+Proposition 4. Under the assumptions above, the gradient and Hessian estimates with Type =
+Heavy-tail returned by Algorithms 4 and 5, respectively, satisfy the conditions of Definitions 1 and 2
+with the following parameters:
+αg = c1
+�
+p log(1.4/δ)
+n
+,
+βg = c2
+�
+p log(1.4/δ)
+n
+,
+αh = 0,
+βh = c3∥Σx∥2p
+�
+log(1.4/δ)
+n
+,
+with probability at least 1 − δ.
+Proof. By Lemma 5, the distribution of the gradients ∇L(θ, z) has bounded fourth moments.
+Moreover,
+∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2
+��
+LΦ,4 + LΦ,2
+�
+∥θ − θ∗∥2
+2
++ C2∥Σx∥2
+�
+BΦ,2 +
+�
+BΦ,4 + c(σ)
+�
+MΦ,2,2 +
+�
+c(σ)3MΦ,4,1
+�
+.
+Plugging this bound into inequality (37) of Lemma 2, we obtain
+∥g(θ) − E[∇L(θ, z)]∥2 ≤ 11
+�
+tr(Cov(∇L(θ, z))) log(1.4/δ)
+n
+≤ 11
+�
+p Cov(∇L(θ, z)) log(1.4/δ)
+n
+≤ c1
+�
+p log(1.4/δ)
+n
+∥θ − θ∗∥2 + c2
+�
+p log(1.4/δ)
+n
+.
+Hence, the gradient estimate returned by Algorithm 4 satisfies Definition 1 with αg = c1
+�
+p log(1.4/δ)
+n
+and βg = c2
+�
+p log(1.4/δ)
+n
+.
+23
+
+By Lemma 6, the distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth
+moments. Moreover, combining Lemma 6 with Lemma 4, we obtain
+∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C3
+�
+tr(Cov(flatten(∇2L(θ, z)))) log(1.4/δ)
+n
+≤ c3∥Σx∥2p
+�
+log(1.4/δ)
+n
+.
+Hence, the Hessian estimate returned by Algorithm 5 satisfies Definition 2 with αh = 0 and βh =
+c3∥Σx∥2
+�
+log(1.4/δ)
+n
+.
+We then have the following result:
+Theorem 4. Let {zi}n
+i=1 be n i.i.d. samples drawn from a heavy-tailed distribution P that satisfies
+the GLM in (3), and suppose the aforementioned assumptions are satisfied. Define (η, c2, ζ, γg, γh)
+as in Theorem 1 and γ as in Theorem 2. Let δ > 0. Suppose n satisfies
+n > C max
+�
+p
+�α2g
+, p
+�β2g
+, p2∥Σx∥2
+2
+�β2
+h
+, 1
+ζ2
+�
+log
+�1.4
+δ
+�
+,
+(51)
+where (�αg, �βg, �βh) are as defined in Lemma 9. Then applying Algorithm 3 on {zi}n
+i=1, with initial-
+ization θ0 ∈ Θ and number of iterations
+T ≥ R(θ0) − R(θ∗)
+γ/2
++ log2 log2
+�6c2L
+m2
+�
+,
+returns an output such that
+∥θT − θ∗∥2 ≤ 12c2
+m
+= O
+��
+p2
+n
+�
+,
+with probability at least 1 − Tδ
+�
+5 +
+� log( m
+M ( 1
+2 −κ1))
+log κ2
+��
+.
+Proof. We will follow a similar outline as in the proof of Theorem 3.
+Using the assumption on n and Proposition 4, it is straightforward to verify that the conditions
+of Lemma 9 are satisfied. Furthermore, the conditions of Lemma 8 are satisfied by inequality (51),
+as well. Applying Theorem 2, the risk R(θt) is reduced by at least γ
+2 in each step of the damped
+Newton phase of the algorithm. Hence, the number of such iterations cannot exceed Tdamp, defined
+as in equation (49). Applying Theorem 1, we observe that after Tpure iterations (defined as in
+equation (50)) in the pure Newton phase, we have m
+L
+� 1
+2
+�2t
+< 6c2
+m . Therefore, from inequality (13),
+we have ∥�θ − θ∗∥2 ≤ 12c2
+m . Combining inequalities (49) and (50), we obtain the bound on the total
+number of iterations T.
+From the preceding analysis on the robust gradient and Hessian estimators (cf. Proposition 4),
+observe that γg = O
+��
+p
+n
+�
+and γh = O
+��
+p2
+n
+�
+. Hence, c2 is O
+��
+p2
+n
+�
+. From inequality (13), we
+then have ∥θT − θ∗∥2 = O
+��
+p2
+n
+�
+.
+Computing the error probability of the algorithm via a union bound is the same as in Theorem 3
+with the use of appropriate gradient, Hessian, and robust estimates.
+Remark 6. Again, assuming 4-wise independence of the coordinates of the covariate distribution,
+we can reduce the dimension-dependence of the bounds (cf. Remark 1). We then take βh ≍ √p∥Σx∥2
+2,
+to obtain an estimation error bound of the form ∥�θ − θ∗∥2 = O
+��
+p
+n
+�
+.
+24
+
+5
+Simulations
+We note that in our simulations, we have implemented the code from Lai et al. [15] for agnostic mean
+estimation. In particular, the outlier truncation step is slightly different from the one analyzed in
+Prasad et al. [28], and consequently also in our theorems above.
+5.1
+Huber’s Contamination Model
+We begin with simulations for linear and logistic regression in Huber’s contamination model.
+5.1.1
+Linear Regression
+For our simulations, we set the dimension to be p = 10 and the number of data points to be
+n = 1000.
+We simulated the clean covariates as xi ∼ N(0, Ip), with corresponding responses
+yi = xT
+i θ∗ + wi, where wi ∼ N(0, 0.1) is i.i.d. noise and the true parameter is θ∗ =
+1
+√p(1, 1, . . . , 1).
+We simulated the outlier covariates as xi ∼ N(0, p2Ip), with corresponding responses yi = 0.
+Figure 1 shows the results for Robust Newton’s Method (RNM), Robust Gradient Descent
+(RGD), and ordinary least squares (OLS). We used the initialization θ0 = (0.4, . . . , 0.4) + 10w,
+with w ∼ N(0, Ip), for both RNM and RGD. For RNM, we used the backtracking linesearch
+parameters κ1 = 0.01, κ2 = 0.5, and ζ = 10−8. For RGD, we used stepsize η = 0.1. We repeated
+the algorithm three times with contamination fractions ϵ = 0.1, 0.2, and 0.3. As seen in the figure,
+the statistical error indeed decreases quite quickly for RNM in comparison to RGD.
+Figure 1: Error log(∥θt − θ∗∥2) with respect to each iteration of Robust Newton’s Method (RNM),
+Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear regression with Huber
+contamination.
+25
+
+4
+3
+2
+1*6
+0
+一
+ell
+log(
+OLS for e= 0.1
+RGD for  = 0.1
+-2
+RNM for e = 0.1
+OLS for = 0.2
+-3
+RGD for E = 0.2
+RNM for e = 0.2
+OLS for = 0.3
+-4
+RGD for = 0.3
+RNM for  = 0.3
+-5
+0
+10
+20
+30
+40
+50
+60
+70
+80
+t5.1.2
+Logistic Regression
+Next, we generated data from a logistic model with p = 10, n = 1000, and θ∗ = (1/√p, . . . , 1/√p),
+where we sampled the covariates as xi ∼ N(0, Ip) and sampled yi ∈ {0, 1} such that p(yi =
+1|xi) =
+1
+1+e−xT
+i θ∗ . We then randomly changed an ϵ fraction of the labels to be either 0 or 1, with
+equal probability. For various values of ϵ, we ran Robust Gradient Descent (RGD) and Robust
+Newton’s Method (RNM), and plotted the parameter error in Figure 2. For RNM, we used the same
+backtracking linesearch parameters as in the case of linear regression with Huber contamination.
+For RGD, we used a stepsize of η = 3. As seen in the figure, the statistical error again decreases
+more quickly for RNM than for RGD.
+Figure 2: Error log(∥θt − θ∗∥2) with respect to each iteration for Robust Newton’s Method (RNM)
+and Robust Gradient Descent (RGD) for logistic regression with Huber contamination. The behav-
+ior of the non-robust optimizer, found using Newton’s method (NM), is also shown for reference.
+5.2
+Heavy-Tailed Data
+For heavy-tailed data, we took p = 10 and n = 1000. We generated the covariates xi ∼ N(0, Ip) and
+the corresponding responses yi = xT
+i θ∗ + wi, with wi following a Pareto distribution with variance
+σ2 and tail-index parameter β. We set the regression parameter θ∗ =
+1
+√p(1, 1, . . . , 1).
+Figure 3 compares the results of Robust Newton’s Method (RNM), Robust Gradient Descent
+(RGD), and ordinary least squares (OLS). We used the initialization θ0 = (10, 10, . . . , 10) for both
+RNM and RGD. For RNM, we used the backtracking linesearch parameters κ1 = 0.01, κ2 = 0.5,
+and ζ = 1000. For RGD, we used stepsize η = 0.1. We repeated the algorithm three times, for
+σ = 0.5, 1, and 1.5, all with β = 1. As seen in the figure, the statistical error again decreases more
+quickly for RNM than for RGD; on the other hand, in this simulation, the final error of RGD is
+lower than that of RNM.
+26
+
+2.5
+RGD for eps= 0.01
+O
+2
+RNM for eps = 0.01
+9-- NM for eps = 0.01
+RGD for eps= 0.02
+1.5
+RNM for eps = 0.02
+NM for eps = 0.02
+1
+(l* - *
+0.5
+log(II t
+0
+-0.5
+00i0i0i000i00i0i00!0i0i0!0i0i00!0i0i0!0i0i0
+-1
+-1.5
+-2
+0
+5
+10
+15
+20
+25
+30
+tFigure 3: Error log(∥θt −θ∗∥2) with respect to each iteration for Robust Newton’s Method (RNM),
+Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear regression with heavy-
+tailed data.
+6
+Robust Hessian Estimation: The Conjugate Gradient Approach
+In this section, we discuss an alternative to Newton’s method (and present a robust variant thereof)
+which does not involve explicitly computing the Hessian. Inspired by Martens [23], the idea is to
+estimate ∇2f(θ)v, for any vector v, using the approximation
+hv(θ) = ∇f(θ + δv) − ∇f(θ)
+δ
+,
+(52)
+for some small δ > 0. Note that in order to compute the Newton step ∆θnt, we need to solve
+the system ∇2f(θ)∆θ = −∇f(θ), which we will do using the conjugate gradient algorithm, which
+provides an iterative method for solving a linear system of the form Ax = b [30, Chapter 5]. Our
+robust approach will involve using the robust gradient estimate g(θ) in place of ∇f(θ).
+The details of the algorithm are provided in Algorithm 6. Note that we have specified that the
+CGNewtonStep subroutine for finding the Newton direction on each iteration of CGRobustNewton
+runs for p steps, because in the noiseless case, the conjugate gradient method is known to terminate
+in at most p steps.
+27
+
+4
+OLS foro=0.5
+RGD for o = 0.5
+3
+RNM for  = 0.5
+OLS for g = 1
+RGD for  = 1
+2
+RNM for o= 1
+OLSforg=1.5
+RGD for  = 1.5
+RNM for o = 1.5
+-2
+-3
+-4
+0
+10
+20
+30
+40
+50
+60
+70
+80
+tAlgorithm 6 Conjugate Gradient Robust Newton’s Method
+Require: Data samples S = {zi}n
+i=1, Number of iterations T, Initial guess θ0 ∈ Θ, Backtracking
+linesearch parameters κ1 ∈ (0, 0.5), κ2 ∈ (0, 1), and ζ, Tolerance δ
+1:
+2: function CGRobustNewton(S, ξ, θ0, , κ1, κ2, ζ)
+3:
+for t = 0 to T − 1 do
+4:
+Compute losses {L(θt, zi)}n
+i=1 and gradients {∇L(θt, zi)}n
+i=1
+5:
+Compute Newton step ∆θnt = CGNewtonStep(θt)
+6:
+Compute stepsize α = BacktrackingLineSearch(S, θt, ∆θnt, g(θt), κ1, κ2, ζ)
+7:
+Update θt+1 = θt + α∆θnt
+8:
+end for
+return θT
+9: end function
+10:
+11: function CGNewtonStep(θ)
+12:
+Randomly initialize ∆θ(0) ∈ Θ
+13:
+Compute gradient estimate g(θ) = RobustGradientEstimate(S, θ)
+14:
+Compute Hessian-vector product estimate h∆θ(0)(θ) = HVProduct(θ, ∆θ(0))
+15:
+Set r0 = h∆θ(0)(θ) + g(θ)
+16:
+Set p0 = −r0
+17:
+for k = 1 to p − 1 do
+18:
+Compute Hessian-vector product estimate hpk(θ) = HVProduct(θ, pk)
+19:
+Set αk =
+rT
+k rk
+pT
+k hpk(θ)
+20:
+Set ∆θ(k+1) = ∆θ(k) + αkpk
+21:
+Set rk+1 = rk + αkhpk(θ)
+22:
+Set βk+1 =
+rT
+k+1rk+1
+rT
+k rk
+23:
+Set pk+1 = −rk+1 + βk+1pk
+24:
+end for
+return ∆θ(d)
+25: end function
+26:
+27: function HVProduct(θ, v)
+28:
+Compute gradient estimate g(θ) = RobustGradientEstimate(S, θ)
+29:
+Compute gradient estimate g(θ + δv) = RobustGradientEstimate(S, θ + δv)
+return g(θ+δv)−g(θ)
+δ
+30: end function
+31:
+32: function BacktrackingLineSearch((S, θ, ∆θnt), g(θ), κ1, κ2, ζ)
+33:
+Set α = 1
+34:
+while RobustEstimate({L(θ + α∆θnt, zi)}n
+i=1) > RobustEstimate({L(θ, zi)}n
+i=1) +
+κ1αg(θ)∆θnt + ζ do
+35:
+Update α = κ2α
+36:
+end while
+return α
+37: end function
+28
+
+6.1
+Convergence
+We sketch some ideas here; a rigorous proof giving rates of convergence of the robust conjugate
+gradient method is beyond the scope of this work. Focusing on the pure Newton phase, note that
+our analysis of the iterates of robust Newton’s method essentially hinges on the Newton step ∆θnt
+satisfying the equation
+∇f(θt) = −∇2f(θt)∆θnt + χt,
+(53)
+where the next iterate is then defined by θt+1 = θt + ∆θnt and χt is a small, bounded error (cf. in-
+equalities (24) and (25)). In particular, we can bound χt using the fact that ∆θnt = −H(θt)−1g(θt),
+and ∥g(θt) − ∇f(θt)∥2 and ∥H(θt) − ∇2f(θt)∥2 are small. In the case of the robust conjugate gra-
+dient method, we can again think of the conjugate gradient method as providing an approximate
+solution of the form
+∇f(θt) = −∇2f(θt)∆�θnt + �χt,
+(54)
+where successive iterates are then defined by �θt+1 = �θt + ∆�θnt. Thus, the main challenge is to
+understand the propagation of errors when the conjugate gradient method is applied to solve the
+system Ax = b, but the matrix-vector pair (A, b) is replaced by ( �A,�b) on each iteration. To the
+best of our knowledge, this is actually an open question in optimization [12, 13]. We note, however,
+that since our ultimate statistical estimation error bounds are all up to a small radius of, e.g.,
+O(√ϵ), we only need the output of the conjugate gradient method to be correct up to this error. In
+particular, as it is known that the exact conjugate gradient method terminates after p steps [30],
+it would for instance suffice to show that an inexact conjugate gradient method, where the error
+of ( �A,�b) is also O(√ϵ), only accumulates O(√ϵ) error after p steps. Alternatively, one could try
+to derive a geometric rate of convergence (cf. equation (5.36) of Nocedal and Wright [30]), with an
+additional additive error term, for inexact conjugate gradient steps.
+To this end, we also need to understand the error terms introduced to conjugate gradient steps
+due to inexactness. This depends on the increment δ used in the finite-difference approximation of
+the Hessian term (52). Note that by a Taylor expansion, we have
+∇f(θ + δv) = ∇f(θ) + δ∇2f(θ)v + Cδ2,
+for some constant C. Thus, we have the error bounds
+∥hv(θ) − ∇2f(θ)v∥2 =
+����
+g(θ + δv) − g(θ)
+δ
+− ∇2f(θ)v
+����
+2
+=
+����
+g(θ + δv) − g(θ)
+δ
+− ∇f(θ + δv) − ∇f(θ)
+δ
+− Cδ
+����
+2
+≤ ∥g(θ + δv) − ∇f(θ + δv)∥2
+δ
++ ∥g(θ) − ∇f(θ)∥2
+δ
++ Cδ.
+If we had deviations bounds of the form (8), e.g., with αg, βg ≍ √ϵ, the optimal choice of δ would
+be δ ≍ ϵ1/4.
+In summary, we conjecture that the robust conjugate gradient method would allow us to incur
+an overall estimation error of O(ϵ1/4) in the case of Huber’s ϵ-contamination model, again at a
+quadratic convergence rate for the successive Newton iterates. Although this is a slower rate than
+the one derived in Section 4 for GLMs, it may be applicable to a much wider range of settings.
+We also note that for the SEVER algorithm [9], a rate of O(ϵ1/4) is also derived for empirical risk
+minimization for a class of classification problems. If the above discussion could be made rigorous,
+it would then also be extendable to the heavy-tailed setting in a straightforward manner.
+29
+
+6.2
+Simulations
+In Figure 4, we compare the Newton Conjugate Gradient Method (NCGM), Robust Gradient
+Descent (RGD), and ordinary least squares (OLS) on a linear model with Huber ϵ-contaminated
+data, with the same setup as in Section 5.1.1. We used the initial parameter θ0 = (1, . . . , 1) + 2w,
+with w ∼ N(0, Ip), for both NCGM and RGD. For NCGM, we used the backtracking linesearch
+parameters κ1 = 0.01, κ2 = 0.5, and ζ = 0.001. We also used δ = 10−9 for the estimation of
+Hessian-vector products. For RGD, we used stepsize η = 0.02. We repeated the algorithm two
+times, with contamination fractions ϵ = 0.01 and 0.02.
+Figure 4: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient
+Method (NCGM), Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear
+regression with Huber contamination.
+In Figure 5, we compare the Newton Conjugate Gradient Method (NCGM), Robust Gradient
+Descent (RGD), and ordinary least squares (OLS) on a linear model with heavy-tailed data, again
+with the same setup as in Section 5.1.1. We used the initial parameter θ0 = (1.5, . . . , 1.5) + 2w,
+with w ∼ N(0, Ip), for both NCGM and RGD. For NCGM, we used the backtracking linesearch
+parameters κ1 = 0.01, κ2 = 0.5, and ζ = 0.00001. We also used δ = 10−10 for the estimation of
+Hessian-vector products. For RGD, we used stepsize η = 0.2. We repeated the algorithm twice
+with σ = 0.5 and 0.25, with β = 0.7.
+In Figure 6, we compare the Newton Conjugate Gradient Method (NCGM) and Robust Gradi-
+ent Descent (RGD) on a logistic model with Huber ϵ contamination. To generate the contaminated
+logistic data, we used the same procedure outlined in Section 5.1.2. We also used the same hyper-
+parameters for NCGM and RGD as in Section 5.1.2.
+7
+Discussion
+We have presented a novel second-order method for robust parameter estimation, based on an
+adaptation of Newton’s method where gradients and Hessians are computed in a robust manner
+30
+
+2
+0
+-2
+-3
+OLS for E=0.01
+RGD for = 0.01
+NCGM for = 0.01
+-4
+OLS for =0.02
+RGD for = 0.02
+NCGM for = 0.02
+-5
+0
+5
+10
+15
+20
+25
+30
+tFigure 5: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient
+Method (NCGD), Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear
+regression with heavy-tailed data.
+on each iteration.
+In particular, we have shown that a variant of the backtracking linesearch
+algorithm will adaptively choose stepsizes in such a way that a finite number of iterates initially
+lie in a “damped” phase of the algorithm, after which the algorithm enters a “pure” phase where
+it only chooses stepsizes equal to 1 and converges quadratically to a small ball around the true
+parameter. Although our method shows clear advantages in comparison to previously analyzed
+first-order methods, many possible improvements exist which may also be extremely interesting to
+study theoretically. First of all, we have used a rather naive method for computing robust Hessians
+based on vectorizing the Hessian matrix and applying the same subroutine used for robust gradient
+estimation. Obtaining optimal rates for robust matrix estimation under a variety of metrics and
+contamination settings is an active area of research [25, 26, 4], and it is entirely possible that a
+different robust matrix estimator would lead to better overall error bounds for our robust Newton
+algorithm.
+Another plausible extension of our analysis that could be studied under a similar
+theoretical framework would be to use robust gradient and Hessian estimators which employ the
+estimation procedures of Diakonikolas et al. [10] rather than those of Lai et al. [15]; we note that
+this would allow us to also handle the setting of adversarially contaminated data, rather than i.i.d.
+data from either an ϵ-contaminated or heavy-tailed model.
+It would also be interesting and practically important to devise robust second-order algorithms
+appropriate for higher-dimensional data. For moderate to large p (even in settings where p < n),
+implementing the robust version of Newton’s method can become more tedious, since it involves
+robustly computing p × p matrices and then inverting them on each iteration. In the truly high-
+dimensional case (p > n), even the canonical version of Newton’s method must be modified, since
+the Hessian matrix becomes rank-deficient. This raises the question of whether it would be beneficial
+to analyze a robust inexact second-order algorithm, instead, where the Hessian matrix need not be
+approximated as closely. In the truly high-dimensional setting, combining this with regularization
+would be a natural direction for future work.
+31
+
+2
+1.5
+1
+0.5
+1og(1 t - *1)
+0
+0.5
+-1
+OLS for g=0.5
+-1.5
+RGD for  = 0.5
+NCGM for  = 0.5
+OLSforg=0.25
+-2
+RGD for  = 0.25
+NCGM for  = 0.25
+-2.5
+0
+5
+10
+15Figure 6: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient
+Method (NCGM) and Robust Gradient Descent (RGD) for logistic regression with Huber contam-
+ination. The behavior of the non-robust optimizer, found using Newton’s method (NM), is also
+shown for reference.
+Finally, we have proposed the robust conjugate gradient method as an alternative second-order
+algorithm which, though based on Newton’s method, only requires computing robust gradients
+rather than needing to separately compute robust Hessians. This method could potentially enjoy the
+fast convergence benefits of Newton’s method while bypassing some of the computational issues in
+higher dimensions. However, a rigorous analysis of the robust conjugate gradient method is beyond
+the current scope of this paper—in particular, it would involve carefully tracking the propagation
+of errors through iterates of the conjugate gradient method, which has remained a long-standing
+open problem. We note that any error bounds on successive conjugate gradient iterates could then
+easily be plugged into our proofs to obtain quadratic convergence to an appropriate ball around
+the true parameter.
+Acknowledgments
+The work of EI was supported by the Cantab Capital Institute for the Mathematics of Information
+via the Philippa Fawcett Internship programme (Faculty of Mathematics, University of Cambridge).
+References
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+[3] M. Chen, C. Gao, and Z. Ren. A general decision theory for Huber’s ϵ-contamination model.
+Electronic Journal of Statistics, 10(2):3752–3774, 2016.
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+34
+
diff --git a/mNFPT4oBgHgl3EQf4DUP/content/tmp_files/load_file.txt b/mNFPT4oBgHgl3EQf4DUP/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf,len=1075
+page_content='Robust empirical risk minimization via Newton’s method  Eirini Ioannou  University of Edinburgh  ei250@cantab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='uk  Muni Sreenivas Pydi  LAMSADE, Universit´e Paris Dauphine-PSL  muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='pydi@lamsade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='dauphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='fr  Po-Ling Loh  University of Cambridge  pll28@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='uk  Abstract  We study a variant of Newton’s method for empirical risk minimization, where at each  iteration of the optimization algorithm, we replace the gradient and Hessian of the objective  function by robust estimators taken from existing literature on robust mean estimation for  multivariate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' After proving a general theorem about the convergence of successive iterates  to a small ball around the population-level minimizer, we study consequences of our theory in  generalized linear models, when data are generated from Huber’s epsilon-contamination model  and/or heavy-tailed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also propose an algorithm for obtaining robust Newton  directions based on the conjugate gradient method, which may be more appropriate for high-  dimensional settings, and provide conjectures about the convergence of the resulting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Compared to the robust gradient descent algorithm proposed by Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' (2020), our  algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-  order algorithms for convex problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', quadratic convergence in a neighborhood of the  optimum, with a stepsize that may be chosen adaptively via backtracking linesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  1  Introduction  Statistical estimation via classical procedures often depends on strong model assumptions, which  only hold in the absence of outliers and other deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' However, many real-life datasets do not  typically follow these model assumptions, necessitating the use of robust statistical methods [14,  29, 22], which remain reasonably accurate even under deviations from the model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In  this paper, we focus on situations where data are sampled from a small ball around a parametric  distribution, according to Huber’s ϵ-contamination model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In other words, we have samples of the  form zi ∼ (1 − ϵ)Pθ∗ + ϵQ, where Q is an arbitrary distribution and the goal is to estimate the  unknown parameter θ∗ based on an observed data set {zi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also analyze the behavior of the  same algorithms in situations where data are generated from a heavy-tailed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Although  the parameter corresponds to the true data-generating distribution, “outliers” are observed in  the data set due to random sampling, and the goal is to obtain an estimator with similar high-  probability guarantees as in the case of standard parameter estimation techniques for lighter-tailed  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Classical robust statistics [14] suggests the use of M-estimators, which involve optimizing an ap-  propriate loss function over the space of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' More specifically, suppose we wish to estimate  the parameter θ∗ = arg minθ∈Θ R(θ), where the risk R(θ) = E[L(θ, (x, y))] is the expectation of a  loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In practice, one uses an empirical risk minimizer �θ ∈ arg minθ∈Θ 1  n  �n  i=1 L(θ, (xi, yi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='13192v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='ML]  30 Jan 2023  Standard theory of parametric statistics shows that the optimal choice of L corresponds to the log-  likelihood function when data are not contaminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' However, taking into account ϵ-contamination  leads to the use of other losses such as the Huber loss, which can be shown to be optimal in a min-  imax sense when the uncontaminated data are normally distributed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Similarly, while the  least-squares loss corresponds to maximum likelihood for Gaussian errors, minimizing a different  loss function may be advantageous in the case of heavy-tailed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In this paper, we adopt an alternative approach inspired by optimization methods [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Rather  than seeking to design a robust loss, we introduce robustness into the estimation algorithm by  implementing robust updates in an iterative second-order optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our work is di-  rectly inspired by the work of Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28], who proposed and analyzed a first-order version of  this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our algorithm, which we call “robust Newton’s method,” utilizes the AgnosticMean  algorithm from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15] in the Huber contamination setting to obtain robust gradient and  Hessian estimates on each iterate of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Moreover, given appropriate assumptions, we  prove that the rate of convergence of this algorithm is faster than that of robust gradient descent,  and successive iterates converge quadratically to a small ball around θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Furthermore, a suitable  stepsize may be determined adaptively using a robust variant of backtracking linesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our anal-  ysis of the Newton iterates is fairly general, and can be used to derive convergence guarantees when  alternative procedures are employed for gradient/Hessian estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We consequently propose a  method based on the conjugate gradient method [30] for obtaining approximate Newton directions  which may be useful in higher dimensions, and discuss some conjectures about the corresponding  convergence rate on ϵ-contaminated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Related Work  Here, we discuss several other general approaches to robust empirical risk minimization which have  appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A variety of algorithms have been proposed based on median-of-means  estimators, which give a robust alternative to a mean estimator (a more detailed description is  provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Median-of-mean tournaments [21, 20, 19] provide a method for  comparing pairs of candidate regression functions based on the number of blocks in which the em-  pirical mean of the loss function is smaller for one function than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The final estimator is  a function which “wins” the most pairwise matches among other candidate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Another use  of median-of-means estimators derives an estimator by considering a “minimaximization” problem  formed by increments of the objective function, where a median-of-means estimate is used in place  of the expectation appearing in the population-level version of the problem [16, 17, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Finally, and  more similar in spirit to the approach taken in our paper, we mention a method which involves  modifying gradient descent by computing a gradient with respect to a median block on each itera-  tion [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The median block is defined as the block with the smallest empirical mean (with respect  to the objective function value) on the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Excess risk bounds are then derived for a  class of binary classification problems, where a certain fraction of the data are arbitrarily generated  outliers and the rest are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' from the uncontaminated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  The SEVER algorithm [9] also operates via an appropriate modification of an iterative opti-  mization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' It uses any “approximate learner” algorithm, which can find an approximate  critical point of an empirical risk minimization problem, as a subroutine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', gradient descent,  stochastic gradient descent, or Newton’s method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' On successive iterations, the SEVER algorithm  filters out data points by applying the approximate learner to the currently remaining set of data  points and then filtering out any points with outlying gradients computed at the parameter chosen  by the approximate learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Statistical error bounds are derived for the output of the SEVER algo-  rithm on classification and regression problems, where data are drawn from a possibly heavy-tailed  2  model and then corrupted by a small fraction of adversarial outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Finally, our work is most closely related to the work of Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28], which may be seen as  a first-order version of our second-order algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In that paper, the authors propose to perform  parameter estimation by running a variant of gradient descent on the empirical risk objective, where  successive gradients are computed by treating each gradient computation as an approximation of  a population-level mean, and then applying a robust mean estimation procedure for multivariate  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As in our work, they use the mean estimation algorithm by Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15] for their multivariate  estimation procedure in the case of Huber’s ϵ-contamination model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' They also derive statistical  error bounds for successive iterates, which hold with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The main difference with  our work is that we are able to derive faster rates of convergence due to the use of second-order  algorithms, while enjoying the broad applicability of their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Outline  The remainder of our paper is organized as follows: In Section 2, we discuss the setup of the  problem we are aiming to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In Section 3, we introduce our novel robust Newton’s method  and present two theorems concerning its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In Section 4, we discuss applications of our  general theory to generalized linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In Section 5, we provide some illustrative numerical  results and comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In Section 6, we present a version of robust Newton’s method based on  the conjugate gradient method and provide some conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Finally, we conclude our paper with  a discussion of open directions in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3  Notation  For a matrix A ∈ Rp×p, we use ∥A∥2 to denote the spectral norm, λmin(A) to denote the minimum  eigenvalue, and tr(A) to denote the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We use c, C, c1, C1, c2, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' to denote universal positive  constants whose specific values may change from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For functions f(n) and g(n), we write  f(n) = O(g(n)) to mean that f(n) ≤ Cg(n) for some constant C > 0, and also write f(n) ≾ g(n)  and g(n) ≿ f(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We write f(n) ≍ g(n) when both inequalities hold simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  2  Background  We consider a parametric estimation problem, wherein the data {zi}n  i=1 ⊆ Z sampled from a true  distribution P are to be fit to a model with parameter θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A loss function L : Θ × Z → R  measures the goodness of fit of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The optimal parameter θ∗ ∈ Θ minimizes the population  risk of the model, which is the expected loss incurred by the model over the true data distribution:  θ∗ = argmin  θ∈Θ  R(θ) := Ez∼P [L(θ, z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (1)  Given n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' data points {zi}n  i=1 sampled from the true distribution P, the goal in empirical  risk minimization (ERM) is to estimate the parameter �θn that minimizes the empirical risk of the  model, which is the average loss incurred by the model over the n data points:  �θn = argmin  θ∈Θ  �Rn(θ) := 1  n  n  �  i=1  L(θ, zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (2)  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Examples  Linear regression:  In linear regression, data z ∈ Z are of the form z = (x, y) ∈ Rp × R, where  the covariate x and response y are related via  y = xT θ∗ + w,  where w ∈ R is noise that is sampled independently from x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The loss function we use for  this model is the squared loss function,  L(θ, (x, y)) = 1  2(y − xT θ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Generalized linear models:  In a generalized linear model (GLM), data z = (z, y) ∈ Rp × R  are sampled from a true distribution P that satisfies the following relation on the conditional  probability of x given y:  P(y|x) ∝ exp  �yxT θ∗ − Φ(xT θ∗)  c(σ)  �  ,  (3)  where c(σ) is the scale parameter and Φ : R → R is a convex link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The loss function we  use for a GLM is the negative log-likelihood,  L(θ, (x, y)) = −yxT θ + Φ(xT θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Optimization Algorithms  In practice, we seek efficient algorithms for solving the ERM problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A popular algorithm is  gradient descent [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Given an initial guess for the parameter θ0 ∈ Θ and a stepsize η, the gradient  descent algorithm generates a sequence of iterates {θt}∞  t=1, as follows:  θt+1 = θt − η∇R(θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Another popular algorithm is Newton’s method [30, 2], whose iterates are given by the following  update equation:  θt+1 = θt − (∇2R(θt))−1∇R(θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (5)  Whereas gradient descent uses only gradient information at the current iterate θt, Newton’s  method uses both gradient and Hessian information at the current iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3  Huber’s ϵ-Contamination Model  In Huber’s ϵ-contamination model, samples are drawn from a mixture distribution of the form  Pϵ = (1 − ϵ)P + ϵQ,  (6)  where P is the true data distribution and Q is an arbitrary noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The goal is to  estimate a parameter θ∗ ∈ Θ corresponding to the uncontaminated component P, given n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  samples drawn from the corrupted distribution Pϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Huber’s contamination model is a classical model studied in robust statistics [14, 29, 22], with  many exciting theoretical breakthroughs in estimation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  More recently, as robust  4  statistics received renewed attention in the theoretical computer science community, additional  questions were raised, particularly concerning computational tractability for optimal robust esti-  mators in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The contemporaneous work of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15] and Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [8]  studied computationally-tractable mean estimation in multivariate Gaussian settings, where the  former paper studied contamination with respect to Huber’s model and the latter paper studied  a stronger form of “adversarial” contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The subroutine which we call Algorithm 1 comes  from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15]—we state it in the slightly adapted version studied in Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Algorithm 1 Huber Estimator  Require: Samples S = {si}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Corruption level ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Dimension p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Failure probability δ  1: function HuberEstimator(S = {si}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  2:  Set �S = HuberOutlierTruncation(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  3:  if p = 1 then  4:  return mean(�S)  5:  else  6:  Compute Σ�S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' the covariance matrix of �S  7:  Compute V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' the span of the top p/2 principal components of Σ�S and W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' its complement  8:  Set S1 := PV (�S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' where PV is the projection operation onto V  9:  Set �µV := HuberEstimator(S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  10:  Set �µW := mean(PW �S)  11:  Set �µ ∈ Rp such that PV (�µ) = �µV and PW (�µ) = �µW  return �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  12:  end if  13: end function  14:  15: function HuberOutlierGradientTruncation(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  16:  if p = 1 then  17:  Let [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' b] be the smallest interval containing  �  1 − ϵ − C  �  log(|S|/δ)  |S|  �  (1 − ϵ) fraction of  points  18:  �S ← S ∩ [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' b]  19:  return �S  20:  else  21:  Let [S]i be the samples with the ith coordinates only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [S]i = {⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ei⟩ |x ∈ S}  22:  for i = 1 to p do  23:  a[i] = HuberEstimator([S]i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ/p)  24:  end for  25:  Let B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' a) be the ball of smallest radius centered at a containing  (1 − ϵ −  Cp  ��  p  |S| log  �  |S|  pδ  ��  (1 − ϵ) fraction of points in S  26:  �S ← S ∩ B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' a)  27:  return �S  28:  end if  29: end function  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4  Heavy-Tailed Model  In the heavy-tailed model, we assume that data are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' from a distribution with some  number of finite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that the heavy-tailed model does not involve a contaminating  distribution Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' However, the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' data may still appear to have “outlier” points due to random  sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  A popular approach for heavy-tailed mean estimation in the probably approximately correct  (PAC) framework—obtaining high-probability deviation bounds which are as tight as possible under  minimal distributional assumptions—is to use a median-of-means (MOM) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Roughly  speaking, data are randomly partitioned into blocks, the mean of each block is computed, and the  median of all of the block means is returned as the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In multiple dimensions, different  notions of medians exist, leading to different flavors of MOM estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  For a more detailed  overview, see the survey [19] and the references cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The MOM algorithm is summarized  in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In particular, we will employ a version of the algorithm from Minsker [24], which  combines the mean estimates using the geometric median, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', the point which minimizes the sum  of ℓ2-distances to the block means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Algorithm 2 Heavy-Tailed Estimator  Require: Samples S = {si}n  i=1, Failure probability δ  1: function HeavyTailedEstimator(S = {si}n  i=1, δ)  2:  Set b = 1 + ⌊3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5 log 1/δ⌋, the number of buckets  3:  Partition S into b blocks B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , Bb, each of size ⌊n/b⌋  4:  for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' n do  5:  �µi =  1  |Bi|  �  s∈Bi  s  6:  end for  7:  Set �µ = argmin  µ  b  �  i=1  ∥µ − �µi∥2  return �µ  8: end function  3  Robust Newton’s Method  We now present our variant of robust Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' At each iterate, we will use gradient and  Hessian estimates (g(θ), H(θ)) in place of (∇R(θ), ∇2R(θ)) in the update equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We assume  that these estimates satisfy the conditions described in the following definitions:  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' (Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28]) A function g(θ) is a robust gradient estimator for a data set  S = {zi}n  i=1, if for functions αg and βg, with probability at least 1 − δ, at any fixed θ ∈ Θ, the  estimator satisfies the following inequality:  ∥g(θ) − ∇R(θ)∥2 ≤ αg(n, δ)∥θ − θ∗∥2 + βg(n, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (7)  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A function H(θ) is a robust Hessian estimator for a data set S = {zi}n  i=1, if for  functions αh and βh, with probability at least 1 − δ, at any fixed θ ∈ Θ, the estimator satisfies the  following inequality:  ∥H(θ) − ∇2R(θ)∥2 ≤ αh(n, δ)∥θ − θ∗∥2 + βh(n, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (8)  6  Successive iterates then take the form  θt+1 = θt − αtH(θt)−1g(θt),  where αt is chosen via a version of backtracking linesearch [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The exit condition of backtracking  linesearch differs from its non-robust version in that function evaluations are replaced by robust  estimates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Lemmas 7 and 8 below) and an extra tolerance parameter ζ is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The full  algorithm is provided in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Algorithm 3 Robust Newton’s Method  Require: Data samples S = {zi}n  i=1, Number of iterations T, Initial guess θ0 ∈ Θ, Backtracking  linesearch parameters κ1 ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2 ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' and ζ  1: function RobustNewton(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  2:  for t = 0 to T − 1 do  3:  Compute losses {L(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1 and gradients {∇L(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1  4:  Compute gradient estimate g(θt) = RobustGradientEstimate(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θt)  5:  Compute Hessian estimate H(θt) = RobustHessianEstimate(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θt)  6:  Compute Newton step ∆θnt = −H(θt)−1g(θt)  7:  Compute stepsize α = BacktrackingLineSearch(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∆θnt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' g(θt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  8:  Update θt+1 = θt + α∆θnt  9:  end for  return θT  10: end function  11:  12: function BacktrackingLineSearch((S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∆θnt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' g(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  13:  Set α = 1  14:  while RobustEstimate({L(θ + α∆θnt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1) > RobustEstimate({L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1) +  κ1αg(θ)∆θnt + ζ do  15:  Update α = κ2α  16:  end while  return α  17: end function  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  General Analysis for Robust Newton’s Method  For the results of this section, we assume that f(θ) := R(θ) is twice-differentiable and satisfies  the Lipschitz condition ∥∇2f(θ1) − ∇2f(θ2)∥2 ≤ L∥θ1 − θ2∥2, for all θ1, θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also assume that  f satisfies the strong convexity and smoothness conditions mI ⪯ ∇2f(θ) ⪯ MI, for all θ close  enough to the initialization θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' (We will verify these conditions for GLMs in Propositions 1 and 2  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=') Finally, we will assume that at each iterate, the gradient and Hessian estimates g(θ) and  H(θ) satisfy inequalities (7) and (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As demonstrated in Theorems 3 and 4 later, the  last condition can typically be justified w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' via a union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Observe that in this setting, the  unique global minimum of f is the true parameter θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  The first result shows that if ∥∇f(θ0)∥2 is sufficiently small, the backtracking linesearch pro-  cedure will always choose stepsize 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' (This is known as the “pure Newton” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=') Furthermore,  successive iterates converge at a geometric rate to a small ball around θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Recall that the parameters  (κ1, κ2) of backtracking linesearch are defined as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  7  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose ∥∇f(θ0)∥2 < η, where  η := m2  8L · min {3(1 − 2κ1), 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (9)  Suppose the gradient and Hessian errors satisfy the bounds  γg := 2ηαg  m  + βg ≤ η,  and  γh := 2ηαh  m  + βh ≤ m  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (10)  Also suppose the robust estimates satisfy  |RobustEstimate({L(θt + α∆θt, zi)}n  i=1) − f(θt + α∆θt)| ≤ ζ  4,  (11)  for each evaluation of backtracking linesearch, where we set the linesearch parameter to be  ζ ≥ 8γgη  m  + 16γhη2  m2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (12)  Then backtracking linesearch chooses unit steps on all successive iterates, and we have ∥∇f(θt)∥2 <  η and  ∥θt − θ∗∥2 ≤ m  L  �1  2  �2t  + 6c2  m  (13)  for all t ≥ 1, where  c2 = η  �4γgL  m2 + 2γh  m  �  + 2Lγ2  g  m2  + γg + 2γgγh  m  ,  and we further assume that (γg, γh) are small enough so that  c2 ≤ min  �η  2, m2  24L  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our first step is to show that backtracking linesearch chooses unit steps whenever the gra-  dient is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', ∥∇f(θt)∥2 < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In other words, we want to prove that  ˜f(θ + ∆θnt) ≤ ˜f(θ) − κ1˜λ(θ)2 + ζ,  where θ = θt denotes the iterate, ˜f denotes the robust estimate of f, and we have defined the noisy  Newton decrement  ˜λ(θ) :=  �  g(θ)T H−1(θ)g(θ)  �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (15)  Recall that ∆θnt = −H(θt)−1g(θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that since ∥∇f(θt)∥2 < η, we have [2, Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='11)]  ∥θt − θ∗∥2 ≤ 2  m∥∇f(θt)∥2 < 2η  m := γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (16)  In particular, this implies a bound of γg := αgγ0 + βg on the error of the gradient, and a bound of  γh := αhγ0 + βh on the error of the Hessian, according to Definitions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We will show that  f(θ + ∆θnt) ≤ f(θ) − κ1˜λ(θ)2 + ζ  2,  (17)  from which the desired result clearly follows by the accuracy bound (11) on the robust estimates  and the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  8  Note that ˜λ(θ)2 = ∆θT  ntH(θ)∆θnt, implying that  ˜λ(θ)2 ≥ (m − γh)∥∆θnt∥2  2 > m  2 ∥∆θnt∥2  2  (18)  (where we assume γh ≤ m  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Furthermore, by the Lipschitz condition, for u ≥ 0, we have  ∥∇2f(θ + u∆θnt) − ∇2f(θ)∥2 ≤ uL∥∆θnt∥2,  so  ��∆θT  nt  �  ∇2f(θ + u∆θnt) − ∇2f(θ)  �  ∆θnt  �� ≤ uL∥∆θnt∥3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (19)  Defining ¯f(u) := f(θ + u∆θnt), we have ¯f′′(u) = ∆θT  nt∇2f(θ + u∆θnt)∆θnt, so we can rewrite  inequality (19) as  | ¯f′′(u) − ¯f′′(0)| ≤ uL∥∆θnt∥3  2,  implying that  ¯f′′(u) ≤ ¯f′′(0) + uL∥∆θnt∥3  2 ≤ ¯f′′(0) + uL  � 2  m  �3/2  ˜λ(θ)3,  using inequality (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Integrating with respect to u gives  ¯f′(u) ≤ ¯f′(0) + u ¯f′′(0) + u2L  2  � 2  m  �3/2  ˜λ(θ)3,  and a second integration gives  ¯f(u) ≤ ¯f(0) + u ¯f′(0) + u2  2  ¯f′′(0) + u3L  6  � 2  m  �3/2  ˜λ(θ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (20)  Now note that  ¯f′(0) = ∇f(θ)T ∆θnt  = −∇f(θ)T H−1(θ)g(θ)  = −˜λ(θ)2 + (g(θ) − ∇f(θ))H−1(θ)g(θ)  ≤ −˜λ(θ)2 + γg  1  m − γh  (∥∇f(θ)∥2 + γg)  ≤ −˜λ(θ)2 + γg  1  m − γh  (η + γg) ,  (21)  whereas  ¯f′′(0) = ∆θT  nt∇2f(θ)∆θnt  = ˜λ(θ)2 + ∆θT  nt  �  ∇2f(θ) − H(θ)  �  ∆θnt  ≤ ˜λ(θ)2 + γh∥∆θnt∥2  2  ≤ ˜λ(θ)2  �  1 + 2γh  m  �  ,  (22)  using the bound (18) in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Plugging inequalities (21) and (22) into inequality (20)  (with u = 1) then gives  f(θ + ∆θnt) ≤ f(θ) +  �  −˜λ(θ)2 + γg(η + γg)  m − γh  �  +  ˜λ(θ)2  2  �  1 + 2γh  m  �  + L  6  � 2  m  �3/2  ˜λ(θ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  9  Finally, note that  ˜λ(θ) ≤ ∥H−1/2(θ)∥2 · ∥g(θ)∥2 ≤ ∥∇f(θ)∥2 + γg  √m − γh  ≤  η + γg  √m − γh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Since  ζ ≥ 2  �  �γg · 2η  m/2 + γh  m  �  2η  �  m/2  �2�  �  and using the assumptions γh ≤ m  2 and γg ≤ η, we then have  f(θ + ∆θnt) ≤ f(θ) − ˜λ(θ)2  �  1  2 − L  6  � 2  m  �3/2  ˜λ(θ)  �  + ζ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In particular, if ˜λ(θ) ≤  3−6κ1  L(2/m)3/2 , which is guaranteed if η is chosen sufficiently small so  η + γg  √m − γh  ≤  3 − 6κ1  L(2/m)3/2 ,  then inequality (17) is indeed satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We can guarantee this last inequality by taking η ≤  3m2(1−2κ1)  8L  , assuming γh ≤ m  2 and γg ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  To derive the geometric convergence rate (13), we will use induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We first establish an  inequality of the form  ∥∇f(θ + ∆θnt)∥2 ≤ c1∥∇f(θ)∥2  2 + c2,  (23)  assuming ∥∇f(θ)∥2 < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that  ∥∇f(θ + ∆θnt)∥2 = ∥∇f(θ + ∆θnt) − g(θ) − H(θ)∆θnt∥2  ≤ ∥∇f(θ + ∆θnt) − ∇f(θ) − ∇2f(θ)∆θnt∥2 + γg + γh∥∆θnt∥2  =  ����  � 1  0  �  ∇2f(θ + u∆θnt) − ∇2f(θ)  �  ∆θntdu  ����  2  + γg + γh∥∆θnt∥2  ≤ L  2 ∥∆θnt∥2  2 + γg + γh∥∆θnt∥2,  (24)  using the Lipschitz condition in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Next, we use the bound  ∥∆θnt∥2 = ∥H−1(θ)g(θ)∥2 ≤  1  m − γh  (∥∇f(θ)∥2 + γg) ≤ 2  m (∥∇f(θ)∥2 + γg) ,  assuming γh ≤ m  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Plugging back into inequality (24) gives  ∥∇f(θ + ∆θnt)∥2 ≤ L  2  �2 (∥∇f(θ)∥2 + γg)  m  �2  + γg + γh  �2 (∥∇f(θ)∥2 + γg)  m  �  = 2L  m2 ∥∇f(θ)∥2  2 + ∥∇f(θ)∥2  �4γgL  m2 + 2γh  m  �  +  �  2Lγ2  g  m2  + γg + 2γgγh  m  �  ≤ 2L  m2 ∥∇f(θ)∥2  2 + η  �4γgL  m2 + 2γh  m  �  + 2Lγ2  g  m2  + γg + 2γgγh  m  ,  (25)  10  giving inequality (23) with c1 = 2L  m2 and c2 = η  �  4γgL  m2 + 2γh  m  �  +  2Lγ2  g  m2 + γg + 2γgγh  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In particular, c2  can be made small if we choose γg and γh small enough, and we will assume that c2 ≤ η  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We will  also assume that c1c2 ≤ 1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We are now ready for our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Using the notation yt := c1∥∇f(θt)∥2, we will prove that  yt < c1η and yt ≤ y2t  0 + c1c2 for all t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For the base case t = 1, note that  y1 ≤ y2  0 + c1c2 = y21  0 + c1c2,  using inequality (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Furthermore, since y0 < c1η < 1  2 and c2 ≤ η  2 by assumption, we have  y1 ≤ y0  2 + c1η  2  < c1η  2 + c1η  2  = c1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  For the inductive step, suppose t ≥ 1, and we have ys < c1η and ys ≤ y2s  0 + 3c1c2 for all s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Then by inequality (23), we have  yt+1 ≤ y2  t + c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (26)  Furthermore, using the assumption η ≤ m2  4L , we have yt < 1  2, so if c2 ≤ η  2, this implies that  yt+1 ≤ yt  2 + c1η  2  < c1η  2 + c1η  2  ≤ c1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  By inequality (26) and the induction hypothesis, we now write  yt+1 ≤ y2  t + c1c2  ≤  �  y2t  0 + 3c1c2  �2  + c1c2  = y2t+1  0  + 6c1c2y2t  0 + 9c2  1c2  2 + c1c2  ≤ y2t+1  0  + 3  2c1c2 + 3  4c1c2 + c1c2  ≤ y2t+1  0  + 3c1c2,  using the assumption c1c2 ≤ 1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Thus,  c1∥∇f(θt)∥2 ≤  �1  2  �2t  + 3c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Applying inequality (16) then gives the convergence rate  ∥θt+1 − θ∗∥2 ≤ 2  m · m2  2L  ��1  2  �2t+1  + 6L  m2 c2  �  ,  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Next, we show that after a finite number of steps, the iterates will indeed satisfy ∥∇f(θt)∥2 < η,  for an appropriate η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose the parameters η and ζ are as defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define  γ := κ1 · κ2  m  M  �1  2 − κ1  � η2  4  √  2M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  11  Suppose (γg, γh) are chosen small enough such that ζ ≤ γ  2 and conditions (10) and (11) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Also suppose  2αg  m  �  2M (f(θ0) − f(θ∗)) + βg ≤ min  �  η  2,  �m  2 · 1  2  �  η2  4  √  2M  �  ,  (27)  and  2αh  m  �  2M (f(θ0) − f(θ∗)) + βh ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (28)  If ∥∇f(θt)∥2 ≥ η for an iterate t ≥ 0, then f(θt) ≤ f(θ0) and f(θt+1) − f(θt) < − γ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' First, we show that we have an upper bound γ′  0 :=  2  m  �  2M (f(θ0) − f(θ∗)) on ∥θt − θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We can then translate this into upper bounds γ′  g := αgγ′  0 + βg and γ′  h := αhγ′  0 + βh on the gradient  and Hessian deviations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' By the result of Theorem 1, we must have ∥∇f(θs)∥2 ≥ η for  all 0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We now show by induction that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' f(θs) ≤ f(θ0), and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∥θs − θ∗∥2 ≤ γ′  0,  for all 0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For the base case s = 0, note that claim (1) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We can establish claim  (2) by noting that  ∥θs − θ∗∥2 ≤ 2  m∥∇f(θs)∥2 ≤ 2  m  �  2M(f(θs) − f(θ∗)) ≤ 2  m  �  2M(f(θ0) − f(θ∗)) = γ′  0,  (29)  using inequality (16), inequality (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='14) of Boyd and Vandenberghe [2], and claim (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Turning to the inductive step, suppose claims (1) and (2) hold for all s ≤ s′, where 0 ≤ s′ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We wish to establish the claims for s = s′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that if we prove claim (1), then claim (2)  follows by the same chain of inequalities (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Thus, it remains to establish claim (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Assuming γ′  g ≤  η  2, we have ∥g(θs)∥2 ≥  η  2 for all 0 ≤ s ≤ s′ by claim (2), the fact that  ∥∇f(θs)∥2 ≥ η, and the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Using the same notation for the Newton decrement (15),  we note that  ˜λ(θs)2 ≥  1  �  M + γ′  h  ∥g(θs)∥2  2 ≥  η2  4  √  2M  ,  (30)  where we assume γ′  h ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  First, we will prove that the exit condition of BacktrackingLineSearch function will be satis-  fied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', we want to prove that  ˜f(θs′ + α∆θnt) ≤ ˜f(θ) − κ1α˜λ(θs′)2 + ζ  (31)  holds for small enough α, where ˜f is the robust estimate of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For convenience, we use the notation  θ = θs′ in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In fact, we will show that  f(θ + α∆θnt) ≤ f(θ) − κ1α˜λ(θ)2  (32)  for small enough α, which clearly then implies inequality (31) by the triangle inequality and the  condition (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  12  Consider the following:  f(θ + α∆θnt) ≤ f(θ) + α∇f(θ)T ∆θnt + M  2 ∥∆θnt∥2  2α2  ≤ f(θ) + αg(θ)∆θnt + αγ′  g∥∆θnt∥2 + M  2  ˜λ(θ)2 2  mα2  = f(θ) − α˜λ(θ)2 + αγ′  g∥∆θnt∥2 + M  m  ˜λ(θ)2α2  ≤ f(θ) − α˜λ(θ)2 + αγ′  g˜λ(θ)  �  2  m + M  m  ˜λ(θ)2α2,  where we use the relation −˜λ(θ)2 = g(θ)T ∆θnt and inequality (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Assuming γ′  g  �  2  m ≤ 1  2  �  η2  4  √  2M  and using inequality (30), the last expression is upper-bounded as  f(θ + α∆θnt) ≤ f(θ) − α  2  ˜λ(θ)2 + M  m  ˜λ(θ)2α2  = f(θ) − ˜λ(θ)2α  �1  2 − M  m α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (33)  Hence, the condition (32) is indeed satisfied for sufficiently small α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', α ≤ m  M  � 1  2 − κ1  �  , and in  particular, the linesearch procedure must return a stepsize satisfying α ≥ κ2 m  M  � 1  2 − κ1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Plugging  such a stepsize into inequality (31), we have  f(θs′+1) ≤ f(θs′) − κ1 · κ2  m  M  �1  2 − κ1  � η2  4  √  2M  + ζ ≤ f(θ0) − γ + ζ ≤ f(θ0) − γ  2,  (34)  using inequality (30) and the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This implies that claim (1) is true, completing  the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Finally, note that the inequality f(θt+1)−f(θt) < − γ  2 follows by the same argument in inequal-  ity (34) with θ = θt, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  The preceding theorem directly implies that after a finite number of steps (known as the  “damped Newton” phase), all successive iterates of the algorithm satisfy ∥∇f(θt)∥2 < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Robust Estimation of Gradients and Hessians  In this subsection, we explain how robust estimators for gradients and Hessians can be obtained  under two models of contamination, namely the Huber ϵ-contamination model and the heavy-tailed  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Robust Gradient Estimation  For the ϵ-contamination model, we obtain a robust gradient estimate by applying Algorithm 1 to  the gradients computed on each of the n sampled data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Similarly, for the heavy-tailed model,  we use Algorithm 2 to obtain a robust gradient estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For completeness, we summarize this  procedure in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  13  Algorithm 4 Robust Gradient Estimator  Require: Samples S = {zi}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Parameter θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Contamination type Type  Require: (If Type = Huber) Corruption Level ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Dimension p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Failure probability δ  Require: (If Type = Heavy-tail) Failure probability δ  1: function RobustGradientEstimator(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  2:  Compute {∇L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' the gradient of the loss at each data point in S  3:  if Type = Huber then  return HuberEstimator({∇L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  4:  end if  5:  if Type = Heavy-tail then  return HeavyTailedEstimator({∇L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  6:  end if  7: end function  The following lemmas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' borrowed from Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28], show that Algorithm 4 returns a robust  gradient estimator that satisfies Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 1 (Lemma 1 of Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a Huber  ϵ-contaminated distribution (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let the true distribution of gradients ∇L(θ, z), with z drawn from  P, have bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then Algorithm 4 with S = {zi}n  i=1, Type = Huber, and any  θ ∈ Θ returns a gradient estimate g(θ) that satisfies  ∥g(θ) − E[∇L(θ, z)]∥2 ≤ C1(√ϵ + γ(n, p, δ, ϵ))  �  ∥ Cov(∇L(θ, z))∥2 log p,  (35)  with probability at least 1 − δ, where C1 > 0 is a constant and γ is given by  γ(n, p, δ, ϵ) =  �p log(p) log(n/(pδ))  n  �3/8  +  �ϵp2 log(p) log(p log(p)/δ)  n  �1/4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (36)  Lemma 2 (Lemma 2 of Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a heavy-tailed  distribution P such that the true distribution of gradients ∇L(θ, z) has bounded second moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Then Algorithm 4 with S = {zi}n  i=1, Type = Heavy-tail, and any θ ∈ Θ returns a gradient estimate  g(θ) that satisfies  ∥g(θ) − E[∇L(θ, z)]∥2 ≤ 11  �  tr(Cov(∇L(θ, z))) log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ,  (37)  with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Robust Hessian Estimation: The Vectorizing Approach  The procedure for obtaining a robust Hessian, summarized in Algorithm 5, is similar to that  of Algorithm 4, except that the appropriate multivariate estimation procedure is applied to a  vectorized version of the Hessian matrix (where we use flatten(A) to denote a vectorized version of  the matrix A, and use unflatten() to denote the inverse function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  14  Algorithm 5 Robust Hessian Estimator  Require: Samples S = {zi}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Parameter θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Contamination type Type  Require: (If Type = Huber) Corruption Level ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Dimension p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Failure probability δ  Require: (If Type = Heavy-tail) Failure probability δ  1: function RobustHessianEstimator(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ)  2:  Compute {∇2L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' the Hessian of the loss at each data point in S  3:  if Type = Huber then  return unflatten(HuberEstimator({flatten(∇2L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi))}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ))  4:  end if  5:  if Type = Heavy-tail then  return unflatten(HeavyTailedEstimator({flatten(∇2L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi))}n  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' δ))  6:  end if  7: end function  The next two lemmas follow immediately from the arguments used to derive Lemmas 1 and 2:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a Huber ϵ-contaminated distribution (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Suppose Cov(flatten(∇2L(θ, z))) is finite and flatten(∇2L(θ, z))) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Then Algorithm 4 with S = {zi}n  i=1, Type = Huber, and any θ ∈ Θ returns a Hessian estimate  H(θ) that satisfies  ∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C2(√ϵ + γ(n, p, δ, ϵ))  �  ∥ Cov(flatten(∇2L(θ, z)))∥2 log p,  (38)  with probability at least 1 − δ, where C2 > 0 is a constant and γ is given by equation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a heavy-tailed distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Suppose  Cov(flatten(∇2L(θ, z))) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then Algorithm 4 with S = {zi}n  i=1, Type = Heavy-tail, and any  θ ∈ Θ returns a Hessian estimate H(θ) that satisfies  ∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C3  �  tr(Cov(flatten(∇2L(θ, z)))) log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ,  (39)  with probability at least 1 − δ, where C3 > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  4  Application to GLMs  In this section, we apply the robust Newton method to parametric estimation in GLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We consider  the Huber ϵ-contamination model in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1, and we consider the heavy-tailed contamination  model in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Throughout this section, we will assume that the uncontaminated model is a GLM of the  form (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Consider the loss function in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We assume that the link function Φ of the  GLM satisfies the following bounds:  E  �  |Φ′(xT  i θ) − Φ′(xT  i θ∗)|2k�  ≤ LΦ,2k∥θ − θ∗∥2  2 + BΦ,2k,  ∀θ ∈ Θ,  (40)  E  �  |Φ(t)(xT  i θ∗)|k�  ≤ MΦ,t,k,  (41)  and  ∥Φ(t)∥∞ ≤ MΦ,t,  (42)  15  for pairs (k, t) to be specified in the sequel, where Φ(t) is the tth derivative of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We also make assumptions on the boundedness of moments of xi ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We say that xi has  bounded 2kth moments if there is a constant �C2k > 0 such that for every unit vector v ∈ Rd, we  have E[(xT  i v)2k] ≤ �C2k  �  E[(xT  i v)2]  �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We assume that the covariate distribution has bounded eighth  moments and a finite covariance matrix Σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  From Lemmas 1 and 2, we see that the term Cov(∇L(θ, z)) plays a crucial role in proving  that our gradient estimates are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Likewise, Lemmas 3 and 4 show the importance of the  term Cov(flatten(∇2L(θ, z))) in proving that the Hessian estimates are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The following two  lemmas provide upper bounds on these two terms for the specific case of GLMs:  Lemma 5 (Lemma 4 in Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a distribution  that satisfies the GLM model (3) with the assumptions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Let the link function Φ satisfy  inequalities (40) and (41) for k ∈ {1, 2} and t ∈ {2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then the true distribution of gradients  ∇L(θ, z) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Moreover,  ∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2  ��  LΦ,4 + LΦ,2  �  ∥θ − θ∗∥2  2  + C2∥Σx∥2  �  BΦ,2 +  �  BΦ,4 + c(σ)  �  MΦ,2,2 +  �  c(σ)3MΦ,4,1  �  ,  (43)  where C1, C2 > 0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a distribution that satisfies the GLM  model (3) with the assumptions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let the link function Φ satisfy inequality (42) for t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then  the distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Moreover,  we have  tr(Cov(flatten(∇2L(θ, z)))) ≤ M  2  Φ,2 �C4p2∥Σx∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (44)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' From the definition of the loss function (4), we have E[∇2L(θ, z)] = E[Φ′′(xT  i θ)xixT  i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' By  our assumptions on the boundedness of Φ′′ and bounded eighth moments of xi, we see that the  distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We then  write  tr(Cov(flatten(∇2L(θ, z)))) = tr(Cov(flatten(Φ′′(xT  i θ)xixT  i )))  ≤ tr(E[flatten(Φ′′(xT  i θ)xixT  i ) flatten(Φ′′(xT  i θ)xixT  i )T ])  ≤ M  2  Φ,2 tr(E[flatten(xixT  i ) flatten(xixT  i )T ])  = M  2  Φ,2  p  �  j,k=1  E[x2  ijx2  ik]  ≤ M  2  Φ,2E  �  �  �  �  p  �  j=1  xij  �  �  4�  �  ≤ M  2  Φ,2E  ��  xT  i 1  �4�  ,  where 1 denotes the all-ones vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Finally, note that  E  ��  xT  i 1  �4�  ≤ �C4E  ��  xT  i 1  �2�2  ≤ �C4p2∥Σx∥2  2,  implying the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  16  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that under additional assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', 4-wise independence of the components  of the xi’s), we can prove that  ∥ Cov(flatten(∇2L(θ, z)))∥2 ≤ C �C4∥Σx∥2  2,  for some constant C > 0, which avoids an extra dimension-dependent factor in comparison to  inequality (44) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2 in Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15]) for the Huber contamination setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Indeed,  only the spectral norm of the covariance of the flattened Hessian appears in the deviation bound  of Lemma 3 (Huber’s ϵ-contamination model);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' the trace of the covariance appears in Lemma 4  (heavy-tailed model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In order to apply Theorems 1 and 2 to GLMs, we need the Hessian ∇2R(θ) to be Lipschitz  smooth and satisfy mI ⪯ ∇2R(θ) ⪯ MI for all θ ∈ Θ close enough to the initialization θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We now  verify these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let the link function Φ satisfy inequality (42) for t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then the Hessian ∇2R(θ)  is L-Lipschitz and satisfies ∇2R(θ) ⪯ MI with L :=  �  �C4MΦ,3∥Σx∥2 and M := MΦ,2  �  �C4∥Σx∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For the Lipschitz condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' note that for any θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ2 ∈ Rp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' we have  ∥∇2R(θ1) − ∇2R(θ2)∥2 = ∥∇2L(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' z) − ∇2L(θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' z)∥2  =  ��E  �  xixT  i  �  Φ′′(xT  i θ1) − Φ′′(xT  i θ2)  ����  2  =  sup  u∈Sp−1 uT E  �  xixT  i  �  Φ′′(xT  i θ1) − Φ′′(xT  i θ2)  ��  u  =  sup  u∈Sp−1 E  �  (uT xi)2 �  Φ′′(xT  i θ1) − Φ′′(xT  i θ2)  ��  ≤  sup  u∈Sp−1 E  �  (uT xi)4� 1  2 E  ��  Φ′′(xT  i θ1) − Φ′′(xT  i θ2)  �2� 1  2  ≤  sup  u∈Sp−1  �  �C4E  �  (uT xi)2� �  MΦ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3∥θ1 − θ2∥2  ≤  �  �C4MΦ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3∥Σx∥2∥θ1 − θ2∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  where we use the mean value theorem to upper-bound the expectation in the second-to-last in-  equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  For any θ ∈ Rp, we have  ∥∇2R(θ)∥2 = ∥∇2L(θ, z)∥2  =  ��E  �  xixT  i  �  Φ′′(xT  i θ)  ����  2  =  sup  u∈Sp−1 uT E  �  xixT  i  �  Φ′′(xT  i θ)  ��  u  =  sup  u∈Sp−1 E  �  (uT xi)2 �  Φ′′(xT  i θ)  ��  ≤  sup  u∈Sp−1 E  �  (uT xi)4� 1  2 E  ��  Φ′′(xT  i θ)  �2� 1  2  ≤  sup  u∈Sp−1  �  �C4E  �  (uT xi)2�  MΦ,2  ≤ MΦ,2  �  �C4∥Σx∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  17  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose there exist constants B, τ > 0 such that for any θ ∈ Rp such that ∥θ∥2 ≤  B, we have  �C4∥Σx∥2  2 · P(|xT  i θ| > τ) ≤ 1  4λ2  min(Σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (45)  Define bτ := inf|u|≤τ Φ′′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then bτ  2 λmin(Σx)I ⪯ ∇2R(θ) for all θ ∈ Rp such that ∥θ∥2 ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose v ∈ Rp is a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We write  vT ∇2R(θ)v = E  �  (vT xi)2 · Φ′′(xT  i θ)  �  ≥ E  �  (vT xi)2 · bτ1{|xT  i θ| ≤ τ}  �  = bτ  �  E  �  (vT xi)2�  − E  �  (vT xi)2 · 1{|xT  i θ| > τ}  ��  ≥ bτ  �  λmin(Σx) −  �  E [(vT xi)4] · P  �  |xT  i θ| > τ  ��  ≥ bτ  �  λmin(Σx) −  �  �C4∥Σx∥2  2 · P  �  |xT  i θ| > τ  ��  ≥ bτ  2 λmin(Σx),  where we have used the fact that Φ′′ is always nonnegative in the first inequality, applied Cauchy-  Schwarz in the second inequality, and used the assumption (45) in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that when the covariates are sub-Gaussian, we can certainly guarantee that the  tail condition (45) is satisfied for sufficiently large τ, since xT  i θ is sub-Gaussian with parameter  scaling with B and the sub-Gaussian parameter σ2  x of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Thus, we have  P(|xT  i θ| > τ) ≤ c1 exp  �  − c2τ 2  B2σ2x  �  ,  and it suffices to take τ = c3Bσx log1/2 � c4∥Σx∥2  2  λ2  min(Σx)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Furthermore, in the proofs of Theorems 1 and 2  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' inequalities (16) and (29), respectively), we show that ∥θt − θ∗∥2 remains bounded (where the  bound depends on θ0 and the problem parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In the case of logistic regression, we have Φ′′(u) =  eu  (1+eu)2 , and it is easy to see that bτ > 0 for  any value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  For applying Theorems 1 and 2, we also need the robust estimate of the losses to be close to the  population risk, as in inequality (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In the following two lemmas, we show that this assumption  holds with high probability for the robust estimates obtained by applying Algorithms 1 and 2 on  the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Further note that the following lemmas require boundedness of higher-order moments  of L(θ, z), which can be justified in our scenario if θ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As mentioned in Remark 2, we  can indeed assume that the iterates {θt}, to which Lemmas 7 and 8 are applied in the sequel, are  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a Huber ϵ-contaminated distribution (6),  where the true distribution satisfies the GLM model (3) with the assumptions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let L(θ, z)  have bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then with probability at least 1 − δ, the robust estimate returned by  Algorithm 1 satisfies  |HuberEstimate({L(θ, zi)}n  i=1) − R(θ)|  ≤ C1  �  ϵ +  �  log(n/δ)  n  � 3  4  + C2  �  ϵ +  �  log(n/δ)  n  � 1  2 log(1/δ)  n  ,  18  where C1, C2 > 0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The desired result follows from an application of Lemma 14 in Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a heavy-tailed distribution P that satisfies  the GLM model (3) with the assumptions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let L(θ, z) have bounded second moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then  with probability at least 1 − δ, the robust estimate returned by Algorithm 2 satisfies  |HeavyTailedEstimate({L(θ, zi)}n  i=1) − R(θ)| ≤ C  �  log  � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4  δ  �  n  ,  with probability at least 1 − δ, where C > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This result follows from similar arguments to those in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  The following lemma shows how small the parameters (αg, βg, αh, βh) in the robust gradient  and Hessian estimates need to be in order to satisfy the assumptions of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define  �αg := min  �  m  64,  ηm  8  �  2M(f(θ0) − f(θ∗))  ,  �  η2m  8  √  2M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  m  8  �  2M(f(θ0) − f(θ∗))  �  ,  �βg := min  �  η  32,  ηm  8  �  2M(f(θ0) − f(θ∗))  , 1  4  �  η2m  8  √  2M  �  ,  �αh := min  �  m2  256η,  mM  4  �  2M(f(θ0) − f(θ∗))  �  ,  �βh := min  � m  128, M  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Suppose αg ≤ �αg, βg ≤ �βg, αh ≤ �αh, and βh ≤ �βh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then the bounds (10) and (14) of Theorem 1,  as well as the bounds (27) and (28) of Theorem 2, are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Under the assumptions, we have  γg = 2ηαg  m  + βg ≤ η  32 + η  32 = η  16,  γh = 2ηαh  m  + βh ≤ m  128 + m  128 = m  64,  2αg  m  �  2M (f(θ0) − f(θ∗)) + βg ≤ min  �  η  2,  �m  2 · 1  2  �  η2  4  √  2M  �  ,  2αh  m  �  2M (f(θ0) − f(θ∗)) + βh ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Hence, inequalities (10), (27), and (28) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Using the fact that η := m2  8L ·min {3(1 − 2κ1), 2} ≤  19  m2  4L , we have  L  m2 ≤  1  4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then  c2 = η  �4γgL  m2 + 2γh  m  �  + 2Lγ2  g  m2  + γg + 2γgγh  m  ≤ η  �4η  16 · 1  4η + 2  m · m  64  �  + 2η2  256 · 1  4η + η  16 + 2  m · η  16 · m  64  < η  6  ≤ m2  24L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Hence, inequality (14) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In Propositions 3 and 4, we will derive expressions for (αg, βg, αh, βh) for the Huber contami-  nation and heavy-tailed models, which will then allow us to translate the conditions of Lemma 9  into assumptions involving the contamination level and/or minimum sample size required for our  theoretical results to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Huber Contamination  In this subsection, we assume that the link function Φ satisfies inequality (40) for k ∈ {1, 2, 4},  inequality (41) for t ∈ {2, 4}, and inequality (42) for t ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We assume that the population  risk R(θ) is m-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define L and M as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Under the assumptions above, the gradient and Hessian estimates with Type =  Huber returned by Algorithms 4 and 5, respectively, satisfy the conditions of Definitions 1 and 2  with the following parameters:  αg = c1(√ϵ + γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p,  βg = c2(√ϵ + γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p,  αh = 0,  βh = c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p  �  log p,  with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' By Lemma 5, the true distribution of the gradients ∇L(θ, z) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Moreover,  ∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2  ��  LΦ,4 + LΦ,2  �  ∥θ − θ∗∥2  2  + C2∥Σx∥2  �  BΦ,2 +  �  BΦ,4 + c(σ)  �  MΦ,2,2 +  �  c(σ)3MΦ,4,1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Plugging the above bound into inequality (35) of Lemma 1, we obtain  ∥g(θ) − E[∇L(θ, z)]∥2 ≤ C′  1(√ϵ + γ(n, p, δ, ϵ))  �  ∥ Cov(∇L(θ, z))∥2 log p  ≤ c1(√ϵ + γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p · ∥θ − θ∗∥2  + c2(√ϵ + γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  20  Hence, the gradient estimate returned by Algorithm 4 satisfies Definition 1 with αg = c1(√ϵ +  γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p and βg = c2(√ϵ + γ(n, p, δ, ϵ))  �  ∥Σx∥2 log p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  By Lemma 6, the true distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded  fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Moreover, combining Lemma 6 with Lemma 3, we obtain  ∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C′  2(√ϵ + γ(n, p, δ, ϵ))  �  ∥ Cov(flatten(∇2L(θ, z))))∥2 log p  ≤ c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p  �  log p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Hence, the Hessian estimate returned by Algorithm 5 satisfies Definition 2 with αh = 0 and βh =  c3(√ϵ + γ(n, p, δ, ϵ))∥Σx∥2p√log p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We then have the following result:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a Huber ϵ-contaminated distribution (6),  where the true distribution satisfies the GLM model (3) and the aforementioned assumptions are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define (η, c2, γg, γh) as in Theorem 1 and γ as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose  ζ = C max  �  �  �  γgη  m + γhη2  m2 ,  �  ϵ +  �  log(n/δ)  n  � 3  4  +  �  ϵ +  �  log(n/δ)  n  � 1  2 log(1/δ)  n  �  �  � ≤ γ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (46)  Let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define  �γ := min  �  �αg  c1  �  ∥Σx∥2 log p  ,  �βg  c2  �  ∥Σx∥2 log p  ,  �βh  c3∥Σx∥2p√log p  �  ,  (47)  where (�αg, �βh, �βg) are as defined in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose n and ϵ are such that  √ϵ + γ (n, p, δ, ϵ) < �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (48)  Then applying Algorithm 3 on {zi}n  i=1 with initialization θ0 ∈ Θ and number of iterations  T ≥ R(θ0) − R(θ∗)  γ/2  + log2 log2  �6c2L  m2  �  returns an output such that  ∥θT − θ∗∥2 ≤ 12c2  m  = O  �  p2�  ϵ log p  �  ,  with probability at least 1 − Tδ  �  5 +  � log( m  M ( 1  2 −κ1))  log κ2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We will apply Theorems 1 and 2 to show that Algorithm 5 returns �θT such that ∥θT −θ∗∥2 ≤  12c2  m = O  �  ϵ log(p) +  �  ϵ log(p)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Under the assumption that √ϵ + γ (n, p, δ, ϵ) < �γ, and using Proposition 3, the assumptions  of Lemma 9 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that the assumptions of Lemma 7 are likewise satisfied by the  condition (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Applying Theorem 2, the risk R(θt) is reduced by at least γ  2 in each step of the  damped Newton phase of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Hence, the number of such iterations cannot exceed  Tdamp := R(θ0) − R(θ∗)  γ/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (49)  21  Define  Tpure := log2 log2  �6c2L  m2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  (50)  Applying Theorem 1, we observe that after Tpure iterations in the pure Newton phase, we have  m  L  � 1  2  �2t  < 6c2  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Therefore, from inequality (13), we have ∥�θ − θ∗∥2 ≤ 12c2  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Combining inequali-  ties (49) and (50), we obtain the bound on the total number of iterations T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  From the preceding analysis on the robust gradient and Hessian estimators, observe that η, L,  and m are independent of ϵ and p, while γg is O(√ϵ log p) and γh is O(p√ϵ log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Hence, from  inequality (13), we have ∥θT − θ∗∥2 = O  �  p√ϵ log p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We now compute the error probability of the algorithm via a union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For each of the  T gradient and Hessian calculations, we have a possible error of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Furthermore, each call of  backtracking linesearch incurs a possible error from the robust estimates, by Lemma 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' once at θt  and once for each value of α used in the linesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This is a total of 2Tpure evaluations for the  pure Newton steps, and a maximum of Tdamp  �  1 +  � log( m  M ( 1  2 −κ1))  log κ2  ��  evaluations for the damped  Newton steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Thus, the overall probability of error is at most  2Tδ + 2Tpureδ + Tdamp  �  1 +  �  log  � m  M  � 1  2 − κ1  ��  log κ2  ��  δ ≤ Tδ  �  5 +  �  log  � m  M  � 1  2 − κ1  ��  log κ2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Examining the condition (48), we see that (assuming (m, M, L, η, ∥Σx∥2) are all con-  stants) we have a required upper bound of �γ2 ≍  1  log p on the contamination proportion ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Further-  more, from the expression (36), we have (ignoring log factors) n ≿ max  �  p, ϵp2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The condition (46)  likewise gives a minimum sample size requirement on n in terms of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' It is instructive to compare the result of Theorem 3 to Theorem 4 in Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28],  which gives a convergence statement of the form  ∥θt − θ∗∥2 ≤ κt∥θ0 − θ∗∥2 + C∥Σx∥1/2  2  √log p  1 − κ  �√ϵ + γ(n, p, δ, ϵ)  �  for iterates {θt}t≥0 of robust gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For sufficiently large t, the second term dominates,  leaving an error term of O(√ϵ log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our theorem has a dominant factor of O(p√ϵ log p), which  can be reduced to O(√ϵ log p) if we assume 4-wise independence of the coordinates of the covariate  distribution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Remark 1 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In terms of the convergence rate of the optimization procedure,  however, we just need T ≍ log log 1  ϵ, compared to T ≍ log 1  ϵ in the case of robust gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Linear regression is of course a special case of GLMs, for which Theorem 3 readily applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  On the other hand, note that a much more direct way to obtain a robust estimator for linear  regression would be to directly robustify the estimator �θOLS =  �  XT X  n  �−1 �  XT y  n  �  , where we apply  Algorithm 1 to obtain robust estimates of E[yixi] and E[xixT  i ] (the latter matrix being vectorized  before applying the agnostic mean algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Indeed, in the non-robust case, applying Newton’s  method to the ordinary least squares objective converges in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A careful analysis of this  so-called “robust plug-in estimator” would also give an error of O(√ϵ) in the robust case, but a  direct analysis would provide an error bound which depends on ∥θ∗∥2, since Cov(xi, yi) would scale  with ∥θ∗∥2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Corollary 3 in Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' On the other hand, the guarantee of Theorem 3  for the full robust Newton’s method does not involve ∥θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  22  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' A natural question is whether the estimation error upper bounds in Theorem 3 are tight:  For i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples from a GLM with Huber ϵ-contamination, is it possible to derive estimators with  error smaller than C√ϵ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For the case of linear regression, this problem has been studied quite  carefully, and it has been established that when the uncontaminated data are Gaussian with an  isotropic covariance, the rate should be Θ(ϵ) [3, 11, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In the case when the covariates only follow  a bounded fourth moment assumption, the rate improves to Θ(√ϵ) [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We are not aware of  existing lower bounds in the literature for more general GLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Heavy-Tailed Distributions  In this subsection, we assume that the link function Φ satisfies inequality (40) for k ∈ {1, 2},  inequality (41) for t ∈ {2, 4}, and inequality (42) for t ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As in the case of Huber contam-  ination, we assume that the population risk R(θ) is m-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define L and M as in  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Under the assumptions above, the gradient and Hessian estimates with Type =  Heavy-tail returned by Algorithms 4 and 5, respectively, satisfy the conditions of Definitions 1 and 2  with the following parameters:  αg = c1  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ,  βg = c2  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ,  αh = 0,  βh = c3∥Σx∥2p  �  log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ,  with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' By Lemma 5, the distribution of the gradients ∇L(θ, z) has bounded fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Moreover,  ∥ Cov(∇L(θ, z))∥2 ≤ C1∥Σx∥2  ��  LΦ,4 + LΦ,2  �  ∥θ − θ∗∥2  2  + C2∥Σx∥2  �  BΦ,2 +  �  BΦ,4 + c(σ)  �  MΦ,2,2 +  �  c(σ)3MΦ,4,1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Plugging this bound into inequality (37) of Lemma 2, we obtain  ∥g(θ) − E[∇L(θ, z)]∥2 ≤ 11  �  tr(Cov(∇L(θ, z))) log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ≤ 11  �  p Cov(∇L(θ, z)) log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ≤ c1  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ∥θ − θ∗∥2 + c2  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Hence, the gradient estimate returned by Algorithm 4 satisfies Definition 1 with αg = c1  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  and βg = c2  �  p log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  23  By Lemma 6, the distribution of the flattened Hessian flatten(∇2L(θ, z)) has bounded fourth  moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Moreover, combining Lemma 6 with Lemma 4, we obtain  ∥H(θ) − E[∇2L(θ, z)]∥2 ≤ C3  �  tr(Cov(flatten(∇2L(θ, z)))) log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  ≤ c3∥Σx∥2p  �  log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Hence, the Hessian estimate returned by Algorithm 5 satisfies Definition 2 with αh = 0 and βh =  c3∥Σx∥2  �  log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4/δ)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We then have the following result:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let {zi}n  i=1 be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' samples drawn from a heavy-tailed distribution P that satisfies  the GLM in (3), and suppose the aforementioned assumptions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Define (η, c2, ζ, γg, γh)  as in Theorem 1 and γ as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Suppose n satisfies  n > C max  �  p  �α2g  , p  �β2g  , p2∥Σx∥2  2  �β2  h  , 1  ζ2  �  log  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4  δ  �  ,  (51)  where (�αg, �βg, �βh) are as defined in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Then applying Algorithm 3 on {zi}n  i=1, with initial-  ization θ0 ∈ Θ and number of iterations  T ≥ R(θ0) − R(θ∗)  γ/2  + log2 log2  �6c2L  m2  �  ,  returns an output such that  ∥θT − θ∗∥2 ≤ 12c2  m  = O  ��  p2  n  �  ,  with probability at least 1 − Tδ  �  5 +  � log( m  M ( 1  2 −κ1))  log κ2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We will follow a similar outline as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Using the assumption on n and Proposition 4, it is straightforward to verify that the conditions  of Lemma 9 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Furthermore, the conditions of Lemma 8 are satisfied by inequality (51),  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Applying Theorem 2, the risk R(θt) is reduced by at least γ  2 in each step of the damped  Newton phase of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Hence, the number of such iterations cannot exceed Tdamp, defined  as in equation (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Applying Theorem 1, we observe that after Tpure iterations (defined as in  equation (50)) in the pure Newton phase, we have m  L  � 1  2  �2t  < 6c2  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Therefore, from inequality (13),  we have ∥�θ − θ∗∥2 ≤ 12c2  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Combining inequalities (49) and (50), we obtain the bound on the total  number of iterations T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  From the preceding analysis on the robust gradient and Hessian estimators (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Proposition 4),  observe that γg = O  ��  p  n  �  and γh = O  ��  p2  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Hence, c2 is O  ��  p2  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' From inequality (13), we  then have ∥θT − θ∗∥2 = O  ��  p2  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Computing the error probability of the algorithm via a union bound is the same as in Theorem 3  with the use of appropriate gradient, Hessian, and robust estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Again, assuming 4-wise independence of the coordinates of the covariate distribution,  we can reduce the dimension-dependence of the bounds (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We then take βh ≍ √p∥Σx∥2  2,  to obtain an estimation error bound of the form ∥�θ − θ∗∥2 = O  ��  p  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  24  5  Simulations  We note that in our simulations, we have implemented the code from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15] for agnostic mean  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In particular, the outlier truncation step is slightly different from the one analyzed in  Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [28], and consequently also in our theorems above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Huber’s Contamination Model  We begin with simulations for linear and logistic regression in Huber’s contamination model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Linear Regression  For our simulations, we set the dimension to be p = 10 and the number of data points to be  n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We simulated the clean covariates as xi ∼ N(0, Ip), with corresponding responses  yi = xT  i θ∗ + wi, where wi ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' noise and the true parameter is θ∗ =  1  √p(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We simulated the outlier covariates as xi ∼ N(0, p2Ip), with corresponding responses yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Figure 1 shows the results for Robust Newton’s Method (RNM), Robust Gradient Descent  (RGD), and ordinary least squares (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We used the initialization θ0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='4) + 10w,  with w ∼ N(0, Ip), for both RNM and RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RNM, we used the backtracking linesearch  parameters κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01, κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, and ζ = 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RGD, we used stepsize η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We repeated  the algorithm three times with contamination fractions ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As seen in the figure,  the statistical error indeed decreases quite quickly for RNM in comparison to RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Figure 1: Error log(∥θt − θ∗∥2) with respect to each iteration of Robust Newton’s Method (RNM),  Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear regression with Huber  contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  25  4  3  2  1*6  0  一  ell  log(  OLS for e= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  RGD for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  2  RNM for e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  OLS for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  3  RGD for E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  RNM for e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  OLS for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3  4  RGD for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3  RNM for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='3  5  0  10  20  30  40  50  60  70  80  t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Logistic Regression  Next, we generated data from a logistic model with p = 10, n = 1000, and θ∗ = (1/√p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 1/√p),  where we sampled the covariates as xi ∼ N(0, Ip) and sampled yi ∈ {0, 1} such that p(yi =  1|xi) =  1  1+e−xT  i θ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We then randomly changed an ϵ fraction of the labels to be either 0 or 1, with  equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For various values of ϵ, we ran Robust Gradient Descent (RGD) and Robust  Newton’s Method (RNM), and plotted the parameter error in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RNM, we used the same  backtracking linesearch parameters as in the case of linear regression with Huber contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  For RGD, we used a stepsize of η = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As seen in the figure, the statistical error again decreases  more quickly for RNM than for RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Figure 2: Error log(∥θt − θ∗∥2) with respect to each iteration for Robust Newton’s Method (RNM)  and Robust Gradient Descent (RGD) for logistic regression with Huber contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The behav-  ior of the non-robust optimizer, found using Newton’s method (NM), is also shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Heavy-Tailed Data  For heavy-tailed data, we took p = 10 and n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We generated the covariates xi ∼ N(0, Ip) and  the corresponding responses yi = xT  i θ∗ + wi, with wi following a Pareto distribution with variance  σ2 and tail-index parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We set the regression parameter θ∗ =  1  √p(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Figure 3 compares the results of Robust Newton’s Method (RNM), Robust Gradient Descent  (RGD), and ordinary least squares (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We used the initialization θ0 = (10, 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 10) for both  RNM and RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RNM, we used the backtracking linesearch parameters κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01, κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5,  and ζ = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RGD, we used stepsize η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We repeated the algorithm three times, for  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, 1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, all with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' As seen in the figure, the statistical error again decreases more  quickly for RNM than for RGD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' on the other hand, in this simulation, the final error of RGD is  lower than that of RNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RGD for eps= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  O  2  RNM for eps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  9-- NM for eps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  RGD for eps= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RNM for eps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  NM for eps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  1  (l* - *  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  log(II t  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  00i0i0i000i00i0i00!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='0i0i0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='0i0i00!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='0i0i0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='0i0i0  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  2  0  5  10  15  20  25  30  tFigure 3: Error log(∥θt −θ∗∥2) with respect to each iteration for Robust Newton’s Method (RNM),  Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear regression with heavy-  tailed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  6  Robust Hessian Estimation: The Conjugate Gradient Approach  In this section, we discuss an alternative to Newton’s method (and present a robust variant thereof)  which does not involve explicitly computing the Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Inspired by Martens [23], the idea is to  estimate ∇2f(θ)v, for any vector v, using the approximation  hv(θ) = ∇f(θ + δv) − ∇f(θ)  δ  ,  (52)  for some small δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that in order to compute the Newton step ∆θnt, we need to solve  the system ∇2f(θ)∆θ = −∇f(θ), which we will do using the conjugate gradient algorithm, which  provides an iterative method for solving a linear system of the form Ax = b [30, Chapter 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Our  robust approach will involve using the robust gradient estimate g(θ) in place of ∇f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  The details of the algorithm are provided in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that we have specified that the  CGNewtonStep subroutine for finding the Newton direction on each iteration of CGRobustNewton  runs for p steps, because in the noiseless case, the conjugate gradient method is known to terminate  in at most p steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  27  4  OLS foro=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RGD for o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  3  RNM for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  OLS for g = 1  RGD for  = 1  2  RNM for o= 1  OLSforg=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RGD for  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RNM for o = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  2  3  4  0  10  20  30  40  50  60  70  80  tAlgorithm 6 Conjugate Gradient Robust Newton’s Method  Require: Data samples S = {zi}n  i=1, Number of iterations T, Initial guess θ0 ∈ Θ, Backtracking  linesearch parameters κ1 ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2 ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' and ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Tolerance δ  1:  2: function CGRobustNewton(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  3:  for t = 0 to T − 1 do  4:  Compute losses {L(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1 and gradients {∇L(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1  5:  Compute Newton step ∆θnt = CGNewtonStep(θt)  6:  Compute stepsize α = BacktrackingLineSearch(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∆θnt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' g(θt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  7:  Update θt+1 = θt + α∆θnt  8:  end for  return θT  9: end function  10:  11: function CGNewtonStep(θ)  12:  Randomly initialize ∆θ(0) ∈ Θ  13:  Compute gradient estimate g(θ) = RobustGradientEstimate(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ)  14:  Compute Hessian-vector product estimate h∆θ(0)(θ) = HVProduct(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∆θ(0))  15:  Set r0 = h∆θ(0)(θ) + g(θ)  16:  Set p0 = −r0  17:  for k = 1 to p − 1 do  18:  Compute Hessian-vector product estimate hpk(θ) = HVProduct(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' pk)  19:  Set αk =  rT  k rk  pT  k hpk(θ)  20:  Set ∆θ(k+1) = ∆θ(k) + αkpk  21:  Set rk+1 = rk + αkhpk(θ)  22:  Set βk+1 =  rT  k+1rk+1  rT  k rk  23:  Set pk+1 = −rk+1 + βk+1pk  24:  end for  return ∆θ(d)  25: end function  26:  27: function HVProduct(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' v)  28:  Compute gradient estimate g(θ) = RobustGradientEstimate(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ)  29:  Compute gradient estimate g(θ + δv) = RobustGradientEstimate(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ + δv)  return g(θ+δv)−g(θ)  δ  30: end function  31:  32: function BacktrackingLineSearch((S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ∆θnt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' g(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' ζ)  33:  Set α = 1  34:  while RobustEstimate({L(θ + α∆θnt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1) > RobustEstimate({L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' zi)}n  i=1) +  κ1αg(θ)∆θnt + ζ do  35:  Update α = κ2α  36:  end while  return α  37: end function  28  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1  Convergence  We sketch some ideas here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' a rigorous proof giving rates of convergence of the robust conjugate  gradient method is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Focusing on the pure Newton phase, note that  our analysis of the iterates of robust Newton’s method essentially hinges on the Newton step ∆θnt  satisfying the equation  ∇f(θt) = −∇2f(θt)∆θnt + χt,  (53)  where the next iterate is then defined by θt+1 = θt + ∆θnt and χt is a small, bounded error (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' in-  equalities (24) and (25)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In particular, we can bound χt using the fact that ∆θnt = −H(θt)−1g(θt),  and ∥g(θt) − ∇f(θt)∥2 and ∥H(θt) − ∇2f(θt)∥2 are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In the case of the robust conjugate gra-  dient method, we can again think of the conjugate gradient method as providing an approximate  solution of the form  ∇f(θt) = −∇2f(θt)∆�θnt + �χt,  (54)  where successive iterates are then defined by �θt+1 = �θt + ∆�θnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Thus, the main challenge is to  understand the propagation of errors when the conjugate gradient method is applied to solve the  system Ax = b, but the matrix-vector pair (A, b) is replaced by ( �A,�b) on each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' To the  best of our knowledge, this is actually an open question in optimization [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We note, however,  that since our ultimate statistical estimation error bounds are all up to a small radius of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=',  O(√ϵ), we only need the output of the conjugate gradient method to be correct up to this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In  particular, as it is known that the exact conjugate gradient method terminates after p steps [30],  it would for instance suffice to show that an inexact conjugate gradient method, where the error  of ( �A,�b) is also O(√ϵ), only accumulates O(√ϵ) error after p steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Alternatively, one could try  to derive a geometric rate of convergence (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='36) of Nocedal and Wright [30]), with an  additional additive error term, for inexact conjugate gradient steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  To this end, we also need to understand the error terms introduced to conjugate gradient steps  due to inexactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This depends on the increment δ used in the finite-difference approximation of  the Hessian term (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Note that by a Taylor expansion, we have  ∇f(θ + δv) = ∇f(θ) + δ∇2f(θ)v + Cδ2,  for some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Thus, we have the error bounds  ∥hv(θ) − ∇2f(θ)v∥2 =  ����  g(θ + δv) − g(θ)  δ  − ∇2f(θ)v  ����  2  =  ����  g(θ + δv) − g(θ)  δ  − ∇f(θ + δv) − ∇f(θ)  δ  − Cδ  ����  2  ≤ ∥g(θ + δv) − ∇f(θ + δv)∥2  δ  + ∥g(θ) − ∇f(θ)∥2  δ  + Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  If we had deviations bounds of the form (8), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=', with αg, βg ≍ √ϵ, the optimal choice of δ would  be δ ≍ ϵ1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In summary, we conjecture that the robust conjugate gradient method would allow us to incur  an overall estimation error of O(ϵ1/4) in the case of Huber’s ϵ-contamination model, again at a  quadratic convergence rate for the successive Newton iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Although this is a slower rate than  the one derived in Section 4 for GLMs, it may be applicable to a much wider range of settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  We also note that for the SEVER algorithm [9], a rate of O(ϵ1/4) is also derived for empirical risk  minimization for a class of classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' If the above discussion could be made rigorous,  it would then also be extendable to the heavy-tailed setting in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  29  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2  Simulations  In Figure 4, we compare the Newton Conjugate Gradient Method (NCGM), Robust Gradient  Descent (RGD), and ordinary least squares (OLS) on a linear model with Huber ϵ-contaminated  data, with the same setup as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We used the initial parameter θ0 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 1) + 2w,  with w ∼ N(0, Ip), for both NCGM and RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For NCGM, we used the backtracking linesearch  parameters κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01, κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also used δ = 10−9 for the estimation of  Hessian-vector products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RGD, we used stepsize η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We repeated the algorithm two  times, with contamination fractions ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Figure 4: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient  Method (NCGM), Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear  regression with Huber contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In Figure 5, we compare the Newton Conjugate Gradient Method (NCGM), Robust Gradient  Descent (RGD), and ordinary least squares (OLS) on a linear model with heavy-tailed data, again  with the same setup as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We used the initial parameter θ0 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5) + 2w,  with w ∼ N(0, Ip), for both NCGM and RGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For NCGM, we used the backtracking linesearch  parameters κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01, κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5, and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='00001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also used δ = 10−10 for the estimation of  Hessian-vector products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For RGD, we used stepsize η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We repeated the algorithm twice  with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='25, with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In Figure 6, we compare the Newton Conjugate Gradient Method (NCGM) and Robust Gradi-  ent Descent (RGD) on a logistic model with Huber ϵ contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' To generate the contaminated  logistic data, we used the same procedure outlined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We also used the same hyper-  parameters for NCGM and RGD as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  7  Discussion  We have presented a novel second-order method for robust parameter estimation, based on an  adaptation of Newton’s method where gradients and Hessians are computed in a robust manner  30  2  0  2  3  OLS for E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  RGD for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  NCGM for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='01  4  OLS for =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  RGD for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  NCGM for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='02  5  0  5  10  15  20  25  30  tFigure 5: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient  Method (NCGD), Robust Gradient Descent (RGD), and ordinary least squares (OLS) for linear  regression with heavy-tailed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  on each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  In particular, we have shown that a variant of the backtracking linesearch  algorithm will adaptively choose stepsizes in such a way that a finite number of iterates initially  lie in a “damped” phase of the algorithm, after which the algorithm enters a “pure” phase where  it only chooses stepsizes equal to 1 and converges quadratically to a small ball around the true  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Although our method shows clear advantages in comparison to previously analyzed  first-order methods, many possible improvements exist which may also be extremely interesting to  study theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' First of all, we have used a rather naive method for computing robust Hessians  based on vectorizing the Hessian matrix and applying the same subroutine used for robust gradient  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' Obtaining optimal rates for robust matrix estimation under a variety of metrics and  contamination settings is an active area of research [25, 26, 4], and it is entirely possible that a  different robust matrix estimator would lead to better overall error bounds for our robust Newton  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Another plausible extension of our analysis that could be studied under a similar  theoretical framework would be to use robust gradient and Hessian estimators which employ the  estimation procedures of Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [10] rather than those of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' we note that  this would allow us to also handle the setting of adversarially contaminated data, rather than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  data from either an ϵ-contaminated or heavy-tailed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  It would also be interesting and practically important to devise robust second-order algorithms  appropriate for higher-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' For moderate to large p (even in settings where p < n),  implementing the robust version of Newton’s method can become more tedious, since it involves  robustly computing p × p matrices and then inverting them on each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In the truly high-  dimensional case (p > n), even the canonical version of Newton’s method must be modified, since  the Hessian matrix becomes rank-deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This raises the question of whether it would be beneficial  to analyze a robust inexact second-order algorithm, instead, where the Hessian matrix need not be  approximated as closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' In the truly high-dimensional setting, combining this with regularization  would be a natural direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  31  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  1og(1 t - *1)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  1  OLS for g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  RGD for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  NCGM for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  OLSforg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='25  2  RGD for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='25  NCGM for  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='5  0  5  10  15Figure 6: Error log(∥θn − θ∗∥2) with respect to each iteration for the Newton Conjugate Gradient  Method (NCGM) and Robust Gradient Descent (RGD) for logistic regression with Huber contam-  ination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' The behavior of the non-robust optimizer, found using Newton’s method (NM), is also  shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Finally, we have proposed the robust conjugate gradient method as an alternative second-order  algorithm which, though based on Newton’s method, only requires computing robust gradients  rather than needing to separately compute robust Hessians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' This method could potentially enjoy the  fast convergence benefits of Newton’s method while bypassing some of the computational issues in  higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' However, a rigorous analysis of the robust conjugate gradient method is beyond  the current scope of this paper—in particular, it would involve carefully tracking the propagation  of errors through iterates of the conjugate gradient method, which has remained a long-standing  open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content=' We note that any error bounds on successive conjugate gradient iterates could then  easily be plugged into our proofs to obtain quadratic convergence to an appropriate ball around  the true parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
+page_content='  Acknowledgments  The work of EI was supported by the Cantab Capital Institute for the Mathematics of Information  via the Philippa Fawcett Internship programme (Faculty of Mathematics, University of Cambridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFPT4oBgHgl3EQf4DUP/content/2301.13192v1.pdf'}
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diff --git a/odAyT4oBgHgl3EQfY_eZ/content/tmp_files/2301.00215v1.pdf.txt b/odAyT4oBgHgl3EQfY_eZ/content/tmp_files/2301.00215v1.pdf.txt
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--- /dev/null
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@@ -0,0 +1,1029 @@
+1
+Nuclear Magnetic Resonance Measurements in
+High Flat-top Pulsed Magnetic Field up to 40 T at
+WHMFC
+Wenqi Wei, Qinying Liu, Le Yuan, Jian Zhang, Shiyu Liu, Rui Zhou, Yongkang Luo, and Xiaotao Han
+Abstract—Nuclear magnetic resonance (NMR) technique ben-
+efits from high magnetic field not only due to the field-enhanced
+measurement sensitivity and resolution, but also because it is
+a powerful tool to investigate field-induced physics in modern
+material science. In this study, we successfully performed NMR
+measurements in high flat-top pulsed magnetic field (FTPMF) up
+to 40 T. A two-stage corrected FTPMF with fluctuation less than
+10 mT and duration longer than 9 ms was established. Besides,
+a Giga-Hz NMR spectrometer and a sample probe suitable for
+pulsed-field condition were developed. Both free-induction-decay
+and spin-echo protocols were exploited for the measurements.
+The derived
+93Nb NMR results show that the stability and
+homogeneity of the FTPMF reach an order of 102 ppm / 10
+ms and 102 ppm / 10 mm3 respectively, which is approaching
+a degree of maturity for some researches on condensed matter
+physics.
+Index Terms—Nuclear magnetic resonance (NMR), high field,
+flat-top pulsed magnetic field (FTPMF), NMR spectrometer.
+I. INTRODUCTION
+E
+VER since discovered in 1940s, nuclear magnetic res-
+onance (NMR) has manifested itself as one of the
+most significant tools to derive detailed information on the
+atomic scale about material properties. Nowadays, the trend
+of performing NMR experiments in high magnetic field is
+actively driven not only by higher sensitivity and resolution,
+but also by the desirable exploration of field-induced physics
+in modern material science [1]–[18]. For instance, researches
+on half-integer quadrupolar nuclei such as
+27Al and
+17O
+notably benefit from the high magnetic field, because the
+quadrupolar interaction induced shift and broadening of the
+central transition are inversely proportional to the square of
+the magnetic field strength. The reduction in second-order
+quadrupolar broadening brings resolution enhancement and
+additional information, which provides new opportunities for
+measurements of nuclei not probed before [1]–[5], [8], [10].
+This work was supported in part by the National Key Research and
+Development Plan Project of China under Grant 2021YFA1600301 and
+2022YFA1602602, and in part by the National Natural Science Foundation
+of China under Grant U21A20458 and 51821005. (Corresponding author:
+Xiaotao Han.)
+Wenqi Wei, Le Yuan, Jian Zhang, Shiyu Liu, Yongkang Luo, and Xiaotao
+Han are with the Wuhan National High Magnetic Field Center, Huazhong
+University of Science and Technology, Wuhan 430074, China (e-mail:
+vinkey7wei@hust.edu.cn; mpzslyk@gmail.com; xthan@hust.edu.cn).
+Qinying Liu is with the Electric Power Research Institute, China Southern
+Power Grid Company Limited, Guangzhou 510623, China.
+Rui Zhou is with the Institute of Physics, Chinese Academy of Sciences, and
+Beijing National Laboratory for Condensed Matter Physics, Beijing 100190,
+China (e-mail: rzhou@iphy.ac.cn).
+On the other side, since the high magnetic field often causes
+electronic and structural transitions of materials, the interests
+in field-induced physics like high-Tc cuprate superconductors
+have been increasing in the community of modern condensed
+matter physics. The NMR spectroscopy can provide more fun-
+damental insights in the properties of materials, hence people
+have been making efforts to carry out NMR experiments in
+higher magnetic field [6], [7], [9], [11]–[18].
+At present, the high magnetic field greater than 30 T can
+be generated by continuous-operation (also called as “steady-
+state” or “DC”) or pulsed magnets. The DC magnets with
+operation time of more than seconds include a variety of water-
+cooled resistive magnets (up to 41.5 T [19]), superconducting
+magnets (up to 32.35 T [20]) and hybrid magnets (up to
+45.5 T [21]). The continuous magnetic field is applicable to
+perform NMR experiments, because there is enough time to
+correct the field both in stability and homogeneity, as well as
+to execute various sequences for measurements and achieve
+signal averaging. However, large construction and operation
+costs and limited field strength of superconducting conductors
+hinder the development of NMR experiments to higher DC
+magnetic field.
+Currently, only pulsed resistive magnets are practically
+available to produce the high magnetic field exceeded 45 T
+and even up to 100 T for timescales typically in the range of
+1-100 ms [22], [23]. Although some NMR experiments were
+successfully optimized and achieved in the short and time-
+varying pulsed field [6]–[16], it is still desirable to perform
+NMR measurements in a stable magnetic field [17], [18],
+[24]. Firstly, a large real-time bandwidth up to tens of MHz
+or higher is required to excite and detect NMR signals due
+to the low repeatability and time-varying of the pulse field,
+which brings great difficulties to hardware design and signal
+search [25]. Besides, the instability of the pulsed magnetic
+field causes the violent fluctuation of the resonance frequency
+and leads to the distortion of the NMR spectra. Despite the
+application of the complex signal post correction algorithms,
+the signal quality is still limited [13], [15]. Last but not least,
+some highly accurate parameter measurements such as the
+relaxation time T1 and T2 demand that the time dependence of
+the magnetic field should be reduced to zero during the long
+enough sequence durations [24]. Hence, it is still challenging
+to carry out NMR measurements in the pulsed magnetic field.
+Fortunately, lots of efforts have been being made to create a
+quasi-stable stage (usually called “flat-top”) near the peak of
+the pulsed magnetic field [26]–[32], opening a new avenue
+arXiv:2301.00215v1  [physics.ins-det]  31 Dec 2022
+
+2
+for the pulsed magnetic field NMR measurements [18]. In
+the early stage, Weickert et al. [26] tried to use a long-pulse
+magnet powered by high-power capacitor bank to creat an
+approximate 44 T flat-top for NMR experiments. However,
+the unregulated long pulsed magnetic field is inefficient in
+improving the flat-top stability and has a long repetition
+time more than 8 hours. Recently, Ihara and Kohama et al.
+[17], [24] systematically peformed NMR measurements in a
+dynamically controlled flat-top field, showing great advantages
+of the flat-top pulsed magnetic field (FTPMF). Nevertheless,
+their measurements are limited in the low field of 13 T, and a
+higher FTPMF is desired in the community.
+In this work, we present a practically available FTPMF for
+NMR measurements up to 40 T at Wuhan National High
+Magnetic Field Center (WHMFC). In Sec. II, we briefly
+introduce basic concepts of NMR measurements. Furthermore,
+experimental setups are presented in Sec. III including a two-
+stage corrected flat-top pulsed magnetic field, a modular GHz
+NMR spectrometer and a sample probe. In Sec. IV, NMR
+measurements of 93Nb carried out in the DC field and FTPMF
+are described and discussed. Finally, the paper is concluded
+in Sec. V.
+II. BASIC CONCEPTS OF NMR
+To understand basic concepts of NMR measurements, a
+brief description is presented here and shown in Fig. 1. The
+interaction between an external magnetic field B0 and nuclear
+spin I (I > 0) is described by the Zeeman effect, in which
+the spectral line splits into multiple discrete levels. For the
+simplest case of I = 1
+2, the Zeeman energy levels are given
+by
+EZee = ±1
+2hγB0
+(1)
+where γ is gyromagnetic ratio (in MHz/T), and h is Planck’s
+constant. The difference of energy levels can be written as
+∆E = hfL = hγB0
+(2)
+where the fL is called the Larmor frequency and fL = γB0.
+If a radio-frequency (RF) field B1 (t) = B1 cos (2πfLt) per-
+pendicular to the B0 excites the nucleus, transitions between
+these energy levels can occur and NMR phenomenon can be
+observed.
+Commonly, a π/2 RF pulse is used to irradiate the NMR
+free induction decay (FID) signals which have the form of
+sFID (t) ∼ cos (2πfLt) · e
+−
+t
+T ∗
+2
+(3)
+where T ∗
+2 is the decay time. In the corresponding Fourier
+transform spectrum, the standard NMR spectrum is of the
+following form
+S (f) ∼
+1/T ∗
+2
+(1/T ∗
+2 )2 + (2πf − 2πfL)2
+(4)
+which is called the absorption Lorentzian. It is obvious that
+the peak of the spectrum is at the Larmor frequency fL. When
+the external magnetic field B0 changes with the time during
+the period of the FID, the fL also changes proportionally and
+this leads to the distortion of the NMR spectrum (an example
+Fig. 1.
+Basic principles of NMR measurements. (a) The Zeeman energy
+levels of a nucleus with spin 1/2 in the presence of an external magnetic
+field B0. NMR transition occurs at the Larmor frequency fL = γB0.
+(b) FID signals irradiated by a π/2 RF pulse. (c) A standard absorption
+Lorentzian NMR spectrum. the FWHM of 1/
+�
+πT ∗
+2
+�
+includes the intrinsic
+homogeneous broadening quantified by the transverse relaxation time T2 and
+the inhomogeneous broadening due to the spatial inhomogeneity ∆B0 (r) of
+the B0. (d) The distortion of the NMR spectrum due to the temporal instability
+of the B0. The ∆B0 (t) reflects the degree of the instability.
+is shown in Fig. 1(d)). Hence, high stability of B0 with time is
+one of the key prerequisites for NMR measurements. Another
+essential parameter in the NMR spectrum is the full width
+at half maxima (FWHM) parameterized by 1/ (πT ∗
+2 ), which
+represents the spectral resolution. In fact, the FWHM includes
+the intrinsic homogeneous broadening quantified by the trans-
+verse relaxation time T2 and the inhomogeneous broadening
+due to the spatial inhomogeneity of the B0. Consequently, B0
+with less spatial inhomogeneity ∆B0 (r) is desired for higher
+spectral resolution.
+III. EXPERIMENTAL SETUPS
+A. Flat-top Pulsed Magnetic Field
+For decades, the FTPMF technique has been pursuing to
+improve the stability and duration of the flat-top stage while
+maintaining the high field strength [26]–[32]. The economical
+and reliable high-voltage capacitor bank is the most commonly
+used main power source for generating the high pulsed mag-
+netic field, but it is difficult to regulate the discharge current for
+controlling the magnetic field. Despite all this, some solutions
+have been proposed, such as sequentially discharging multiple
+capacitors [27], [28], [32], or using a small compensation coil
+built into the bore of the main magnet to finely regulate the
+magnetic field [29]. The former method can achieve a long
+flat-top duration of more than 10 ms but has high instability
+beyond 103 ppm, while the latter stands the opposite. Hence,
+we combined the advantages of the two schemes, and the flat-
+top realization is divided into two processes as shown in Fig.
+2.
+In terms of concrete implementation, the main magnetic
+field was produced by a standard 60 T pulsed magnet with the
+height of 150 mm and bore of 20 mm, which has an inductance
+of 3.5 mH and a resistance of 30 mΩ at 77 K (the magnet
+was immersed in the liquid nitrogen for cooling). The main
+
+t
+SFID(t)～ COs(2πfLt) · e
+T2
+(b)
+T
++
+2
+△E=hfL=hBo
+Bo
+△ABo(r
+(c)
+元
+元1
+fL
+(p)
+0()-0()-0()8-3
+Fig. 2.
+Scheme of the two-stage corrected flat-top pulsed magnetic field
+for NMR measurements. (a) Diagram of connection of main components
+in the two-stage corrected FTPMF system. (b) Illustration of magnetic field
+waveforms generated by the two-stage corrected FTPMF system.
+magnet was connected in series with the primary winding of
+a pulse transformer (refer to Ref. [28] for specific parameters)
+and powered by 8 parallel capacitor banks (3.84 mF, 1.2 MJ).
+The secondary winding of the pulse transformer was powered
+by 6 parallel capacitor banks (3.84 mF, 1.2 MJ) and formed the
+auxiliary circuit. Before the current of the main circuit reached
+its peak, the auxiliary circuit was discharged to achieve the
+first correction. According to the different optimized discharge
+voltages of both circuits and trigger time of the auxiliary
+circuit, the flat-top pulsed magnetic field up to 45 T after the
+first correction were produced, whose stability was within 0.5
+T, and duration was within 12–15 ms depending on the field
+strength.
+The compensation coil used for the second correction
+consists of a main regulating coil with the height of 36
+mm at the center of the main magnet and two oppositely
+wound decoupling coils with the height of 18 mm at the
+ends of the main regulating coil. Due to the strict radial
+space limitation in the main magnet bore of 20 mm, the
+copper compensation coil was double wound around a 17
+mm Helium cryostat tail with only 0.4 mm wire diameter
+and reinforced by a Zylon/epoxy composite layer of 0.5 mm
+thickness. The compensation coil was powered by twelve 12-
+V lead-acid batteries, and its current can be linearly regulated
+by insulated gate bipolar transistor (IGBT) modules with a
+homemade driver to generate an adjustable magnetic field from
+0 to 1 T beyond 10 ms. magnetic field signals at the sample
+area were sensed by a pick-up coil and the pick-up voltage
+was acquired at a sample rate of 100 kS/s by a 16-Bit analog-
+to-digital converter equipped with a field programmable gate
+array (FPGA) module. Digital proportional integral derivative
+(PID) control was applied to achieve the fine compensation
+of the magnetic field for the second correction. Finally, the
+magnetic field fluctuation after the two-stage correction was
+limited less than 10 mT, which is shown in Fig. 3. Due to
+the establishment time of the PID control and the decrease
+of the regulation capability of the compensation coil caused
+Fig. 3. Magnetic field profiles up to 45 T generated by the two-stage corrected
+flat-top system and measured by the pick-up coil. The insert shows that the
+magnetic field fluctuation of the flat-top was limited less than 10 mT (integral
+drift had been corrected).
+by the main magnet heating, the flat-top time after the second
+correction was reduced by about 4 ms compared with that of
+the first-order correction. It is noteworthy that both centers
+of the main magnet and compensation coil should be aligned
+accurately to provide a high magnetic field homogeneity area
+for NMR experiments. The axial locating processes were
+realized by using the pick-up coil and a lock-in amplifier with
+an error less than 2 mm. Considering the aging of the main
+magnet, we would perform NMR experiments in the FTPMF
+up to 40 T for safety.
+B. Modular GHz NMR Spectrometer
+A flexible modular GHz NMR spectrometer was developed,
+in which a series of stand-alone instruments were combined
+for use. The advantages of the modular spectrometer are
+convenient for testing and maintenance, as well as further
+upgrading and expansion. According to the properties of the
+FTPMF, several requirements of the NMR spectrometer are as
+follows: (1) Since the pulsed field is transient and its duration
+is typically less than 100 ms, the spectrometer should be
+controlled by a sophisticated timing system and has excellent
+transient responses. (2) The spectrometer should support high
+radio frequency up to several GHz, due to the fact that the
+resonance frequency is proportional to the field strength. For
+example, 2.6 GHz is needed for 1H at 60 T. (3) Low noise
+is required because the electromagnetic environment of the
+pulsed field is far worse than that of the DC field, and the
+number of times of the signal averaging is limited in the
+FTPMF.
+A diagram of the self-built modular NMR spectrometer is
+presented in Fig. 4. The construction of the spectrometer was
+based on a National Instruments (NI) PXI system including a
+chassis and an embedded computer, which was controlled via
+a self-written LabVIEW interface. Currently, the PXI system
+hosted three main modules, consisting of a RF generator
+NI PXIe-5651, a vector signal analyzer NI PXI-5661 and
+a pulse programmer NI PXI-6542. The RF generator can
+
+(a)
+0～20 kA
+Cap. Banks
+0～0.2 kA
+Batteries
+3.84 mF
+Main Circuit
+12 X12
+Compensation Circuit
+X8
+Pri.
+Tansformer
+IGBT Modules
+Sec. *
+Cap. Banks
+PID Control
+dB(t)
+3.84 mF
+Auxiliary Circuit
+P
+Controller & Driver
+X6
+Z 0 ~ 20 kA
+TT
+telose = tm + tdelay
+Main (
+Compensation Pick- up
+Coil
+Coil
+Magnet
+1st Correction Part
+2nd Correction Part
+(b)
+B
+Original Pulsed Magnetic Field
+FTPMF after 1st Correction
+FTPMF after 1st & 2nd Correction
+0 ～ 45 T
+Compensated Magnetic Field
+0～1T
+ät
+tm
+tm + tdelay50
+10 mT/ 8 ms
+45
+45.204
+Field
+45.202
+40
+10 mT/ 9 ms
+45.200
+Magnetic Field (T)
+35
+45.198
+45.196
+10 mT/ 10 ms
+30
+45.194
+13 14 15 16 17 18 19 20 21
+25
+10 mT/ 11 ms
+Time (ms)
+20
+15
+10
+5
+0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Time (ms)4
+produce RF signals with frequency ranges from 500 kHz to 3.3
+GHz. Correspondingly, the vector signal analyzer can support
+intermediate frequency (IF) down-conversion of NMR signals
+up to 2.7 GHz with a 20 MHz real-time bandwidth, and the
+NMR signals are finally recorded by a 100 MS/s and 14-Bit
+oscilloscope with using quadrature digital down-conversion.
+The 100 MHz digital pulse programmer was used to receive
+an external trigger when the flat-top field arrived, and then
+trigger each independent module in the spectrometer. The
+all components were synchronously clocked by a 10 MHz
+reference clock from the RF generator.
+Fig. 4. Diagram of the modular NMR spectrometer.
+In the preceding stage circuit for excitation, the continuous
+RF signals were modulated into desired NMR sequences by a
+pulse gate (Analog Devices, HMC427ALP3E) with switching
+time smaller than 10 ns and isolation up to 40 dB. Then, the
+power of the modulated signals was amplified by a Tomco
+100 W RF amplifier with band 5-650 MHz. The duplexer
+is a key three ports component for switching between the
+excitation of the high-power RF signals and the reception of
+the weak NMR signals. A 1 GHz bandwidth homemade active
+duplexer based on PIN diodes and various impedance lines was
+developed with 0.56 dB insert loss, 500 W power capacity, 37
+dB isolation and 1 µs switching time. To protect the receiving
+part from the crosstalk of the high-power excitation signals, a
+70 dB isolation switch (Mini-Circuits, ZASWA-2-50DRA+)
+was applied with switching time of 20 ns. A lower noise
+amplifier N141-306CB up to 1 GHz with gain greater than 30
+dB and noise figure smaller than 0.8 dB was used to amplifier
+the weak NMR signals.
+C. Sample Probe
+Compared with the conventional NMR probe for the DC
+field, the probe for the pulsed field is challenging to design
+because of the strict space limitation in the bore of the pulsed
+magnet, as well as serious electromagnetic interference and
+mechanical vibration environment. In practice, the bore space
+of the magnet available for the probe design is only 10 mm,
+but the probe needs to have multiple functional components,
+for example, including a sample chamber with enough space, a
+well tuned and matched resonance circuit, a temperature sensor
+and a heater, as well as a magnetic field pick-up coil. Closed
+loops of the conducting materials leading to eddy current have
+to be avoid, and all components should be well fixed.
+Fig. 5. (a) Schematic illustration of the sample probe designed for the pulsed
+field. The partial photo of the sample probe and connection of the resonance
+circuit are also shown. (b) Photo of the experimental station layout.
+Fig. 5(a) shows the schematic illustration of the sample
+probe designed for the pulsed field. The probe was located
+in a cryostat that can use liquid helium or nitrogen to per-
+form low-temperature experiments. The volume of the sample
+chamber was about several tens of cubic millimeters and it
+was surrounded by a micro RF coil to excite and detect
+NMR signals. A tuning capacitor and a matching coil were
+connected to the resonant circuit which had been preliminarily
+calculated and tested at room temperature. Although there
+was deviation about several MHz from the expected resonant
+frequency in the working condition, precise tuning was not
+very necessary in the pulsed field experiments. The return
+loss of the resonance circuit at the resonance frequency was
+adjusted to be less than -10 dB, meaning that more than 90%
+of the power was absorbed by the RF coil. The magnetic field
+pick-up coil was wounded on the outer sleeve of the probe
+and at the same axial height as the center of the sample to
+ensure that the magnetic field at the sample were accurate
+collected. The heater and temperature sensor were located
+near the resonance circuit to control the local temperature
+around the sample. Most components were arranged vertically
+to improve the space utilization. Based on the establishment
+of the above experimental setups, a photo of the experimental
+station layout is presented in Fig. 5(b).
+93Nb powder (99.999% purity) uniformly mixed by resin
+
+Probe
+Low Noise
+Isolation
+Power
+Switch
+Amplifier
+Amplifier
+Duplexer
+LNA
+PA
+Pulse Gate
+Vector Signal
+Pulse Programmer
+Analyzer
+RF Generator
+Computer and Chassis
+External Trigger(a)
+NMR Signal
+Duplexer
+Excitation and
+Detection Channel
+Detection
+Excitation
+Heating and
+Magnetic Field
+ Matching Coil
+Temperature
+Detection Channel
+RM
+LM
+Detection Channel
+W
+333
+CT :
+5333
+Tuning
+Rrf
+Lrf
+Capacitor
+RF Coil
+Cryostat
+Resonance Circuit 
+Compensation Coil
+Main Magnet
+Liquid Nitrogen
+Heater and
+Magnetic Field
+Heater and
+Tuning
+Matching
+RF
+Sample
+Sample
+Temperature
+Pick-up Coil
+Temperature Sensor
+Capacitor
+Coil
+Coil
+Sensor
+(b)
+Spectrometer
+Probe
+FTPMF
+Second Correction
+System
+ Main Magnet5
+was selected as the testing sample. 93Nb is ideal for NMR
+measurements under the pulsed field, because it has the natural
+abundance of 100%, a large gyromagnetic ratio γ=10.405
+MHz/T, a short longitudinal relaxation time T1 of 4.7 ms at 77
+K, and a high relative sensitivity of 48% compared with 1H.
+The mixed sample was cut into a cylinder and placed in the RF
+coil wound by copper wire (about 2 mm in diameter, 4 mm
+in length, 0.3 mm wire diameter with 6 turns). Considering
+the Knight shift (K), the resonance frequency of 93Nb is
+γ
+�
+1 +
+93K
+�
+B0, where 93K = 0.87%.
+IV. RESULTS AND DISCUSSION
+A. NMR Measurements in the DC Field and FTPMF
+Since the pick-up coil was not well calibrated for measuring
+the pulsed magnetic field with an estimated error lager than
+1% and the repetition time for the pulsed magnet cooling was
+usually longer than 30 minutes, a pre-optimization process in
+the DC field was necessary to achieve the maximum sensi-
+tivity of NMR signals in a single measurement as possible.
+This process was carried out with our self-built spectrometer
+and probe in a superconducting magnet (Oxford) with high
+stability (better than 10 ppm/h) and homogeneity (better than
+10 ppm/cm3). The FID measurements of 93Nb in the DC field
+were performed at 14.55 T and 77 K, and the corresponding
+resonance frequency is 152.7 MHz. In order to irradiate the
+FID signals with both high sensitivity and bandwidth, the
+power and duration of the π/2 RF pulse were optimized.
+Finally, the 50 W and 1.5 µs RF pulse was implemented
+with the maximum signal-to-noise ratio (SNR) of 20 dB and
+bandwidth of 600 kHz. The FID signals after quadrature down-
+conversion are shown in Fig. 6(a), which were measured at
+the given RF of 152.6 MHz for the clarity of the oscillation.
+Ignoring the switching dead time of 1 µs, the decay time T ∗
+2
+of the FID signals about 8 µs was observed, which can be
+approximately regarded as the T2 due to the high stability and
+homogeneity of the DC field.
+For quickly locating the NMR signals and calibrating the
+pick-up coil in the FTPMF, a field-sweep mode with slope-top
+pulse was performed. According to the bandwidth of excitation
+and longitudinal relaxation time T1 of 93Nb, the sweep slope
+rate was set to 10 mT/ms and the interval of excitation was
+1 ms. After several sweep tests, the NMR measurements can
+be normally carried out in the FTPMF. The FID signals at the
+field of 23.24 T and excited RF of 244 MHz after quadrature
+down-conversion are shown in Fig. 6(b). It is obvious that the
+decay time of the FID signals reduces from 8 µs to 3 µs in the
+FTPMF compared to the DC field, which was mainly caused
+by the inhomogeneity of the FTPMF and will be discussed
+later. Although the field strength of the FTPMF is 1.6 times
+higher than that of the DC field, the sensitivity of FID signals
+do not significantly increase due to the short polarization time
+of nuclei, inhomogeneity of the magnetic field and so on.
+These 93Nb FID results can be further confirmed by a spin-
+echo (SE) technique. A standard SE measurement (π/2−τ−π,
+1.5-20-3 µs in this case) was performed in the FTPMF, the
+result of which is shown in Fig. 7. It is clear that the SE
+signals refocus after waiting for 20 µs, as expected.
+Fig. 6. FID signals after quadrature down-conversion: (a) at the DC field of
+14.55 T and excited RF of 152.6 MHz; (b) at the FTPMF of 23.24 T and
+excited RF of 244 MHz. (c) at the FTPMF of 39.54 T and excited RF of 415
+MHz. All measurements were carried out at temperature of 77 K.
+Fig. 7.
+NMR RF pulse sequence followed by the spin-echo signals at the
+FTPMF of 23.26 T and excited RF of 244 MHz. The given SE sequence is
+1.5-20-3.0 µs.
+The FID signals in the FTPMF of 39.54 T and excited RF
+of 415 MHz were measured and are presented in Fig. 6(c).
+It is noteworthy that the given π/2 RF pulse was reoptimized
+with the duration of 9 µs to achieve the highest possible signal
+sensitivity. The results show that the signal sensitivity in the
+high field of 39.54 T increases about 1.6 times in proportion
+to the magnetic field strength, compared with that in the low
+field of 23.24 T. However, the decay time of FID further
+decreases to 2 µs due to the degradation of the magnetic field
+homogeneity in the absolute value.
+
+20
+Amplitude (a.u.)
+(a)
+15
+Real Part
+ Imaginary Part
+10
+Magnitude
+5
+0
+-5
+DC field @14.55 T & 152.6 MHz
+10
+sn
+-15
+0
+2
+4
+6
+8 10 12 14 16 18 20 22 24 26 28 30 32 34
+Time (us)
+20
+Amplitude (a.u.)
+(b)
+15
+Real Part
+Imaginary Part
+10
+ Magnitude
+5
+0
+-5
+FTPMF @23.24 T & 244 MHz
+sn
+10
+-15
+0
+2
+4
+6
+8 10 12 14 16 18 20 22 24 26 28 30 32 34
+Time (μs)
+20
+Amplitude (a.u.)
+C
+Real Part
+15
+Imaginary Part
+10
+Magnitude
+5
+0
+-5
+FTPMF @39.54 T & 415 MHz
+us
+10
+-15
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+32
+34
+Time (us)10
+Real Part
+8
+Imaginary Part
+Magnitude
+6
+Amplitude (a.u.)
+4
+2
+0
+-2
+-4
+-6
+T
+SE
+-8
+2
+-10
+10
+20
+30
+40
+50
+60
+70
+80
+Time (μs)6
+B. Stability of the FTPMF
+The NMR technique itself is also an accurate method for
+measuring the magnetic field, hence the FID signals were
+excited several times to check the stability of the FTPMF
+during the period of the flat-top. Fig. 8(a) shows the nine
+excitation points with an interval of 1 ms during the FTPMF
+at 39.55 T measured by the pick-up coil. The corresponding
+NMR spectra at the excited RF of 415 MHz are presented
+in Fig. 8(b). The central frequency offset of the nine spectra
+was limited within 100 kHz, meaning the field fluctuation was
+less than 10 mT. It well corresponds to the result measured
+by the pick-up coil. Overall, the current stability of the
+FTPMF in the timescale about 10 ms would potentially find
+some applications for NMR experiments in condensed matter
+physics [33]. A higher resolution ADC for the acquisition of
+the magnetic field and smaller feedback control cycle for the
+second correction system are desired to further improve the
+stability of the FTPMF.
+Fig. 8.
+Stability of the FTPMF checked by the NMR spectra. (a) Nine
+excitation points of the FID signals with an interval of 1 ms during the
+FTPMF at 39.55 T measured by the pick-up coil. (b) The corresponding
+NMR spectra at the excited RF of 415 MHz. The central frequency offset of
+the nine spectra was limited within 100 kHz, which corresponds well to the
+magnetic field fluctuation of 10 mT measured by the pick-up coil.
+C. Homogeneity of the FTPMF
+In the section II, we revealed that the decay time T ∗
+2 of the
+FID signals decreases with the increase of the magnetic field
+spatial inhomogeneity. Besides, the inhomogeneity can also
+lead to the broadening of the NMR spectra, which is harmful
+to the high resolution NMR measurements. Compared with
+the commercial superconducting magnet with sophisticated
+shimming techniques, the pulsed magnet has great magnetic
+field inhomogeneity up to 1000 ppm or higher in 1 cm3
+space of the magnet center. Ignoring the winding error and
+magnet deformation, the axial distribution of the magnetic
+field generated by the solenoid-type magnet is approximately a
+quadratic function of B0 (z) = B0−az2 , where the coefficient
+a > 0. Hence, small volume and highly accurate axial
+positioning of the sample in the pulsed magnet are important
+to weaken the influence of the magnetic field inhomogeneity.
+Fig. 9. The NMR spectra of 93Nb under different conditions. Relative FWHM
+is marked for comparison.
+The NMR spectra measured in the DC field of 14.55 T
+and measured in the FTPMF of 23.25 T are compared in Fig.
+9. The axial position of the sample in the FTPMF was well
+optimized with an accuracy less than 1 mm (labelled at 0 mm).
+Because of the high homogeneity better than 10 ppm/cm3 in
+the DC field, the FWHM of 300 ppm can be approximately
+attributed to the intrinsic homogeneous broadening. Although
+the strength of the FTPMF is higher than that of the DC field,
+the FWHM increases to 550 ppm due to the magnetic field
+inhomogeneity. Because the magnetic field inhomogeneity was
+caused by both the compensation coil and main magnet, the
+original pulsed magnetic field at the peak with change rate
+less than 1 mT/10 µs was applied to check the inhomogeneity
+induced by the main magnet. The results show that the main
+magnet is the main source of the magnetic field inhomogeneity
+and the compensation coil has little impact. The spectrum of
+the sample positioned at 3 mm away the optimized position
+was also measured and is shown in Fig. 9. It is obvious that
+the FWHM increases to 1440 ppm due the heavier magnetic
+field inhomogeneity.
+The NMR spectrum in the FTPMF of 39.55 T is also
+presented in Fig. 9. Despite greater absolute broadening of the
+spectrum in 39.55 T compared with that in the low field of
+23.25 T, the relative broadening almost remains unchanged due
+to the increasing of the resonance frequency from 244 MHz
+up to 415 MHz. On balance, the inhomogeneity effect of the
+pulsed magnet can be reduced to the level of 102 ppm with a
+sample volume of 10 mm3 by the location optimization of the
+sample. A main magnet after specially designed with higher
+magnetic field homogeneity is needed to further promote the
+resolution of NMR measurements [34].
+V. CONCLUSION
+In this paper, we report the nuclear magnetic resonance
+measurements in the high flat-top pulsed magnetic field up
+40 T. The scheme of FTPMF with two-stage correction was
+proposed, whose magnetic field fluctuation was limited within
+10 mT and duration was beyond 9 ms. Besides, the NMR
+
+39.560
+(a)
+#8
+#9
+39.555
+#7
+1
+msi
+#6
+#5
+39.550
+#4
+:#1
+#3
+#2
+39.545
+39.540
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+Time (ms)
+Magnetic field Offset (mT)
+-48
+-38
+-29
+-19
+-10
+0
+10
+19
+29
+38
+48
+Amplitude (a.u.)
+1.5
+(b)
+#1
+#2
+#3
+1.0
+#4
+#5
+0.5
+#6
+#7
+#8
+0.0
+#9
+500 -400 -300 -200 -100
+0
+100
+200
+300
+400
+500
+Freguency Offset (kHz)Magnetic Field Offset (mT)
+-38
+-29
+-19
+-10
+0
+10
+19
+29
+38
+DC Field @ 14.55 T
+5
+FTPMF @ 23.25 T & 0 mm
+Pulsed Field @ 23.25 T & 0 mm
+Pulsed Field @ 23.25 T & 3 mm
+4
+FWHM: 300 ppm
+Amplitude (a.u.)
+FTPMF @ 39.55 T & 0 mm
+3
+FWHM: 560 ppm
+FWHM: 560 ppm
+2
+FwHM: 550 ppm
+1
+0
+FWHM: 1440 ppm
+-1
+-400
+-300
+-200
+-100
+0
+100
+200
+300
+400
+Freguency Offset (kHz)7
+spectrometer and probe suitable for the pulsed field condition
+were also developed. The NMR measurements of 93Nb were
+preoptimized in the DC field and finally carried out in the
+FTMPF. The results show that the stability and homogeneity
+of the FTPMF can reach an order of 102 ppm / 10 ms and 102
+ppm / 10 mm3 respectively, which is sufficient for exploring
+microscopic structures of some condensed matters in the high
+magnetic field. Crucial avenues of research in the future are
+the improvements of the FTPMF in many aspects including
+field strength, duration, stability, homogeneity and so on.
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+
diff --git a/odAyT4oBgHgl3EQfY_eZ/content/tmp_files/load_file.txt b/odAyT4oBgHgl3EQfY_eZ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..16cfcd481007cbd455d7f3697e02fef7ef657c5b
--- /dev/null
+++ b/odAyT4oBgHgl3EQfY_eZ/content/tmp_files/load_file.txt
@@ -0,0 +1,898 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf,len=897
+page_content='1  Nuclear Magnetic Resonance Measurements in  High Flat-top Pulsed Magnetic Field up to 40 T at  WHMFC  Wenqi Wei, Qinying Liu, Le Yuan, Jian Zhang, Shiyu Liu, Rui Zhou, Yongkang Luo, and Xiaotao Han  Abstract—Nuclear magnetic resonance (NMR) technique ben-  efits from high magnetic field not only due to the field-enhanced  measurement sensitivity and resolution, but also because it is  a powerful tool to investigate field-induced physics in modern  material science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In this study, we successfully performed NMR  measurements in high flat-top pulsed magnetic field (FTPMF) up  to 40 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A two-stage corrected FTPMF with fluctuation less than  10 mT and duration longer than 9 ms was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Besides,  a Giga-Hz NMR spectrometer and a sample probe suitable for  pulsed-field condition were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Both free-induction-decay  and spin-echo protocols were exploited for the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The derived  93Nb NMR results show that the stability and  homogeneity of the FTPMF reach an order of 102 ppm / 10  ms and 102 ppm / 10 mm3 respectively, which is approaching  a degree of maturity for some researches on condensed matter  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Index Terms—Nuclear magnetic resonance (NMR), high field,  flat-top pulsed magnetic field (FTPMF), NMR spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' INTRODUCTION  E  VER since discovered in 1940s, nuclear magnetic res-  onance (NMR) has manifested itself as one of the  most significant tools to derive detailed information on the  atomic scale about material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Nowadays, the trend  of performing NMR experiments in high magnetic field is  actively driven not only by higher sensitivity and resolution,  but also by the desirable exploration of field-induced physics  in modern material science [1]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' For instance, researches  on half-integer quadrupolar nuclei such as  27Al and  17O  notably benefit from the high magnetic field, because the  quadrupolar interaction induced shift and broadening of the  central transition are inversely proportional to the square of  the magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The reduction in second-order  quadrupolar broadening brings resolution enhancement and  additional information, which provides new opportunities for  measurements of nuclei not probed before [1]–[5], [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  This work was supported in part by the National Key Research and  Development Plan Project of China under Grant 2021YFA1600301 and  2022YFA1602602, and in part by the National Natural Science Foundation  of China under Grant U21A20458 and 51821005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (Corresponding author:  Xiaotao Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  Wenqi Wei, Le Yuan, Jian Zhang, Shiyu Liu, Yongkang Luo, and Xiaotao  Han are with the Wuhan National High Magnetic Field Center, Huazhong  University of Science and Technology, Wuhan 430074, China (e-mail:  vinkey7wei@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' mpzslyk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' xthan@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Qinying Liu is with the Electric Power Research Institute, China Southern  Power Grid Company Limited, Guangzhou 510623, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Rui Zhou is with the Institute of Physics, Chinese Academy of Sciences, and  Beijing National Laboratory for Condensed Matter Physics, Beijing 100190,  China (e-mail: rzhou@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  On the other side, since the high magnetic field often causes  electronic and structural transitions of materials, the interests  in field-induced physics like high-Tc cuprate superconductors  have been increasing in the community of modern condensed  matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The NMR spectroscopy can provide more fun-  damental insights in the properties of materials, hence people  have been making efforts to carry out NMR experiments in  higher magnetic field [6], [7], [9], [11]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  At present, the high magnetic field greater than 30 T can  be generated by continuous-operation (also called as “steady-  state” or “DC”) or pulsed magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The DC magnets with  operation time of more than seconds include a variety of water-  cooled resistive magnets (up to 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5 T [19]), superconducting  magnets (up to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='35 T [20]) and hybrid magnets (up to  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5 T [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The continuous magnetic field is applicable to  perform NMR experiments, because there is enough time to  correct the field both in stability and homogeneity, as well as  to execute various sequences for measurements and achieve  signal averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' However, large construction and operation  costs and limited field strength of superconducting conductors  hinder the development of NMR experiments to higher DC  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Currently, only pulsed resistive magnets are practically  available to produce the high magnetic field exceeded 45 T  and even up to 100 T for timescales typically in the range of  1-100 ms [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Although some NMR experiments were  successfully optimized and achieved in the short and time-  varying pulsed field [6]–[16], it is still desirable to perform  NMR measurements in a stable magnetic field [17], [18],  [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Firstly, a large real-time bandwidth up to tens of MHz  or higher is required to excite and detect NMR signals due  to the low repeatability and time-varying of the pulse field,  which brings great difficulties to hardware design and signal  search [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Besides, the instability of the pulsed magnetic  field causes the violent fluctuation of the resonance frequency  and leads to the distortion of the NMR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Despite the  application of the complex signal post correction algorithms,  the signal quality is still limited [13], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Last but not least,  some highly accurate parameter measurements such as the  relaxation time T1 and T2 demand that the time dependence of  the magnetic field should be reduced to zero during the long  enough sequence durations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Hence, it is still challenging  to carry out NMR measurements in the pulsed magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fortunately, lots of efforts have been being made to create a  quasi-stable stage (usually called “flat-top”) near the peak of  the pulsed magnetic field [26]–[32], opening a new avenue  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='00215v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='ins-det]  31 Dec 2022  2  for the pulsed magnetic field NMR measurements [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In  the early stage, Weickert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' [26] tried to use a long-pulse  magnet powered by high-power capacitor bank to creat an  approximate 44 T flat-top for NMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' However,  the unregulated long pulsed magnetic field is inefficient in  improving the flat-top stability and has a long repetition  time more than 8 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Recently, Ihara and Kohama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  [17], [24] systematically peformed NMR measurements in a  dynamically controlled flat-top field, showing great advantages  of the flat-top pulsed magnetic field (FTPMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Nevertheless,  their measurements are limited in the low field of 13 T, and a  higher FTPMF is desired in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  In this work, we present a practically available FTPMF for  NMR measurements up to 40 T at Wuhan National High  Magnetic Field Center (WHMFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' II, we briefly  introduce basic concepts of NMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Furthermore,  experimental setups are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' III including a two-  stage corrected flat-top pulsed magnetic field, a modular GHz  NMR spectrometer and a sample probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' IV, NMR  measurements of 93Nb carried out in the DC field and FTPMF  are described and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Finally, the paper is concluded  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' BASIC CONCEPTS OF NMR  To understand basic concepts of NMR measurements, a  brief description is presented here and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The  interaction between an external magnetic field B0 and nuclear  spin I (I > 0) is described by the Zeeman effect, in which  the spectral line splits into multiple discrete levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' For the  simplest case of I = 1  2, the Zeeman energy levels are given  by  EZee = ±1  2hγB0  (1)  where γ is gyromagnetic ratio (in MHz/T), and h is Planck’s  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The difference of energy levels can be written as  ∆E = hfL = hγB0  (2)  where the fL is called the Larmor frequency and fL = γB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  If a radio-frequency (RF) field B1 (t) = B1 cos (2πfLt) per-  pendicular to the B0 excites the nucleus, transitions between  these energy levels can occur and NMR phenomenon can be  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Commonly, a π/2 RF pulse is used to irradiate the NMR  free induction decay (FID) signals which have the form of  sFID (t) ∼ cos (2πfLt) · e  −  t  T ∗  2  (3)  where T ∗  2 is the decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In the corresponding Fourier  transform spectrum, the standard NMR spectrum is of the  following form  S (f) ∼  1/T ∗  2  (1/T ∗  2 )2 + (2πf − 2πfL)2  (4)  which is called the absorption Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It is obvious that  the peak of the spectrum is at the Larmor frequency fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' When  the external magnetic field B0 changes with the time during  the period of the FID, the fL also changes proportionally and  this leads to the distortion of the NMR spectrum (an example  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Basic principles of NMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (a) The Zeeman energy  levels of a nucleus with spin 1/2 in the presence of an external magnetic  field B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' NMR transition occurs at the Larmor frequency fL = γB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  (b) FID signals irradiated by a π/2 RF pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (c) A standard absorption  Lorentzian NMR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' the FWHM of 1/  �  πT ∗  2  �  includes the intrinsic  homogeneous broadening quantified by the transverse relaxation time T2 and  the inhomogeneous broadening due to the spatial inhomogeneity ∆B0 (r) of  the B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (d) The distortion of the NMR spectrum due to the temporal instability  of the B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The ∆B0 (t) reflects the degree of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Hence, high stability of B0 with time is  one of the key prerequisites for NMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Another  essential parameter in the NMR spectrum is the full width  at half maxima (FWHM) parameterized by 1/ (πT ∗  2 ), which  represents the spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In fact, the FWHM includes  the intrinsic homogeneous broadening quantified by the trans-  verse relaxation time T2 and the inhomogeneous broadening  due to the spatial inhomogeneity of the B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Consequently, B0  with less spatial inhomogeneity ∆B0 (r) is desired for higher  spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' EXPERIMENTAL SETUPS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Flat-top Pulsed Magnetic Field  For decades, the FTPMF technique has been pursuing to  improve the stability and duration of the flat-top stage while  maintaining the high field strength [26]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The economical  and reliable high-voltage capacitor bank is the most commonly  used main power source for generating the high pulsed mag-  netic field, but it is difficult to regulate the discharge current for  controlling the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Despite all this, some solutions  have been proposed, such as sequentially discharging multiple  capacitors [27], [28], [32], or using a small compensation coil  built into the bore of the main magnet to finely regulate the  magnetic field [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The former method can achieve a long  flat-top duration of more than 10 ms but has high instability  beyond 103 ppm, while the latter stands the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Hence,  we combined the advantages of the two schemes, and the flat-  top realization is divided into two processes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  In terms of concrete implementation, the main magnetic  field was produced by a standard 60 T pulsed magnet with the  height of 150 mm and bore of 20 mm, which has an inductance  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5 mH and a resistance of 30 mΩ at 77 K (the magnet  was immersed in the liquid nitrogen for cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The main  t  SFID(t)～ COs(2πfLt) · e  T2  (b)  T  +  2  △E=hfL=hBo  Bo  △ABo(r  (c)  元  元1  fL  (p)  0()-0()-0()8-3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Scheme of the two-stage corrected flat-top pulsed magnetic field  for NMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (a) Diagram of connection of main components  in the two-stage corrected FTPMF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (b) Illustration of magnetic field  waveforms generated by the two-stage corrected FTPMF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  magnet was connected in series with the primary winding of  a pulse transformer (refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' [28] for specific parameters)  and powered by 8 parallel capacitor banks (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='84 mF, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='2 MJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The secondary winding of the pulse transformer was powered  by 6 parallel capacitor banks (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='84 mF, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='2 MJ) and formed the  auxiliary circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Before the current of the main circuit reached  its peak, the auxiliary circuit was discharged to achieve the  first correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' According to the different optimized discharge  voltages of both circuits and trigger time of the auxiliary  circuit, the flat-top pulsed magnetic field up to 45 T after the  first correction were produced, whose stability was within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5  T, and duration was within 12–15 ms depending on the field  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The compensation coil used for the second correction  consists of a main regulating coil with the height of 36  mm at the center of the main magnet and two oppositely  wound decoupling coils with the height of 18 mm at the  ends of the main regulating coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Due to the strict radial  space limitation in the main magnet bore of 20 mm, the  copper compensation coil was double wound around a 17  mm Helium cryostat tail with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='4 mm wire diameter  and reinforced by a Zylon/epoxy composite layer of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5 mm  thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The compensation coil was powered by twelve 12-  V lead-acid batteries, and its current can be linearly regulated  by insulated gate bipolar transistor (IGBT) modules with a  homemade driver to generate an adjustable magnetic field from  0 to 1 T beyond 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' magnetic field signals at the sample  area were sensed by a pick-up coil and the pick-up voltage  was acquired at a sample rate of 100 kS/s by a 16-Bit analog-  to-digital converter equipped with a field programmable gate  array (FPGA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Digital proportional integral derivative  (PID) control was applied to achieve the fine compensation  of the magnetic field for the second correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Finally, the  magnetic field fluctuation after the two-stage correction was  limited less than 10 mT, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Due to  the establishment time of the PID control and the decrease  of the regulation capability of the compensation coil caused  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Magnetic field profiles up to 45 T generated by the two-stage corrected  flat-top system and measured by the pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The insert shows that the  magnetic field fluctuation of the flat-top was limited less than 10 mT (integral  drift had been corrected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  by the main magnet heating, the flat-top time after the second  correction was reduced by about 4 ms compared with that of  the first-order correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It is noteworthy that both centers  of the main magnet and compensation coil should be aligned  accurately to provide a high magnetic field homogeneity area  for NMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The axial locating processes were  realized by using the pick-up coil and a lock-in amplifier with  an error less than 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Considering the aging of the main  magnet, we would perform NMR experiments in the FTPMF  up to 40 T for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Modular GHz NMR Spectrometer  A flexible modular GHz NMR spectrometer was developed,  in which a series of stand-alone instruments were combined  for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The advantages of the modular spectrometer are  convenient for testing and maintenance, as well as further  upgrading and expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' According to the properties of the  FTPMF, several requirements of the NMR spectrometer are as  follows: (1) Since the pulsed field is transient and its duration  is typically less than 100 ms, the spectrometer should be  controlled by a sophisticated timing system and has excellent  transient responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (2) The spectrometer should support high  radio frequency up to several GHz, due to the fact that the  resonance frequency is proportional to the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' For  example, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 GHz is needed for 1H at 60 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (3) Low noise  is required because the electromagnetic environment of the  pulsed field is far worse than that of the DC field, and the  number of times of the signal averaging is limited in the  FTPMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  A diagram of the self-built modular NMR spectrometer is  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The construction of the spectrometer was  based on a National Instruments (NI) PXI system including a  chassis and an embedded computer, which was controlled via  a self-written LabVIEW interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Currently, the PXI system  hosted three main modules, consisting of a RF generator  NI PXIe-5651, a vector signal analyzer NI PXI-5661 and  a pulse programmer NI PXI-6542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The RF generator can  (a)  0～20 kA  Cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Banks  0～0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='2 kA  Batteries  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='84 mF  Main Circuit  12 X12  Compensation Circuit  X8  Pri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Tansformer  IGBT Modules  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' *  Cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Banks  PID Control  dB(t)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='84 mF  Auxiliary Circuit  P  Controller & Driver  X6  Z 0 ~ 20 kA  TT  telose = tm + tdelay  Main (  Compensation Pick- up  Coil  Coil  Magnet  1st Correction Part  2nd Correction Part  (b)  B  Original Pulsed Magnetic Field  FTPMF after 1st Correction  FTPMF after 1st & 2nd Correction  0 ～ 45 T  Compensated Magnetic Field  0～1T  ät  tm  tm + tdelay50  10 mT/ 8 ms  45  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='204  Field  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='202  40  10 mT/ 9 ms  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='200  Magnetic Field (T)  35  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='198  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='196  10 mT/ 10 ms  30  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='194  13 14 15 16 17 18 19 20 21  25  10 mT/ 11 ms  Time (ms)  20  15  10  5  0  0  10  20  30  40  50  60  70  80  90  Time (ms)4  produce RF signals with frequency ranges from 500 kHz to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='3  GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Correspondingly, the vector signal analyzer can support  intermediate frequency (IF) down-conversion of NMR signals  up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='7 GHz with a 20 MHz real-time bandwidth, and the  NMR signals are finally recorded by a 100 MS/s and 14-Bit  oscilloscope with using quadrature digital down-conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The 100 MHz digital pulse programmer was used to receive  an external trigger when the flat-top field arrived, and then  trigger each independent module in the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The  all components were synchronously clocked by a 10 MHz  reference clock from the RF generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Diagram of the modular NMR spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  In the preceding stage circuit for excitation, the continuous  RF signals were modulated into desired NMR sequences by a  pulse gate (Analog Devices, HMC427ALP3E) with switching  time smaller than 10 ns and isolation up to 40 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Then, the  power of the modulated signals was amplified by a Tomco  100 W RF amplifier with band 5-650 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The duplexer  is a key three ports component for switching between the  excitation of the high-power RF signals and the reception of  the weak NMR signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A 1 GHz bandwidth homemade active  duplexer based on PIN diodes and various impedance lines was  developed with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='56 dB insert loss, 500 W power capacity, 37  dB isolation and 1 µs switching time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' To protect the receiving  part from the crosstalk of the high-power excitation signals, a  70 dB isolation switch (Mini-Circuits, ZASWA-2-50DRA+)  was applied with switching time of 20 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A lower noise  amplifier N141-306CB up to 1 GHz with gain greater than 30  dB and noise figure smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='8 dB was used to amplifier  the weak NMR signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Sample Probe  Compared with the conventional NMR probe for the DC  field, the probe for the pulsed field is challenging to design  because of the strict space limitation in the bore of the pulsed  magnet, as well as serious electromagnetic interference and  mechanical vibration environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In practice, the bore space  of the magnet available for the probe design is only 10 mm,  but the probe needs to have multiple functional components,  for example, including a sample chamber with enough space, a  well tuned and matched resonance circuit, a temperature sensor  and a heater, as well as a magnetic field pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Closed  loops of the conducting materials leading to eddy current have  to be avoid, and all components should be well fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (a) Schematic illustration of the sample probe designed for the pulsed  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The partial photo of the sample probe and connection of the resonance  circuit are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (b) Photo of the experimental station layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 5(a) shows the schematic illustration of the sample  probe designed for the pulsed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The probe was located  in a cryostat that can use liquid helium or nitrogen to per-  form low-temperature experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The volume of the sample  chamber was about several tens of cubic millimeters and it  was surrounded by a micro RF coil to excite and detect  NMR signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A tuning capacitor and a matching coil were  connected to the resonant circuit which had been preliminarily  calculated and tested at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Although there  was deviation about several MHz from the expected resonant  frequency in the working condition, precise tuning was not  very necessary in the pulsed field experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The return  loss of the resonance circuit at the resonance frequency was  adjusted to be less than -10 dB, meaning that more than 90%  of the power was absorbed by the RF coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The magnetic field  pick-up coil was wounded on the outer sleeve of the probe  and at the same axial height as the center of the sample to  ensure that the magnetic field at the sample were accurate  collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The heater and temperature sensor were located  near the resonance circuit to control the local temperature  around the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Most components were arranged vertically  to improve the space utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Based on the establishment  of the above experimental setups, a photo of the experimental  station layout is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  93Nb powder (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='999% purity) uniformly mixed by resin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Probe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Low Noise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Isolation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Switch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Amplifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Amplifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Duplexer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='LNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='PA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Pulse Gate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Vector Signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Pulse Programmer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Analyzer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='RF Generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Computer and Chassis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='External Trigger(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='NMR Signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Duplexer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Excitation and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Detection Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Detection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Excitation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Heating and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Magnetic Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Matching Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Temperature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Detection Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='RM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='LM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Detection Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='W  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='333  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='CT :  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5333  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Tuning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Rrf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Lrf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Capacitor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='RF Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Cryostat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Resonance Circuit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Compensation Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Main Magnet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Liquid Nitrogen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Heater and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Magnetic Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Heater and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Tuning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Matching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='RF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Sample  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Sample  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Temperature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Pick-up Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Temperature Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Capacitor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Coil  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Spectrometer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Probe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='FTPMF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Second Correction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='System  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='Main Magnet5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='was selected as the testing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 93Nb is ideal for NMR  measurements under the pulsed field, because it has the natural  abundance of 100%, a large gyromagnetic ratio γ=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='405  MHz/T, a short longitudinal relaxation time T1 of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='7 ms at 77  K, and a high relative sensitivity of 48% compared with 1H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The mixed sample was cut into a cylinder and placed in the RF  coil wound by copper wire (about 2 mm in diameter, 4 mm  in length, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='3 mm wire diameter with 6 turns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Considering  the Knight shift (K), the resonance frequency of 93Nb is  γ  �  1 +  93K  �  B0, where 93K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='87%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' NMR Measurements in the DC Field and FTPMF  Since the pick-up coil was not well calibrated for measuring  the pulsed magnetic field with an estimated error lager than  1% and the repetition time for the pulsed magnet cooling was  usually longer than 30 minutes, a pre-optimization process in  the DC field was necessary to achieve the maximum sensi-  tivity of NMR signals in a single measurement as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  This process was carried out with our self-built spectrometer  and probe in a superconducting magnet (Oxford) with high  stability (better than 10 ppm/h) and homogeneity (better than  10 ppm/cm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The FID measurements of 93Nb in the DC field  were performed at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T and 77 K, and the corresponding  resonance frequency is 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='7 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' In order to irradiate the  FID signals with both high sensitivity and bandwidth, the  power and duration of the π/2 RF pulse were optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Finally, the 50 W and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5 µs RF pulse was implemented  with the maximum signal-to-noise ratio (SNR) of 20 dB and  bandwidth of 600 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The FID signals after quadrature down-  conversion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 6(a), which were measured at  the given RF of 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 MHz for the clarity of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Ignoring the switching dead time of 1 µs, the decay time T ∗  2  of the FID signals about 8 µs was observed, which can be  approximately regarded as the T2 due to the high stability and  homogeneity of the DC field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  For quickly locating the NMR signals and calibrating the  pick-up coil in the FTPMF, a field-sweep mode with slope-top  pulse was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' According to the bandwidth of excitation  and longitudinal relaxation time T1 of 93Nb, the sweep slope  rate was set to 10 mT/ms and the interval of excitation was  1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' After several sweep tests, the NMR measurements can  be normally carried out in the FTPMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The FID signals at the  field of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='24 T and excited RF of 244 MHz after quadrature  down-conversion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It is obvious that the  decay time of the FID signals reduces from 8 µs to 3 µs in the  FTPMF compared to the DC field, which was mainly caused  by the inhomogeneity of the FTPMF and will be discussed  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Although the field strength of the FTPMF is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 times  higher than that of the DC field, the sensitivity of FID signals  do not significantly increase due to the short polarization time  of nuclei, inhomogeneity of the magnetic field and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  These 93Nb FID results can be further confirmed by a spin-  echo (SE) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A standard SE measurement (π/2−τ−π,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5-20-3 µs in this case) was performed in the FTPMF, the  result of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It is clear that the SE  signals refocus after waiting for 20 µs, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' FID signals after quadrature down-conversion: (a) at the DC field of  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T and excited RF of 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (b) at the FTPMF of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='24 T and  excited RF of 244 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (c) at the FTPMF of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='54 T and excited RF of 415  MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' All measurements were carried out at temperature of 77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  NMR RF pulse sequence followed by the spin-echo signals at the  FTPMF of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='26 T and excited RF of 244 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The given SE sequence is  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5-20-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='0 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The FID signals in the FTPMF of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='54 T and excited RF  of 415 MHz were measured and are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  It is noteworthy that the given π/2 RF pulse was reoptimized  with the duration of 9 µs to achieve the highest possible signal  sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The results show that the signal sensitivity in the  high field of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='54 T increases about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 times in proportion  to the magnetic field strength, compared with that in the low  field of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='24 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' However, the decay time of FID further  decreases to 2 µs due to the degradation of the magnetic field  homogeneity in the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  20  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  (a)  15  Real Part  Imaginary Part  10  Magnitude  5  0  5  DC field @14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T & 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='6 MHz  10  sn  15  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 34  Time (us)  20  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  (b)  15  Real Part  Imaginary Part  10  Magnitude  5  0  5  FTPMF @23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='24 T & 244 MHz  sn  10  15  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 34  Time (μs)  20  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  C  Real Part  15  Imaginary Part  10  Magnitude  5  0  5  FTPMF @39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='54 T & 415 MHz  us  10  15  12  14  16  18  20  22  24  26  28  30  32  34  Time (us)10  Real Part  8  Imaginary Part  Magnitude  6  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  4  2  0  2  4  6  T  SE  8  2  10  10  20  30  40  50  60  70  80  Time (μs)6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Stability of the FTPMF  The NMR technique itself is also an accurate method for  measuring the magnetic field, hence the FID signals were  excited several times to check the stability of the FTPMF  during the period of the flat-top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 8(a) shows the nine  excitation points with an interval of 1 ms during the FTPMF  at 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T measured by the pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The corresponding  NMR spectra at the excited RF of 415 MHz are presented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The central frequency offset of the nine spectra  was limited within 100 kHz, meaning the field fluctuation was  less than 10 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It well corresponds to the result measured  by the pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Overall, the current stability of the  FTPMF in the timescale about 10 ms would potentially find  some applications for NMR experiments in condensed matter  physics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A higher resolution ADC for the acquisition of  the magnetic field and smaller feedback control cycle for the  second correction system are desired to further improve the  stability of the FTPMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Stability of the FTPMF checked by the NMR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (a) Nine  excitation points of the FID signals with an interval of 1 ms during the  FTPMF at 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T measured by the pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' (b) The corresponding  NMR spectra at the excited RF of 415 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The central frequency offset of  the nine spectra was limited within 100 kHz, which corresponds well to the  magnetic field fluctuation of 10 mT measured by the pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Homogeneity of the FTPMF  In the section II, we revealed that the decay time T ∗  2 of the  FID signals decreases with the increase of the magnetic field  spatial inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Besides, the inhomogeneity can also  lead to the broadening of the NMR spectra, which is harmful  to the high resolution NMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Compared with  the commercial superconducting magnet with sophisticated  shimming techniques, the pulsed magnet has great magnetic  field inhomogeneity up to 1000 ppm or higher in 1 cm3  space of the magnet center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Ignoring the winding error and  magnet deformation, the axial distribution of the magnetic  field generated by the solenoid-type magnet is approximately a  quadratic function of B0 (z) = B0−az2 , where the coefficient  a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Hence, small volume and highly accurate axial  positioning of the sample in the pulsed magnet are important  to weaken the influence of the magnetic field inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The NMR spectra of 93Nb under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Relative FWHM  is marked for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The NMR spectra measured in the DC field of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T  and measured in the FTPMF of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='25 T are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The axial position of the sample in the FTPMF was well  optimized with an accuracy less than 1 mm (labelled at 0 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  Because of the high homogeneity better than 10 ppm/cm3 in  the DC field, the FWHM of 300 ppm can be approximately  attributed to the intrinsic homogeneous broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Although  the strength of the FTPMF is higher than that of the DC field,  the FWHM increases to 550 ppm due to the magnetic field  inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Because the magnetic field inhomogeneity was  caused by both the compensation coil and main magnet, the  original pulsed magnetic field at the peak with change rate  less than 1 mT/10 µs was applied to check the inhomogeneity  induced by the main magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The results show that the main  magnet is the main source of the magnetic field inhomogeneity  and the compensation coil has little impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The spectrum of  the sample positioned at 3 mm away the optimized position  was also measured and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' It is obvious that  the FWHM increases to 1440 ppm due the heavier magnetic  field inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  The NMR spectrum in the FTPMF of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T is also  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Despite greater absolute broadening of the  spectrum in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T compared with that in the low field of  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='25 T, the relative broadening almost remains unchanged due  to the increasing of the resonance frequency from 244 MHz  up to 415 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' On balance, the inhomogeneity effect of the  pulsed magnet can be reduced to the level of 102 ppm with a  sample volume of 10 mm3 by the location optimization of the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' A main magnet after specially designed with higher  magnetic field homogeneity is needed to further promote the  resolution of NMR measurements [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' CONCLUSION  In this paper, we report the nuclear magnetic resonance  measurements in the high flat-top pulsed magnetic field up  40 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The scheme of FTPMF with two-stage correction was  proposed, whose magnetic field fluctuation was limited within  10 mT and duration was beyond 9 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Besides, the NMR  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='560  (a)  #8  #9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='555  #7  1  msi  #6  #5  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='550  #4  :#1  #3  #2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='545  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='540  13  14  15  16  17  18  19  20  21  22  Time (ms)  Magnetic field Offset (mT)  48  38  29  19  10  0  10  19  29  38  48  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5  (b)  #1  #2  #3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='0  #4  #5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='5  #6  #7  #8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='0  #9  500 -400 -300 -200 -100  0  100  200  300  400  500  Freguency Offset (kHz)Magnetic Field Offset (mT)  38  29  19  10  0  10  19  29  38  DC Field @ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T  5  FTPMF @ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='25 T & 0 mm  Pulsed Field @ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='25 T & 0 mm  Pulsed Field @ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='25 T & 3 mm  4  FWHM: 300 ppm  Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=')  FTPMF @ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content='55 T & 0 mm  3  FWHM: 560 ppm  FWHM: 560 ppm  2  FwHM: 550 ppm  1  0  FWHM: 1440 ppm  1  400  300  200  100  0  100  200  300  400  Freguency Offset (kHz)7  spectrometer and probe suitable for the pulsed field condition  were also developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The NMR measurements of 93Nb were  preoptimized in the DC field and finally carried out in the  FTMPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' The results show that the stability and homogeneity  of the FTPMF can reach an order of 102 ppm / 10 ms and 102  ppm / 10 mm3 respectively, which is sufficient for exploring  microscopic structures of some condensed matters in the high  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
+page_content=' Crucial avenues of research in the future are  the improvements of the FTPMF in many aspects including  field strength, duration, stability, homogeneity and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAyT4oBgHgl3EQfY_eZ/content/2301.00215v1.pdf'}
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diff --git a/otE0T4oBgHgl3EQf9AIJ/content/tmp_files/2301.02794v1.pdf.txt b/otE0T4oBgHgl3EQf9AIJ/content/tmp_files/2301.02794v1.pdf.txt
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+Single crystal growths and magnetic properties of hexagonal polar semimetals RAuGe
+(R = Y, Gd−Tm, and Lu)
+Takashi Kurumaji,1 Masaki Gen,1, 2 Shunsuke Kitou,2 Kazuhiko
+Ikeuchi,3 Mitsutaka Nakamura,4 Akihiko Ikeda,5 and Taka-hisa Arima1, 2
+1Department of Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan
+2RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
+3Neutron Science and Technology Center, Comprehensive Research Organization
+for Science and Society (CROSS), Tokai, Ibaraki, 319-1106, Japan
+4J-PARC Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
+5Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
+(Dated: January 10, 2023)
+We study structural and magnetic properties of rare-earth based semimetals RAuGe (R = Y,
+Gd−Tm, and Lu) using flux-grown single crystals. These compounds belong to the noncentrosym-
+metric polar space group P63mc. We confirm the systematic structural evolution at room tempera-
+ture as a function of ionic radius of rare earths to clarify the isopointal crossover between two polar
+structures: three-dimensional LiGaGe-type and quasi-two-dimensional NdPtSb-type. Magnetism
+shows a characteristic anisotropy in reasonable agreement with the crystal electric field (CEF) the-
+ory; the easy-plane-type anisotropy for R = Tb and Dy turns into the Ising-type anisotropy for
+R = Er and Tm. We evaluate the CEF parameters based on the Stevens operators to reasonably
+reproduce the temperature dependence of magnetic susceptibilities and specific heat for RAuGe
+(R = Tb−Tm). The estimated energy scale of the Ising gap (∼ 11 meV) in TmAuGe is consistent
+with an excitation observed in an inelastic neutron scattering experiment. These findings suggest an
+opportunity for interplay between conduction electrons and nontrivial spin structures in the family
+of magnetic polar semimetals RAuGe.
+INTRODUCTION
+Polar metals are defined as the conductive materials
+with symmetry breaking allowing electric polarization,
+which provide a possible route to ferroelectric metals
+[1, 2] and to hyperferroelectricity [3, 4]. These materials
+are potentially applied to electric-field-switchable spin-
+tronic devices [5, 6], and a metallic counterpart of multi-
+ferroics [7–9] with topological band crossings induced by
+magnetic correlation [10, 11] and/or spin textures [12–
+15].
+Rare-earth-based semimetals RAuGe (R = Sc, Y,
+La−Nd, Sm, Gd−Tm, and Lu) belong to the equiatomic
+ternary RTX (T = transition metals, X = p-block ele-
+ments) intermetallic phases [16] of the hexagonal polar
+space group P63mc (No.
+186) [17, 18].
+Their crystal
+structure is shown in Fig. 1(a), where the positions of
+Au and Ge are shifted upward and downward, respec-
+tively, along the c axis from the aristotype ZrBeSi struc-
+ture (space group P63/mmc, No. 194) to break the inver-
+sion symmetry, which is in common with a prototypical
+polar metal considered in theories [3, 7–11, 19–22].
+CeAuGe and TmAuGe are known to show ferromag-
+netism below TC = 10 K [23–25] and TC = 4 K [26], re-
+spectively. The compounds for other R (Nd, and Gd−Er)
+are antiferromagnetic with a transition temperature TN
+ranging between 6 K and 17 K [25, 27–32].
+A previ-
+ous neutron study detected a commensurate modula-
+tion vector ( 1
+2, 0, 0) in the ground state for R = Tb−Er
+[27, 29–31]. Incommensurate modulations were also ob-
+served to coexist at low temperatures or just below TN,
+suggesting inherent magnetic frustration in triangular-
+lattice layers of R. The density of state has a pseudo-
+gap nature near the Fermi level due to the overlap of
+electron and hole bands, which is enabled by nomi-
+nal valence configuration R3+Au+Ge4− [18, 33, 34] and
+contributes to semimetallic transport properties of non-
+mangetic compounds (R = Sc, Y, La, and Lu) [25, 33].
+Resistivities of magnetic compounds, RAuGe (R = Ce,
+Nd, Gd, Ho, and Tm) [23, 26, 31, 32], are known to
+show nonmonotonic temperature dependences indicating
+a coupling with magnetic orders. These features suggest
+an opportunity to encounter unconventional properties
+of polar semimetals intertwinned with frustrated mag-
+netism in magnetic RAuGe. The above-mentioned phys-
+ical properties were identified using polycrystals, and sin-
+gle crystal growths are only known for CeAuGe using
+Czochlarski and floating-zone methods [27, 35], and for
+epitaxial thin-film form of LaAuGe [36].
+In this study, we report the single-crystal growths and
+physical properties of RAuGe (R = Y, Gd−Tm, and Lu).
+We analyzed the crystal structure using x-ray radiation
+to confirm the systematic evolution of polar structure
+as a function of R-ionic radius.
+We measured specific
+heat, magnetization, and thermal expansion to reveal the
+anisotropic magnetism significantly affected by the crys-
+tal electric field (CEF) as well as magnetic transitions
+at low temperatures.
+We estimate the CEF potential
+parameters to reproduce the magnetic susceptibility and
+specific heat and obtain a systematic R dependence rea-
+sonably agreed with the Stevens theory [37]. The esti-
+arXiv:2301.02794v1  [cond-mat.mtrl-sci]  7 Jan 2023
+
+2
+FIG. 1. (a) Crystal structure of RAuGe. z(Au) and z(Ge) are
+fractional coordinates along the c axis of Au and Ge atoms
+at 2b sites ( 1
+3, 2
+3, z), which are shifted upward from 0.75 and
+downward from 0.25, respectively, producing the polarity in
+RAuGe. (b) Photograph of single crystals of LuAuGe. Black
+bar is 1 mm. (c)-(d) Lattice constants, a and c, and inter
+and intralayer Au-Ge bond lengths in a unit cell at room
+temperature as a function of the R3+ ionic radius for six-
+coordination. The values for LaAuGe are excerpts from Ref.
+[33], and those for CeAuGe and ScAuGe are from Ref. [18].
+(e) Side views of two isopointal [18] structures, LiGaGe and
+NdPtSb types, with distinct interlayer Au-Ge bonding (cyan
+arrow).
+mated Ising gap in TmAuGe is 11 meV, which is con-
+sistent with the excitation at 13 meV observed in the
+inelastic neutron scattering (INS) experiment. These re-
+sults provide basic understanding on the structural and
+magnetic properties of the family of polar semimetals
+RAuGe.
+FIG. 2. (a)-(b) Temperature dependence of magnetic suscep-
+tibility (M/H) in H ⊥ c and the specific heat (Cp) of RAuGe
+(R = Y and Lu) at zero field. The inset of (b) plots Cp/T vs.
+T 2. The dashed lines are linear fit.
+EXPERIMENTAL METHODS
+Single crystals of RAuGe (R = Y, Gd−Tm, and Lu)
+were grown using the Au-Ge self-flux method. The mo-
+lar ratio of R:Au:Ge was 1:2:2 for R = Y, Tb−Tm, and
+Lu and 1:3:3 for R = Gd. Starting materials were R in-
+got (99.9%), Au wire (99.5%), and Ge chunks (99.999%).
+They were loaded into an alumina crucible, which was
+sealed in an evacuated quartz tube. The tube was heated
+to 1100 ◦C and kept at this temperature for 24 hours be-
+fore being slowly cooled to 800 ◦C in 200 hours. After
+being kept at 800 ◦C for 4 days, the tube was taken out
+from the furnace and centrifuged to remove excess flux.
+The obtained crystals were of platelet shape with typical
+dimensions 3×3×0.5 mm3 as shown in Fig. 1(b). The
+atomic composition and the phase were checked by en-
+ergy dispersive x-ray spectroscopy (EDS, JEOL model
+JSM-6010LA) and a powder x-ray diffraction (Rigaku
+SmartLab diffractometer) using Cu Kα radiation, respec-
+tively. Single-crystal X-ray diffraction data for RAuGe
+were collected on Rigaku AFC-8 diffractometer equipped
+with a HyPix-6000 detector using Mo Kα radiation at
+room temperature. The intensities of Bragg reflections
+were collected by CrysAlisPro program [38]. The crystal
+structures were solved by SUPERFLIP [39] and refined
+by Jana2006 [40]. The details of the structural analy-
+
+a
+b
+LuAuGe
+z(Au)
+z(Ge)
+R
+b
+a
+Sc
+Gd
+CeLa
+Lu
+6.3
+c
+9.0
+6.0
+5.7
+8.0
+C
+5.4
+A
+A
+5.1
+a
+7.0
+4.8
+4.5
+0 0000 0
+6.0
+4.2
+0.7
+0.8
+0.9
+1.0
+1.1
+d
+4.0
+3.6
+(A)
+Au-Ge
+3.2
+inter
+2.8
+0.7
+0.8
+0.9
+1.0
+1.1
+R
+3+
+ionic radius （A)
+e
+intra
+inter
+LiGaGe type (ScAuGe)
+NdPtSb type (CeAuGe)emu/mol)
+RAuGe
+20
+a
+MoH= 1 T
+10
+R=Y
+HIc
+0
+G-
+(10°
+-10
+Lu
+MIH (
+-20
+1
+0
+100
+200
+300
+b
+80
+MoH= O T
+Lu
+60
+Y
+ (J/molK)
+8
+K
+40
+low/rw),
+S
+4
+20
+0
+10
+20
+0
+T (K)
+0
+0
+100
+200
+300
+Temperature (K)3
+FIG. 3. (a)-(b) Temperature dependence of magnetic part of
+specific heat (Cmag) and that of entropy (Smag) of RAuGe (R
+= Gd−Tm) in zero field. LuAuGe data was used to subtract
+the phonon background.
+sis results are shown in supplementary materials.
+At-
+tempts to grow the P63mc phase for YbAuGe [17] were
+unsuccessful and resulted in the orthorhombic phases,
+consistent with α-, β- (space group Pnma, No. 62) or
+γ-YbAuGe (space group Imma, No. 74) as reported in
+Ref. [41].
+Specific heat Cp was measured using a commercial sys-
+tem (heat capacity option of a Quantum Design PPMS).
+Magnetization was measured using a superconducting
+quantum interference device magnetometer (Quantum
+Design MPMS-XL). Thermal expansion was measured by
+the fiber-Bragg-grating (FBG) technique using an optical
+sensing instrument (Hyperion si155, LUNA) in a cryostat
+equipped with a superconducting magnet (Oxford Spec-
+tromag).
+Optical fibers were glued using epoxy (Sty-
+cast1266) on a (001) surface of as-grown crystals to mea-
+sure the elongation or compression along the ab plane.
+INS experiments were performed on a time-of-flight
+(TOF) spectrometer (4SEASONS) at the Materials and
+Life Science Experimental Facility (MLF) of J-PARC in
+Japan [42]. Polycrystalline TmAuGe sample of 5 g was
+prepared by arc melting of elements. A boule of TmAuGe
+was annealed at 500 ◦C for 21 days in an evacuated quartz
+tube, and sliced into pieces with thickness of 3 mm to re-
+duce the absorption of neutrons. The chopper frequency
+was set to 200 Hz, and the incident energy of Ei = 41
+meV was used to observe low-energy excitations.
+The
+FIG. 4. (a)-(f) Temperature dependence of the inverse mag-
+netic susceptibility (H/M) of RAuGe (R = Gd−Tm) for
+H ∥ c (red) and H ⊥ c (blue). Dashed lines are Curie-Weiss
+fit to the data in T = 150-300 K. (g)-(h) R dependence of (g)
+effective magnetic moment (peff), and (h) Weiss temperature.
+Dashed line in (a) is free ions values.
+sample was set in a GM refrigerator to control the tem-
+perature down to 5 K. The aquired data were analyzed
+with the UTSUSEMI software package [43].
+RESULTS
+The crystal structures of RAuGe are systematically de-
+pendent on the ionic radius of R.
+Single-crystal x-ray
+diffraction confirms the polar P63mc structure for all R,
+consistent with the previous reports [17, 18, 33]. Figures
+1(c) and (d) display the R3+ ionic radius dependence
+of the lattice constants and inter- and intralayer Au-Ge
+bond lengths. Larger R ions lead to the expansion of the
+unit-cell volume especially resulting in the elongation of
+the c axis. This tendency correlates with the weaken-
+ing of puckered distortion of Au-Ge honeycomb layer,
+where the out-of-plane (interlayer) Au-Ge bond becomes
+longer (see Fig. 1(d)). ScAuGe has the smallest R radius
+(Sc3+(VI):0.745 ˚A) with the interlayer Au-Ge bond of
+275.2 pm, while it elongates to 385.8 pm in LaAuGe with
+
+RAuGe
+a
+30
+R = Gd, Tb, Dy
+25
+Ho, Er, Tm
+(J/molK)
+20
+15
+10
+5
+0
+2
+3
+4.
+56
+2
+3
+4
+5
+6
+2
+1
+10
+100
+Temperature (K)
+30
+b
+(J/molK)
+R = Gd, Tb, Dy
+I0=H
+25
+Ho, Er, Tm
+20
+RIn8
+15
+10
+RIn2
+mag
+5
+S
+0
+2
+3 
+456
+2
+3
+4
+56
+2
+1
+10
+100
+Temperature (K)60
+d
+60
+a
+R= Gd
+R= Ho
+oH= 0.1 T
+1/x (mol/emu)
+oH= 0.1 T.
+ (mol/emu)
+40
+HIc
+40
+HIIc-
+20
+Hic
+20
+X/ 
+HIc
+0
+0
+0
+100
+200
+300
+0
+100
+200
+300
+b
+60
+60
+e
+R= Tb uoH= 1 T(lc)
+ (mol/emu)
+1/x (mol/emu)
+oH= 0.1 T.
+R= Er
+= 0.1 T (Ic)
+40
+40
+HIIc
+HIc
+20
+20
+HIIc
+Hc
+0
+0
+100
+200
+300
+100
+200
+300
+0
+0
+60
+f
+60
+C
+1/x (mol/emu)
+R= Dy
+uoH= 0.1 T
+1/x (mol/emu)
+R=Tm
+uoH= 0.1 T
+40
+40
+HIC
+Hc
+HIIc
+20
+20
+HIc
+0
+0
+0
+100
+200
+300
+0
+100
+200
+300
+Temperature (K)
+Temperature (K)
+15
+K
+100
+g
+h
+HIc
+Temperature
+HIIc
+0
+(/ar)
+10
+-100
+HIC
+Peff
+HIIc
+200
+Neiss
+5
+-300
+1
+1
+Gd
+Tb
+Dy
+Ho Er
+Tm
+Gd
+Tb
+Dy
+Ho Er
+Tm4
+the largest R-ion radius (La3+(VI):1.032 ˚A) [44].
+The
+materials for R = Y, Gd−Tm, and Lu smoothly interpo-
+late the two extremes to verify the isopointal crossover
+of the crystal-structure type between NdPtSb type and
+LiGaGe type (Fig. 1(e)), both of which belong to the
+space group P63mc [18, 23, 45]. In other words, as the
+ionic R3+ becomes larger, Au-Ge networks evolve from
+three-dimensional ZnO-type to staggered stacking of two-
+dimensional puckered binary honeycomb layers, asymp-
+totically towards hBN-type network in the centrosym-
+metric ZrBeSi-type structure (space group P63/mmc,
+No. 194).
+Next, we look through the physical properties of non-
+magnetic YAuGe and LuAuGe.
+Magnetic susceptibil-
+ity (Fig. 2(a)) shows diamagnetic nature for both com-
+pounds at high temperatures, which is consistent with
+the previous report [33]. The low-temperature rise is as-
+cribable to magnetic impurities. Specific heat Cp (Fig.
+2(b)) is dominated by lattice contribution while Cp for
+LuAuGe is larger than that of YAuGe in the entire tem-
+perature range because of the lower Debye temperature
+[33]. Small electronic contribution, γT, to Cp is deduced
+by Cp/T-vs.-T 2 plot (see inset of Fig. 2(b)). Densities of
+states at Fermi energy per a formula unit for YAuGe and
+LuAuGe are 0.175 eV−1 (γ = 0.4 mJ/mol K2) and 0.085
+eV−1 (γ = 0.2 mJ/mol K2), respectively, the order of
+magnitude of which agrees with the previous study [33].
+Using Cp of LuAuGe as the reference for the lattice
+contribution, we obtain the magnetic specific heat (Cmag)
+for R = Gd−Tm (Fig. 3(a)). The double peak structure
+at T = 15 and 17 K for R = Gd reproduces the previ-
+ous measurement [25]. The peaks below 10 K for R =
+Dy−Tm represent magnetic transitions at low tempera-
+tures while a Schottky peak around 50-60 K for R = Er
+and Tm, for example, suggests the significant contribu-
+tion of crystal electric field (CEF) excitations. To gain
+insight on the CEF energy level, we integrate Cmag/T
+with respect to T to estimate the magnetic entropy Smag
+(Fig. 3(b)). For R = Gd, the most of Smag for free-spin
+value, R ln 8, is released below the transition tempera-
+tures owing to the Heisenberg nature of Gd spins. The
+release of Smag between 4 and 20 K for ErAuGe and
+TmAuGe only approaches R ln 2, suggesting the contri-
+bution of the lowest-lying (quasi-)doublet well isolated
+from excited states. This feature reflects on the Ising-
+type anisotropy of their magnetic moments.
+For R =
+Tb−Ho, in contrast, the T-dependence of Smag does not
+show any saturation behavior below 100 K, suggesting
+overlapped contributions of CEF excitations.
+The systematic R-dependence of magnetic anisotropy
+can be seen in the inverse magnetic susceptibility (H/M)
+summarized in Figs. 4(a)-(f). Magnetic properties for
+R = Gd and Ho are roughly isotropic showing similar in-
+and out-of-plane M/H. For R = Tb and Dy, easy-plane-
+type anisotropy is observed, while ErAuGe and TmAuGe
+host easy-axis-type anisotropy. All the compounds show
+a Curie-Weiss behavior at high temperatures, where the
+Curie constants and the Weiss temperatures (ΘW) are
+obtained by a linear fit of the H/M-T curve between 150
+and 300 K. The effective moment peff is evaluated from
+the Curie constant. The R dependences of peff and ΘW
+are summarized in Figs. 4(g) and (h). Except for R =
+Tb, peff agrees well with the free-ion value (dashed curve
+in Fig. 4(g)), suggesting that the energy scale of CEF
+is lower than the thermal energy at room temperature.
+ΘW for H ∥ c is larger than that for H ⊥ c in the case
+of R = Er and Tm. The systematic anisotropy change is
+associated with the CEF parameters [37].
+We discuss the effect of CEF to clarify the R depen-
+dence of magnetic anisotropy in RAuGe. The 4f electrons
+on R sites in RAuGe are affected by the CEF potential
+(VCEF) for the site symmetry of C3v. VCEF can be ex-
+pressed by the Stevens operators ( ˆOm
+n ) as
+VCEF = B20 ˆO0
+2 + B40 ˆO0
+4 + B60 ˆO0
+6
++B43 ˆO3
+4 + B63 ˆO3
+6 + B66 ˆO6
+6.
+(1)
+Here, we assume B43 = B63 = B66 = 0 for simplic-
+ity of calculations, which is reasonable when the axial
+anisotropy of magnetic properties mainly arises from B20,
+B40, and B60. In such a case the |Jz = n⟩ state is also an
+eigenstate of VCEF. In the high temperature limit, the
+ˆO0
+2 = [3 ˆJ2
+z − J(J + 1)] term dominates the anisotropy
+of magnetic susceptibilities [46, 47]. The difference be-
+tween the inverse of magnetic susceptibility for H ∥ c and
+H ⊥ c, χ−1
+∥c and χ−1
+⊥c, respectively, is approximated to be
+a constant as
+χ−1
+∥c − χ−1
+⊥c = 3
+2
+(2J − 1)(2J + 3)
+5Cave
+B20,
+(2)
+where J is the total angular momentum J = L ± S and
+Cave is the Curie constant obtained by their average for
+H ∥ c and H ⊥ c. Assuming the identical radial distri-
+bution of 4f electron wave function for different R, B20
+is predicted to be scaled with a Stevens factor αJ [37],
+which smoothly evolves from R = Tb to Tm and changes
+the sign through Ho to Er. This feature of B20 agrees well
+with the R-dependence of magnetic susceptibility captur-
+ing the origin of the anisotropy switch in RAuGe, which
+will be discussed more quantitatively later.
+To evaluate the systematic evolution of the CEF of
+R3+ ions, we simulate the temperature dependence of
+χ∥c, χ⊥c, and Cmag on the basis of the formulae (see e.g.
+Ref. [48]):
+χ∥c = NA(gJµB)2
+Z
+×
+1
+kBT
+J
+�
+n=−J
+J2
+z e−En/kBT ,
+(3)
+
+5
+FIG. 5.
+(a)-(e) Temperature dependence of M/H for H ∥ c (red) and H ⊥ c (blue), and Cmag (black) of RAuGe (R = Tb−Tm).
+Magnetic susceptibility of ErAuGe and TmAuGe for H ∥ c (red curves in (d) and (e)) is corrected by the demagnetization
+factor, Nd, where M is divided by the internal magnetic field, Hint(= H − NdM). Cyan, orange, and gray curves are obtained
+from Eqs. (3)-(5), respectively (see text) with optimized parameters, B20, B40, and B60 (see supplementary materials). Inset
+of (e) is a color map of intensity vs. |Q| of the polycrystalline INS profile of TmAuGe with the incident neutron energy Ei =
+41 meV at T = 5 K. (f) R dependence of B20, B40, and B60 (open circles). Dashed lines are Stevens factors, αJ, βJ, and γJ
+[37], normalized by the values for R = Tm. Closed orange circles in (f) is obtained from the anisotropy of Curie-Weiss curves
+using Eq. (2).
+χ⊥c = 2NA(gJµB)2
+Z
+×
+J−1
+�
+n=−J
+| ⟨n + 1|Jx|n⟩|2 e−En/kBT − e−En+1/kBT
+En+1 − En
+,
+(4)
+Cmag =
+NA
+kBT 2
+��
+1
+Z
+�
+n
+E2
+ne−En/kBT
+�
+−
+�
+1
+Z
+�
+n
+Ene−En/kBT
+�2�
+� ,
+(5)
+where NA is Avogadro constant, kB is Boltzmann con-
+stant, µB is Bohr magneton, gJ is Lande’s g factor, En
+is the energy for a state, |Jz = n⟩. Z = �
+n e−En/kBT is
+the partition function. We summarize the results of sim-
+ulation of magnetic susceptibilities and magnetic specific
+heat in Figs. 5(a)-(e), and the deduced Stevens parame-
+ters in Fig. 5(f). We note that all the Stevens parameters
+systematically scale with the Stevens factors, αJ, βJ, and
+γJ, confirming the validity of the simulations. We also
+show the B20 values estimated from the formula of high-T
+limit Eq. (2) (closed orange circle) in Fig. 5(f), agreeing
+well with the all-data analysis.
+To double check the CEF gap in TmAuGe, we perform
+INS and observe the energy transfer as shown in the inset
+of Fig. 5(e). Crystal field around a Tm3+ ion in C3v
+symmetry splits the J = 6 manifold of Tm3+ 4f12 levels
+into five singlets and four doublets. The lowest quasi-
+
+RAuGe
+10
+20
+b
+10
+20
+c
+10
+20
+a
+R= Dy
+R= Ho
+R= Tb
+-Eq. (3)
+-Eq. (4)
+15
+15
+15
+Eq. (5)→
+MIH (emu/mol)
+ (emu/mol)
+MIH (emu/mol)
+0
+10
+10°
+10
+(J/molK)
+(J/molK)
+10
+10
+10
+MIH
+10
+HIC
+10
+10
+5
+5
+5
+HiIc
+-2 
+2
+2
+0
+0
+10°
+10°
+10°
+0
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+Temperature (K)
+Temperature (K)
+Temperature (K)
+d
+10
+20
+32101
+e
+10
+f
+R= Tm 30
+R= Er
+8
+Intensity
+0
+neV)
+20
+Eq. (2)
+15
+M/Hint (emu/mol)
+2
+E10
+1
+2
+(10°
+0
+0
+024(
+B40 
+8
+6
+-1
+10°
+10
+IQI(A
+10
+5
+5
+5
+(J/molK)
+G-
+J
+(10°
+0
+5
+0
+-2
+-2
+0
+-10
+0
+10
+10 
+100
+0
+200
+300
+0
+100
+200
+300
+Tb
+Gd
+Dy
+Ho Er
+Tm
+Temperature (K)
+Temperature (K)6
+TABLE I. Structural and magnetic parameters of RAuGe (R = Y, Gd−Tm, and Lu). a, c: lattice constants, and zAu, zGe:
+fractional coordinate for Au and Ge atoms along the c axis (see supplementary materials).
+TC: ferromagnetic transition
+temperature. TN: antiferromagnetic transition temperature. peff: effective magnetic moment, and ΘW: Weiss temperature for
+H ∥ c and H ⊥ c.
+R
+Y
+Gd
+Tb
+Dy
+Ho
+Er
+Tm
+Lu
+Structural parameters
+a (˚A)
+4.4061(2) 4.42799(1) 4.4177(2)
+4.4105(3)
+4.4045(2)
+4.3996(2)
+4.3879(3)
+4.3772(3)
+c (˚A)
+7.3011(4)
+7.4176(2)
+7.3300(3)
+7.2801(4)
+7.2402(4)
+7.2018(3)
+7.1594(5)
+7.1175(3)
+zAu
+0.78435(5) 0.77968(6) 0.78308(7) 0.78614(11) 0.78703(6) 0.78931(6) 0.79083(7) 0.79415(9)
+zGe
+0.2088(2)
+0.2114(3)
+0.2097(3)
+0.2088(6)
+0.2075(3)
+0.2069(3)
+0.2068(3)
+0.2071(5)
+Magnetic properties
+TC (K)
+-
+-
+-
+-
+-
+-
+4.1
+-
+TN (K)
+-
+17.0
+6.0
+4.4
+4.9
+3.2
+-
+-
+peff⊥c
+-
+7.9
+9.8
+10.6
+10.6
+9.5
+7.8
+-
+ΘW⊥c (K)
+-
+-11.6
+17.6
+9.0
+-2.6
+-31.8
+-72.3
+-
+peff∥c
+-
+8.3
+11.4
+10.4
+10.7
+9.5
+7.4
+-
+ΘW∥c (K)
+-
+-33.8
+-191
+-45.0
+-16.6
+33.0
+51.6
+-
+doublet is mainly composed of Jz = ±6, and the CEF
+splitting between the quasi-doublet and Jz = ±5 doublet
+provides Ising nature of the magnetic moment. At 5 K,
+dispersionless excitations are observed at around 13 meV,
+which is consistent with the calculation on the basis of
+Eqs. (3)-(5) giving the energy splitting of 11 meV (see
+supplementary materials). We note that the lowest quasi-
+doublet should have a tiny gap due to the non-Kramers
+nature of a Tm3+(4f12) ion [49, 50], if one considers the
+effect of CEF potentials ˆO±3
+n
+(n = 4, 6) in Eq. (1), which
+has remained unresolved in the present study.
+Established the magnetic anisotropy originating from
+the CEF in RAuGe, we focus on the magnetic transitions
+at low temperatures. Figures 6(a)-(l) show the suscepti-
+bility at µ0H = 0.1 T as well as specific heat and thermal
+expansion at zero field near the transition temperatures.
+Together with the structural parameters, we summarize
+the magnetic properties in Table I.
+Figures 6(a) and (b) compare the temperature depen-
+dences of physical properties of GdAuGe. All the mea-
+surements were performed by using a unique single crys-
+tal. There are anomalies in M/H for both H ∥ c and
+H ⊥ c (Fig.
+6(a)) indicating two successive magnetic
+transitions at TN = 16.9 K and Tmag = 10.0 K. In the spe-
+cific heat measurement (Fig. 6(b)), on the other hand, we
+observed two anomalies at TN = 17.0 K and THC = 14.8
+K, reproducing the previous study [25]. Note that the
+observed lattice-length change below TN can be mainly
+attributed to the exchange striction because the CEF
+anisotropy is absent [51]. No obvious anomaly at Tmag
+or THC potentially suggests that these transitions are re-
+lated with spin-orientation change keeping Si · Sj for lo-
+cal Gd spin moments (Si) unchanged. Aside from the
+clear antiferromagnetic transition at 17 K, we observed
+secondary anomalies at different temperatures, Tmag and
+THC, through different probes because each anomaly is
+too subtle to be seen through the other properties. It
+is a possible future study to develop the magnetic phase
+diagram and to identify the evolution of the magnetic
+structures at low temperatures.
+The magnetic phase transition of TbAuGe takes place
+at TN = 6 K as seen as a peak of M/H for H ⊥ c (Fig.
+6(c)) and a tiny shoulder-type anomaly in Cmag (Fig.
+6(d)). For H ∥ c, M/H shows a kink at TN and is well
+suppressed in contrast to that for H ⊥ c owing to the
+strong easy-plane type anisotropy.
+Thermal expansion
+shows an anomaly at TN. We note that the lowest CEF
+level for the non-Kramers Tb3+ ion is a non-magnetic sin-
+glet isolated from the lowest excited state. The energy
+scale of the separation is estimated to be 15 K on the basis
+of the simulation (see supplementary materials)). Mag-
+netic ordering in such a singlet-ground-state system is en-
+abled by induced-moment instability upon the exchange
+interaction exceeding the CEF gap [52, 53]. This is con-
+sistent with the suppressed Cmag peak at TN as observed
+in Pr-based compounds [54–56]. An additional anomaly
+associated with an incommensurate-commensurate tran-
+sition is not identified down to T = 1.8 K in our measure-
+ments. A diffraction study is to be reported in future.
+The antiferromagnetic transition in DyAuGe is clearer
+than that in TbAuGe as shown as a drop of M/H at
+TN = 4 K (Fig.
+6(e)).
+For H ⊥ c, the M-T curve
+shows a hysteresis at lower temperatures, suggesting that
+the in-plane magnetic field induces rearrangement of the
+magnetic domains breaking C6 symmetry of the crystal.
+This is consistent with a hysteresis also seen in the in-
+plane thermal expansion (Fig. 6(f)). The M/H shows
+a drop below TN for both H ∥ c and H ⊥ c, suggesting
+that the magnetic moments can have both in- and out-of-
+plane components. The temperature derivative dM/dT
+for H ⊥ c shows a peak clarifying TN (see the orange
+arrow in Fig.
+6(e)).
+The magnetic transition appears
+as a broad peak of specific heat (Fig.
+6(f)) while the
+successive nature is not clearly resolved.
+
+7
+FIG. 6.
+(a)-(l) Temperature dependence of magnetic and thermodynamic properties of RAuGe (R = Gd−Tm) at low
+temperatures. Red and blue curves represent M/H for H ∥ c and H ⊥ c at µ0H = 0.1 T, respectively. Black and gray
+curves represent Cmag and in-plane ∆L/L0 in zero field. In (e) and (g), temperature derivative of M (dM/dT) for H ⊥ c and
+H ∥ c, respectively, are shown. Closed (open) symbols and solid (dashed) lines are measured in temperature-decrease (increase)
+processes. In (k), demagnetization field is corrected by using the internal magnetic field, Hint = Hext − NdM. Black curve in
+(k) is the Curie-Weiss fit (χ∥c =
+Clow
+T −Θ∥c,low ) for 4 K < T < 30 K.
+The magnetic susceptibility of HoAuGe is isotropic
+(with a subtle easy-plane nature) at high temperatures
+while an easy-axis nature is enhanced at low tempera-
+tures (Fig. 6(g)). The antiferromagnetic transition can
+be seen as a weak drop of M/H at TN1 = 5 K and a
+tiny peak in Cmag (black arrow in Fig. 6(h)). The main
+peak in the specific heat is observed at TN2 = 4 K, which
+is accompanied by a thermal-expansion jump and a kink
+in M/H for H ∥ c. The latter is more clearly seen in
+dM/dT (orange arrow in Fig. 6(g)). These anomalies are
+not discernible in M/H for H ⊥ c, potentially indicating
+the predominant c-axis component of the spin moment
+in the magnetic ordered states.
+The successive tran-
+sitions support the previous neutron diffraction study,
+where a mixture of commensurate and incommensurate
+orders was identified below TN1 = 6 K, which eventually
+developed to the pure commensurate order [29–31]. We
+do not observe multi-step anomalies at 5.6, 4, or 2 K in
+the specific heat measurement in contrast to a previous
+study [31] potentially because of the difference in sample
+quality or annealing condition.
+ErAuGe shows an antiferromagnetic transition at
+TN = 3.2 K. M-T curves (Fig. 6(i)) are hysteretic for
+both H ∥ c and H ⊥ c at low temperatures, suggest-
+ing a glassy nature of the magnetic order. Large easy-
+axis type anisotropy indicates that the magnetic mo-
+ments are aligned along the c axis. The transition can be
+clearly identified as a specific-heat peak and a thermal-
+expansion kink (Fig.
+6(j)).
+This is in contrast to the
+successive transitions at 3.7 K and 2.8 K associated with
+
+TbAuGe
+DyAuGe
+a
+GdAuGe
+c
+e
+0.25
+0.04
+0.9
+6
+(emu/mol)
+MIH (emu/mol)
+MIH 
+ (emu/mol)
+oH = 0.1 1
+UoH= 0.1
+MoH = 0.1
+dM/dT (a.u.)
+0.20
+4
+HIC
+(emu/mol)
+HIc
+HIc
+0.03
+0.8
+2
+0.15
+Hc
+MIH 
+MIH
+Hic
+HIIc
+0
+0.10
+0.02
+0.7
+0
+2
+6
+10
+2
+3
+0
+5
+10
+15
+20
+0
+4
+8
+4
+5
+6
+b
+d
+f
+20
+10
+10
+1.5
+20
+30
+ALIL
+(J/molK)
+MoH=O T
+(>low/r)
+oH= O T
+(>low/r)
+MoH=oT
+15
+15
+20
+-10 K
+10
+5
+5
+10
+0.5
+10
+５
+5
+0.0
+0
+0
+0
+0
+0
+5
+10
+15
+20
+0
+2
+4
+6
+8
+10
+2
+3
+4
+5
+6
+1
+Temperature (K)
+Temperature (K)
+Temperature (K)
+HoAuGe
+i
+ErAuGe
+k
+TmAuGe
+g
+2
+2.5
+ (emu/mol)
+10°
+MIH (emu/mol)
+(emu/mol)
+dMIdT (a.u.)
+oH= 0.1 ↑
+MoH = 0.1 T
+2
+uoH= (
+HC
+10°
+Hc
+HIc
+2.0
+MIHint (
+HIW
+HIc
+HIc
+HIc
+x30
+0
+1.5
+2
+0
+10
+0
+2
+4
+6
+8
+2
+1
+3
+4
+0
+2
+4
+6
+8
+10
+h
+j
+15
+15
+15
+1.5
+8
+1.0
+△LIL
+△LIL8 k (
+LoH= O T
+(J/molK)
+MoH= O T
+(>low/r)
+MoH= o T
+10
+10
+1.0
+10
+0.5
+V
+(10
+5
+5
+b.5
+5
+0.0
+0
+0
+6.0
+0
+1
+0
+6
+8
+2
+4.
+2
+3
+4
+5
+0
+2
+4
+6
+8
+10
+1
+Temperature (K)
+Temperature (K)
+Temperature (K)8
+FIG. 7.
+R dependence of the transition temperatures.
+Blue (red) markers represent antiferromagnetic (ferromag-
+netic) transitions.
+Dashed line is the de Gennes factor
+(gJ − 1)2J(J + 1) normalized by the Gd value.
+the incommensurate-commensurate transition observed
+in previous neutron diffraction [29, 30].
+TmAuGe is the only ferromagnetic system among
+RAuGe for R = Gd−Tm, as identified in Ref. [26]. To see
+the divergence of magnetic susceptibility, we correct the
+demagnetization field for H ∥ c (red curve in Fig. 6(k)).
+Corrected magnetic susceptibility is obtained by M/Hint.
+We fit the M/Hint curve for H ∥ c at low temperatures
+to the Curie-Weiss law (χ∥c =
+Clow
+T −Θ∥c,low ) as shown by
+the black curve in Fig. 6(k). The Weiss temperature is
+Θ∥c,low = 4.1 K, consistent with TC. We obtain the effec-
+tive moment pIsing = 7.07µB by the formula for the Curie
+constant Clow = NAp2
+Ising/kB in the case of the Ising-type
+magnetic moments. The obtained pIsing agrees well with
+the expected value pIsing = gJJ = 7µB for the lowest
+quasi-doublet composed of Jz = ±6 well separated from
+the lowest excited state, which is consistent with the CEF
+analysis.
+Finally, we discuss the R dependence of magnetic tran-
+sition temperature summarized in Fig. 7. On the basis
+of the de Gennes scaling rule [57], the transition temper-
+atures among isostructural R compounds are supposed
+to be scaled with the de Gennes factor, (gJ −1)2J(J +1)
+(see the dashed curve in Fig. 7). This argument is valid
+as long as the CEF effect and anisotropy of exchange in-
+teraction are negligible and Heisenberg-type RKKY in-
+teraction (Jex(qmag)) stabilizes a magnetic order with
+the identical magnetic modulation vector qmag among
+RAuGe. We observe the breakdown of the de Gennes
+scaling as it overestimates TN for Tb and Dy and under-
+estimates TC for Tm. Indeed, one of the above condi-
+tions is violated as the Tm compound is ferromagnetic
+(qmag = (0, 0, 0)) in contrast that the antiferromagnetic
+order with qmag ∼ (0.4−0.5, 0, 0) in the reciprocal lattice
+unit is favored for the other compounds [27, 29, 30]. We
+have to consider the modification of Jex(qmag), which is
+potentially sensitive to subtle details of the band struc-
+ture in the present semimetallic materials. We also note
+that the suppression of TN for R = Tb is reasonable be-
+cause of the singlet ground state [58] while that of R =
+Dy is nontrivial. Theories predict that the CEF effect is
+basically supposed to enhance the transition temperature
+in the case of the easy-axis type anisotropy (B20 < 0, not
+valid for R = Dy) [59, 60], but the suppression of TN is
+possible when the easy-plane type anisotropy (B20 > 0)
+surpasses the exchange interaction [61].
+The observed
+thermal expansion (∆L/L ∼ 1.7 × 10−4) in DyAuGe is
+the largest in the present RAuGe family, which may re-
+flect the magnetoelastic strain originating from the mag-
+netic anisotropy rather than the exchange striction. We
+might also have to consider the effect of magnetic frus-
+tration in DyAuGe to potentially promote the compe-
+tition among noncollinear/noncoplanar magnetic struc-
+tures suppressing TN.
+In summary, we succeeded in growing single crystals
+of RAuGe for R = Y, Gd−Tm, and Lu and revealed
+the thermodynamic and magnetic properties. Magnetic
+anisotropy is systematically determined by the CEF ef-
+fect of R ions, and low-temperature transitions are dom-
+inated by the lowest CEF levels.
+Continuous struc-
+tural evolution of the polarity and its potential inter-
+play with nontrivial magnetism would be viewed as a
+promising platform for studying metallic multiferroics
+[62, 63]. Findings in the present study provide basic un-
+derstanding of single crystalline properties of a family
+of polar semimetal RAuGe. Transport properties inter-
+twinned with frustrated magnetism is a desired future
+study, which will be reported elsewhere.
+T.K. was financially supported by Ministry of Edu-
+cation Culture Sports Science and Technology (MEXT)
+Leading Initiative for Excellent Young Researchers (JP-
+MXS0320200135), Japan Society for the Promotion of
+Science (JSPS) KAKENHI Grant-in-Aid for Young Sci-
+entists B (No.
+21K13874).
+M.G. was supported by
+the JSPS KAKENHI Grant-in-Aid for Scientific Re-
+search (No.
+20J10988).
+S.K. was supported by JSPS
+KAKENHI Grant-in-Aid for Early-Career Scientists (No.
+22K14010).
+This work was partly supported by JSPS
+KAKENHI Grant-in-Aid for Scientific Research on Inno-
+vative Areas Quantum Liquid Crystals (No. JP19H05826
+and No. 19H01835). The neutron experiment at the Ma-
+terials and Life Science Experimental Facility of the J-
+PARC was performed under a user program (Proposal
+No. 2022A0045). This work was partly performed using
+the facilities of the Materials Design and Characteriza-
+tion Laboratory in the Institute for Solid State Physics,
+the University of Tokyo. S.K. thanks K. Adachi and D.
+Hashizume for the support of the in-house XRD experi-
+ments.
+
+20
+Transition Temperature (K)
+-(gJ-1)J(J+1)
+o Tc
+15
+TN
+10
+5
+0
+0
+Gd
+Tb
+Dy
+Ho
+Er
+Tm9
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+
diff --git a/otE0T4oBgHgl3EQf9AIJ/content/tmp_files/load_file.txt b/otE0T4oBgHgl3EQf9AIJ/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf,len=1101
+page_content='Single crystal growths and magnetic properties of hexagonal polar semimetals RAuGe  (R = Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Gd−Tm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' and Lu)  Takashi Kurumaji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 Masaki Gen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2 Shunsuke Kitou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2 Kazuhiko  Ikeuchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3 Mitsutaka Nakamura,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4 Akihiko Ikeda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5 and Taka-hisa Arima1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2  1Department of Advanced Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Kashiwa 277-8561,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan  2RIKEN Center for Emergent Matter Science (CEMS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Wako 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan  3Neutron Science and Technology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Comprehensive Research Organization  for Science and Society (CROSS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Tokai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Ibaraki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 319-1106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan  4J-PARC Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan Atomic Energy Agency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Tokai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Ibaraki 319-1195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan  5Department of Engineering Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' University of Electro-Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Chofu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Tokyo 182-8585,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Japan  (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2023)  We study structural and magnetic properties of rare-earth based semimetals RAuGe (R = Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Gd−Tm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' and Lu) using flux-grown single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These compounds belong to the noncentrosym-  metric polar space group P63mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We confirm the systematic structural evolution at room tempera-  ture as a function of ionic radius of rare earths to clarify the isopointal crossover between two polar  structures: three-dimensional LiGaGe-type and quasi-two-dimensional NdPtSb-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Magnetism  shows a characteristic anisotropy in reasonable agreement with the crystal electric field (CEF) the-  ory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' the easy-plane-type anisotropy for R = Tb and Dy turns into the Ising-type anisotropy for  R = Er and Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We evaluate the CEF parameters based on the Stevens operators to reasonably  reproduce the temperature dependence of magnetic susceptibilities and specific heat for RAuGe  (R = Tb−Tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The estimated energy scale of the Ising gap (∼ 11 meV) in TmAuGe is consistent  with an excitation observed in an inelastic neutron scattering experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These findings suggest an  opportunity for interplay between conduction electrons and nontrivial spin structures in the family  of magnetic polar semimetals RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  INTRODUCTION  Polar metals are defined as the conductive materials  with symmetry breaking allowing electric polarization,  which provide a possible route to ferroelectric metals  [1, 2] and to hyperferroelectricity [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These materials  are potentially applied to electric-field-switchable spin-  tronic devices [5, 6], and a metallic counterpart of multi-  ferroics [7–9] with topological band crossings induced by  magnetic correlation [10, 11] and/or spin textures [12–  15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Rare-earth-based semimetals RAuGe (R = Sc, Y,  La−Nd, Sm, Gd−Tm, and Lu) belong to the equiatomic  ternary RTX (T = transition metals, X = p-block ele-  ments) intermetallic phases [16] of the hexagonal polar  space group P63mc (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  186) [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Their crystal  structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 1(a), where the positions of  Au and Ge are shifted upward and downward, respec-  tively, along the c axis from the aristotype ZrBeSi struc-  ture (space group P63/mmc, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 194) to break the inver-  sion symmetry, which is in common with a prototypical  polar metal considered in theories [3, 7–11, 19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  CeAuGe and TmAuGe are known to show ferromag-  netism below TC = 10 K [23–25] and TC = 4 K [26], re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The compounds for other R (Nd, and Gd−Er)  are antiferromagnetic with a transition temperature TN  ranging between 6 K and 17 K [25, 27–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  A previ-  ous neutron study detected a commensurate modula-  tion vector ( 1  2, 0, 0) in the ground state for R = Tb−Er  [27, 29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Incommensurate modulations were also ob-  served to coexist at low temperatures or just below TN,  suggesting inherent magnetic frustration in triangular-  lattice layers of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The density of state has a pseudo-  gap nature near the Fermi level due to the overlap of  electron and hole bands, which is enabled by nomi-  nal valence configuration R3+Au+Ge4− [18, 33, 34] and  contributes to semimetallic transport properties of non-  mangetic compounds (R = Sc, Y, La, and Lu) [25, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Resistivities of magnetic compounds, RAuGe (R = Ce,  Nd, Gd, Ho, and Tm) [23, 26, 31, 32], are known to  show nonmonotonic temperature dependences indicating  a coupling with magnetic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These features suggest  an opportunity to encounter unconventional properties  of polar semimetals intertwinned with frustrated mag-  netism in magnetic RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The above-mentioned phys-  ical properties were identified using polycrystals, and sin-  gle crystal growths are only known for CeAuGe using  Czochlarski and floating-zone methods [27, 35], and for  epitaxial thin-film form of LaAuGe [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  In this study, we report the single-crystal growths and  physical properties of RAuGe (R = Y, Gd−Tm, and Lu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  We analyzed the crystal structure using x-ray radiation  to confirm the systematic evolution of polar structure  as a function of R-ionic radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  We measured specific  heat, magnetization, and thermal expansion to reveal the  anisotropic magnetism significantly affected by the crys-  tal electric field (CEF) as well as magnetic transitions  at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  We estimate the CEF potential  parameters to reproduce the magnetic susceptibility and  specific heat and obtain a systematic R dependence rea-  sonably agreed with the Stevens theory [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The esti-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='02794v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='mtrl-sci]  7 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (a) Crystal structure of RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' z(Au) and z(Ge) are  fractional coordinates along the c axis of Au and Ge atoms  at 2b sites ( 1  3, 2  3, z), which are shifted upward from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='75 and  downward from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='25, respectively, producing the polarity in  RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (b) Photograph of single crystals of LuAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Black  bar is 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (c)-(d) Lattice constants, a and c, and inter  and intralayer Au-Ge bond lengths in a unit cell at room  temperature as a function of the R3+ ionic radius for six-  coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The values for LaAuGe are excerpts from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  [33], and those for CeAuGe and ScAuGe are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  (e) Side views of two isopointal [18] structures, LiGaGe and  NdPtSb types, with distinct interlayer Au-Ge bonding (cyan  arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  mated Ising gap in TmAuGe is 11 meV, which is con-  sistent with the excitation at 13 meV observed in the  inelastic neutron scattering (INS) experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These re-  sults provide basic understanding on the structural and  magnetic properties of the family of polar semimetals  RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (a)-(b) Temperature dependence of magnetic suscep-  tibility (M/H) in H ⊥ c and the specific heat (Cp) of RAuGe  (R = Y and Lu) at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The inset of (b) plots Cp/T vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The dashed lines are linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  EXPERIMENTAL METHODS  Single crystals of RAuGe (R = Y, Gd−Tm, and Lu)  were grown using the Au-Ge self-flux method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The mo-  lar ratio of R:Au:Ge was 1:2:2 for R = Y, Tb−Tm, and  Lu and 1:3:3 for R = Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Starting materials were R in-  got (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9%), Au wire (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5%), and Ge chunks (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='999%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  They were loaded into an alumina crucible, which was  sealed in an evacuated quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The tube was heated  to 1100 ◦C and kept at this temperature for 24 hours be-  fore being slowly cooled to 800 ◦C in 200 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' After  being kept at 800 ◦C for 4 days, the tube was taken out  from the furnace and centrifuged to remove excess flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The obtained crystals were of platelet shape with typical  dimensions 3×3×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5 mm3 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The  atomic composition and the phase were checked by en-  ergy dispersive x-ray spectroscopy (EDS, JEOL model  JSM-6010LA) and a powder x-ray diffraction (Rigaku  SmartLab diffractometer) using Cu Kα radiation, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Single-crystal X-ray diffraction data for RAuGe  were collected on Rigaku AFC-8 diffractometer equipped  with a HyPix-6000 detector using Mo Kα radiation at  room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The intensities of Bragg reflections  were collected by CrysAlisPro program [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The crystal  structures were solved by SUPERFLIP [39] and refined  by Jana2006 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The details of the structural analy-  a  b  LuAuGe  z(Au)  z(Ge)  R  b  a  Sc  Gd  CeLa  Lu  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3  c  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  C  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4  A  A  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1  a  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  0 0000 0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1  d  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  (A)  Au-Ge  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2  inter  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1  R  3+  ionic radius （A)  e  intra  inter  LiGaGe type (ScAuGe)  NdPtSb type (CeAuGe)emu/mol)  RAuGe  20  a  MoH= 1 T  10  R=Y  HIc  0  G-  (10°  10  Lu  MIH (  20  1  0  100  200  300  b  80  MoH= O T  Lu  60  Y  (J/molK)  8  K  40  low/rw),  S  4  20  0  10  20  0  T (K)  0  0  100  200  300  Temperature (K)3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (a)-(b) Temperature dependence of magnetic part of  specific heat (Cmag) and that of entropy (Smag) of RAuGe (R  = Gd−Tm) in zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' LuAuGe data was used to subtract  the phonon background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  sis results are shown in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  At-  tempts to grow the P63mc phase for YbAuGe [17] were  unsuccessful and resulted in the orthorhombic phases,  consistent with α-, β- (space group Pnma, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 62) or  γ-YbAuGe (space group Imma, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 74) as reported in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Specific heat Cp was measured using a commercial sys-  tem (heat capacity option of a Quantum Design PPMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Magnetization was measured using a superconducting  quantum interference device magnetometer (Quantum  Design MPMS-XL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Thermal expansion was measured by  the fiber-Bragg-grating (FBG) technique using an optical  sensing instrument (Hyperion si155, LUNA) in a cryostat  equipped with a superconducting magnet (Oxford Spec-  tromag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Optical fibers were glued using epoxy (Sty-  cast1266) on a (001) surface of as-grown crystals to mea-  sure the elongation or compression along the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  INS experiments were performed on a time-of-flight  (TOF) spectrometer (4SEASONS) at the Materials and  Life Science Experimental Facility (MLF) of J-PARC in  Japan [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Polycrystalline TmAuGe sample of 5 g was  prepared by arc melting of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' A boule of TmAuGe  was annealed at 500 ◦C for 21 days in an evacuated quartz  tube, and sliced into pieces with thickness of 3 mm to re-  duce the absorption of neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The chopper frequency  was set to 200 Hz, and the incident energy of Ei = 41  meV was used to observe low-energy excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (a)-(f) Temperature dependence of the inverse mag-  netic susceptibility (H/M) of RAuGe (R = Gd−Tm) for  H ∥ c (red) and H ⊥ c (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Dashed lines are Curie-Weiss  fit to the data in T = 150-300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (g)-(h) R dependence of (g)  effective magnetic moment (peff), and (h) Weiss temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Dashed line in (a) is free ions values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  sample was set in a GM refrigerator to control the tem-  perature down to 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The aquired data were analyzed  with the UTSUSEMI software package [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  RESULTS  The crystal structures of RAuGe are systematically de-  pendent on the ionic radius of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Single-crystal x-ray  diffraction confirms the polar P63mc structure for all R,  consistent with the previous reports [17, 18, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Figures  1(c) and (d) display the R3+ ionic radius dependence  of the lattice constants and inter- and intralayer Au-Ge  bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Larger R ions lead to the expansion of the  unit-cell volume especially resulting in the elongation of  the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This tendency correlates with the weaken-  ing of puckered distortion of Au-Ge honeycomb layer,  where the out-of-plane (interlayer) Au-Ge bond becomes  longer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' ScAuGe has the smallest R radius  (Sc3+(VI):0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='745 ˚A) with the interlayer Au-Ge bond of  275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2 pm, while it elongates to 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8 pm in LaAuGe with  RAuGe  a  30  R = Gd, Tb, Dy  25  Ho, Er, Tm  (J/molK)  20  15  10  5  0  2  3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  56  2  3  4  5  6  2  1  10  100  Temperature (K)  30  b  (J/molK)  R = Gd, Tb, Dy  I0=H  25  Ho, Er, Tm  20  RIn8  15  10  RIn2  mag  5  S  0  2  3  456  2  3  4  56  2  1  10  100  Temperature (K)60  d  60  a  R= Gd  R= Ho  oH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T  1/x (mol/emu)  oH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  (mol/emu)  40  HIc  40  HIIc-  20  Hic  20  X/  HIc  0  0  0  100  200  300  0  100  200  300  b  60  60  e  R= Tb uoH= 1 T(lc)  (mol/emu)  1/x (mol/emu)  oH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  R= Er  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T (Ic)  40  40  HIIc  HIc  20  20  HIIc  Hc  0  0  100  200  300  100  200  300  0  0  60  f  60  C  1/x (mol/emu)  R= Dy  uoH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T  1/x (mol/emu)  R=Tm  uoH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T  40  40  HIC  Hc  HIIc  20  20  HIc  0  0  0  100  200  300  0  100  200  300  Temperature (K)  Temperature (K)  15  K  100  g  h  HIc  Temperature  HIIc  0  (/ar)  10  100  HIC  Peff  HIIc  200  Neiss  5  300  1  1  Gd  Tb  Dy  Ho Er  Tm  Gd  Tb  Dy  Ho Er  Tm4  the largest R-ion radius (La3+(VI):1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='032 ˚A) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The  materials for R = Y, Gd−Tm, and Lu smoothly interpo-  late the two extremes to verify the isopointal crossover  of the crystal-structure type between NdPtSb type and  LiGaGe type (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 1(e)), both of which belong to the  space group P63mc [18, 23, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In other words, as the  ionic R3+ becomes larger, Au-Ge networks evolve from  three-dimensional ZnO-type to staggered stacking of two-  dimensional puckered binary honeycomb layers, asymp-  totically towards hBN-type network in the centrosym-  metric ZrBeSi-type structure (space group P63/mmc,  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 194).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Next, we look through the physical properties of non-  magnetic YAuGe and LuAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Magnetic susceptibil-  ity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2(a)) shows diamagnetic nature for both com-  pounds at high temperatures, which is consistent with  the previous report [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The low-temperature rise is as-  cribable to magnetic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Specific heat Cp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  2(b)) is dominated by lattice contribution while Cp for  LuAuGe is larger than that of YAuGe in the entire tem-  perature range because of the lower Debye temperature  [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Small electronic contribution, γT, to Cp is deduced  by Cp/T-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='-T 2 plot (see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Densities of  states at Fermi energy per a formula unit for YAuGe and  LuAuGe are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='175 eV−1 (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4 mJ/mol K2) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='085  eV−1 (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2 mJ/mol K2), respectively, the order of  magnitude of which agrees with the previous study [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Using Cp of LuAuGe as the reference for the lattice  contribution, we obtain the magnetic specific heat (Cmag)  for R = Gd−Tm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The double peak structure  at T = 15 and 17 K for R = Gd reproduces the previ-  ous measurement [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The peaks below 10 K for R =  Dy−Tm represent magnetic transitions at low tempera-  tures while a Schottky peak around 50-60 K for R = Er  and Tm, for example, suggests the significant contribu-  tion of crystal electric field (CEF) excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' To gain  insight on the CEF energy level, we integrate Cmag/T  with respect to T to estimate the magnetic entropy Smag  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' For R = Gd, the most of Smag for free-spin  value, R ln 8, is released below the transition tempera-  tures owing to the Heisenberg nature of Gd spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The  release of Smag between 4 and 20 K for ErAuGe and  TmAuGe only approaches R ln 2, suggesting the contri-  bution of the lowest-lying (quasi-)doublet well isolated  from excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This feature reflects on the Ising-  type anisotropy of their magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  For R =  Tb−Ho, in contrast, the T-dependence of Smag does not  show any saturation behavior below 100 K, suggesting  overlapped contributions of CEF excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The systematic R-dependence of magnetic anisotropy  can be seen in the inverse magnetic susceptibility (H/M)  summarized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 4(a)-(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Magnetic properties for  R = Gd and Ho are roughly isotropic showing similar in-  and out-of-plane M/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' For R = Tb and Dy, easy-plane-  type anisotropy is observed, while ErAuGe and TmAuGe  host easy-axis-type anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' All the compounds show  a Curie-Weiss behavior at high temperatures, where the  Curie constants and the Weiss temperatures (ΘW) are  obtained by a linear fit of the H/M-T curve between 150  and 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The effective moment peff is evaluated from  the Curie constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The R dependences of peff and ΘW  are summarized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 4(g) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Except for R =  Tb, peff agrees well with the free-ion value (dashed curve  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 4(g)), suggesting that the energy scale of CEF  is lower than the thermal energy at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  ΘW for H ∥ c is larger than that for H ⊥ c in the case  of R = Er and Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The systematic anisotropy change is  associated with the CEF parameters [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  We discuss the effect of CEF to clarify the R depen-  dence of magnetic anisotropy in RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The 4f electrons  on R sites in RAuGe are affected by the CEF potential  (VCEF) for the site symmetry of C3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' VCEF can be ex-  pressed by the Stevens operators ( ˆOm  n ) as  VCEF = B20 ˆO0  2 + B40 ˆO0  4 + B60 ˆO0  6  +B43 ˆO3  4 + B63 ˆO3  6 + B66 ˆO6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  (1)  Here, we assume B43 = B63 = B66 = 0 for simplic-  ity of calculations, which is reasonable when the axial  anisotropy of magnetic properties mainly arises from B20,  B40, and B60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In such a case the |Jz = n⟩ state is also an  eigenstate of VCEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In the high temperature limit, the  ˆO0  2 = [3 ˆJ2  z − J(J + 1)] term dominates the anisotropy  of magnetic susceptibilities [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The difference be-  tween the inverse of magnetic susceptibility for H ∥ c and  H ⊥ c, χ−1  ∥c and χ−1  ⊥c, respectively, is approximated to be  a constant as  χ−1  ∥c − χ−1  ⊥c = 3  2  (2J − 1)(2J + 3)  5Cave  B20,  (2)  where J is the total angular momentum J = L ± S and  Cave is the Curie constant obtained by their average for  H ∥ c and H ⊥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Assuming the identical radial distri-  bution of 4f electron wave function for different R, B20  is predicted to be scaled with a Stevens factor αJ [37],  which smoothly evolves from R = Tb to Tm and changes  the sign through Ho to Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This feature of B20 agrees well  with the R-dependence of magnetic susceptibility captur-  ing the origin of the anisotropy switch in RAuGe, which  will be discussed more quantitatively later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  To evaluate the systematic evolution of the CEF of  R3+ ions, we simulate the temperature dependence of  χ∥c, χ⊥c, and Cmag on the basis of the formulae (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' [48]):  χ∥c = NA(gJµB)2  Z  ×  1  kBT  J  �  n=−J  J2  z e−En/kBT ,  (3)  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  (a)-(e) Temperature dependence of M/H for H ∥ c (red) and H ⊥ c (blue), and Cmag (black) of RAuGe (R = Tb−Tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Magnetic susceptibility of ErAuGe and TmAuGe for H ∥ c (red curves in (d) and (e)) is corrected by the demagnetization  factor, Nd, where M is divided by the internal magnetic field, Hint(= H − NdM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Cyan, orange, and gray curves are obtained  from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (3)-(5), respectively (see text) with optimized parameters, B20, B40, and B60 (see supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Inset  of (e) is a color map of intensity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' |Q| of the polycrystalline INS profile of TmAuGe with the incident neutron energy Ei =  41 meV at T = 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (f) R dependence of B20, B40, and B60 (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Dashed lines are Stevens factors, αJ, βJ, and γJ  [37], normalized by the values for R = Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Closed orange circles in (f) is obtained from the anisotropy of Curie-Weiss curves  using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  χ⊥c = 2NA(gJµB)2  Z  ×  J−1  �  n=−J  | ⟨n + 1|Jx|n⟩|2 e−En/kBT − e−En+1/kBT  En+1 − En  ,  (4)  Cmag =  NA  kBT 2  ��  1  Z  �  n  E2  ne−En/kBT  �  −  �  1  Z  �  n  Ene−En/kBT  �2�  � ,  (5)  where NA is Avogadro constant, kB is Boltzmann con-  stant, µB is Bohr magneton, gJ is Lande’s g factor, En  is the energy for a state, |Jz = n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Z = �  n e−En/kBT is  the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We summarize the results of sim-  ulation of magnetic susceptibilities and magnetic specific  heat in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 5(a)-(e), and the deduced Stevens parame-  ters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 5(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We note that all the Stevens parameters  systematically scale with the Stevens factors, αJ, βJ, and  γJ, confirming the validity of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We also  show the B20 values estimated from the formula of high-T  limit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (2) (closed orange circle) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 5(f), agreeing  well with the all-data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  To double check the CEF gap in TmAuGe, we perform  INS and observe the energy transfer as shown in the inset  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Crystal field around a Tm3+ ion in C3v  symmetry splits the J = 6 manifold of Tm3+ 4f12 levels  into five singlets and four doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The lowest quasi-  RAuGe  10  20  b  10  20  c  10  20  a  R= Dy  R= Ho  R= Tb  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (3)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (4)  15  15  15  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (5)→  MIH (emu/mol)  (emu/mol)  MIH (emu/mol)  0  10  10°  10  (J/molK)  (J/molK)  10  10  10  MIH  10  HIC  10  10  5  5  5  HiIc  2  2  2  0  0  10°  10°  10°  0  0  100  200  300  0  100  200  300  0  100  200  300  Temperature (K)  Temperature (K)  Temperature (K)  d  10  20  32101  e  10  f  R= Tm 30  R= Er  8  Intensity  0  neV)  20  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (2)  15  M/Hint (emu/mol)  2  E10  1  2  (10°  0  0  024(  B40  8  6  1  10°  10  IQI(A  10  5  5  5  (J/molK)  G-  J  (10°  0  5  0  2  2  0  10  0  10  10  100  0  200  300  0  100  200  300  Tb  Gd  Dy  Ho Er  Tm  Temperature (K)  Temperature (K)6  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Structural and magnetic parameters of RAuGe (R = Y, Gd−Tm, and Lu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' a, c: lattice constants, and zAu, zGe:  fractional coordinate for Au and Ge atoms along the c axis (see supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  TC: ferromagnetic transition  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' TN: antiferromagnetic transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' peff: effective magnetic moment, and ΘW: Weiss temperature for  H ∥ c and H ⊥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  R  Y  Gd  Tb  Dy  Ho  Er  Tm  Lu  Structural parameters  a (˚A)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4061(2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='42799(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4177(2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4105(3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4045(2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3996(2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3879(3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3772(3)  c (˚A)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3011(4)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4176(2)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3300(3)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2801(4)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2402(4)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2018(3)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1594(5)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1175(3)  zAu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='78435(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='77968(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='78308(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='78614(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='78703(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='78931(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='79083(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='79415(9)  zGe  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2088(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2114(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2097(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2088(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2075(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2069(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2068(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2071(5)  Magnetic properties  TC (K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 TN (K) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2 peff⊥c 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8 ΘW⊥c (K) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3 peff∥c 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4 ΘW∥c (K) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  191  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6 doublet is mainly composed of Jz = ±6, and the CEF  splitting between the quasi-doublet and Jz = ±5 doublet  provides Ising nature of the magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' At 5 K,  dispersionless excitations are observed at around 13 meV,  which is consistent with the calculation on the basis of  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (3)-(5) giving the energy splitting of 11 meV (see  supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We note that the lowest quasi-  doublet should have a tiny gap due to the non-Kramers  nature of a Tm3+(4f12) ion [49, 50], if one considers the  effect of CEF potentials ˆO±3  n  (n = 4, 6) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' (1), which  has remained unresolved in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Established the magnetic anisotropy originating from  the CEF in RAuGe, we focus on the magnetic transitions  at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Figures 6(a)-(l) show the suscepti-  bility at µ0H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T as well as specific heat and thermal  expansion at zero field near the transition temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Together with the structural parameters, we summarize  the magnetic properties in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Figures 6(a) and (b) compare the temperature depen-  dences of physical properties of GdAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' All the mea-  surements were performed by using a unique single crys-  tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' There are anomalies in M/H for both H ∥ c and  H ⊥ c (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(a)) indicating two successive magnetic  transitions at TN = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9 K and Tmag = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In the spe-  cific heat measurement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(b)), on the other hand, we  observed two anomalies at TN = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0 K and THC = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  K, reproducing the previous study [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Note that the  observed lattice-length change below TN can be mainly  attributed to the exchange striction because the CEF  anisotropy is absent [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' No obvious anomaly at Tmag  or THC potentially suggests that these transitions are re-  lated with spin-orientation change keeping Si · Sj for lo-  cal Gd spin moments (Si) unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Aside from the  clear antiferromagnetic transition at 17 K, we observed  secondary anomalies at different temperatures, Tmag and  THC, through different probes because each anomaly is  too subtle to be seen through the other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' It  is a possible future study to develop the magnetic phase  diagram and to identify the evolution of the magnetic  structures at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The magnetic phase transition of TbAuGe takes place  at TN = 6 K as seen as a peak of M/H for H ⊥ c (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(c)) and a tiny shoulder-type anomaly in Cmag (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' For H ∥ c, M/H shows a kink at TN and is well  suppressed in contrast to that for H ⊥ c owing to the  strong easy-plane type anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Thermal expansion  shows an anomaly at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We note that the lowest CEF  level for the non-Kramers Tb3+ ion is a non-magnetic sin-  glet isolated from the lowest excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The energy  scale of the separation is estimated to be 15 K on the basis  of the simulation (see supplementary materials)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Mag-  netic ordering in such a singlet-ground-state system is en-  abled by induced-moment instability upon the exchange  interaction exceeding the CEF gap [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This is con-  sistent with the suppressed Cmag peak at TN as observed  in Pr-based compounds [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' An additional anomaly  associated with an incommensurate-commensurate tran-  sition is not identified down to T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8 K in our measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' A diffraction study is to be reported in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The antiferromagnetic transition in DyAuGe is clearer  than that in TbAuGe as shown as a drop of M/H at  TN = 4 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  For H ⊥ c, the M-T curve  shows a hysteresis at lower temperatures, suggesting that  the in-plane magnetic field induces rearrangement of the  magnetic domains breaking C6 symmetry of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  This is consistent with a hysteresis also seen in the in-  plane thermal expansion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The M/H shows  a drop below TN for both H ∥ c and H ⊥ c, suggesting  that the magnetic moments can have both in- and out-of-  plane components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The temperature derivative dM/dT  for H ⊥ c shows a peak clarifying TN (see the orange  arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The magnetic transition appears  as a broad peak of specific heat (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(f)) while the  successive nature is not clearly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  (a)-(l) Temperature dependence of magnetic and thermodynamic properties of RAuGe (R = Gd−Tm) at low  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Red and blue curves represent M/H for H ∥ c and H ⊥ c at µ0H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Black and gray  curves represent Cmag and in-plane ∆L/L0 in zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In (e) and (g), temperature derivative of M (dM/dT) for H ⊥ c and  H ∥ c, respectively, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Closed (open) symbols and solid (dashed) lines are measured in temperature-decrease (increase)  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' In (k), demagnetization field is corrected by using the internal magnetic field, Hint = Hext − NdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Black curve in  (k) is the Curie-Weiss fit (χ∥c =  Clow  T −Θ∥c,low ) for 4 K < T < 30 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The magnetic susceptibility of HoAuGe is isotropic  (with a subtle easy-plane nature) at high temperatures  while an easy-axis nature is enhanced at low tempera-  tures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The antiferromagnetic transition can  be seen as a weak drop of M/H at TN1 = 5 K and a  tiny peak in Cmag (black arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The main  peak in the specific heat is observed at TN2 = 4 K, which  is accompanied by a thermal-expansion jump and a kink  in M/H for H ∥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The latter is more clearly seen in  dM/dT (orange arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' These anomalies are  not discernible in M/H for H ⊥ c, potentially indicating  the predominant c-axis component of the spin moment  in the magnetic ordered states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The successive tran-  sitions support the previous neutron diffraction study,  where a mixture of commensurate and incommensurate  orders was identified below TN1 = 6 K, which eventually  developed to the pure commensurate order [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We  do not observe multi-step anomalies at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='6, 4, or 2 K in  the specific heat measurement in contrast to a previous  study [31] potentially because of the difference in sample  quality or annealing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  ErAuGe shows an antiferromagnetic transition at  TN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' M-T curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(i)) are hysteretic for  both H ∥ c and H ⊥ c at low temperatures, suggest-  ing a glassy nature of the magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Large easy-  axis type anisotropy indicates that the magnetic mo-  ments are aligned along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The transition can be  clearly identified as a specific-heat peak and a thermal-  expansion kink (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  6(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  This is in contrast to the  successive transitions at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7 K and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8 K associated with  TbAuGe  DyAuGe  a  GdAuGe  c  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='9  6  (emu/mol)  MIH (emu/mol)  MIH  (emu/mol)  oH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 1  UoH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1  MoH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1  dM/dT (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='20  4  HIC  (emu/mol)  HIc  HIc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='15  Hc  MIH  MIH  Hic  HIIc  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7  0  2  6  10  2  3  0  5  10  15  20  0  4  8  4  5  6  b  d  f  20  10  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  20  30  ALIL  (J/molK)  MoH=O T  (>low/r)  oH= O T  (>low/r)  MoH=oT  15  15  20  10 K  10  5  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  10  ５  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  0  0  0  0  0  5  10  15  20  0  2  4  6  8  10  2  3  4  5  6  1  Temperature (K)  Temperature (K)  Temperature (K)  HoAuGe  i  ErAuGe  k  TmAuGe  g  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  (emu/mol)  10°  MIH (emu/mol)  (emu/mol)  dMIdT (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=')  oH= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 ↑  MoH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 T  2  uoH= (  HC  10°  Hc  HIc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  MIHint (  HIW  HIc  HIc  HIc  x30  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  2  0  10  0  2  4  6  8  2  1  3  4  0  2  4  6  8  10  h  j  15  15  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  △LIL  △LIL8 k (  LoH= O T  (J/molK)  MoH= O T  (>low/r)  MoH= o T  10  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  V  (10  5  5  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  0  0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='0  0  1  0  6  8  2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  2  3  4  5  0  2  4  6  8  10  1  Temperature (K)  Temperature (K)  Temperature (K)8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  R dependence of the transition temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Blue (red) markers represent antiferromagnetic (ferromag-  netic) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Dashed line is the de Gennes factor  (gJ − 1)2J(J + 1) normalized by the Gd value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  the incommensurate-commensurate transition observed  in previous neutron diffraction [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  TmAuGe is the only ferromagnetic system among  RAuGe for R = Gd−Tm, as identified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' To see  the divergence of magnetic susceptibility, we correct the  demagnetization field for H ∥ c (red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Corrected magnetic susceptibility is obtained by M/Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  We fit the M/Hint curve for H ∥ c at low temperatures  to the Curie-Weiss law (χ∥c =  Clow  T −Θ∥c,low ) as shown by  the black curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 6(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The Weiss temperature is  Θ∥c,low = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='1 K, consistent with TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We obtain the effec-  tive moment pIsing = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='07µB by the formula for the Curie  constant Clow = NAp2  Ising/kB in the case of the Ising-type  magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The obtained pIsing agrees well with  the expected value pIsing = gJJ = 7µB for the lowest  quasi-doublet composed of Jz = ±6 well separated from  the lowest excited state, which is consistent with the CEF  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Finally, we discuss the R dependence of magnetic tran-  sition temperature summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' On the basis  of the de Gennes scaling rule [57], the transition temper-  atures among isostructural R compounds are supposed  to be scaled with the de Gennes factor, (gJ −1)2J(J +1)  (see the dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This argument is valid  as long as the CEF effect and anisotropy of exchange in-  teraction are negligible and Heisenberg-type RKKY in-  teraction (Jex(qmag)) stabilizes a magnetic order with  the identical magnetic modulation vector qmag among  RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We observe the breakdown of the de Gennes  scaling as it overestimates TN for Tb and Dy and under-  estimates TC for Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Indeed, one of the above condi-  tions is violated as the Tm compound is ferromagnetic  (qmag = (0, 0, 0)) in contrast that the antiferromagnetic  order with qmag ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='5, 0, 0) in the reciprocal lattice  unit is favored for the other compounds [27, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We  have to consider the modification of Jex(qmag), which is  potentially sensitive to subtle details of the band struc-  ture in the present semimetallic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We also note  that the suppression of TN for R = Tb is reasonable be-  cause of the singlet ground state [58] while that of R =  Dy is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Theories predict that the CEF effect is  basically supposed to enhance the transition temperature  in the case of the easy-axis type anisotropy (B20 < 0, not  valid for R = Dy) [59, 60], but the suppression of TN is  possible when the easy-plane type anisotropy (B20 > 0)  surpasses the exchange interaction [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  The observed  thermal expansion (∆L/L ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='7 × 10−4) in DyAuGe is  the largest in the present RAuGe family, which may re-  flect the magnetoelastic strain originating from the mag-  netic anisotropy rather than the exchange striction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' We  might also have to consider the effect of magnetic frus-  tration in DyAuGe to potentially promote the compe-  tition among noncollinear/noncoplanar magnetic struc-  tures suppressing TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  In summary, we succeeded in growing single crystals  of RAuGe for R = Y, Gd−Tm, and Lu and revealed  the thermodynamic and magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Magnetic  anisotropy is systematically determined by the CEF ef-  fect of R ions, and low-temperature transitions are dom-  inated by the lowest CEF levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  Continuous struc-  tural evolution of the polarity and its potential inter-  play with nontrivial magnetism would be viewed as a  promising platform for studying metallic multiferroics  [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Findings in the present study provide basic un-  derstanding of single crystalline properties of a family  of polar semimetal RAuGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' Transport properties inter-  twinned with frustrated magnetism is a desired future  study, which will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' was financially supported by Ministry of Edu-  cation Culture Sports Science and Technology (MEXT)  Leading Initiative for Excellent Young Researchers (JP-  MXS0320200135), Japan Society for the Promotion of  Science (JSPS) KAKENHI Grant-in-Aid for Young Sci-  entists B (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  21K13874).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' was supported by  the JSPS KAKENHI Grant-in-Aid for Scientific Re-  search (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  20J10988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' was supported by JSPS  KAKENHI Grant-in-Aid for Early-Career Scientists (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  22K14010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='  This work was partly supported by JSPS  KAKENHI Grant-in-Aid for Scientific Research on Inno-  vative Areas Quantum Liquid Crystals (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' JP19H05826  and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 19H01835).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' The neutron experiment at the Ma-  terials and Life Science Experimental Facility of the J-  PARC was performed under a user program (Proposal  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' 2022A0045).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' This work was partly performed using  the facilities of the Materials Design and Characteriza-  tion Laboratory in the Institute for Solid State Physics,  the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQf9AIJ/content/2301.02794v1.pdf'}
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+MNRAS 000, 1–16 (2022)
+Preprint 12 January 2023
+Compiled using MNRAS LATEX style file v3.0
+On the functional form of the radial acceleration relation
+Harry Desmond1⋆, Deaglan J. Bartlett2,3† and Pedro G. Ferreira3
+1Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK
+2CNRS & Sorbonne Université, Institut d’Astrophysique de Paris (IAP), UMR 7095, 98 bis bd Arago, F-75014 Paris, France
+3Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
+12 January 2023
+ABSTRACT
+We apply a new method for learning equations from data—Exhaustive Symbolic Regression (ESR)—to late-type
+galaxy dynamics as encapsulated in the radial acceleration relation (RAR). Relating the centripetal acceleration due
+to baryons, gbar, to the total dynamical acceleration, gobs, the RAR has been claimed to manifest a new law of
+nature due to its regularity and tightness, in agreement with Modified Newtonian Dynamics (MOND). Fits to this
+relation have been restricted by prior expectations to particular functional forms, while ESR affords an exhaustive
+and nearly prior-free search through functional parameter space to identify the equations optimally trading accuracy
+with simplicity. Working with the SPARC data, we find the best functions typically satisfy gobs ∝ gbar at high gbar,
+although the coefficient of proportionality is not clearly unity and the deep-MOND limit gobs ∝ √gbar as gbar → 0 is
+little evident at all. By generating mock data according to MOND with or without the external field effect, we find
+that symbolic regression would not be expected to identify the generating function or reconstruct successfully the
+asymptotic slopes. We conclude that the limited dynamical range and significant uncertainties of the SPARC RAR
+preclude a definitive statement of its functional form, and hence that this data alone can neither demonstrate nor
+rule out law-like gravitational behaviour.
+Key words: galaxies: kinematics and dynamics – dark matter – methods: data analysis
+1 INTRODUCTION
+Kinematic measurements of galaxies relate their visible and
+dynamical masses, affording constraints on the distribution
+of dark matter and/or the behaviour of gravity. These mea-
+surements are simplest to perform for late-type galaxies sup-
+ported predominantly by rotation, as the enclosed dynami-
+cal mass may be calculated from the centripetal acceleration
+and the law of gravity. Such studies have revealed a strik-
+ing correlation between the enclosed baryonic and total dy-
+namical mass assuming Newtonian gravity, dubbed the mass
+discrepancy–acceleration (Sanders 1990; McGaugh 2004) or
+radial acceleration relation (RAR; Lelli et al. 2017). It has
+been claimed that the RAR indicates that at high acceler-
+ations the Newtonian dynamical mass follows the baryonic
+mass (indicating little dark matter and the validity of New-
+tonian mechanics), while as acceleration drops below a new
+constant of nature g0 ≈ 10−10 ms−2 the dynamical mass in-
+creasingly exceeds the baryonic mass in a regular way.
+One may attempt to understood these observations from ei-
+ther a dark matter or modified gravity perspective. In ΛCDM
+the difference between the dynamical and baryonic mass is
+due to the dark matter that makes up most of the mass of
+⋆ harry.desmond@port.ac.uk
+† deaglan.bartlett@physics.ox.ac.uk
+the galaxy. The RAR must therefore be explained by the rela-
+tive distributions of dark and visible mass established by the
+process of galaxy formation. Interactionless cold dark mat-
+ter is influenced only gravitationally by the baryonic mass
+so the emergence of the RAR must be somewhat fortuitous;
+it is not established directly by a baryon–dark matter cou-
+pling (although see Blanchet & Le Tiec 2008; Berezhiani &
+Khoury 2015; Famaey et al. 2018 for alternative ideas). In
+contrast, the modified gravity (or modified inertia) interpre-
+tation posits a breakdown of Newtonian mechanics at low
+acceleration so that the dynamical mass inferred by a New-
+tonian analysis is not the true dynamical mass of the galaxy.
+The prototypical instantiation of this idea is Modified New-
+tonian Dynamics (MOND; Milgrom 1983a,c,b), in which the
+kinematic acceleration gobs follows the square root of the
+Newtonian acceleration gbar in the weak-field regime. This
+enables the total dynamical mass of the galaxy to remain
+equal to the baryonic mass across galaxies’ rotation curves,
+eliminating the need for dark matter in them. MOND has cos-
+mologically viable relativistic extensions (most recently Sko-
+rdis & Złośnik 2021), and is reviewed in Famaey & McGaugh
+(2012) and Banik & Zhao (2022).
+Central to the dark matter–modified gravity debate in the
+context of galaxy dynamics is the functional form of the
+RAR. This is because MOND makes a very specific predic-
+tion (absent the external field effect: gobs = gbar in the high-
+© 2022 The Authors
+arXiv:2301.04368v1  [astro-ph.GA]  11 Jan 2023
+
+2
+Desmond, Bartlett & Ferreira
+acceleration “Newtonian regime” and gobs ∝ g1/2
+bar in the low-
+acceleration “deep-MOND regime”) while dark matter could
+accommodate a range of possibilities depending on the effect
+of galaxy formation on halo density profiles, which remains
+highly uncertain (e.g. Duffy et al. 2010; Macciò et al. 2012;
+Grudić et al. 2020; Tenneti et al. 2018; Ludlow et al. 2017;
+Navarro et al. 2017; Keller & Wadsley 2017). The only po-
+tentially unambiguous prediction is that the RAR tends to
+gobs = Ωm/Ωb gbar at radii sufficiently large to encompass the
+cosmic baryon fraction, but it is unclear where or even if this
+occurs in galaxies. Thus, while the ΛCDM prediction for the
+full RAR can be tested only by applying potentially restric-
+tive priors on galaxy formation effects (Di Cintio & Lelli 2016;
+Desmond 2017; Paranjape & Sheth 2021), a more direct route
+towards informing the dark matter–modified gravity debate
+is to test the MOND prediction, specifically the limiting be-
+haviour at g ≪ g0 and g ≫ g0, the small intrinsic scatter and
+the lack of residual correlations.
+Here we focus on the asymptotic behaviour. This can be
+assessed to some extent by fitting a functional form with
+free power-law slopes at both ends (Lelli et al. 2017), but
+this assumes that the slope tends to a constant at each end
+and restricts to a specific part of the functional parameter
+space for which this is the case. These are in question when
+assessing the accuracy of the MOND prescription. A fully
+satisfactory fit should therefore make no such assumptions,
+eliminating potential confirmation bias and testing without
+any priors the assertion that the RAR implies no dynami-
+cally relevant dark matter at high g and the deep-MOND
+limit at low g. We accomplish this here by means of a novel
+regression algorithm dubbed Exhaustive Symbolic Regression
+(ESR; Bartlett et al. 2022b), and hence assess the degree to
+which the RAR supports the tenets of MOND. Within the
+MOND paradigm, this method also enables optimisation of
+the “interpolating function” (IF) gobs = F(gbar) between the
+two stipulated limits.
+The structure of the paper is as follows. In Sec. 2 we de-
+scribe the RAR data that we use, and in Sec. 3 our algo-
+rithm for generating functions and assessing their aptitude
+for describing the data. Sec. 4 presents the results. In Sec. 5
+we discuss the broader ramifications, potential remaining un-
+certainties and ways in which the programme could be fur-
+thered in the future. Sec. 6 concludes. Full details on ESR
+are given in Bartlett et al. (2022b). Units not explicitly given
+are 10−10 ms−2, and all logarithms are natural.
+2 OBSERVATIONAL DATA
+We use the SPARC data set (Lelli et al. 2016),1 a compila-
+tion of 175 rotation curves from the literature combined with
+Spitzer 3.6µm photometry. We apply the same quality cuts as
+the RAR study of Lelli et al. (2017), removing galaxies with
+quality flag 3 (indicating large asymmetries, non-circular mo-
+tions and/or offsets between stellar and HI distributions) and
+those with inclinations i < 30 deg, and points for which the
+quoted fractional uncertainty on the observed rotation veloc-
+ity is greater than 10 per cent. This leaves 2,696 points from
+147 galaxies.
+1 http://astroweb.cwru.edu/SPARC/
+3 METHOD
+We describe our method for generating and assessing trial
+functions in Sec. 3.1, and our likelihood function in Sec. 3.2.
+In Sec. 3.3 we outline our criteria for assessing whether a
+function displays MOND-like behaviour.
+3.1 Exhaustive Symbolic Regression
+While algorithms for symbolic regression (SR)—the search
+for good functional descriptions of a dataset—are becoming
+mature, they remain fallible (La Cava et al. 2021). Unless
+the generating function of the data is known at the outset
+(in which case SR is not required), it is not possible to de-
+termine whether any SR algorithm has uncovered the best
+function. This motivated us to develop “Exhaustive Symbolic
+Regression” (ESR) which, given a set of basis functions, pro-
+duces and evaluates every possible function up to a given
+complexity of equation, defined here as the number of nodes
+in its tree representation. This enables a brute-force solu-
+tion to relatively simple problems and provides a touchstone
+for assessing the results of stochastic algorithms at higher
+complexity. ESR, presented in full in Bartlett et al. (2022b),
+has two main steps: i) generating, and optimising the pa-
+rameters of, all functions up to a given complexity, and ii)
+ranking these functions using an information-theoretic metric
+combining accuracy and simplicity.
+For part i, the steps are:
+(1) Generate all possible trees containing a given number of
+nodes (equal to the complexity of functions considered).
+(2) Generate the complete set of such functions by decorat-
+ing these trees with all permutations of the operators
+from the operator list specified in advance, utilising the
+constraints on the arity of the operator that can occupy
+a given node.
+(3) Simplify the functions and remove duplicates. Variants of
+the same function (e.g. x(x+θ0) and x2+θ0x) are however
+retained as these may have different model complexities
+(used in step ii, below). For each unique function the
+variant is retained that minimises this.
+(4) Determine the values of the free parameters appearing
+in the functions that maximise the likelihood of the data
+(see Sec. 3.2).
+(5) Repeat for all complexities under consideration.
+The only degrees of freedom in this procedure are the max-
+imum complexity considered (here set at 9 as higher complex-
+ity is computationally prohibitive) and the set of operators
+of which the functions are composed. Here we choose:2
+• Nullary: gbar, θ
+• Unary: exp, sqrt, square, inv
+• Binary: +, −, ∗, /, pow
+where θ is a free parameter. We implicitly take the absolute
+value of the argument of any square root or power.
+The result of this procedure is a list of all functions up
+to the maximum complexity (of which there are 2.24×107),
+2 Many of these operators can manifestly be constructed from
+combinations of others. We include these repetitions to simplify
+the function trees, allowing a greater range of expressions up to
+the maximum complexity.
+MNRAS 000, 1–16 (2022)
+
+On the functional form of the radial acceleration relation
+3
+along with the parameter values that maximise the likelihood
+of the RAR data. As in regular regression, using the maxi-
+mum likelihood as the model selection criterion would favour
+overfitting, whereby a function fits the data near-perfectly
+but generalises or extrapolates poorly. To remedy this, SR
+typically uses two-objective optimisation, where the second
+objective is the “simplicity” of the function. In the absence of
+a metric for trading accuracy (the first objective) with com-
+plexity, optimal functions form a “Pareto front” where ac-
+curacy cannot be increased without reducing simplicity and
+vice versa. Simplicity has been defined analogously to model
+complexity (the number of nodes in the tree representation;
+e.g. in PySR; Cranmer et al. 2020), among others, but such
+definitions are typically arbitrary and thus compromise the
+objectivity of the regression results.
+To remedy this, part ii of ESR implements the minimum
+description length principle (MDL; Rissanen 1978; Grünwald
+& Roos 2019; Grunwald 2007) as a model selection criterion,
+which has an information-theoretic motivation and provides a
+natural framework for making commensurable the two objec-
+tives. MDL states that functions are preferred to the extent
+that they compress the data, i.e. minimise the number of bits
+required to communicate the data with the aid of the func-
+tion. We implement this with a two-step code in which the
+description length (also called codelength) is comprised of a
+component describing the function and a component describ-
+ing the residuals of the data around the function’s expecta-
+tion. We use the Shannon–Fano coding scheme for the latter
+(Cover & Thomas 1991), and for the former include contri-
+butions both from the structure of the function (penalising
+those employing more operators) and from the free parame-
+ters (penalising more parameters, especially ones that must
+be specified to high precision to achieve a high likelihood).
+The overall codelength of the compressed data, L(D), is de-
+rived in sec. 3 of Bartlett et al. (2022b):
+L(D) = L(D|H) + L(H)
+= − log( ˆL) + k log(n) − p
+2 log(3)
++
+p
+�
+i
+�1
+2 log(ˆIii) + log(|ˆθi|)
+�
++
+�
+j
+log(cj),
+(1)
+where L is the description length, D the dataset, H the hy-
+pothesis (i.e. function in question), L the likelihood, θ a free
+parameter of the function, k the number of nodes in the func-
+tion’s tree representation, n the number of unique operators
+involved, p the total number of free parameters, I the Fisher
+information matrix of the parameters and cj any constant
+natural numbers generated by simplifications. A hat denotes
+evaluation at the maximum-likelihood point. With all loga-
+rithms natural, this is the number of nats required to commu-
+nicate the data with the aid of the function. L(D) supports
+a probabilistic interpretation over function space that gener-
+alises the likelihood: the relative probability of a function is
+exp(−L(D)) (Grunwald 2007).
+The structure of the function alone determines the k log(n)
+term, but the remaining terms require the free parameters to
+be numerically optimised to maximise the likelihood (which
+we use interchangeably with minimising the loss).3 We now
+describe our choice of likelihood for the RAR data.
+3 The p log(3)/2 term is only affected by the numerical optimisa-
+3.2 Loss function
+As is typical (e.g. Lelli et al. 2017), we assume that gbar, gobs
+and their uncertainties are uncorrelated across the dataset.
+We further assume that the true gbar and gobs values, de-
+noted gt
+bar and gt
+obs, generate the observed values with lognor-
+mal probability distributions centred at the true values with
+widths given by their uncertainties δgbar and δgobs. Following
+Lelli et al. (2017), we fix the mass-to-light ratios Υgas = 1.33,
+Υdisk = 0.5 and Υbulge = 0.7 and assign them 10, 25 and 25
+per cent uncertainties respectively, summing these in quadra-
+ture to estimate δVbar and hence δgbar (assuming no uncer-
+tainty in radial position). We likewise assume the uncertain-
+ties on distance D and inclination i to be statistical and hence
+sum their contributions in quadrature with the quoted sta-
+tistical uncertainty on Vobs according to Lelli et al. (2017, eq.
+2) to estimate δgbar.
+The likelihood of an observation given the function in ques-
+tion, f(gt
+bar), is then:
+L(log(gobs)) =
+� ∞
+−∞
+L(log(gobs)| log(gt
+bar)) L(log(gt
+bar)) d log(gt
+bar)
+=
+1
+2π δ log(gbar) δ log(gobs)
+� ∞
+−∞
+exp
+�
+−
+�
+log(gobs) − log(f(gt
+bar))
+�2
+2 δ log(gobs)2
+�
+× exp
+�
+−
+�
+log(gt
+bar) − log(gbar)
+�2
+2 δ log(gbar)2
+�
+d log(gt
+bar)
+≈
+1
+�
+2πσ2
+tot
+exp
+�
+−(log(gobs) − log(f(gbar)))2
+2σ2
+tot
+�
+,
+(2)
+where
+σ2
+tot ≡ δ log(gobs)2 +
+�d log(f(gbar))
+d log(gbar)
+�2
+δ log(gbar)2.
+(3)
+This is derived by keeping the leading-order term in the Tay-
+lor expansion of log(f(gt
+bar)) around log(f(gbar)) and there-
+fore assumes this to be small relative to the rate of change
+of f. An advantage of working in log(gbar) − log(gobs) space
+as opposed to gbar − gobs is that, the RAR being roughly a
+product of power-laws, this minimises the error due to the
+first-order approximation. We find this to be good for all of
+the best functions. The likelihood is then the product over
+all data points. We discuss in Sec. A the limitations of this
+likelihood model and how it could be improved.
+3.3 Assessing MOND
+The core of the MOND paradigm is that gobs = gbar at g ≫ g0
+and gobs = √gbarg0 at g ≪ g0 (Milgrom 1983a,c,b). This
+implies
+s ≡ d log(gobs)
+d log(gbar) =
+�
+1,
+gbar → ∞
+1/2,
+gbar → 0.
+(4)
+Common choices for the IF covering the intermediate region
+g ≈ g0 include
+tion if any parameters are set to 0 due to their maximum-likelihood
+values being less than 1 precision unit from 0 (see sec. 3 of Bartlett
+et al. 2022b).
+MNRAS 000, 1–16 (2022)
+
+4
+Desmond, Bartlett & Ferreira
+• “Simple”: gobs = gbar/2+
+�
+g2
+bar/4 + gbarg0 (Famaey & Bin-
+ney 2005),
+• “Standard”: gobs =
+1
+√
+2
+�
+g2
+bar +
+�
+g2
+bar(g2
+bar + 4g2
+0) (Mil-
+grom 1983c),
+• “RAR”: gobs = gbar/(1 − exp(−
+�
+gbar/g0)) (Lelli et al.
+2017).
+Fig. 1 plots these functions on top of the RAR data for the
+best-fit values on the SPARC data (shown in the lower rows
+of Table 1 in Sec. 4.1). The Simple and RAR IFs are distin-
+guished from the Standard IF principally by a more gradual
+transition between the Newtonian and deep-MOND regimes,
+although the Standard IF also prefers a significantly higher
+value of g0. While the basic MOND framework is not commit-
+ted to any particular IF, it is committed to Eq. 4 providing an
+optimal description of the data. Our assessment of the theory
+will therefore be based on the extent to which the best func-
+tions (those with lowest description length) conform to these
+limits: any function that does so, in addition to possessing a
+coefficient of proportionality of unity in gobs ∝ gbar at high
+gbar, may be considered a new MOND IF. (The low-gbar co-
+efficient of proportionality is √g0 but g0 is unknown a priori,
+so this does not supply an additional requirement.) Following
+Lelli et al. (2017), we will also consider a double power law
+fit:
+gobs = θ1
+�
+1 + gbar
+θ0
+�θ2−θ3 �gbar
+θ0
+�θ3
+(5)
+which has limiting logarithmic slopes of θ3 and θ2, and plot
+the best-fit in Fig. 1. We define s− ≡ limgbar→0 s and s+ ≡
+limgbar→∞ s.
+The MOND interpretation of the RAR is complicated by
+the possibility of the external field effect (EFE), a break-
+down of the strong equivalence principle due to the nonlin-
+ear, acceleration-based modification to Newtonian mechan-
+ics (Milgrom 1983a). The EFE implies that otherwise iden-
+tical galaxies in different external gravitational fields have
+different dynamics, which is a function of the external field
+strength gex relative to g0 and the internal field gin. In the
+quasi-Newtonian regime gin < gex < g0, Kepler’s laws are re-
+covered with dynamical masses scaled by gex/g0, while in the
+external field-dominated regime gin < g0 < gex, Newtonian
+mechanics are fully recovered (Famaey & McGaugh 2012).
+This steepens the RAR at low gbar.
+The precise effect of the EFE is difficult to calculate in gen-
+eral because it depends both on the underlying MOND theory
+and on a galaxy’s morphology and orientation with respect
+to the external field direction. The most sophisticated fitting
+functions to MOND simulations are currently to be found
+in Zonoozi et al. (2021) for QUMOND (Milgrom 2010) and
+Chae & Milgrom (2022) for AQUAL (Bekenstein & Milgrom
+1984). Chae et al. (2022) tested these expectations by fit-
+ting the SPARC rotation curves for the average external field
+strength, finding this to be in good agreement with indepen-
+dent estimates based on the baryonic mass surrounding the
+SPARC galaxies (Chae et al. 2021) for AQUAL, but less so
+for QUMOND. We will therefore use AQUAL to explore the
+effect of the EFE on the expected low-gbar slope and func-
+tional form more generally. Chae & Milgrom (2022) eq. 15
+gives
+Figure 1. The Simple, Standard and RAR IFs, a double power
+law, and Simple IF with a global external field strength in AQUAL,
+overlaid on the SPARC data (blue points). The parameters are set
+to their maximum-likelihood values shown in Table 1. The dashed
+black line shows the one-to-one relation (Newtonian limit) and the
+cross in the lower right shows the average uncertainty size.
+gobs = gbar
+�
+�
+� 1
+2 +
+�
+� 1
+4 +
+�� gbar
+g0
+�2
++ (1.1eN)2
+�− 1
+2
+�
+�
+1
+2 �
+�
+� ×
+�
+1 + tanh
+� 1.1eN
+gbar/g0
+�1.2
+×
+�
+− 1
+3
+�
+×
+(6)
+���
+gbar
+g0
+�2
++ (1.1eN)2
+�− 1
+2
+� �
+1
+4 +
+��
+gbar
+g0
+�2
++ (1.1eN)2
+�− 1
+2
+�− 1
+2
+1 +
+�
+1
+2 + 2
+��
+gbar
+g0
+�2
++ (1.1eN)2
+�− 1
+2
+� 1
+2
+�
+.
+This allows for variable disk thickness and scale length, and is
+azimuthally averaged to reduce sensitivity to the orientation
+of the field relative to the disk axis. It recovers the Simple IF
+as eN ≡ gex/g0 → 0 and hence we refer to it as “Simple IF +
+EFE”.
+Note that, while some form of the EFE is generically pre-
+dicted by MOND, in modified inertia formulations it may
+be very different (e.g. a function of the entire past trajec-
+tory of an object) or negligible (Milgrom 2011). While there
+is evidence for the EFE in many systems (McGaugh & Mil-
+grom 2013; Haghi et al. 2019; Chae et al. 2020b), in others
+it appears conspicuously absent (Hernandez et al. 2019; Fre-
+undlich et al. 2022). The black curve in Fig. 1 shows the best
+fit to the data using Eq. 6.
+3.4 Mock data generation
+To shed light on the significance of our results we apply
+ESR also to two stacks of mock data sets. We generate each
+mock data set using exactly the same number of points as
+the SPARC data, and with identical log gbar, δ log gbar and
+δ log gobs values, but with log gobs generated using a MON-
+Dian function. This assumes gt
+bar equal to the SPARC gbar
+(the maximum a priori estimate), log gbar for each mock re-
+alisation drawn from N(log gt
+bar, δ log gbar), and log gobs from
+MNRAS 000, 1–16 (2022)
+
+102
+Simple IF
+Standard IF
+RAR IF
+Double power law
+101
+Simple IF + EFE
+100
+10-1
++
+10-3
+10-2
+10-1
+100
+101
+102
+9bar / 10-10 ms-2On the functional form of the radial acceleration relation
+5
+N(F(log gt
+bar), δ log gobs). To reduce the impact of noise in
+the mock data we apply ESR to a stack of 10 independent
+realisations.4 The only terms in the description length that
+depend on the dataset size are log(L) and the Fisher matrix
+I, both of which scale linearly. To make the results compat-
+ible with the real data we therefore divide these terms by
+10.
+The two mock data set stacks differ in the generating func-
+tion F. For the first, we use the RAR IF with the best-fit
+value on the data g0 = 1.127 (see Sec. 4). This function is
+already known to describe the RAR well (Lelli et al. 2017)
+and has low enough complexity to be included (as a special
+case of a more general function, see below) in our function
+list. Since Eq. 4 is satisfied by construction in this case, eval-
+uating it on the best functions from ESR will address the
+question of whether the dynamic range of the data is suffi-
+ciently high—and the uncertainties sufficiently low—to pick
+out unambiguously a correctly MONDian solution, as only in
+this case could one expect to obtain such behaviour for the
+real data were it generated by MOND.
+The second stack is created using Eq. 6. We adopt g0 = 1.2
+and ⟨gex⟩ = 1.2 × 10−2 (eN = 0.01), corresponding roughly
+to maximal clustering of unobserved baryons (as expected
+in a MOND cosmology and maximising agreement with the
+rotation curve fits; Chae et al. 2021) and hence providing
+an upper bound on the impact of the EFE. This is similar
+to the value inferred in Chae (2022) and Chae et al. (2022)
+from fits to the SPARC rotation curves, and from our fit to
+the SPARC data in Table 1.5
+4 RESULTS
+4.1 SPARC data
+We show in Table 1 the statistics of the best functions found
+by ESR on the SPARC data. We split the codelength of
+Eq. 1 into terms describing the residuals of the data around
+the functional expectation, the functional form and the pa-
+rameter values as shown in the table footnotes. Below the
+horizontal line we give the results of the three MOND IFs,
+for which the free parameter corresponds to g0, the dou-
+ble power law (Eq. 5) and the Simple IF + EFE (Eq. 6).
+P(f) ≡ exp(−L(D))/ �(exp(−L(D))) is the probability of
+4 The reason not to do more is that the time required for param-
+eter optimisation scales with the number of data points, and this
+is already expensive at complexity 9. We assess convergence by
+fitting the complexity 4-6 equations to an independent stack of 10
+realisations. We find that the order of equations when sorted by
+L(D) is identical between the two stacks and that the difference
+between the description lengths of particular equations falls with
+complexity, from ∼ 40 at complexity 4 to ∼ 5 at complexity 6.
+This implies that the best equations, mostly at complexity 8 and
+9, are not sensitive to the random number generation.
+5 Chae & Milgrom (2022) argue that Eq. 6 is only reliable with
+the inner points of galaxies’ rotation curves removed. As we are in-
+terested only in the approximate effect of the EFE on the low-gbar
+slope of the RAR, mainly sourced by outer rotation curve points,
+we do not apply a cut. The somewhat larger value eN ≈ 0.017
+obtained when all points are fitted (Chae 2022, Fig. 7) would have
+a greater effect on the low-gbar slope, but is in clear disagreement
+with the large-scale structure calculations of Chae et al. (2021).
+Figure 2. The top 20 functions found by ESR overlaid on the
+SPARC data (blue points), colour-coded by their relative proba-
+bility in the full function list. The top panel fits the SPARC data,
+the middle panel mock data generated by the RAR IF, and the
+bottom panel mock data generated by the Simple IF with uni-
+versal external field strength gex = 1.2 × 10−12 ms−2. The mock
+datasets are 10 times larger than SPARC, although this is factored
+out in the description length calculation.
+the function given its description length, where the sum is
+over all functions up to complexity 9. (Note that these values
+would be changed by low-L(D) functions at higher complex-
+ity.) For reference, L(D) for the raw data (corresponding to
+the hypothesis log gobs = 0) is 53471, showing that significant
+compression is possible.
+We find the best-fit g0 value for the RAR IF to be 1.13,
+somewhat lower than the 1.20 quoted by Lelli et al. (2017)
+although the data are the same. This is because Lelli et al.
+(2017) used scipy.odr to perform the optimisation rather
+than using the full first-order likelihood Eqs. 2-3, and also
+did the analysis in the log(gobs)−gbar rather than log(gbar)−
+log(gobs) plane (F. Lelli private communication). The double
+power law fit has much higher maximum likelihood than the
+MOND IFs, outweighing its increased codelength due to its
+four free parameters. Although the RAR IF has complexity
+9 it is not explicitly produced by ESR due to the constant
+MNRAS 000, 1–16 (2022)
+
+SPARC data
+103.
+log(Prel)
+-10
+101.
+-15
+10-1
+-20
+RAR IE mock
+103
+-2
+log(Prel)
+101.
+4
+10-1
+-6
+Simple IF + EFE mock
+103
+-2
+-4
+log(Prel)
+101.
+-6
+-8
+10-1
+-10
+10-3
+10-1
+101
+103
+bar /10-10ms-26
+Desmond, Bartlett & Ferreira
+Figure 3. The logarithmic slopes s ≡ d log(gobs)
+d log(gbar) of the top 10 ESR functions on each dataset, for comparison with the low- and high-gbar
+MONDian expectations 1/2 and 1 respectively (blue and red vertical dashed lines). The blue and red points are the limiting slopes
+s− ≡ limgbar→0+ s and s+ ≡ limgbar→∞ s, while cyan and magenta indicate the slopes at the minimum and maximum gbar of the SPARC
+data (0.0083 and 65.4). In case a slope depends on a parameter value we show the 95% confidence interval as a bar (often very thin),
+obtained from an MCMC fit. Arrowheads indicate points or bars beyond the range of the plot.
+Figure 4. The Pareto fronts identified by ESR for the SPARC, RAR IF mock and Simple IF + EFE mock datasets, for both log(L)
+(blue) and total description length L(D) (red). The quantities plotted have the minimum values subtracted so that the best results appear
+at 0. Also shown are the results of the RAR, Simple and Standard IFs, Simple IF + EFE and double power law fits. ESR significantly
+outperforms these “by eye” guesses, even for mock data generated from them. Short diagonal lines on the x-axis indicate breaks. In the
+left and right panels both red and blue points for the Standard IF at complexity 14 lie above the top of the plot.
+“1” appearing,6 a generalised form in which this is replaced
+by a free parameter appears at rank 17, with a probability
+4×1010 times lower than the top-ranked function (∆L(D) =
+24.5). When θ0 ̸= 0 the low-gbar logarithmic slope s− of this
+function is 1 rather than 1/2, so it does not function as a
+MOND IF. We refer to it as the “generalised RAR IF”.
+The best ESR functions are clearly superior to the MOND
+6 This will be changed in a future version of ESR that performs
+“integer snap” of parameters where this reduces L(D).
+functions or double power law. While the best metric for this
+is L(D) (or equivalently P(f)), other statistics lead to the
+same conclusion. There are many functions more accurate
+(lower − log(L)) than even the double power law. Although
+the functions at rank 1-9 have more free parameters than
+the IFs this is more than compensated for by their greater
+accuracy: as an alternative metric, the Bayesian Information
+Criterion (BIC) of the rank 1 function is 108 lower than the
+Simple IF and even that of the rank 3 function with 4 free
+parameters is 94.5 lower, corresponding to a very strong pref-
+MNRAS 000, 1–16 (2022)
+
+SPARC data
+RAR IF mock
+Simple IF + EFE mock
+s(gbar, min)
+2
+s+
+s(gbar, max)
+3 -
+rank
+4
+5
+Function
+6
+7
+8 -
+9 -
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Logarithmic slope
+Logarithmic slope
+Logarithmic slopeSPARC data
+RAR IF mock
+Simple IF + EFE mock
+160
+Pareto front
+160
+★
+RAR IF
+140
+140
+Simple IF
+120
+Standard IF
+120
+Simple IF + EFE
+100
+100
+Double power law
+Alog
+80
+80
+60 
+60
+40
+40
+*
+20 
+20
+★
+0
+.
+0
+5
+6
+7
+8
+9
+10
+11
+14
+59 5
+6
+7
+8
+9
+10
+11
+14
+595
+6
+7
+8
+9
+10
+11
+14
+59
+Complexity
+Complexity
+ComplexityOn the functional form of the radial acceleration relation
+7
+erence. In agreement with Chae et al. (2020b, 2021, 2022),
+when fitting the Simple IF + EFE we find that eN > 0 is
+clearly preferred, and recovers a value around the large-scale
+structure expectation. Although this function is significantly
+more accurate than any IF on its own, more than compensat-
+ing for its additional free parameter (∆BIC = −35 compared
+to the Simple IF), it has a poor description length due to the
+large functional contribution. The complexity is 59 using our
+current basis set of operators, although this would fall to 45 if
+tanh were explicitly included. In general, the great improve-
+ment in accuracy and simplicity of the ESR functions demon-
+strates the advantage of this method over guessing functions
+“by eye”.
+In the top panel of Fig. 2 we plot the best 20 functions
+on top of the SPARC data. The functions are colour-coded
+by P(f), with darker colouring indicating functions favoured
+by MDL. The top 6 functions have discontinuities in s at
+gbar ≈ 0.02. We have checked that this is not due to outlying
+points and does not invalidate the first-order approximation
+of Eq. 2; instead it is likely due to the relative simplicity (i.e.
+complexity ≤ 9) of the functions considered, as we discuss
+further in Sec. 5. While such behaviour could be excluded by
+requiring no discontinuity in the derivative within the range
+of the data (or more generally weighting functions with an
+s-dependent prior), we see no principled reason to do so. The
+first function without a discontinuity is at rank 7, which has
+the Newtonian limit and s− = 0.52 at maximum likelihood.
+Although this is 9.3 σ from s− = 1/2, and hence this equation
+cannot function as a pure-MOND IF, the EFE leads to the
+expectation that s− > 1/2 as discussed in more detail below.
+Several of the remaining highly ranked functions have similar
+quasi-MONDian behaviour.
+While some of the best functions have MONDian limits
+around gbar = 10±3 others do not, and in particular s− < 1/2
+is common. To explore this further we calculate s− and s+
+analytically for the top 10 equations as a function of their
+free parameters, showing the results in Table B1. For each
+of the functions where s− and/or s+ depend on θ (i.e. are
+not fixed by the functional form alone) we perform a Markov
+Chain Monte Carlo (MCMC) inference using the numpyro
+sampler (Phan et al. 2019; Bingham et al. 2019) with broad
+flat priors to constrain the parameters and hence derive the
+posterior predictive distributions of the limiting slopes. Fig. 3
+(left panel) shows the results for the top 10 functions, using
+a dot to indicate a limit fixed by the functional form, a bar
+to show the 95 per cent confidence interval in cases where the
+slope depends on the parameters, and an arrowhead to indi-
+cate a limit outside of the plotting range (including ±∞). In
+many cases the 95% confidence interval is extremely narrow.
+Although it is s− and s+ that directly relate to the MOND
+hypothesis, they require extrapolation far beyond the range
+of the data. To understand how the slopes behave near the
+limits of the SPARC data we also calculate s at the minimum,
+gbar, min = 8.32 × 10−13 ms−2, and maximum, gbar, max =
+6.54 × 10−9 ms−2, measured baryonic accelerations. These
+are plotted in cyan and magenta respectively in Fig. 3. At
+gbar, max the logarithmic slope is ≲ 1 for almost all functions,
+as expected for the Newtonian limit as only at gbar → ∞
+does s become 1. However, we find that the gbar,min slopes of
+the top 5 functions are not ∼ 1/2 but actually < 0 due to the
+aforementioned discontinuity. The remaining top functions
+have low-acceleration slopes typically slightly larger than 1/2.
+These results show that the SPARC data do not unambigu-
+ously favour s− = 1/2 and s+ = 1. The requirement for an
+interpolating function to be MONDian is in fact even more
+stringent than this, since the coefficient of proportionality in
+the limiting high-gbar power-law relation must be unity, i.e.
+gobs = gbar. We find that among the functions in the top 10
+for which s+ = 1, four have such a coefficient (at rank 2, 4,
+7, 10) while for the rank 1 function this is 0.84±0.006 and at
+rank 8 it is 0.72 ± 0.01, where the uncertainties are obtained
+by fitting the functions with MCMC. At low gbar the coeffi-
+cient in gobs ∝ g1/2
+bar should be √g0 ≈ 1. The only function
+with s− = 1/2 (rank 6) has the coefficient 1.12 (close to the
+1.10 expected from the canonical g0 = 1.2), while those fur-
+ther down with sgbar,min ≈ 1/2 have a coefficient of 1. The
+relative simplicity of the functions we consider here should
+preference a coefficient of 1, and hence again it is not clear
+to what extent the data may be said to be MONDian. The
+double power law has the limits gobs = 0.81 g1.03
+bar at high gbar
+and gobs = 1.57 g0.60
+bar at low gbar.
+Next, we show in the left panel of Fig. 4 the separate Pareto
+fronts of description length and negative log-likelihood, with
+the second (“simplicity”) objective measured by functional
+complexity. Unlike the Pareto fronts produced by traditional
+SR algorithms, those of ESR are guaranteed to be optimal.
+L(D) and − log(L) are minimised separately at each com-
+plexity, and have their minimum values over all complexity
+subtracted so that the globally best functions appear at 0.
+We show the MONDian functions and double power law as
+separate symbols, all of which we find to be strongly Pareto-
+dominated by the best ESR functions at lower complexity.
+Note that while the “knee” of the Pareto front (where L(D)
+or − log(L) turns over) would appear to be at complexity 6-7,
+there is a significant improvement in going from complexity 8
+to 9. This cautions against automatically selecting functions
+at the knee (the default for example in PySR), and indicates
+that further improvement would likely be achievable by go-
+ing beyond complexity 9. This is beyond the scope of the
+present work; we are content here to have discovered simple
+functional forms for the RAR surpassing any that have been
+considered heretofore.
+4.2 Mock data
+The above results suggest a Newtonian limit (gobs → gbar
+as gbar → ∞) is somewhat favoured by the data while a
+deep-MOND limit (gobs → √g0 gbar as gbar → 0) is question-
+able. However, given the limited dynamical range and signif-
+icant uncertainties of the data it is unclear to what extent
+we should expect to find these limits even if the generating
+function were MONDian. In addition, the EFE may lead to
+the expectation that s > 1/2 at low gbar. To investigate these
+issues we now apply ESR to the mock data of Sec. 3.4.
+4.2.1 RAR IF generating function
+Table 2 shows the best functions found by ESR for the RAR
+IF mock data, along with the results for the IFs and dou-
+ble power law. The RAR IF is by construction a good fit to
+this data, but there are 15 functions with lower L(D), in-
+cluding several at lower complexity. This indicates that the
+characteristics of the SPARC data (dynamic range and un-
+certainties) are insufficient to pick out the true generating
+MNRAS 000, 1–16 (2022)
+
+8
+Desmond, Bartlett & Ferreira
+Rank
+Function
+Comp.
+P(f)
+Parameters
+Description length
+θ0
+θ1
+θ2
+θ3
+Resid.1
+Func.2
+Param.3
+Total
+1
+θ0
+�
+|θ1 + x|θ2 + x
+�
+9
+9.3×10−1
+0.84
+-0.02
+0.38
+—
+-1279.1
+14.5
+14.0
+-1250.6
+2
+||θ1|x + θ0|θ2 + x
+9
+6.4×10−2
+-0.99
+0.64
+0.36
+—
+-1279.9
+12.5
+19.6
+-1247.9
+3
+|θ0||θ1−x|θ2 −θ3
+9
+2.0×10−3
+-1.4×102
+0.02
+0.14
+0.89
+-1276.4
+12.5
+19.5
+-1244.4
+4
+|θ0(θ1 + x)|θ2 + x
+9
+1.4×10−4
+0.35
+-0.02
+0.34
+—
+-1268.9
+14.5
+12.7
+-1241.7
+5
+��θ0 − |θ1 − x|θ2��θ3
+9
+1.0×10−5
+-0.30
+0.02
+0.42
+2.14
+-1271.1
+12.5
+19.5
+-1239.1
+6
+√x exp
+�
+|θ0+x|θ1
+2
+�
+9
+1.5×10−9
+-0.02
+0.36
+—
+—
+-1257.9
+17.5
+10.0
+-1230.3
+7
+� |θ0|x
+x
+�θ1 + x
+9
+2.4×10−10
+1.87
+-0.52
+—
+—
+-1250.6
+14.5
+7.6
+-1228.5
+8
+�
+|θ0 + x| + θ1x
+8
+1.8×10−10
+-1.8×10−3
+0.72
+—
+—
+-1245.6
+12.9
+4.5
+-1228.2
+9
+���θ0 +
+1
+4√x
+���
+θ1
+8
+9.6×10−11
+-0.22
+-2.14
+—
+—
+-1251.1
+14.3
+9.2
+-1227.6
+10
+�√x + 1
+x
+�θ0 + x
+9
+8.2×10−11
+-0.53
+—
+—
+—
+-1248.3
+16.1
+4.8
+-1227.4
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+17
+x/(exp(θ0) − |θ1|
+√x)
+9
+2.2×10−11
+0.03
+0.44
+—
+—
+-1250.9
+17.5
+7.3
+-1226.1
+—
+Double power law
+11
+9.7×10−16
+4.65
+3.96
+1.03
+0.60
+-1252.3
+17.7
+18.5
+-1216.1
+—
+Simple IF
+10
+5.5×10−25
+1.11
+—
+—
+—
+-1217.3
+18.6
+3.9
+-1194.8
+—
+RAR IF
+9
+6.7×10−26
+1.13
+—
+—
+—
+-1212.8
+16.1
+3.9
+-1192.7
+—
+Simple IF + EFE
+59
+5.0×10−69
+1.16
+6.8×10−3
+—
+—
+-1238.9
+139.9
+5.6
+-1093.4
+—
+Standard IF
+14
+9×10−150
+1.54
+—
+—
+—
+-939.5
+27.9
+4.1
+-907.5
+1 − log L(ˆθ)
+2k log(n) + �
+j log(cj)
+3 − p
+2 log(3) + �p
+i (log(Iii)1/2 + log(|ˆθi|))
+Table 1. Top functions found by ESR applied to the SPARC data, ranked by total description length, compared to four MOND IFs and
+a double power law below the horizontal line. x ≡ gbar/10−10 ms−2. We include also the “generalised RAR IF” at rank 17, although this
+does not have a deep-MOND limit. The IFs and double power law are not produced explicitly by our implementation of ESR and hence
+their ranks are unknown, although they are clearly worse than the best ESR functions in both description length and likelihood. For the
+Simple, RAR and Standard IFs the parameter is g0, while for the “Simple IF + EFE” (Eq. 6), the first is g0 and the second eN.
+function: ESR prefers simpler functions which may achieve
+slightly higher likelihoods. Since the − log(L) term in L(D)
+becomes dominant at large dataset size, simply increasing the
+number of (mock) observations with otherwise identical prop-
+erties would not be sufficient to push the generating function
+to the top of the list. For this dataset we find that the gener-
+alised RAR IF, appearing at rank 41, has higher L(D) than
+the RAR IF despite slightly higher likelihood, a success of
+MDL’s penalisation of more complex functions. Note that the
+best-fit generalised RAR IF is x/(0.995 − exp(−
+�
+x/1.068)),
+somewhat offset from the ground-truth values {1, 1.127}.
+This results from a combination of the limited dataset size
+(introducing random noise) and a small bias in the maximum-
+likelihood estimator which we discuss further in Appendix A.
+At rank 27 we find a close cousin of the RAR IF in which the
+“1” is free and g0 is pinned to 1. This performs only slightly
+worse than the RAR IF itself because the true g0 is close to
+1.
+The highest ranked ESR function is better by ∆L(D) = 6.3
+than the RAR IF. Although the relative probability between
+the best function and the best RAR-like function is smaller
+than in the observations (∼ 1600 compared to 4 × 1010), it is
+interesting that the best RAR-like function appears further
+up the list in the real data (rank 17 vs 27). The double power
+law is disfavoured relative to the RAR IF and many ESR
+functions despite having the highest likelihood shown in the
+table, another success of the complexity penalisation. There
+are a few functions overall with lower − log(L), the lowest
+being −2048.0 at rank 51 ((θ0 + |θ1 + θ2/x|1/2)−1, L(D) =
+−2018.4). Also as expected, the Simple IF + EFE has eN
+snapped to 0 and hence behaves identically to the Simple IF,
+albeit with larger functional complexity.
+The top 20 ESR functions for the RAR IF mock are over-
+plotted on that data in the middle panel of Fig. 2. We find a
+slightly reduced spread in s at both the high-gbar and low-gbar
+ends compared to the real data, without any discontinuities
+within the data range. However there is still significant un-
+certainty beyond the range of the data. This is quantified in
+the middle panel of Fig. 3, where the top 10 functions are all
+observed to have slopes of approximately 1/2 at gbar,min and
+1 at gbar,max, although only in two cases is s− = 1/2: for the
+others it is lower. This indicates that constraining the slope
+to be near 1/2 at gbar,min is insufficient to conclude that s−
+takes a similar value, at least up to complexity 9. One would
+need to reduce the uncertainties, or, preferably, lower gbar,min.
+That this applies to a lesser extent at high-gbar is shown by
+the functions at rank 4 and 7 with s+ = ∞.
+Adding to the conclusion that the mock data characteristics
+are insufficient to pick out a MONDian generating function,
+we find that the coefficient in gobs ∝ gbar at high gbar is only
+unity for one of the top-10 functions for which s+ = 1 (at
+rank 6). For all the others it is 0.64, with the exception of
+that at rank 1 where it is 0.63. These values have uncertain-
+ties ∼ 0.003 when constrained by MCMC, and vary by ∼ 0.02
+MNRAS 000, 1–16 (2022)
+
+On the functional form of the radial acceleration relation
+9
+Rank
+Function
+Comp.
+P(f)
+Parameters
+Description length
+θ0
+θ1
+θ2
+θ3
+Resid.1
+Func.2
+Param.3
+Total
+1
+θ0 + θ1x + √x
+8
+5.6×10−1
+9.1×10−3
+0.63
+—
+—
+-2045.2
+12.9
+4.9
+-2027.4
+2
+�
+|θ0 + x| + θ1x
+8
+2.8×10−1
+3.0×10−3
+0.64
+—
+—
+-2044.4
+12.9
+4.8
+-2026.7
+3
+θ0x + xθ1
+7
+8.2×10−2
+0.64
+0.49
+—
+—
+-2045.2
+11.3
+8.5
+-2025.5
+4
+√x exp
+�
+xθ0
+2
+�
+7
+3.5×10−2
+0.36
+—
+—
+—
+-2040.7
+12.5
+3.5
+-2024.7
+5
+(θ0 + x)
+�
+θ1 +
+1
+√x
+�
+9
+1.1×10−2
+1.3×10−3
+0.64
+—
+—
+-2044.5
+16.1
+4.8
+-2023.5
+6
+1
+�
+|θ0+ 1
+x |
++ x
+8
+8.8×10−3
+1.74
+—
+—
+—
+-2038.5
+12.9
+2.3
+-2023.3
+7
+(x|θ0|)(x|θ1|)θ2
+9
+3.1×10−3
+-2.09
+-1.4×10−4
+0.04
+—
+-2045.3
+12.5
+10.6
+-2022.2
+8
+θ0x + |θ1 + x|θ2
+9
+2.4×10−3
+0.64
+1.4×10−3
+0.49
+—
+-2045.4
+14.5
+8.9
+-2022.0
+9
+x
+�
+|θ0 − x|θ1 − θ2
+�
+9
+2.3×10−3
+1.2×10−3
+-0.51
+-0.64
+—
+-2045.3
+14.5
+8.9
+-2021.9
+10
+(θ0 − x)
+�
+θ1 − xθ2�
+9
+2.2×10−3
+-6.5×10−4
+-0.64
+-0.51
+—
+-2045.4
+14.5
+9.0
+-2021.9
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+27
+x/(exp(θ0) − exp(−√x))
+9
+3.2×10−4
+-0.01
+—
+—
+—
+-2039.3
+17.5
+1.9
+-2020.0
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+41
+x/(exp(θ0) − |θ1|
+√x)
+9
+1.1×10−4
+-5.0×10−3
+0.38
+—
+—
+-2042.1
+17.5
+5.7
+-2018.9
+—
+RAR IF
+9
+1.0×10−3
+1.14
+—
+—
+—
+-2041.1
+16.1
+3.9
+-2021.1
+—
+Double power law
+11
+3.4×10−8
+1.25
+1.47
+0.90
+0.54
+-2047.2
+17.7
+18.7
+-2010.8
+—
+Simple IF
+10
+2.8×10−11
+1.12
+—
+—
+—
+-2026.2
+18.6
+3.9
+-2003.7
+—
+Standard IF
+14
+2.9×10−55
+1.54
+—
+—
+—
+-1934.4
+27.9
+4.1
+-1902.4
+—
+Simple IF + EFE
+59
+5.9×10−64
+1.12
+0
+—
+—
+-2026.2
+139.9
+3.9
+-1882.4
+1 − log L(ˆθ)
+2k log(n) + �
+j log(cj)
+3 − p
+2 log(3) + �p
+i (log(Iii)1/2 + log(|ˆθi|))
+Table 2. As Table 1 but for the RAR IF mock data. We find the generalised RAR IF at rank 41, and a closely related function at rank
+27 in which the free parameter appears in the other term in the denominator. For this dataset the RAR IF itself is superior to both of
+these modified forms, as would be expected given that it generated the data.
+over mock datasets differing only in the random seed. Thus
+the best functions almost always fail to recover the Newto-
+nian limit even when it is the truth, presumably due to an
+insufficient gbar,max. The origin of 0.64 is unclear, but presum-
+ably results from the way the lower-gbar behaviour is filtered
+through the forms of the functions found to be optimal. In
+cases where s− = 1/2, the coefficient of proportionality is 1
+(to be compared to √g0 in MOND), and the double power law
+limits are gobs = 1.20 g0.90
+bar at high gbar and gobs = 1.30 g0.54
+bar
+at low gbar.
+The middle panel of Fig. 4 shows the Pareto front for these
+data. We find more smooth behaviour than for the real data,
+with the optimum solution achieved already by complexity
+8, reflecting the relatively simple nature of the generating
+function. This shows that if the RAR IF was generating
+the real data (and our likelihood and mock data generation
+method were accurate), we would have achieved the L(D)
+minimum on those data too. However, the MONDian func-
+tions (including the RAR IF itself) and double power law are
+Pareto-dominated by the ESR results even on these mock
+data, showing that one should not expect to be able to re-
+cover unambiguously even this simplest of MOND generating
+functions.
+4.2.2 Simple IF + EFE generating function
+Analogous results for the mock data generated using the Sim-
+ple IF with inclusion of the EFE are shown in Table 3 and the
+bottom/right panels of Figs. 2-4. This dataset behaves more
+similarly to the real data in terms of the relative ordering of
+the IFs, double power law and ESR functions, including the
+generalised RAR IF. Indeed, the best-fit parameters of all
+three non-EFE IFs are identical to the SPARC data to two
+decimal places, while those of the generalised RAR IF and
+the high-gbar slope of the double power law are the same to
+one. Here the IFs provide a significantly worse compression of
+the data than the best ESR functions, and the double power
+law also performs relatively poorly due to the curvature at
+low gbar (see Fig. 1). There is again a small bias between
+the maximum-likelihood (1.19 and 8.56×10−3) and true (1.2
+and 1×10−2) g0 and eN values for the Simple IF + EFE
+fit. Although this function has among the highest likelihoods
+achievable by ESR up to complexity 9, its functional com-
+plexity makes it a poor compression of its own SPARC-like
+mock data. This reinforces the conclusion that the charac-
+teristics of these data are insufficient to identify a MONDian
+generating function: this one in particular would require far
+more data than the RAR IF to be favoured by MDL.
+For the Simple IF + EFE mock all top-10 functions have
+s+ = 1 and s(gbar,max) > 0.9. However, only 6 of them re-
+MNRAS 000, 1–16 (2022)
+
+10
+Desmond, Bartlett & Ferreira
+Rank
+Function
+Comp.
+P(f)
+Parameters
+Description length
+θ0
+θ1
+θ2
+θ3
+Resid.1
+Func.2
+Param.3
+Total
+1
+θ0 +
+√
+x2 + 2x
+9
+8.9×10−1
+-0.06
+—
+—
+—
+-2017.7
+14.5
+3.1
+-2000.0
+2
+θ0 +
+�
+x|θ1 + x|
+8
+9.3×10−2
+-0.06
+1.97
+—
+—
+-2017.9
+12.9
+7.3
+-1997.8
+3
+−|θ0|
+√x + θ1 + x
+8
+5.6×10−3
+0.26
+0.95
+—
+—
+-2017.9
+12.9
+10.1
+-1995.0
+4
+(θ0 − x)
+�
+θ1 − xθ2�
+9
+3.3×10−3
+3.1×10−3
+-0.71
+-0.53
+—
+-2019.7
+14.5
+10.7
+-1994.4
+5
+xθ0 − θ1(θ2 − x)
+9
+2.4×10−3
+0.39
+0.79
+0.12
+—
+-2020.9
+14.5
+12.3
+-1994.1
+6
+|θ0 − x|θ1 − θ2x
+9
+2.0×10−3
+5.5×10−3
+0.48
+-0.71
+—
+-2019.1
+14.5
+10.6
+-1994.0
+7
+x|θ0|−|θ1|xθ2
+9
+1.7×10−3
+0.04
+-0.16
+0.33
+—
+-2018.1
+12.5
+11.9
+-1993.8
+8
+x
+�
+θ0 + |θ1 + x|θ2�
+9
+1.5×10−3
+0.71
+0.01
+-0.53
+—
+-2018.7
+14.5
+10.6
+-1993.7
+9
+|θ0||θ1|xθ2 + x
+9
+6.5×10−4
+7.0×10−6
+0.03
+0.17
+—
+-2016.7
+12.5
+11.4
+-1992.8
+10
+exp
+�
+θ0 −
+1
+4√x
+�
++ x
+9
+5.5×10−4
+0.57
+—
+—
+—
+-2014.0
+17.5
+3.9
+-1992.6
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+21
+x/(exp(θ0) − |θ1|
+√x)
+9
+1.8×10−5
+0.03
+0.44
+—
+—
+-2014.2
+17.5
+7.4
+-1989.3
+—
+Double power law
+11
+3.4×10−11
+3.53
+3.31
+0.98
+0.60
+-2012.3
+17.7
+18.6
+-1976.0
+—
+Simple IF
+10
+1.2×10−22
+1.11
+—
+—
+—
+-1972.1
+18.6
+3.9
+-1949.6
+—
+RAR IF
+9
+7.0×10−24
+1.13
+—
+—
+—
+-1966.9
+16.1
+3.9
+-1946.8
+—
+Simple IF + EFE
+59
+3.8×10−57
+1.19
+8.6×10−3
+—
+—
+-2016.0
+139.9
+5.9
+-1870.2
+—
+Standard IF
+14
+2×10−141
+1.54
+—
+—
+—
+-1708.3
+27.9
+4.1
+-1676.3
+1 − log L(ˆθ)
+2k log(n) + �
+j log(cj)
+3 − p
+2 log(3) + �p
+i (log(Iii)1/2 + log(|ˆθi|))
+Table 3. As Table 1 but for the Simple IF + EFE mock data.
+cover the true Newtonian limit gobs = gbar: the others find
+gobs = 0.71 gbar or gobs = 0.79 gbar, again with sub-percent
+uncertainty from MCMC. Thus under this model too one
+would not expect the Newtonian limit to be identified ro-
+bustly. While s(gbar,min) > 1/2, indicating the significant im-
+pact of the EFE, s− is typically 0. Thus gbar,min is too high
+to constrain s− reliably, although this may also be a reflec-
+tion of the relative simplicity of the functions we consider.
+The double power law limits are gobs = 0.96g0.98
+bar at high gbar
+and gobs = 1.55 g0.60
+bar at low gbar. The Pareto front indicates
+this dataset to be somewhat more complex than the RAR IF
+mock, as L(D) continues to fall to complexity 9, although the
+smoothness shows it to be simpler than the real data.
+All functions considered achieve considerably higher likeli-
+hood on the mock datasets than the real data, showing that
+the mocks are simpler. This could be because the data do not
+conform to the MOND expectation—one would expect any
+given gobs = f(gbar) to be relatively inaccurate in a chaotic
+ΛCDM galaxy formation scenario—or because the model for
+scattering the mock data points is overly simplistic. This is
+discussed further in Sec. 5. Relatedly, the P(f) values of the
+top functions are very closely spaced in the mock datasets,
+indicating that there is little to distinguish them. On the
+contrary, on the real data the integrated probability of all
+functions besides the top 5 is ≲10−8, suggesting that these
+functions perhaps ought not to be considered at all. The lim-
+iting slopes of the top 10 equations as a function of the pa-
+rameters can be found in Tables B2 and B3 for the RAR and
+EFE mocks respectively.
+5 DISCUSSION
+Our main conclusion is that the SPARC data are insufficient
+to determine robustly the limiting behaviour of the RAR, and
+hence cannot verify or refute the MOND hypothesis. This
+is reached by studying mock data generated by MOND; in
+particular, generating data according to the RAR IF, not
+only are we unable to identify that as the generating function,
+but, more seriously, we cannot reconstruct s− = 1/2. At the
+high-gbar end, the logarithmic slope of the Newtonian limit
+(s+ = 1) is typically well recovered, although the coefficient
+of proportionality in gobs ∝ gbar is not: in the RAR IF mock
+data this takes a values ∼ 0.64 far more often than 1.
+Improving this situation requires increasing the dynamical
+range of the RAR. At the low-gbar end this may be achieved
+by studying ultra-diffuse galaxies (e.g. Freundlich et al. 2022),
+or local dwarf spheroidals (e.g. McGaugh & Wolf 2010; Mc-
+Gaugh & Milgrom 2013), some of which seem seem to in-
+dicate s− ≈ 0, as found by many of the best ESR func-
+tions (Lelli et al. 2017). Alternatively, one may attempt to
+probe the outer regions of galaxies including the Milky Way
+(e.g. Oman et al. 2020). Particularly promising for a large
+gain is to use stacked weak lensing to probe galaxy outskirts
+that would have insufficient signal-to-noise on an individual-
+object basis (Brouwer et al. 2021). This appears to indicate
+s− ≈ 1/2. Increasing gbar,max requires probing the central re-
+gions of high-mass ellipticals, as well as groups and clusters
+of galaxies (Chae et al. 2020a, 2019; Gopika & Desai 2021;
+Chan & Del Popolo 2020; Tian et al. 2020; Pradyumna &
+Desai 2021). Such data already exists and may readily be
+folded into our framework to increase its constraining power.
+MNRAS 000, 1–16 (2022)
+
+On the functional form of the radial acceleration relation
+11
+A smaller information gain may be achieved by reducing the
+uncertainties in gbar and gobs: in the limit of no uncertainty
+any generating function will be assigned P(f) = 1 by MDL.
+By generating mock data with different characteristics one
+could ascertain the requirements for various features of the
+functional form to be unambiguously determined.
+It is likely that there exist functions at higher complexity
+superior to those of Tables 1-3, especially for the real data
+where L(D) drops significantly from complexity 8 to 9. This
+would be computationally demanding using the ESR algo-
+rithm, and thus a stochastic search (e.g. using a genetic al-
+gorithm) may be required. This search may be seeded by
+the ESR functions: the fact that many of the best-fitting
+functions have similar features (such as gobs ≈ 0.64 gbar as
+gbar → ∞ for the RAR IF mock) suggests these may be
+useful for higher-complexity functions also. Thus ESR may
+be used to validate the underlying assumption of stochas-
+tic searches—that there exist features of functions responsi-
+ble for their fitness—and the identification of these features
+may be useful for tuning hyperparameters. While uncover-
+ing lower-L(D) functions at higher complexity may update
+the optimal limiting behaviours of the functional form of the
+RAR, and hence its compatibility with MOND, it cannot
+compromise our discovery of simpler and more accurate func-
+tions than the IFs and double power law. Indeed, the fact that
+the (or at least a) knee of the Pareto front is reached around
+complexity 7 in the left panel of Fig. 4 shows that such func-
+tions already offer a powerful compression.
+Our best functions on the real data have a discontinuity
+in s around gbar = 0.02. This is likely due to the limited
+complexity of the equations we consider: a cusp is the sim-
+plest way of reducing s sharply. It is probable that the op-
+timal functions at higher complexity will have a smoothed
+form of this behaviour in which s does not become negative
+and may not tend to 0. We therefore doubt that the s− and
+s(gbar,min) values of the best functions in Table 1 are robust.
+One could attempt to construct more complex functions in-
+spired by the ESR results with similar but not discontinuous
+behaviour and calculate their − log(L) and L(D) separately,
+or feed them into a genetic algorithm as mentioned above.
+On the other hand, below complexity 9, there is only a sin-
+gle low description length function that is discontinuous, the
+third best function at complexity 8 (P(f) = 4.2 × 10−11).
+The best functions at lower complexity more frequently have
+s− = 1/2 and s+ = 1, although again they rarely satisfy
+gobs = gbar as gbar → ∞. For example, the top function at
+complexity 6—marking the (first) knee of the L(D) Pareto
+front in Fig. 4—is gobs = 0.70 gbar + √gbar, exhibiting simi-
+lar high-gbar behaviour to the 1st- and 8th-ranked functions
+overall.
+To generate the mock data we assumed that all gbar values
+are uncorrelated. While this is likely true between galaxies,
+it is not within a single galaxy because the uncertainty in
+gbar is dominated by the mass-to-light ratio, a global galaxy
+parameter in the simplest approximation. This may be seen
+from the data in Fig. 1, where lines of points (e.g. scattering
+low around gbar = 2 or high around gbar = 10) are all from the
+same galaxy. A more robust procedure may be to generate Υ
+values for each mock galaxy by randomly drawing from their
+priors, use this to transform gbar and then add any other,
+random sources of noise (e.g. from the uncertainty in 3.6 µm
+luminosity). By enhancing inter-galaxy variations this may
+increase the complexity of the mock datasets, moving their
+ESR results towards those of the SPARC data. Alternatively,
+one may fit each galaxy separately to assess compatibility of
+their individual RARs (analogously to Li et al. 2018 but not
+just for the RAR IF). The assumption of uncorrelated data
+points is also present in our likelihood, as discussed further
+in Appendix A.
+We have assumed no intrinsic scatter in the RAR, such
+that all deviations from the hypothetical functional expecta-
+tion must come from the observational uncertainties. While
+this is expected in MOND, in ΛCDM the complex process of
+galaxy formation would lead to a significant and parameter-
+dependent effective intrinsic scatter (Desmond 2017). Even
+the EFE would introduce some scatter due to galaxy-by-
+galaxy variation in gex (Chae et al. 2021). It would be
+straightforward to add this (in some direction on the RAR
+plane) as an additional free parameter of all functions, which
+would alter the results. MDL naturally penalises the addi-
+tion of this parameter, allowing one to determine whether it
+is justified for any given function. This would provide further
+evidence concerning the optimality of MOND by assessing
+the extent to which the data implies law-like modified grav-
+ity behaviour.
+Our current implementation of MDL treats the parameter
+values as part of the model and chooses them to maximise the
+likelihood. An alternative would be to treat the hypothesis in
+Eq. 1 as the functional form alone, assigning codelengths and
+probabilities to functions regardless of their parameter val-
+ues. In a Bayesian formulation this corresponds to marginal-
+ising over the parameters, and enables a simpler one-part
+coding scheme. An even higher-level approach would be to
+group functions into sets with specific properties, e.g. limit-
+ing behaviour. This would enable calculation of the posterior
+predictive distribution of any feature of the functional repre-
+sentation of the dataset, and hence enable model comparison
+at any level of generality.
+The relative simplicity of the RAR and conformity to the
+Newtonian and deep-MOND limits are the key differences
+between the expectations of MOND and the more chaotic
+galaxy formation scenario of ΛCDM: it is only under the
+simpler scenario that one would expect to find a simple
+gobs = F(gbar). While our results are therefore not partic-
+ularly supportive of the MOND hypothesis, this is not to say
+either that the data could not plausibly have been generated
+by MOND or that it could plausibly have been generated
+under another hypothesis, as only MOND currently has suf-
+ficient predictivity for a test of this precision. We look to
+future SR studies with more data to establish the functional
+form of the RAR—if it exists—definitively.
+6 CONCLUSION
+The radial acceleration relation (RAR) has become central to
+debates about the mass discrepancy problem on astrophysical
+scales. Its tightness and regularity have been used to argue for
+a violation of Newtonian gravity in accordance with Modified
+Newtonian Dynamics (MOND), but the functions used to fit
+the data have been constructed to conform to this theory.
+As the first detailed application of the brand-new technique
+of Exhaustive Symbolic Regression, we rank objectively all
+simple functions in terms of their aptitude for describing the
+MNRAS 000, 1–16 (2022)
+
+12
+Desmond, Bartlett & Ferreira
+SPARC RAR. We employ the minimum description length
+principle to trade accuracy with simplicity and hence perform
+model selection, and calibrate our method on mock MOND
+data generated both with and without the external field effect
+(EFE). Our conclusions are as follows:
+• ESR discovers functions which are better descriptions, in
+both accuracy and simplicity and for both observed and sim-
+ulated data, than MOND functions or a double power law.
+• While the majority of best-fitting functions on the SPARC
+data recover gobs ∝ gbar at high accelerations, not all have
+a best-fit coefficient of proportionality near unity. Thus the
+Newtonian limit is not clearly evidenced.
+• The SPARC data do not prefer functions with the deep-
+MOND limit of gobs ∝ √gbar as gbar → 0. Instead, we find
+that functions with gbar → const typically compress the data
+more efficiently, albeit with considerable uncertainty.
+• SPARC-like mock data generated assuming the MONDian
+RAR interpolating function do not unambiguously recover
+that function. Moreover, many of the best functions for those
+mock data have gobs ≈ 0.64 gbar rather than gobs = gbar at
+high gbar, and most do not have a deep-MOND limit at all.
+• The EFE in AQUAL greatly increases the logarithmic slope
+of the best-fitting functions at the low-gbar end of the data,
+but does not appreciably impact the limiting slope at gbar →
+0. Incorporating the EFE in the mock data produces more
+generally similar results to the real data, so our analysis
+(within the MOND paradigm) hints at it.
+• We conclude that the data have too small a dynamic
+range (and too large uncertainties) to unambiguously favour
+MOND even if it is in fact generating the data. The SPARC
+RAR alone, therefore, does not supporting that theory un-
+ambiguously. The best prospect for improving this situation
+is to increase the acceleration range of the data, e.g. using
+stacked weak lensing at low gbar and groups and clusters at
+high gbar.
+• Our results are a function of the maximum complexity of
+equation considered. Future symbolic regression algorithms—
+exhaustive or non-exhaustive—will reach the true description
+length minimum and hence uncover the optimal functional
+representation of the RAR and determine whether the rela-
+tion implies novel law-like gravitational behaviour.
+Exhaustive Symbolic Regression provides for the first time
+a guaranteed complete search through functional parameter
+space, making it the ideal tool to determine the analytic
+form of observed relations, extract physics from data theory-
+agnostically, and create fitting functions. We make the ESR
+and RAR codes, full function sets and the best 50 functions
+for each dataset we consider publicly available to facilitate
+future applications.
+7 DATA AVAILABILITY
+The code and data associated with ESR and its applica-
+tion to the RAR are released at � and in Bartlett et al.
+(2022a). The SPARC data is available at http://astroweb.
+cwru.edu/SPARC. Other data may be shared on request to the
+corresponding authors.
+ACKNOWLEDGEMENTS
+We thank Kyu-Hyun Chae, Andrei Constantin, Miles Cran-
+mer, Mario Figueiredo, Gianluca Gregori, Thomas Harvey,
+Mark Kotanchek, Federico Lelli, Stacy McGaugh, Andre
+Lukas, Richard Stiskalek and Tariq Yasin for useful inputs
+and discussion.
+HD is supported by a Royal Society University Research
+Fellowship (grant no. 211046). DJB is supported by the Si-
+mons Collaboration on “Learning the Universe” and was sup-
+ported by STFC and Oriel College, Oxford. PGF acknowl-
+edges support from European Research Council Grant No:
+693024 and the Beecroft Trust.
+This project has received funding from the European Re-
+search Council (ERC) under the European Union’s Horizon
+2020 research and innovation programme (grant agreement
+No 693024).
+This
+work
+used
+the
+DiRAC
+Complexity
+and
+DiRAC@Durham
+facilities,
+operated
+by
+the
+University
+of Leicester IT Services and Institute for Computational
+Cosmology, which form part of the STFC DiRAC HPC
+Facility (www.dirac.ac.uk). This equipment is funded by
+BIS National E-Infrastructure capital grants ST/K000373/1,
+ST/P002293/1, ST/R002371/1 and ST/S002502/1, STFC
+DiRAC Operations grant ST/K0003259/1, and Durham
+University
+and
+STFC
+operations
+grant
+ST/R000832/1.
+DiRAC is part of the National E-Infrastructure.
+For the purpose of open access, the authors have applied
+a Creative Commons Attribution (CC BY) licence to any
+Author Accepted Manuscript version arising.
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+APPENDIX A: A NOTE ON LIKELIHOODS
+A1 Correlation of measurements and their
+uncertainties
+The likelihood of Sec. 3.2 treats the uncertainties induced by
+Υ, D and i as statistical and uncorrelated between points.
+This is clearly incorrect: a scattering up of Υ, for example,
+causes a coherent increase in gbar across the rotation curve,
+generating off-diagonal elements in the covariance matrix. A
+better approach is therefore to calculate the covariance ma-
+trix via Monte Carlo. For a given galaxy, one would indepen-
+dently sample Υgas, Υdisk, Υbulge, D and i many times from
+their assumed-Gaussian prior distributions. One would then
+generate the corresponding gobs and gbar values (now vectors
+across the rotation curve of a given galaxy), scatter them by
+their statistical uncertainty (gbar has only the small δL3.6
+term), and calculate the covariance matrices Σobs and Σbar.
+Let us define u ≡ log (gbar) and v ≡ log (gobs), such that
+u and v are vectors containing u and v for all galaxies at
+all measured points along the rotation curve. We denote the
+“true” values with a superscript t and the observed values
+without a superscript. We assume that ut and vt are drawn
+from multivariate Gaussian distributions with covariance ma-
+trices Σu and Σv, respectively. Since ut and vt are statisti-
+cally independent random variables for fixed u and v, the
+likelihood of a given set of points in the
+�
+ut, vt�
+plane would
+then be
+L (u, v) =
+1
+�
+|2πΣu|
+1
+�
+|2πΣv|
+× exp
+�
+−1
+2
+�
+ut − u
+�T Σ−1
+u
+�
+ut − u
+��
+× exp
+�
+−1
+2
+�
+vt − v
+�T Σ−1
+v
+�
+vt − v
+��
+.
+(A1)
+The equivalent of Eq. 2 for a vector-valued function f, v =
+f (u), is then obtained by marginalising over ut,
+L (v) =
+�
+L
+�
+ut, f
+�
+ut��
+dut.
+(A2)
+Taylor expanding f
+�
+ut�
+about u,
+f
+�
+ut�
+≈ f (u) + D
+�
+ut − u
+�
+,
+(A3)
+where Dij ≡ ∂jfi|u, yields
+L (v) =
+1
+�
+|2πΣtot|
+× exp
+�
+−1
+2 (v − f(u))T Σ−1
+tot (v − f(u))
+�
+,
+(A4)
+where Σtot is the total covariance matrix defined by
+Σtot ≡ Σv + DΣuDT.
+(A5)
+This is equivalent to Eq. 3 if u and v have only a single
+element.
+In principle, the assumption that the joint probability dis-
+tributions of ut and vt are a multivariate Gaussian is un-
+necessary as the full empirical distributions are generated by
+the Monte Carlo sampling described above. Using this di-
+rectly would enable a loss function which, while still inde-
+pendent between galaxies, fully encapsulates the correlated,
+non-Gaussian structure of each galaxy’s measurements and
+hence provides a more accurate description of the expected
+probability distributions of gbar and gobs. This model would
+also allow for more accurate mock data generation as dis-
+cussed in Sec. 5.
+A2 Sampling nuisance parameters from their
+posteriors
+All the methods described above sample Υ, D and i, and the
+true gbar values ut, from their priors, i.e. without adjusting
+MNRAS 000, 1–16 (2022)
+
+14
+Desmond, Bartlett & Ferreira
+them to maximise agreement with the function being fitted.
+In principle, the better procedure is to constrain these nui-
+sance parameters jointly with any parameters of the function,
+assuming that function in the fit. Even the relatively simple
+case where ut is drawn from the prior but Υ, D and i from
+the posterior is however impractical as it requires optimisa-
+tion in a parameter space of dimension 147 × 3 + p if the
+three Υs are coupled (or only one is varied), and 147 × 5 + p
+if they are varied separately, where p is the number of free
+parameters in the function. This would naturally account for
+correlations between the gbar and gobs measurements induced
+by variations in Υ, D and i, and thus provides a more accu-
+rate but expensive alternative to the model of Sec. A1. This
+approach will be applied for the first time to the RAR, out-
+side the context of SR, in upcoming work (Desmond 2023, in
+prep).
+The situation is complicated further by inference of ut.
+At the top level of the hierarchical model for predicting v
+from u are both the parameters of the model, θ, and ut. If
+one wanted to find the true maximum likelihood point, one
+should maximise Eq. A1 for ut as well as θ, instead of Eq. A2
+for θ alone.
+To do this, we start by noting that maximising the likeli-
+hood in Eq. A1 is equivalent to minimising
+h
+�
+ut, v
+�
+= 1
+2
+�
+f
+�
+ut�
+− v
+�T Σ−1
+v
+�
+f
+�
+ut�
+− v
+�
++ 1
+2
+�
+ut − u
+�T Σ−1
+u
+�
+ut − u
+�
+.
+(A6)
+Using Eq. A3 and minimising h with respect to ut, we find
+the maximum-likelihood point, ˆut, to be
+ˆut = u + ΣuDTΣ−1
+tot (v − f (u)) ,
+(A7)
+and we must therefore maximise
+L
+�ˆut, v
+�
+=
+1
+�
+|2πΣu|
+1
+�
+|2πΣv|
+× exp
+�
+−1
+2 (v − f(u))T Σ−1
+tot (v − f(u))
+�
+,
+(A8)
+where Σ−1
+tot is defined in Eq. A5. This will not yield the same
+result as maximising Eq. A4 since Eq. A8 does not contain the
+normalisation term which penalises large gradients. When ap-
+plied to mock data generated assuming vt is linearly related
+to ut, we find that Eq. A4 (the commonly used expression,
+and the one we adopt in Sec. 3.2) induces a small bias that
+Eq. A8 does not. This is the reason why the best-fit RAR and
+generalised RAR IFs do not precisely match the generating
+function in Table 2, or the Simple IF + EFE the generating
+function in Table 3. We should expect all best-fit parameter
+values to be likewise slightly biased by the use of Eq. 2.
+In practice, it is challenging to maximise Eq. A1 since
+the lack of the gradient-penalising determinant means that
+naïvely using Eq. A8 can prefer functions with diverging gra-
+dients at at least one point in the domain of u. This breaks
+the linearity assumption in Eq. A3, making the result un-
+trustworthy. Instead, one should numerically solve the full
+optimisation problem without Taylor-expanding f. This in-
+volves solving a root-finding problem for each trial θ during
+the optimisation of the function’s parameters. Although fea-
+sible for a handful of functions, this is computationally im-
+practical for the full set of ESR functions. We therefore use
+the simpler likelihood here, calibrating our results using mock
+data to measure the magnitude of the bias. We defer further
+discussion of this important issue to future work.
+APPENDIX B: LIMITING SLOPES OF ESR
+FUNCTIONS
+Here we provide the analytic low-gbar (s−) and high-gbar (s+)
+logarithmic slopes of the top 10 functions generated using
+ESR up to complexity 9. We give the slopes for the observed
+SPARC data in Table B1, the mock data assuming the RAR
+IF in Table B2, and the mock data assuming the Simple IF
+with EFE in Table B3.
+This paper has been typeset from a TEX/LATEX file prepared by
+the author.
+MNRAS 000, 1–16 (2022)
+
+On the functional form of the radial acceleration relation
+15
+Rank
+Function
+Low-acceleration slope
+High-acceleration slope
+Value
+Condition
+Value
+Condition
+1
+θ0
+�
+|θ1 + x|θ2 + x
+�
+0
+θ2
+θ2 ≥ 1
+1
+otherwise
+2
+||θ1|x + θ0|θ2 + x
+0
+1
+(|θ1| ≤ 1) or (|θ1| > 1 and θ2 ≤ 0)
+∞
+otherwise
+3
+|θ0||θ1−x|θ2 −θ3
+0
+∞
+|θ0| > 1 and θ2 > 0
+−∞
+0 < |θ0| < 1 and θ2 > 0
+0
+otherwise
+4
+|θ0(θ1 + x)|θ2 + x
+0
+θ2
+θ2 ≥ 1
+1
+otherwise
+5
+��θ0 − |θ1 − x|θ2��θ3
+0
+θ2θ3
+θ2 ≥ 0
+0
+otherwise
+6
+√x exp
+�
+|θ0+x|θ1
+2
+�
+1
+2
+1
+2
+θ1 ≤ 0
+∞
+otherwise
+7
+� |θ0|x
+x
+�θ1 + x
+−θ1
+θ1 ≥ −1
+1
+(|θ0| > 1 and θ1 ≤ 0) or (0 < |θ0| < 1 and θ1 ≥ 0) or (|θ0| = 1 and θ1 ≥ −1)
+1
+otherwise
+−θ1
+|θ0| = 1 and θ1 < −1
+1 + 0θ1θ1(−∞)
+θ0 = 0
+∞
+otherwise
+8
+�
+|θ0 + x| + θ1x
+0
+1
+9
+���θ0 +
+1
+4√x
+���
+θ1
+− θ1
+4
+0
+10
+�√x + 1
+x
+�θ0 + x
+−θ0
+θ0 ≥ −1
+θ0
+2
+θ0 ≥ 2
+1
+otherwise
+1
+otherwise
+Table B1. Functional forms and limiting slopes of the ten best functions found by ESR applied to the SPARC data. The functions in the
+second column give the fitted y = gobs/10−10 ms−2 for input x = gbar/10−10 ms−2. The low-acceleration slope is limx→0+ d log y/d log x
+(denoted s− in the text), and the high-acceleration slope is similarly defined but for x → ∞ (s+). {θi} are real parameters fitted to
+the data to maximise the likelihood (see Table 1). Slopes given without conditions are valid ∀ θi. For comparison, the MOND prediction
+without the external field effect is s− = 1/2 and s+ = 1.
+MNRAS 000, 1–16 (2022)
+
+16
+Desmond, Bartlett & Ferreira
+Rank
+Function
+Low-acceleration slope
+High-acceleration slope
+Value
+Condition
+Value
+Condition
+1
+θ0 + θ1x + √x
+0
+1
+2
+�
+|θ0 + x| + θ1x
+0
+1
+3
+θ0x + xθ1
+θ1
+θ1 ≤ 1
+θ1
+θ1 ≥ 1
+1
+otherwise
+1
+otherwise
+4
+√x exp
+�
+xθ0
+2
+�
+1
+2
+θ0 ≥ 0
+1
+2
+θ0 ≤ 0
+−∞
+otherwise
+∞
+otherwise
+5
+(θ0 + x)
+�
+θ1 +
+1
+√x
+�
+− 1
+2
+1
+6
+1
+�
+|θ0+ 1
+x |
++ x
+1
+2
+1
+7
+(x|θ0|)(x|θ1|)θ2
+∞
+|θ1| > 0 and θ2 < 0
+∞
+|θ1| > 0 and θ2 > 0
+1
+θ2 = 0
+0
+|θ1| > 0 and θ2 < 0
+0
+otherwise
+1
+|θ1| > 0 and θ2 = 0
+0θ2θ2∞
+otherwise
+8
+θ0x + |θ1 + x|θ2
+0
+θ2
+θ2 ≥ 1
+1
+otherwise
+9
+x
+�
+|θ0 − x|θ1 − θ2
+�
+1
+θ1 + 1
+θ1 ≥ 0
+1
+otherwise
+10
+(θ0 − x)
+�
+θ1 − xθ2�
+θ2
+θ2 ≤ 0
+θ2 + 1
+θ2 ≥ 0
+0
+otherwise
+1
+otherwise
+Table B2. As Table B1 but for the mock data generated from the RAR IF.
+Rank
+Function
+Low-acceleration slope
+High-acceleration slope
+Value
+Condition
+Value
+Condition
+1
+θ0 +
+√
+x2 + 2x
+0
+1
+2
+θ0 +
+�
+x|θ1 + x|
+0
+1
+3
+−|θ0|
+√x + θ1 + x
+0
+1
+|θ0| ≤ 1
+∞
+otherwise
+4
+(θ0 − x)
+�
+θ1 − xθ2�
+θ2
+θ2 ≤ 0
+1 + θ2
+θ2 ≥ 0
+0
+otherwise
+1
+otherwise
+5
+xθ0 − θ1(θ2 − x)
+θ0
+θ0 ≤ 0
+θ0
+θ0 ≥ 1
+0
+otherwise
+1
+otherwise
+6
+|θ0 − x|θ1 − θ2x
+0
+θ1
+θ1 ≥ 1
+1
+otherwise
+7
+x|θ0|−|θ1|xθ2
+∞ log |θ0|
+|θ1| > 1 and θ2 < 0
+−∞
+|θ0| > 1 and |θ1| > 1 and θ2 > 0
+1
+otherwise
+∞
+0 < |θ0| < 1 and |θ1| > 1 and θ2 > 0
+∞θ2
+θ0 = 0 and |θ1| > 1
+1
+otherwise
+8
+x
+�
+θ0 + |θ1 + x|θ2�
+1
+1 + θ2
+θ2 ≥ 0
+1
+otherwise
+9
+|θ0||θ1|xθ2 + x
+−∞
+|θ0| > 1 and |θ1| > 1 and θ2 < 0
+∞
+|θ0| > 1 and |θ1| > 1 and θ2 > 0
+1
+(θ0 = 0) or (θ1 = 0) or (|θ0| < 1 and |θ1| > 1 and θ2 < 0)
+1
+otherwise
+0
+otherwise
+10
+exp
+�
+θ0 −
+1
+4√x
+�
++ x
+1
+1
+Table B3. As Table B1 but for the mock data generated using the Simple IF + EFE.
+MNRAS 000, 1–16 (2022)
+
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+page_content=' UK  2CNRS & Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Institut d’Astrophysique de Paris (IAP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' UMR 7095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 98 bis bd Arago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' F-75014 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' France  3Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Denys Wilkinson Building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Keble Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Oxford OX1 3RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' UK  12 January 2023  ABSTRACT  We apply a new method for learning equations from data—Exhaustive Symbolic Regression (ESR)—to late-type  galaxy dynamics as encapsulated in the radial acceleration relation (RAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Relating the centripetal acceleration due  to baryons, gbar, to the total dynamical acceleration, gobs, the RAR has been claimed to manifest a new law of  nature due to its regularity and tightness, in agreement with Modified Newtonian Dynamics (MOND).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Fits to this  relation have been restricted by prior expectations to particular functional forms, while ESR affords an exhaustive  and nearly prior-free search through functional parameter space to identify the equations optimally trading accuracy  with simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Working with the SPARC data, we find the best functions typically satisfy gobs ∝ gbar at high gbar,  although the coefficient of proportionality is not clearly unity and the deep-MOND limit gobs ∝ √gbar as gbar → 0 is  little evident at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' By generating mock data according to MOND with or without the external field effect, we find  that symbolic regression would not be expected to identify the generating function or reconstruct successfully the  asymptotic slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We conclude that the limited dynamical range and significant uncertainties of the SPARC RAR  preclude a definitive statement of its functional form, and hence that this data alone can neither demonstrate nor  rule out law-like gravitational behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Key words: galaxies: kinematics and dynamics – dark matter – methods: data analysis  1 INTRODUCTION  Kinematic measurements of galaxies relate their visible and  dynamical masses, affording constraints on the distribution  of dark matter and/or the behaviour of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' These mea-  surements are simplest to perform for late-type galaxies sup-  ported predominantly by rotation, as the enclosed dynami-  cal mass may be calculated from the centripetal acceleration  and the law of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Such studies have revealed a strik-  ing correlation between the enclosed baryonic and total dy-  namical mass assuming Newtonian gravity, dubbed the mass  discrepancy–acceleration (Sanders 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' McGaugh 2004) or  radial acceleration relation (RAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' It has  been claimed that the RAR indicates that at high acceler-  ations the Newtonian dynamical mass follows the baryonic  mass (indicating little dark matter and the validity of New-  tonian mechanics), while as acceleration drops below a new  constant of nature g0 ≈ 10−10 ms−2 the dynamical mass in-  creasingly exceeds the baryonic mass in a regular way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  One may attempt to understood these observations from ei-  ther a dark matter or modified gravity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In ΛCDM  the difference between the dynamical and baryonic mass is  due to the dark matter that makes up most of the mass of  ⋆ harry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='desmond@port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='uk  † deaglan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='bartlett@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='uk  the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The RAR must therefore be explained by the rela-  tive distributions of dark and visible mass established by the  process of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Interactionless cold dark mat-  ter is influenced only gravitationally by the baryonic mass  so the emergence of the RAR must be somewhat fortuitous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  it is not established directly by a baryon–dark matter cou-  pling (although see Blanchet & Le Tiec 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Berezhiani &  Khoury 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Famaey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2018 for alternative ideas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In  contrast, the modified gravity (or modified inertia) interpre-  tation posits a breakdown of Newtonian mechanics at low  acceleration so that the dynamical mass inferred by a New-  tonian analysis is not the true dynamical mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The prototypical instantiation of this idea is Modified New-  tonian Dynamics (MOND;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Milgrom 1983a,c,b), in which the  kinematic acceleration gobs follows the square root of the  Newtonian acceleration gbar in the weak-field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This  enables the total dynamical mass of the galaxy to remain  equal to the baryonic mass across galaxies’ rotation curves,  eliminating the need for dark matter in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' MOND has cos-  mologically viable relativistic extensions (most recently Sko-  rdis & Złośnik 2021), and is reviewed in Famaey & McGaugh  (2012) and Banik & Zhao (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Central to the dark matter–modified gravity debate in the  context of galaxy dynamics is the functional form of the  RAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is because MOND makes a very specific predic-  tion (absent the external field effect: gobs = gbar in the high-  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='04368v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='GA]  11 Jan 2023  2  Desmond, Bartlett & Ferreira  acceleration “Newtonian regime” and gobs ∝ g1/2  bar in the low-  acceleration “deep-MOND regime”) while dark matter could  accommodate a range of possibilities depending on the effect  of galaxy formation on halo density profiles, which remains  highly uncertain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Duffy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Macciò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Grudić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Tenneti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Ludlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Keller & Wadsley 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The only po-  tentially unambiguous prediction is that the RAR tends to  gobs = Ωm/Ωb gbar at radii sufficiently large to encompass the  cosmic baryon fraction, but it is unclear where or even if this  occurs in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus, while the ΛCDM prediction for the  full RAR can be tested only by applying potentially restric-  tive priors on galaxy formation effects (Di Cintio & Lelli 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Desmond 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Paranjape & Sheth 2021), a more direct route  towards informing the dark matter–modified gravity debate  is to test the MOND prediction, specifically the limiting be-  haviour at g ≪ g0 and g ≫ g0, the small intrinsic scatter and  the lack of residual correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Here we focus on the asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This can be  assessed to some extent by fitting a functional form with  free power-law slopes at both ends (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017), but  this assumes that the slope tends to a constant at each end  and restricts to a specific part of the functional parameter  space for which this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' These are in question when  assessing the accuracy of the MOND prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A fully  satisfactory fit should therefore make no such assumptions,  eliminating potential confirmation bias and testing without  any priors the assertion that the RAR implies no dynami-  cally relevant dark matter at high g and the deep-MOND  limit at low g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We accomplish this here by means of a novel  regression algorithm dubbed Exhaustive Symbolic Regression  (ESR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2022b), and hence assess the degree to  which the RAR supports the tenets of MOND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Within the  MOND paradigm, this method also enables optimisation of  the “interpolating function” (IF) gobs = F(gbar) between the  two stipulated limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2 we de-  scribe the RAR data that we use, and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3 our algo-  rithm for generating functions and assessing their aptitude  for describing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 5  we discuss the broader ramifications, potential remaining un-  certainties and ways in which the programme could be fur-  thered in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Full details on ESR  are given in Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Units not explicitly given  are 10−10 ms−2, and all logarithms are natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  2 OBSERVATIONAL DATA  We use the SPARC data set (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2016),1 a compila-  tion of 175 rotation curves from the literature combined with  Spitzer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6µm photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We apply the same quality cuts as  the RAR study of Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2017), removing galaxies with  quality flag 3 (indicating large asymmetries, non-circular mo-  tions and/or offsets between stellar and HI distributions) and  those with inclinations i < 30 deg, and points for which the  quoted fractional uncertainty on the observed rotation veloc-  ity is greater than 10 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This leaves 2,696 points from  147 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  1 http://astroweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='cwru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='edu/SPARC/  3 METHOD  We describe our method for generating and assessing trial  functions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1, and our likelihood function in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3 we outline our criteria for assessing whether a  function displays MOND-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1 Exhaustive Symbolic Regression  While algorithms for symbolic regression (SR)—the search  for good functional descriptions of a dataset—are becoming  mature, they remain fallible (La Cava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Unless  the generating function of the data is known at the outset  (in which case SR is not required), it is not possible to de-  termine whether any SR algorithm has uncovered the best  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This motivated us to develop “Exhaustive Symbolic  Regression” (ESR) which, given a set of basis functions, pro-  duces and evaluates every possible function up to a given  complexity of equation, defined here as the number of nodes  in its tree representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This enables a brute-force solu-  tion to relatively simple problems and provides a touchstone  for assessing the results of stochastic algorithms at higher  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' ESR, presented in full in Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2022b),  has two main steps: i) generating, and optimising the pa-  rameters of, all functions up to a given complexity, and ii)  ranking these functions using an information-theoretic metric  combining accuracy and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  For part i, the steps are:  (1) Generate all possible trees containing a given number of  nodes (equal to the complexity of functions considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (2) Generate the complete set of such functions by decorat-  ing these trees with all permutations of the operators  from the operator list specified in advance, utilising the  constraints on the arity of the operator that can occupy  a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (3) Simplify the functions and remove duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Variants of  the same function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' x(x+θ0) and x2+θ0x) are however  retained as these may have different model complexities  (used in step ii, below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For each unique function the  variant is retained that minimises this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (4) Determine the values of the free parameters appearing  in the functions that maximise the likelihood of the data  (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (5) Repeat for all complexities under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The only degrees of freedom in this procedure are the max-  imum complexity considered (here set at 9 as higher complex-  ity is computationally prohibitive) and the set of operators  of which the functions are composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Here we choose:2  Nullary: gbar, θ  Unary: exp, sqrt, square, inv  Binary: +, −, ∗, /, pow  where θ is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We implicitly take the absolute  value of the argument of any square root or power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The result of this procedure is a list of all functions up  to the maximum complexity (of which there are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='24×107),  2 Many of these operators can manifestly be constructed from  combinations of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We include these repetitions to simplify  the function trees, allowing a greater range of expressions up to  the maximum complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000, 1–16 (2022)  On the functional form of the radial acceleration relation  3  along with the parameter values that maximise the likelihood  of the RAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' As in regular regression, using the maxi-  mum likelihood as the model selection criterion would favour  overfitting, whereby a function fits the data near-perfectly  but generalises or extrapolates poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To remedy this, SR  typically uses two-objective optimisation, where the second  objective is the “simplicity” of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In the absence of  a metric for trading accuracy (the first objective) with com-  plexity, optimal functions form a “Pareto front” where ac-  curacy cannot be increased without reducing simplicity and  vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Simplicity has been defined analogously to model  complexity (the number of nodes in the tree representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' in PySR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020), among others, but such  definitions are typically arbitrary and thus compromise the  objectivity of the regression results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  To remedy this, part ii of ESR implements the minimum  description length principle (MDL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Rissanen 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Grünwald  & Roos 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Grunwald 2007) as a model selection criterion,  which has an information-theoretic motivation and provides a  natural framework for making commensurable the two objec-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' MDL states that functions are preferred to the extent  that they compress the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' minimise the number of bits  required to communicate the data with the aid of the func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We implement this with a two-step code in which the  description length (also called codelength) is comprised of a  component describing the function and a component describ-  ing the residuals of the data around the function’s expecta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We use the Shannon–Fano coding scheme for the latter  (Cover & Thomas 1991), and for the former include contri-  butions both from the structure of the function (penalising  those employing more operators) and from the free parame-  ters (penalising more parameters, especially ones that must  be specified to high precision to achieve a high likelihood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The overall codelength of the compressed data, L(D), is de-  rived in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3 of Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2022b):  L(D) = L(D|H) + L(H)  = − log( ˆL) + k log(n) − p  2 log(3)  +  p  �  i  �1  2 log(ˆIii) + log(|ˆθi|)  �  +  �  j  log(cj),  (1)  where L is the description length, D the dataset, H the hy-  pothesis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' function in question), L the likelihood, θ a free  parameter of the function, k the number of nodes in the func-  tion’s tree representation, n the number of unique operators  involved, p the total number of free parameters, I the Fisher  information matrix of the parameters and cj any constant  natural numbers generated by simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A hat denotes  evaluation at the maximum-likelihood point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' With all loga-  rithms natural, this is the number of nats required to commu-  nicate the data with the aid of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' L(D) supports  a probabilistic interpretation over function space that gener-  alises the likelihood: the relative probability of a function is  exp(−L(D)) (Grunwald 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The structure of the function alone determines the k log(n)  term, but the remaining terms require the free parameters to  be numerically optimised to maximise the likelihood (which  we use interchangeably with minimising the loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3 We now  describe our choice of likelihood for the RAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  3 The p log(3)/2 term is only affected by the numerical optimisa-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 Loss function  As is typical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017), we assume that gbar, gobs  and their uncertainties are uncorrelated across the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  We further assume that the true gbar and gobs values, de-  noted gt  bar and gt  obs, generate the observed values with lognor-  mal probability distributions centred at the true values with  widths given by their uncertainties δgbar and δgobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Following  Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2017), we fix the mass-to-light ratios Υgas = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='33,  Υdisk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5 and Υbulge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7 and assign them 10, 25 and 25  per cent uncertainties respectively, summing these in quadra-  ture to estimate δVbar and hence δgbar (assuming no uncer-  tainty in radial position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We likewise assume the uncertain-  ties on distance D and inclination i to be statistical and hence  sum their contributions in quadrature with the quoted sta-  tistical uncertainty on Vobs according to Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2017, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  2) to estimate δgbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The likelihood of an observation given the function in ques-  tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' f(gt  bar),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' is then:  L(log(gobs)) =  � ∞  −∞  L(log(gobs)| log(gt  bar)) L(log(gt  bar)) d log(gt  bar)  =  1  2π δ log(gbar) δ log(gobs)  � ∞  −∞  exp  �  −  �  log(gobs) − log(f(gt  bar))  �2  2 δ log(gobs)2  �  × exp  �  −  �  log(gt  bar) − log(gbar)  �2  2 δ log(gbar)2  �  d log(gt  bar)  ≈  1  �  2πσ2  tot  exp  �  −(log(gobs) − log(f(gbar)))2  2σ2  tot  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (2)  where  σ2  tot ≡ δ log(gobs)2 +  �d log(f(gbar))  d log(gbar)  �2  δ log(gbar)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (3)  This is derived by keeping the leading-order term in the Tay-  lor expansion of log(f(gt  bar)) around log(f(gbar)) and there-  fore assumes this to be small relative to the rate of change  of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' An advantage of working in log(gbar) − log(gobs) space  as opposed to gbar − gobs is that, the RAR being roughly a  product of power-laws, this minimises the error due to the  first-order approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find this to be good for all of  the best functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The likelihood is then the product over  all data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A the limitations of this  likelihood model and how it could be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3 Assessing MOND  The core of the MOND paradigm is that gobs = gbar at g ≫ g0  and gobs = √gbarg0 at g ≪ g0 (Milgrom 1983a,c,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This  implies  s ≡ d log(gobs)  d log(gbar) =  �  1,  gbar → ∞  1/2,  gbar → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (4)  Common choices for the IF covering the intermediate region  g ≈ g0 include  tion if any parameters are set to 0 due to their maximum-likelihood  values being less than 1 precision unit from 0 (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3 of Bartlett  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000, 1–16 (2022)  4  Desmond, Bartlett & Ferreira  “Simple”: gobs = gbar/2+  �  g2  bar/4 + gbarg0 (Famaey & Bin-  ney 2005),  “Standard”: gobs =  1  √  2  �  g2  bar +  �  g2  bar(g2  bar + 4g2  0) (Mil-  grom 1983c),  “RAR”: gobs = gbar/(1 − exp(−  �  gbar/g0)) (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1 plots these functions on top of the RAR data for the  best-fit values on the SPARC data (shown in the lower rows  of Table 1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The Simple and RAR IFs are distin-  guished from the Standard IF principally by a more gradual  transition between the Newtonian and deep-MOND regimes,  although the Standard IF also prefers a significantly higher  value of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While the basic MOND framework is not commit-  ted to any particular IF, it is committed to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 providing an  optimal description of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Our assessment of the theory  will therefore be based on the extent to which the best func-  tions (those with lowest description length) conform to these  limits: any function that does so, in addition to possessing a  coefficient of proportionality of unity in gobs ∝ gbar at high  gbar, may be considered a new MOND IF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (The low-gbar co-  efficient of proportionality is √g0 but g0 is unknown a priori,  so this does not supply an additional requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=') Following  Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2017), we will also consider a double power law  fit:  gobs = θ1  �  1 + gbar  θ0  �θ2−θ3 �gbar  θ0  �θ3  (5)  which has limiting logarithmic slopes of θ3 and θ2, and plot  the best-fit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We define s− ≡ limgbar→0 s and s+ ≡  limgbar→∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The MOND interpretation of the RAR is complicated by  the possibility of the external field effect (EFE), a break-  down of the strong equivalence principle due to the nonlin-  ear, acceleration-based modification to Newtonian mechan-  ics (Milgrom 1983a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The EFE implies that otherwise iden-  tical galaxies in different external gravitational fields have  different dynamics, which is a function of the external field  strength gex relative to g0 and the internal field gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In the  quasi-Newtonian regime gin < gex < g0, Kepler’s laws are re-  covered with dynamical masses scaled by gex/g0, while in the  external field-dominated regime gin < g0 < gex, Newtonian  mechanics are fully recovered (Famaey & McGaugh 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This steepens the RAR at low gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The precise effect of the EFE is difficult to calculate in gen-  eral because it depends both on the underlying MOND theory  and on a galaxy’s morphology and orientation with respect  to the external field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The most sophisticated fitting  functions to MOND simulations are currently to be found  in Zonoozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2021) for QUMOND (Milgrom 2010) and  Chae & Milgrom (2022) for AQUAL (Bekenstein & Milgrom  1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2022) tested these expectations by fit-  ting the SPARC rotation curves for the average external field  strength, finding this to be in good agreement with indepen-  dent estimates based on the baryonic mass surrounding the  SPARC galaxies (Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2021) for AQUAL, but less so  for QUMOND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We will therefore use AQUAL to explore the  effect of the EFE on the expected low-gbar slope and func-  tional form more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Chae & Milgrom (2022) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 15  gives  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The Simple, Standard and RAR IFs, a double power  law, and Simple IF with a global external field strength in AQUAL,  overlaid on the SPARC data (blue points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The parameters are set  to their maximum-likelihood values shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The dashed  black line shows the one-to-one relation (Newtonian limit) and the  cross in the lower right shows the average uncertainty size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  gobs = gbar  �  �  � 1  2 +  �  � 1  4 +  �� gbar  g0  �2  + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1eN)2  �− 1  2  �  �  1  2 �  �  � ×  �  1 + tanh  � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1eN  gbar/g0  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ×  �  − 1  3  �  ×  (6)  ���  gbar  g0  �2  + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1eN)2  �− 1  2  � �  1  4 +  ��  gbar  g0  �2  + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1eN)2  �− 1  2  �− 1  2  1 +  �  1  2 + 2  ��  gbar  g0  �2  + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1eN)2  �− 1  2  � 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This allows for variable disk thickness and scale length, and is  azimuthally averaged to reduce sensitivity to the orientation  of the field relative to the disk axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' It recovers the Simple IF  as eN ≡ gex/g0 → 0 and hence we refer to it as “Simple IF +  EFE”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Note that, while some form of the EFE is generically pre-  dicted by MOND, in modified inertia formulations it may  be very different (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' a function of the entire past trajec-  tory of an object) or negligible (Milgrom 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While there  is evidence for the EFE in many systems (McGaugh & Mil-  grom 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Haghi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020b), in others  it appears conspicuously absent (Hernandez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Fre-  undlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The black curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1 shows the best  fit to the data using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4 Mock data generation  To shed light on the significance of our results we apply  ESR also to two stacks of mock data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We generate each  mock data set using exactly the same number of points as  the SPARC data, and with identical log gbar, δ log gbar and  δ log gobs values, but with log gobs generated using a MON-  Dian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This assumes gt  bar equal to the SPARC gbar  (the maximum a priori estimate), log gbar for each mock re-  alisation drawn from N(log gt  bar, δ log gbar), and log gobs from  MNRAS 000, 1–16 (2022)  102  Simple IF  Standard IF  RAR IF  Double power law  101  Simple IF + EFE  100  10-1  +  10-3  10-2  10-1  100  101  102  9bar / 10-10 ms-2On the functional form of the radial acceleration relation  5  N(F(log gt  bar), δ log gobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To reduce the impact of noise in  the mock data we apply ESR to a stack of 10 independent  realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4 The only terms in the description length that  depend on the dataset size are log(L) and the Fisher matrix  I, both of which scale linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To make the results compat-  ible with the real data we therefore divide these terms by  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The two mock data set stacks differ in the generating func-  tion F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For the first, we use the RAR IF with the best-fit  value on the data g0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='127 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This function is  already known to describe the RAR well (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017)  and has low enough complexity to be included (as a special  case of a more general function, see below) in our function  list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 is satisfied by construction in this case, eval-  uating it on the best functions from ESR will address the  question of whether the dynamic range of the data is suffi-  ciently high—and the uncertainties sufficiently low—to pick  out unambiguously a correctly MONDian solution, as only in  this case could one expect to obtain such behaviour for the  real data were it generated by MOND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The second stack is created using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We adopt g0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  and ⟨gex⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 × 10−2 (eN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='01), corresponding roughly  to maximal clustering of unobserved baryons (as expected  in a MOND cosmology and maximising agreement with the  rotation curve fits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2021) and hence providing  an upper bound on the impact of the EFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is similar  to the value inferred in Chae (2022) and Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2022)  from fits to the SPARC rotation curves, and from our fit to  the SPARC data in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  4 RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1 SPARC data  We show in Table 1 the statistics of the best functions found  by ESR on the SPARC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We split the codelength of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1 into terms describing the residuals of the data around  the functional expectation, the functional form and the pa-  rameter values as shown in the table footnotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Below the  horizontal line we give the results of the three MOND IFs,  for which the free parameter corresponds to g0, the dou-  ble power law (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 5) and the Simple IF + EFE (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  P(f) ≡ exp(−L(D))/ �(exp(−L(D))) is the probability of  4 The reason not to do more is that the time required for param-  eter optimisation scales with the number of data points, and this  is already expensive at complexity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We assess convergence by  fitting the complexity 4-6 equations to an independent stack of 10  realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find that the order of equations when sorted by  L(D) is identical between the two stacks and that the difference  between the description lengths of particular equations falls with  complexity, from ∼ 40 at complexity 4 to ∼ 5 at complexity 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This implies that the best equations, mostly at complexity 8 and  9, are not sensitive to the random number generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  5 Chae & Milgrom (2022) argue that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6 is only reliable with  the inner points of galaxies’ rotation curves removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' As we are in-  terested only in the approximate effect of the EFE on the low-gbar  slope of the RAR, mainly sourced by outer rotation curve points,  we do not apply a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The somewhat larger value eN ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='017  obtained when all points are fitted (Chae 2022, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 7) would have  a greater effect on the low-gbar slope, but is in clear disagreement  with the large-scale structure calculations of Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The top 20 functions found by ESR overlaid on the  SPARC data (blue points), colour-coded by their relative proba-  bility in the full function list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The top panel fits the SPARC data,  the middle panel mock data generated by the RAR IF, and the  bottom panel mock data generated by the Simple IF with uni-  versal external field strength gex = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 × 10−12 ms−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The mock  datasets are 10 times larger than SPARC, although this is factored  out in the description length calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  the function given its description length, where the sum is  over all functions up to complexity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (Note that these values  would be changed by low-L(D) functions at higher complex-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=') For reference, L(D) for the raw data (corresponding to  the hypothesis log gobs = 0) is 53471, showing that significant  compression is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  We find the best-fit g0 value for the RAR IF to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='13,  somewhat lower than the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='20 quoted by Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2017)  although the data are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is because Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (2017) used scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='odr to perform the optimisation rather  than using the full first-order likelihood Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2-3, and also  did the analysis in the log(gobs)−gbar rather than log(gbar)−  log(gobs) plane (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Lelli private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The double  power law fit has much higher maximum likelihood than the  MOND IFs, outweighing its increased codelength due to its  four free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although the RAR IF has complexity  9 it is not explicitly produced by ESR due to the constant  MNRAS 000, 1–16 (2022)  SPARC data  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  log(Prel)  10  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  15  10-1  20  RAR IE mock  103  2  log(Prel)  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  4  10-1  6  Simple IF + EFE mock  103  2  4  log(Prel)  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  6  8  10-1  10  10-3  10-1  101  103  bar /10-10ms-26  Desmond, Bartlett & Ferreira  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The logarithmic slopes s ≡ d log(gobs)  d log(gbar) of the top 10 ESR functions on each dataset, for comparison with the low- and high-gbar  MONDian expectations 1/2 and 1 respectively (blue and red vertical dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The blue and red points are the limiting slopes  s− ≡ limgbar→0+ s and s+ ≡ limgbar→∞ s, while cyan and magenta indicate the slopes at the minimum and maximum gbar of the SPARC  data (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0083 and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In case a slope depends on a parameter value we show the 95% confidence interval as a bar (often very thin),  obtained from an MCMC fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Arrowheads indicate points or bars beyond the range of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The Pareto fronts identified by ESR for the SPARC, RAR IF mock and Simple IF + EFE mock datasets, for both log(L)  (blue) and total description length L(D) (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The quantities plotted have the minimum values subtracted so that the best results appear  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Also shown are the results of the RAR, Simple and Standard IFs, Simple IF + EFE and double power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' ESR significantly  outperforms these “by eye” guesses, even for mock data generated from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Short diagonal lines on the x-axis indicate breaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In the  left and right panels both red and blue points for the Standard IF at complexity 14 lie above the top of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  “1” appearing,6 a generalised form in which this is replaced  by a free parameter appears at rank 17, with a probability  4×1010 times lower than the top-ranked function (∆L(D) =  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' When θ0 ̸= 0 the low-gbar logarithmic slope s− of this  function is 1 rather than 1/2, so it does not function as a  MOND IF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We refer to it as the “generalised RAR IF”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The best ESR functions are clearly superior to the MOND  6 This will be changed in a future version of ESR that performs  “integer snap” of parameters where this reduces L(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  functions or double power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While the best metric for this  is L(D) (or equivalently P(f)), other statistics lead to the  same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' There are many functions more accurate  (lower − log(L)) than even the double power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although  the functions at rank 1-9 have more free parameters than  the IFs this is more than compensated for by their greater  accuracy: as an alternative metric, the Bayesian Information  Criterion (BIC) of the rank 1 function is 108 lower than the  Simple IF and even that of the rank 3 function with 4 free  parameters is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5 lower, corresponding to a very strong pref-  MNRAS 000, 1–16 (2022)  SPARC data  RAR IF mock  Simple IF + EFE mock  s(gbar, min)  2  s+  s(gbar, max)  3 -  rank  4  5  Function  6  7  8 -  9 -  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  Logarithmic slope  Logarithmic slope  Logarithmic slopeSPARC data  RAR IF mock  Simple IF + EFE mock  160  Pareto front  160  ★  RAR IF  140  140  Simple IF  120  Standard IF  120  Simple IF + EFE  100  100  Double power law  Alog  80  80  60  60  40  40 20  20  ★  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  0  5  6  7  8  9  10  11  14  59 5  6  7  8  9  10  11  14  595  6  7  8  9  10  11  14  59  Complexity  Complexity  ComplexityOn the functional form of the radial acceleration relation  7  erence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In agreement with Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' (2020b, 2021, 2022),  when fitting the Simple IF + EFE we find that eN > 0 is  clearly preferred, and recovers a value around the large-scale  structure expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although this function is significantly  more accurate than any IF on its own, more than compensat-  ing for its additional free parameter (∆BIC = −35 compared  to the Simple IF), it has a poor description length due to the  large functional contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The complexity is 59 using our  current basis set of operators, although this would fall to 45 if  tanh were explicitly included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In general, the great improve-  ment in accuracy and simplicity of the ESR functions demon-  strates the advantage of this method over guessing functions  “by eye”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  In the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2 we plot the best 20 functions  on top of the SPARC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The functions are colour-coded  by P(f), with darker colouring indicating functions favoured  by MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The top 6 functions have discontinuities in s at  gbar ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We have checked that this is not due to outlying  points and does not invalidate the first-order approximation  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' instead it is likely due to the relative simplicity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  complexity ≤ 9) of the functions considered, as we discuss  further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While such behaviour could be excluded by  requiring no discontinuity in the derivative within the range  of the data (or more generally weighting functions with an  s-dependent prior), we see no principled reason to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The  first function without a discontinuity is at rank 7, which has  the Newtonian limit and s− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='52 at maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Although this is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3 σ from s− = 1/2, and hence this equation  cannot function as a pure-MOND IF, the EFE leads to the  expectation that s− > 1/2 as discussed in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Several of the remaining highly ranked functions have similar  quasi-MONDian behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  While some of the best functions have MONDian limits  around gbar = 10±3 others do not, and in particular s− < 1/2  is common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To explore this further we calculate s− and s+  analytically for the top 10 equations as a function of their  free parameters, showing the results in Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For each  of the functions where s− and/or s+ depend on θ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' are  not fixed by the functional form alone) we perform a Markov  Chain Monte Carlo (MCMC) inference using the numpyro  sampler (Phan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Bingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2019) with broad  flat priors to constrain the parameters and hence derive the  posterior predictive distributions of the limiting slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3  (left panel) shows the results for the top 10 functions, using  a dot to indicate a limit fixed by the functional form, a bar  to show the 95 per cent confidence interval in cases where the  slope depends on the parameters, and an arrowhead to indi-  cate a limit outside of the plotting range (including ±∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In  many cases the 95% confidence interval is extremely narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Although it is s− and s+ that directly relate to the MOND  hypothesis, they require extrapolation far beyond the range  of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To understand how the slopes behave near the  limits of the SPARC data we also calculate s at the minimum,  gbar, min = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='32 × 10−13 ms−2, and maximum, gbar, max =  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54 × 10−9 ms−2, measured baryonic accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' These  are plotted in cyan and magenta respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' At  gbar, max the logarithmic slope is ≲ 1 for almost all functions,  as expected for the Newtonian limit as only at gbar → ∞  does s become 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' However, we find that the gbar,min slopes of  the top 5 functions are not ∼ 1/2 but actually < 0 due to the  aforementioned discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The remaining top functions  have low-acceleration slopes typically slightly larger than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  These results show that the SPARC data do not unambigu-  ously favour s− = 1/2 and s+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The requirement for an  interpolating function to be MONDian is in fact even more  stringent than this, since the coefficient of proportionality in  the limiting high-gbar power-law relation must be unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  gobs = gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find that among the functions in the top 10  for which s+ = 1, four have such a coefficient (at rank 2, 4,  7, 10) while for the rank 1 function this is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='006 and at  rank 8 it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='01, where the uncertainties are obtained  by fitting the functions with MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' At low gbar the coeffi-  cient in gobs ∝ g1/2  bar should be √g0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The only function  with s− = 1/2 (rank 6) has the coefficient 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='12 (close to the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='10 expected from the canonical g0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2), while those fur-  ther down with sgbar,min ≈ 1/2 have a coefficient of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The  relative simplicity of the functions we consider here should  preference a coefficient of 1, and hence again it is not clear  to what extent the data may be said to be MONDian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The  double power law has the limits gobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='81 g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='03  bar at high gbar  and gobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='57 g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='60  bar at low gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Next, we show in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 the separate Pareto  fronts of description length and negative log-likelihood, with  the second (“simplicity”) objective measured by functional  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Unlike the Pareto fronts produced by traditional  SR algorithms, those of ESR are guaranteed to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  L(D) and − log(L) are minimised separately at each com-  plexity, and have their minimum values over all complexity  subtracted so that the globally best functions appear at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  We show the MONDian functions and double power law as  separate symbols, all of which we find to be strongly Pareto-  dominated by the best ESR functions at lower complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Note that while the “knee” of the Pareto front (where L(D)  or − log(L) turns over) would appear to be at complexity 6-7,  there is a significant improvement in going from complexity 8  to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This cautions against automatically selecting functions  at the knee (the default for example in PySR), and indicates  that further improvement would likely be achievable by go-  ing beyond complexity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is beyond the scope of the  present work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' we are content here to have discovered simple  functional forms for the RAR surpassing any that have been  considered heretofore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 Mock data  The above results suggest a Newtonian limit (gobs → gbar  as gbar → ∞) is somewhat favoured by the data while a  deep-MOND limit (gobs → √g0 gbar as gbar → 0) is question-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' However, given the limited dynamical range and signif-  icant uncertainties of the data it is unclear to what extent  we should expect to find these limits even if the generating  function were MONDian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In addition, the EFE may lead to  the expectation that s > 1/2 at low gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' To investigate these  issues we now apply ESR to the mock data of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1 RAR IF generating function  Table 2 shows the best functions found by ESR for the RAR  IF mock data, along with the results for the IFs and dou-  ble power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The RAR IF is by construction a good fit to  this data, but there are 15 functions with lower L(D), in-  cluding several at lower complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This indicates that the  characteristics of the SPARC data (dynamic range and un-  certainties) are insufficient to pick out the true generating  MNRAS 000, 1–16 (2022)  8  Desmond, Bartlett & Ferreira  Rank  Function  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  P(f)  Parameters  Description length  θ0  θ1  θ2  θ3  Resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  Total  1  θ0  �  |θ1 + x|θ2 + x  �  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3×10−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='38  —  1279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  2  ||θ1|x + θ0|θ2 + x  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='36  —  1279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  3  |θ0||θ1−x|θ2 −θ3  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='89  1276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  1244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  4  |θ0(θ1 + x)|θ2 + x  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='34  —  1268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  5  ��θ0 − |θ1 − x|θ2��θ3  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='14  1271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  1239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  6  √x exp  �  |θ0+x|θ1  2  �  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5×10−9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='36  —  —  1257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  7  � |θ0|x  x  �θ1 + x  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='52  —  —  1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  8  �  |θ0 + x| + θ1x  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='72  —  —  1245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  9  ���θ0 +  1  4√x  ���  θ1  8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6×10−11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='22  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='14  —  —  1251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  10  �√x + 1  x  �θ0 + x  9  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2×10−11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='53  —  —  —  1248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  17  x/(exp(θ0) − |θ1|  √x)  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2×10−11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='44  —  —  1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  1226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  —  Double power law  11  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7×10−16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='65  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='60  1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  1216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  —  Simple IF  10  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5×10−25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='11  —  —  —  1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  —  RAR IF  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7×10−26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='13  —  —  —  1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  —  Simple IF + EFE  59  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−3  —  —  1238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  1093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  —  Standard IF  14  9×10−150  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54  —  —  —  939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  1 − log L(ˆθ)  2k log(n) + �  j log(cj)  3 − p  2 log(3) + �p  i (log(Iii)1/2 + log(|ˆθi|))  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Top functions found by ESR applied to the SPARC data, ranked by total description length, compared to four MOND IFs and  a double power law below the horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' x ≡ gbar/10−10 ms−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We include also the “generalised RAR IF” at rank 17, although this  does not have a deep-MOND limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The IFs and double power law are not produced explicitly by our implementation of ESR and hence  their ranks are unknown, although they are clearly worse than the best ESR functions in both description length and likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For the  Simple, RAR and Standard IFs the parameter is g0, while for the “Simple IF + EFE” (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 6), the first is g0 and the second eN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  function: ESR prefers simpler functions which may achieve  slightly higher likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Since the − log(L) term in L(D)  becomes dominant at large dataset size, simply increasing the  number of (mock) observations with otherwise identical prop-  erties would not be sufficient to push the generating function  to the top of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For this dataset we find that the gener-  alised RAR IF, appearing at rank 41, has higher L(D) than  the RAR IF despite slightly higher likelihood, a success of  MDL’s penalisation of more complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Note that the  best-fit generalised RAR IF is x/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='995 − exp(−  �  x/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='068)),  somewhat offset from the ground-truth values {1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='127}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This results from a combination of the limited dataset size  (introducing random noise) and a small bias in the maximum-  likelihood estimator which we discuss further in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  At rank 27 we find a close cousin of the RAR IF in which the  “1” is free and g0 is pinned to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This performs only slightly  worse than the RAR IF itself because the true g0 is close to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The highest ranked ESR function is better by ∆L(D) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  than the RAR IF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although the relative probability between  the best function and the best RAR-like function is smaller  than in the observations (∼ 1600 compared to 4 × 1010), it is  interesting that the best RAR-like function appears further  up the list in the real data (rank 17 vs 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The double power  law is disfavoured relative to the RAR IF and many ESR  functions despite having the highest likelihood shown in the  table, another success of the complexity penalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' There  are a few functions overall with lower − log(L), the lowest  being −2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0 at rank 51 ((θ0 + |θ1 + θ2/x|1/2)−1, L(D) =  −2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Also as expected, the Simple IF + EFE has eN  snapped to 0 and hence behaves identically to the Simple IF,  albeit with larger functional complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The top 20 ESR functions for the RAR IF mock are over-  plotted on that data in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find a  slightly reduced spread in s at both the high-gbar and low-gbar  ends compared to the real data, without any discontinuities  within the data range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' However there is still significant un-  certainty beyond the range of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is quantified in  the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3, where the top 10 functions are all  observed to have slopes of approximately 1/2 at gbar,min and  1 at gbar,max, although only in two cases is s− = 1/2: for the  others it is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This indicates that constraining the slope  to be near 1/2 at gbar,min is insufficient to conclude that s−  takes a similar value, at least up to complexity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' One would  need to reduce the uncertainties, or, preferably, lower gbar,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  That this applies to a lesser extent at high-gbar is shown by  the functions at rank 4 and 7 with s+ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Adding to the conclusion that the mock data characteristics  are insufficient to pick out a MONDian generating function,  we find that the coefficient in gobs ∝ gbar at high gbar is only  unity for one of the top-10 functions for which s+ = 1 (at  rank 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For all the others it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64, with the exception of  that at rank 1 where it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' These values have uncertain-  ties ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='003 when constrained by MCMC, and vary by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02  MNRAS 000, 1–16 (2022)  On the functional form of the radial acceleration relation  9  Rank  Function  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  P(f)  Parameters  Description length  θ0  θ1  θ2  θ3  Resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  Total  1  θ0 + θ1x + √x  8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6×10−1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='63  —  —  2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  2  �  |θ0 + x| + θ1x  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64  —  —  2044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  3  θ0x + xθ1  7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='49  —  —  2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  4  √x exp  �  xθ0  2  �  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='36  —  —  —  2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  5  (θ0 + x)  �  θ1 +  1  √x  �  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1×10−2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64  —  —  2044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  6  1  �  |θ0+ 1  x |  + x  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='74  —  —  —  2038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  7  (x|θ0|)(x|θ1|)θ2  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1×10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='04  —  2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  8  θ0x + |θ1 + x|θ2  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  41  x/(exp(θ0) − |θ1|  √x)  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1×10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='38  —  —  2042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  —  RAR IF  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='14  —  —  —  2041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  —  Double power law  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54  2047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  —  Simple IF  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='12  —  —  —  2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  —  Standard IF  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9×10−55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54  —  —  —  1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  —  Simple IF + EFE  59  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9×10−64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='12  0  —  —  2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  1 − log L(ˆθ)  2k log(n) + �  j log(cj)  3 − p  2 log(3) + �p  i (log(Iii)1/2 + log(|ˆθi|))  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' As Table 1 but for the RAR IF mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find the generalised RAR IF at rank 41, and a closely related function at rank  27 in which the free parameter appears in the other term in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For this dataset the RAR IF itself is superior to both of  these modified forms, as would be expected given that it generated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  over mock datasets differing only in the random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus  the best functions almost always fail to recover the Newto-  nian limit even when it is the truth, presumably due to an  insufficient gbar,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The origin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64 is unclear, but presum-  ably results from the way the lower-gbar behaviour is filtered  through the forms of the functions found to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In  cases where s− = 1/2, the coefficient of proportionality is 1  (to be compared to √g0 in MOND), and the double power law  limits are gobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='20 g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='90  bar at high gbar and gobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='30 g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54  bar  at low gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 shows the Pareto front for these  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We find more smooth behaviour than for the real data,  with the optimum solution achieved already by complexity  8, reflecting the relatively simple nature of the generating  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This shows that if the RAR IF was generating  the real data (and our likelihood and mock data generation  method were accurate), we would have achieved the L(D)  minimum on those data too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' However, the MONDian func-  tions (including the RAR IF itself) and double power law are  Pareto-dominated by the ESR results even on these mock  data, showing that one should not expect to be able to re-  cover unambiguously even this simplest of MOND generating  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 Simple IF + EFE generating function  Analogous results for the mock data generated using the Sim-  ple IF with inclusion of the EFE are shown in Table 3 and the  bottom/right panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This dataset behaves more  similarly to the real data in terms of the relative ordering of  the IFs, double power law and ESR functions, including the  generalised RAR IF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Indeed, the best-fit parameters of all  three non-EFE IFs are identical to the SPARC data to two  decimal places, while those of the generalised RAR IF and  the high-gbar slope of the double power law are the same to  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Here the IFs provide a significantly worse compression of  the data than the best ESR functions, and the double power  law also performs relatively poorly due to the curvature at  low gbar (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' There is again a small bias between  the maximum-likelihood (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='19 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='56×10−3) and true (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  and 1×10−2) g0 and eN values for the Simple IF + EFE  fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although this function has among the highest likelihoods  achievable by ESR up to complexity 9, its functional com-  plexity makes it a poor compression of its own SPARC-like  mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This reinforces the conclusion that the charac-  teristics of these data are insufficient to identify a MONDian  generating function: this one in particular would require far  more data than the RAR IF to be favoured by MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  For the Simple IF + EFE mock all top-10 functions have  s+ = 1 and s(gbar,max) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' However, only 6 of them re-  MNRAS 000, 1–16 (2022)  10  Desmond, Bartlett & Ferreira  Rank  Function  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  P(f)  Parameters  Description length  θ0  θ1  θ2  θ3  Resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  Total  1  θ0 +  √  x2 + 2x  9  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9×10−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='06  —  —  —  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  2  θ0 +  �  x|θ1 + x|  8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3×10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='97  —  —  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  3  −|θ0|  √x + θ1 + x  8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='95  —  —  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='3  1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  6  |θ0 − x|θ1 − θ2x  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='6  1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  7  x|θ0|−|θ1|xθ2  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='33  —  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  8  x  �  θ0 + |θ1 + x|θ2�  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='53  —  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  9  |θ0||θ1|xθ2 + x  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5×10−4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='17  —  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  10  exp  �  θ0 −  1  4√x  �  + x  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='57  —  —  —  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  21  x/(exp(θ0) − |θ1|  √x)  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='44  —  —  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  —  Double power law  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4×10−11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='53  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='60  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  —  Simple IF  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2×10−22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='11  —  —  —  1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  —  RAR IF  9  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0×10−24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='13  —  —  —  1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  —  Simple IF + EFE  59  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8×10−57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='19  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6×10−3  —  —  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  —  Standard IF  14  2×10−141  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='54  —  —  —  1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  1676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  1 − log L(ˆθ)  2k log(n) + �  j log(cj)  3 − p  2 log(3) + �p  i (log(Iii)1/2 + log(|ˆθi|))  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' As Table 1 but for the Simple IF + EFE mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  cover the true Newtonian limit gobs = gbar: the others find  gobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='71 gbar or gobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='79 gbar, again with sub-percent  uncertainty from MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus under this model too one  would not expect the Newtonian limit to be identified ro-  bustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While s(gbar,min) > 1/2, indicating the significant im-  pact of the EFE, s− is typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus gbar,min is too high  to constrain s− reliably, although this may also be a reflec-  tion of the relative simplicity of the functions we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The double power law limits are gobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='96g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='98  bar at high gbar  and gobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='55 g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='60  bar at low gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The Pareto front indicates  this dataset to be somewhat more complex than the RAR IF  mock, as L(D) continues to fall to complexity 9, although the  smoothness shows it to be simpler than the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  All functions considered achieve considerably higher likeli-  hood on the mock datasets than the real data, showing that  the mocks are simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This could be because the data do not  conform to the MOND expectation—one would expect any  given gobs = f(gbar) to be relatively inaccurate in a chaotic  ΛCDM galaxy formation scenario—or because the model for  scattering the mock data points is overly simplistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is  discussed further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Relatedly, the P(f) values of the  top functions are very closely spaced in the mock datasets,  indicating that there is little to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' On the  contrary, on the real data the integrated probability of all  functions besides the top 5 is ≲10−8, suggesting that these  functions perhaps ought not to be considered at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The lim-  iting slopes of the top 10 equations as a function of the pa-  rameters can be found in Tables B2 and B3 for the RAR and  EFE mocks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  5 DISCUSSION  Our main conclusion is that the SPARC data are insufficient  to determine robustly the limiting behaviour of the RAR, and  hence cannot verify or refute the MOND hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This  is reached by studying mock data generated by MOND;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' in  particular, generating data according to the RAR IF, not  only are we unable to identify that as the generating function,  but, more seriously, we cannot reconstruct s− = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' At the  high-gbar end, the logarithmic slope of the Newtonian limit  (s+ = 1) is typically well recovered, although the coefficient  of proportionality in gobs ∝ gbar is not: in the RAR IF mock  data this takes a values ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64 far more often than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Improving this situation requires increasing the dynamical  range of the RAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' At the low-gbar end this may be achieved  by studying ultra-diffuse galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Freundlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2022),  or local dwarf spheroidals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' McGaugh & Wolf 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Mc-  Gaugh & Milgrom 2013), some of which seem seem to in-  dicate s− ≈ 0, as found by many of the best ESR func-  tions (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Alternatively, one may attempt to  probe the outer regions of galaxies including the Milky Way  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Oman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Particularly promising for a large  gain is to use stacked weak lensing to probe galaxy outskirts  that would have insufficient signal-to-noise on an individual-  object basis (Brouwer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This appears to indicate  s− ≈ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Increasing gbar,max requires probing the central re-  gions of high-mass ellipticals, as well as groups and clusters  of galaxies (Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020a, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Gopika & Desai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Chan & Del Popolo 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Pradyumna &  Desai 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Such data already exists and may readily be  folded into our framework to increase its constraining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000, 1–16 (2022)  On the functional form of the radial acceleration relation  11  A smaller information gain may be achieved by reducing the  uncertainties in gbar and gobs: in the limit of no uncertainty  any generating function will be assigned P(f) = 1 by MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  By generating mock data with different characteristics one  could ascertain the requirements for various features of the  functional form to be unambiguously determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  It is likely that there exist functions at higher complexity  superior to those of Tables 1-3, especially for the real data  where L(D) drops significantly from complexity 8 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This  would be computationally demanding using the ESR algo-  rithm, and thus a stochastic search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' using a genetic al-  gorithm) may be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This search may be seeded by  the ESR functions: the fact that many of the best-fitting  functions have similar features (such as gobs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64 gbar as  gbar → ∞ for the RAR IF mock) suggests these may be  useful for higher-complexity functions also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus ESR may  be used to validate the underlying assumption of stochas-  tic searches—that there exist features of functions responsi-  ble for their fitness—and the identification of these features  may be useful for tuning hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While uncover-  ing lower-L(D) functions at higher complexity may update  the optimal limiting behaviours of the functional form of the  RAR, and hence its compatibility with MOND, it cannot  compromise our discovery of simpler and more accurate func-  tions than the IFs and double power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Indeed, the fact that  the (or at least a) knee of the Pareto front is reached around  complexity 7 in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4 shows that such func-  tions already offer a powerful compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Our best functions on the real data have a discontinuity  in s around gbar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is likely due to the limited  complexity of the equations we consider: a cusp is the sim-  plest way of reducing s sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' It is probable that the op-  timal functions at higher complexity will have a smoothed  form of this behaviour in which s does not become negative  and may not tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We therefore doubt that the s− and  s(gbar,min) values of the best functions in Table 1 are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  One could attempt to construct more complex functions in-  spired by the ESR results with similar but not discontinuous  behaviour and calculate their − log(L) and L(D) separately,  or feed them into a genetic algorithm as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  On the other hand, below complexity 9, there is only a sin-  gle low description length function that is discontinuous, the  third best function at complexity 8 (P(f) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 × 10−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The best functions at lower complexity more frequently have  s− = 1/2 and s+ = 1, although again they rarely satisfy  gobs = gbar as gbar → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For example, the top function at  complexity 6—marking the (first) knee of the L(D) Pareto  front in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 4—is gobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='70 gbar + √gbar, exhibiting simi-  lar high-gbar behaviour to the 1st- and 8th-ranked functions  overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  To generate the mock data we assumed that all gbar values  are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While this is likely true between galaxies,  it is not within a single galaxy because the uncertainty in  gbar is dominated by the mass-to-light ratio, a global galaxy  parameter in the simplest approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This may be seen  from the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1, where lines of points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' scattering  low around gbar = 2 or high around gbar = 10) are all from the  same galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A more robust procedure may be to generate Υ  values for each mock galaxy by randomly drawing from their  priors, use this to transform gbar and then add any other,  random sources of noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' from the uncertainty in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6 µm  luminosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' By enhancing inter-galaxy variations this may  increase the complexity of the mock datasets, moving their  ESR results towards those of the SPARC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Alternatively,  one may fit each galaxy separately to assess compatibility of  their individual RARs (analogously to Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2018 but not  just for the RAR IF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The assumption of uncorrelated data  points is also present in our likelihood, as discussed further  in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  We have assumed no intrinsic scatter in the RAR, such  that all deviations from the hypothetical functional expecta-  tion must come from the observational uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While  this is expected in MOND, in ΛCDM the complex process of  galaxy formation would lead to a significant and parameter-  dependent effective intrinsic scatter (Desmond 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Even  the EFE would introduce some scatter due to galaxy-by-  galaxy variation in gex (Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' It would be  straightforward to add this (in some direction on the RAR  plane) as an additional free parameter of all functions, which  would alter the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' MDL naturally penalises the addi-  tion of this parameter, allowing one to determine whether it  is justified for any given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This would provide further  evidence concerning the optimality of MOND by assessing  the extent to which the data implies law-like modified grav-  ity behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Our current implementation of MDL treats the parameter  values as part of the model and chooses them to maximise the  likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' An alternative would be to treat the hypothesis in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1 as the functional form alone, assigning codelengths and  probabilities to functions regardless of their parameter val-  ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' In a Bayesian formulation this corresponds to marginal-  ising over the parameters, and enables a simpler one-part  coding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' An even higher-level approach would be to  group functions into sets with specific properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' limit-  ing behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This would enable calculation of the posterior  predictive distribution of any feature of the functional repre-  sentation of the dataset, and hence enable model comparison  at any level of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The relative simplicity of the RAR and conformity to the  Newtonian and deep-MOND limits are the key differences  between the expectations of MOND and the more chaotic  galaxy formation scenario of ΛCDM: it is only under the  simpler scenario that one would expect to find a simple  gobs = F(gbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' While our results are therefore not partic-  ularly supportive of the MOND hypothesis, this is not to say  either that the data could not plausibly have been generated  by MOND or that it could plausibly have been generated  under another hypothesis, as only MOND currently has suf-  ficient predictivity for a test of this precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We look to  future SR studies with more data to establish the functional  form of the RAR—if it exists—definitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  6 CONCLUSION  The radial acceleration relation (RAR) has become central to  debates about the mass discrepancy problem on astrophysical  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Its tightness and regularity have been used to argue for  a violation of Newtonian gravity in accordance with Modified  Newtonian Dynamics (MOND), but the functions used to fit  the data have been constructed to conform to this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  As the first detailed application of the brand-new technique  of Exhaustive Symbolic Regression, we rank objectively all  simple functions in terms of their aptitude for describing the  MNRAS 000, 1–16 (2022)  12  Desmond, Bartlett & Ferreira  SPARC RAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We employ the minimum description length  principle to trade accuracy with simplicity and hence perform  model selection, and calibrate our method on mock MOND  data generated both with and without the external field effect  (EFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Our conclusions are as follows:  ESR discovers functions which are better descriptions, in  both accuracy and simplicity and for both observed and sim-  ulated data, than MOND functions or a double power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  While the majority of best-fitting functions on the SPARC  data recover gobs ∝ gbar at high accelerations, not all have  a best-fit coefficient of proportionality near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Thus the  Newtonian limit is not clearly evidenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The SPARC data do not prefer functions with the deep-  MOND limit of gobs ∝ √gbar as gbar → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Instead, we find  that functions with gbar → const typically compress the data  more efficiently, albeit with considerable uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  SPARC-like mock data generated assuming the MONDian  RAR interpolating function do not unambiguously recover  that function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Moreover, many of the best functions for those  mock data have gobs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='64 gbar rather than gobs = gbar at  high gbar, and most do not have a deep-MOND limit at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The EFE in AQUAL greatly increases the logarithmic slope  of the best-fitting functions at the low-gbar end of the data,  but does not appreciably impact the limiting slope at gbar →  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Incorporating the EFE in the mock data produces more  generally similar results to the real data, so our analysis  (within the MOND paradigm) hints at it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  We conclude that the data have too small a dynamic  range (and too large uncertainties) to unambiguously favour  MOND even if it is in fact generating the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The SPARC  RAR alone, therefore, does not supporting that theory un-  ambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The best prospect for improving this situation  is to increase the acceleration range of the data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' using  stacked weak lensing at low gbar and groups and clusters at  high gbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Our results are a function of the maximum complexity of  equation considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Future symbolic regression algorithms—  exhaustive or non-exhaustive—will reach the true description  length minimum and hence uncover the optimal functional  representation of the RAR and determine whether the rela-  tion implies novel law-like gravitational behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Exhaustive Symbolic Regression provides for the first time  a guaranteed complete search through functional parameter  space, making it the ideal tool to determine the analytic  form of observed relations, extract physics from data theory-  agnostically, and create fitting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We make the ESR  and RAR codes, full function sets and the best 50 functions  for each dataset we consider publicly available to facilitate  future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  7 DATA AVAILABILITY  The code and data associated with ESR and its applica-  tion to the RAR are released at � and in Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The SPARC data is available at http://astroweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  cwru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='edu/SPARC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Other data may be shared on request to the  corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank Kyu-Hyun Chae, Andrei Constantin, Miles Cran-  mer, Mario Figueiredo, Gianluca Gregori, Thomas Harvey,  Mark Kotanchek, Federico Lelli, Stacy McGaugh, Andre  Lukas, Richard Stiskalek and Tariq Yasin for useful inputs  and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  HD is supported by a Royal Society University Research  Fellowship (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 211046).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' DJB is supported by the Si-  mons Collaboration on “Learning the Universe” and was sup-  ported by STFC and Oriel College, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' PGF acknowl-  edges support from European Research Council Grant No:  693024 and the Beecroft Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This project has received funding from the European Re-  search Council (ERC) under the European Union’s Horizon  2020 research and innovation programme (grant agreement  No 693024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This  work  used  the  DiRAC  Complexity  and  DiRAC@Durham  facilities,  operated  by  the  University  of Leicester IT Services and Institute for Computational  Cosmology, which form part of the STFC DiRAC HPC  Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This equipment is funded by  BIS National E-Infrastructure capital grants ST/K000373/1,  ST/P002293/1, ST/R002371/1 and ST/S002502/1, STFC  DiRAC Operations grant ST/K0003259/1, and Durham  University  and  STFC  operations  grant  ST/R000832/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  DiRAC is part of the National E-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  For the purpose of open access, the authors have applied  a Creative Commons Attribution (CC BY) licence to any  Author Accepted Manuscript version arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content=', 2021,  MNRAS, 506, 5468  APPENDIX A: A NOTE ON LIKELIHOODS  A1 Correlation of measurements and their  uncertainties  The likelihood of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2 treats the uncertainties induced by  Υ, D and i as statistical and uncorrelated between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This is clearly incorrect: a scattering up of Υ, for example,  causes a coherent increase in gbar across the rotation curve,  generating off-diagonal elements in the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A  better approach is therefore to calculate the covariance ma-  trix via Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For a given galaxy, one would indepen-  dently sample Υgas, Υdisk, Υbulge, D and i many times from  their assumed-Gaussian prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' One would then  generate the corresponding gobs and gbar values (now vectors  across the rotation curve of a given galaxy), scatter them by  their statistical uncertainty (gbar has only the small δL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  term), and calculate the covariance matrices Σobs and Σbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  Let us define u ≡ log (gbar) and v ≡ log (gobs), such that  u and v are vectors containing u and v for all galaxies at  all measured points along the rotation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We denote the  “true” values with a superscript t and the observed values  without a superscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We assume that ut and vt are drawn  from multivariate Gaussian distributions with covariance ma-  trices Σu and Σv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Since ut and vt are statisti-  cally independent random variables for fixed u and v, the  likelihood of a given set of points in the  �  ut, vt�  plane would  then be  L (u, v) =  1  �  |2πΣu|  1  �  |2πΣv|  × exp  �  −1  2  �  ut − u  �T Σ−1  u  �  ut − u  ��  × exp  �  −1  2  �  vt − v  �T Σ−1  v  �  vt − v  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (A1)  The equivalent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2 for a vector-valued function f, v =  f (u), is then obtained by marginalising over ut,  L (v) =  �  L  �  ut, f  �  ut��  dut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (A2)  Taylor expanding f  �  ut�  about u,  f  �  ut�  ≈ f (u) + D  �  ut − u  �  ,  (A3)  where Dij ≡ ∂jfi|u, yields  L (v) =  1  �  |2πΣtot|  × exp  �  −1  2 (v − f(u))T Σ−1  tot (v − f(u))  �  ,  (A4)  where Σtot is the total covariance matrix defined by  Σtot ≡ Σv + DΣuDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (A5)  This is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3 if u and v have only a single  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  In principle, the assumption that the joint probability dis-  tributions of ut and vt are a multivariate Gaussian is un-  necessary as the full empirical distributions are generated by  the Monte Carlo sampling described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Using this di-  rectly would enable a loss function which, while still inde-  pendent between galaxies, fully encapsulates the correlated,  non-Gaussian structure of each galaxy’s measurements and  hence provides a more accurate description of the expected  probability distributions of gbar and gobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This model would  also allow for more accurate mock data generation as dis-  cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  A2 Sampling nuisance parameters from their  posteriors  All the methods described above sample Υ, D and i, and the  true gbar values ut, from their priors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' without adjusting  MNRAS 000, 1–16 (2022)  14  Desmond, Bartlett & Ferreira  them to maximise agreement with the function being fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  In principle, the better procedure is to constrain these nui-  sance parameters jointly with any parameters of the function,  assuming that function in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Even the relatively simple  case where ut is drawn from the prior but Υ, D and i from  the posterior is however impractical as it requires optimisa-  tion in a parameter space of dimension 147 × 3 + p if the  three Υs are coupled (or only one is varied), and 147 × 5 + p  if they are varied separately, where p is the number of free  parameters in the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This would naturally account for  correlations between the gbar and gobs measurements induced  by variations in Υ, D and i, and thus provides a more accu-  rate but expensive alternative to the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This  approach will be applied for the first time to the RAR, out-  side the context of SR, in upcoming work (Desmond 2023, in  prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  The situation is complicated further by inference of ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  At the top level of the hierarchical model for predicting v  from u are both the parameters of the model, θ, and ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' If  one wanted to find the true maximum likelihood point, one  should maximise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A1 for ut as well as θ, instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A2  for θ alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  To do this, we start by noting that maximising the likeli-  hood in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A1 is equivalent to minimising  h  �  ut, v  �  = 1  2  �  f  �  ut�  − v  �T Σ−1  v  �  f  �  ut�  − v  �  + 1  2  �  ut − u  �T Σ−1  u  �  ut − u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  (A6)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A3 and minimising h with respect to ut, we find  the maximum-likelihood point, ˆut, to be  ˆut = u + ΣuDTΣ−1  tot (v − f (u)) ,  (A7)  and we must therefore maximise  L  �ˆut, v  �  =  1  �  |2πΣu|  1  �  |2πΣv|  × exp  �  −1  2 (v − f(u))T Σ−1  tot (v − f(u))  �  ,  (A8)  where Σ−1  tot is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This will not yield the same  result as maximising Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A4 since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A8 does not contain the  normalisation term which penalises large gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' When ap-  plied to mock data generated assuming vt is linearly related  to ut, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A4 (the commonly used expression,  and the one we adopt in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2) induces a small bias that  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A8 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This is the reason why the best-fit RAR and  generalised RAR IFs do not precisely match the generating  function in Table 2, or the Simple IF + EFE the generating  function in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We should expect all best-fit parameter  values to be likewise slightly biased by the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  In practice, it is challenging to maximise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A1 since  the lack of the gradient-penalising determinant means that  naïvely using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A8 can prefer functions with diverging gra-  dients at at least one point in the domain of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This breaks  the linearity assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' A3, making the result un-  trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Instead, one should numerically solve the full  optimisation problem without Taylor-expanding f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' This in-  volves solving a root-finding problem for each trial θ during  the optimisation of the function’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Although fea-  sible for a handful of functions, this is computationally im-  practical for the full set of ESR functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We therefore use  the simpler likelihood here, calibrating our results using mock  data to measure the magnitude of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We defer further  discussion of this important issue to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  APPENDIX B: LIMITING SLOPES OF ESR  FUNCTIONS  Here we provide the analytic low-gbar (s−) and high-gbar (s+)  logarithmic slopes of the top 10 functions generated using  ESR up to complexity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' We give the slopes for the observed  SPARC data in Table B1, the mock data assuming the RAR  IF in Table B2, and the mock data assuming the Simple IF  with EFE in Table B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  This paper has been typeset from a TEX/LATEX file prepared by  the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1–16 (2022)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='On the functional form of the radial acceleration relation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Rank  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Low-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='High-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ1 + x|θ2 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='||θ1|x + θ0|θ2 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='(|θ1| ≤ 1) or (|θ1| > 1 and θ2 ≤ 0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0||θ1−x|θ2 −θ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0| > 1 and θ2 > 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0 < |θ0| < 1 and θ2 > 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0(θ1 + x)|θ2 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='��θ0 − |θ1 − x|θ2��θ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2θ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 ≥ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='√x exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0+x|θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 ≤ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='� |θ0|x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�θ1 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 ≥ −1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='(|θ0| > 1 and θ1 ≤ 0) or (0 < |θ0| < 1 and θ1 ≥ 0) or (|θ0| = 1 and θ1 ≥ −1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0| = 1 and θ1 < −1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1 + 0θ1θ1(−∞)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0 + x| + θ1x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='���θ0 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4√x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='− θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�√x + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�θ0 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−θ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≥ −1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≥ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Functional forms and limiting slopes of the ten best functions found by ESR applied to the SPARC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The functions in the  second column give the fitted y = gobs/10−10 ms−2 for input x = gbar/10−10 ms−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' The low-acceleration slope is limx→0+ d log y/d log x  (denoted s− in the text), and the high-acceleration slope is similarly defined but for x → ∞ (s+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' {θi} are real parameters fitted to  the data to maximise the likelihood (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Slopes given without conditions are valid ∀ θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' For comparison, the MOND prediction  without the external field effect is s− = 1/2 and s+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' 1–16 (2022)  16  Desmond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' Bartlett & Ferreira  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Rank  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Low-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='High-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 + θ1x + √x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0 + x| + θ1x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0x + xθ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 ≤ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='√x exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='xθ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≥ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≤ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='(θ0 + x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='√x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x |  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='+ x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='(x|θ0|)(x|θ1|)θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='|θ1| > 0 and θ2 < 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='|θ1| > 0 and θ2 > 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ1| > 0 and θ2 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0θ2θ2∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0x + |θ1 + x|θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='θ2 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0 − x|θ1 − θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 ≥ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='(θ0 − x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 − xθ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 ≤ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ2 ≥ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='otherwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content=' As Table B1 but for the mock data generated from the RAR IF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Rank  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Low-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='High-acceleration slope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x2 + 2x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='θ0 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='x|θ1 + x|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='−|θ0|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='√x + θ1 + x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ0| ≤ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='(θ0 − x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ1 − xθ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='θ2 ≤ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='xθ0 − θ1(θ2 − x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≤ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='|θ0 − x|θ1 − θ2x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='θ1 ≥ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='x|θ0|−|θ1|xθ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='∞ log |θ0|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='|θ1| > 1 and θ2 < 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='|θ0| > 1 and |θ1| > 1 and θ2 > 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='(θ0 = 0) or (θ1 = 0) or (|θ0| < 1 and |θ1| > 1 and θ2 < 0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='θ0 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='4√x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='+ x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='Table B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content=' As Table B1 but for the mock data generated using the Simple IF + EFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
+page_content='  MNRAS 000, 1–16 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE3T4oBgHgl3EQfMAkk/content/2301.04368v1.pdf'}
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+arXiv:2301.03922v1  [math.PR]  10 Jan 2023
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING
+PARTICLE SYSTEMS
+BENEDIKT JAHNEL AND JONAS K ¨OPPL
+Abstract. A classical approach for the analysis of the longtime behavior of Markov processes is to
+consider suitable Lyapunov functionals like the variance or more generally Φ-entropies.
+Via purely
+analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite
+general assumptions on the process. In this paper, we complement these classical results via a more
+probabilistic approach and show that dissipation is already present on the level of individual trajectories
+for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry
+and compact local spin spaces. This extends previous results from the setting of finite-state Markov
+chains or diffusions in Rn to an infinite-dimensional setting with weak assumptions on the dynamics.
+1. Introduction
+There are many different techniques to study the long-time behavior of Markov processes that excel
+in different situations. One very common and powerful technique is the use of Lyapunov functionals, i.e.,
+functionals that are monotone in time. An example of such a functional is the variance
+Varµ(f) := Eµ[f 2] − Eµ[f]2,
+f ∈ L2(µ),
+where µ is an invariant measure for some Markov process with semigroup (Pt)t≥0. If we now fix an
+observable f and consider the function
+[0, ∞) ∋ t �→ Varµ(Ptf) ∈ [0, ∞),
+then it is easy to see that this is non-increasing and under some further assumptions one can even
+show that it is strictly decreasing for all non-constant observables f. This whole viewpoint, however, is
+purely based on functional analytic arguments and one does not even need to speak about the underlying
+Markov process itself to carry out the corresponding calculations. From a probabilistic point of view, this
+is somewhat unsatisfactory and we therefore want to specify this coarse and non-probabilistic approach
+with a finer, more probabilistic technique that allows us to obtain trajectorial versions of these results.
+Here, by trajectorial we mean results on the level of single realizations of a stochastic process, as opposed
+to averaged quantities. Thereby our goal is to uncover more of the underlying probabilistic mechanisms
+behind the decay of variance, or more generally the decay of Φ-entropies. For this, we will first briefly
+recall the notion of Φ-entropies and then explain our main results and ideas with the help of the simple
+example of a continuous-time Markov chain on a finite state space. The rest of the article is then devoted
+to extending these ideas to the setting of spatially extended systems of infinitely many interacting particles
+as e.g. considered in [Lig05].
+1.1. Φ-entropies and their decay under Markovian dynamics. Let Φ : I → R be a smooth and
+convex function defined on a not necessarily bounded interval I ⊂ R. Let (E, B(E)) be a Polish space
+equipped with its Borel σ-algebra and assume that µ is a probability measure on (E, B(E)). The Φ-
+entropy functional is then defined on the set of µ-integrable functions f : E → I by
+EntΦ
+µ(f) :=
+�
+E
+Φ(f)dµ − Φ
+��
+E
+fdµ
+�
+= Eµ [Φ(f)] − Φ (Eµ [f]) .
+By Jensen’s inequality one can immediately deduce that the Φ-entropy functional takes its values in
+R+ ∪ {+∞}. Moreover, EntΦ
+µ(f) vanishes if its argument is constant and if Φ is strictly convex, then the
+converse is also true. For special choices of Φ one can recover the classical variance and relative entropy
+functionals since we have
+Entu�→u2
+µ
+= Varµ,
+Entu�→u log u
+µ
+= h(·|µ).
+Now let (X(t))t≥0 be a Markov process on our Polish space E with associated semigroup (Pt)t≥0 acting
+on Cb(E; R), the space of continuous and bounded real-valued functions on E. Let us assume that there
+Date: January 11, 2023.
+2020 Mathematics Subject Classification. Primary 82C20; Secondary 60K35.
+Key words and phrases. Interacting particle systems, phi-entropy, time-reversal, martingale representation.
+1
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+2
+exists an invariant probability measure µ and denote by L the generator of the semigroup (Pt)t≥0 with
+domain dom(L ) ⊂ Cb(E; R).
+By invariance of µ and Jensen’s inequality one can now deduce that for all f ∈ Cb(E; R)
+EntΦ
+µ(Ptf) = Eµ [Φ(Ptf)] − Φ (Eµ [Ptf]) ≤ Eµ [PtΦ(f)] − Φ (Eµ [f]) = EntΦ
+µ(f).
+This tells us that the Φ-entropy is non-increasing as a function of t and can be used as a Lyapunov
+function. More precisely, with purely analytic arguments, one can even deduce the following general
+result about the decay of Φ-entropies.
+Proposition 1.1 (DeBruijn like property for Markov semigroups, [Cha04]). Let (X(t))t≥0 be a Markov
+process on a Polish space E equipped with its Borel σ-algebra B(E) and let (Pt)t≥0 be the associated
+Markov semigroup with generator L . Assume that µ is an invariant probability measure. Then, for any
+continuous and bounded function f : E → I and any t > 0, it holds that
+∂t EntΦ
+µ(Ptf) = Eµ [Φ′(Ptf)L (Ptf)] ≤ 0.
+This result is classical, but we nevertheless recall its short analytic proof. We will later also provide a
+more probabilistic argument to obtain the same result in the context of interacting particle systems.
+Proof. The chain rule and the definition of the generator L directly imply that
+∂t EntΦ
+µ(Ptf) = Eµ
+�
+Φ′(Ptf) d
+dt(Ptf)
+�
+= Eµ [Φ′(Ptf)L (Ptf)] .
+To see that the left-hand side is actually non-positive, it suffices to observe that the convexity of Φ implies
+via Jensen’s inequality for conditional expectations
+Φ(Pt+sg) ≤ Pt(Φ(Psg))
+for any s, t ≥ 0 and hence for all f we have
+EntΦ
+µ(Pt+sf) ≤ EntΦ
+µ(Psf),
+by invariance of µ.
+□
+By integrating one obtains the following classical corollary, which links exponential decay of Φ-entropies
+and functional inequalities involving Φ-entropies.
+Corollary 1.2. In the setting of Proposition 1.1, the following two statements are equivalent.
+i. There exists a constant c > 0 such that for all f ∈ dom(L )
+EntΦ
+µ(f) ≤ −cEµ [Φ′(f)L f] .
+ii. There exists a constant c > 0 such that for all continuous and bounded f : E → I
+EntΦ
+µ(Ptf) ≤ e− t
+c EntΦ
+µ(f).
+Note that, in the special case Φ : u �→ u2, one recovers the Poincar´e inequality
+Varµ(f) ≤ − c
+2 ⟨f, L f⟩L2(µ),
+which is well-known to be equivalent to exponential L2 ergodicity, see e.g. [HS76]. For a more detailed
+review of Φ-entropies and further results we refer the interested reader to [Cha04].
+1.2. A finite state-space example for the trajectorial approach. As one can see, the results above
+can be obtained without even mentioning the underlying stochastic process and just dealing with the
+semigroup and its generator. We want to supplement this viewpoint with a more probabilistic approach
+that allows us to obtain a somewhat finer result on a trajectorial level, which also implies the classical
+results by taking expectations.
+For simplicity, we will discuss the main ideas for the example of a continuous-time Markov chain on
+a finite state space. More precisely, let (Xt)t≥0 be a Markov chain on a finite set E with irreducible
+generator L and strictly positive invariant measure µ. Hence, the corresponding Markov semigroup is
+given by the matrix exponential (etL )t≥0. Denote the underlying probability space by (Ω, A, P) and
+assume that X0 ∼ µ under P.
+It is easy to check that for all bounded f : [0, ∞)×E → R such that for all x ∈ E the partial derivatives
+∂tf(·, x) are continuous and bounded, the process defined by
+f(t, Xt) −
+� t
+0
+(∂s + L )f(s, Xs)ds,
+t ≥ 0,
+(1.1)
+is a martingale w.r.t. the canonical filtration generated by (Xt)t≥0, see e.g. [RW00, Lemma IV.4.20].
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+3
+If we now fix a finite time horizon T > 0 and consider the time-reversal ( ˆXt)0≤t≤T of (Xt)t≥0, where
+ˆXt = XT −t, then under P the time-reversed process is again a time-homogeneous Markov process with
+generator
+ˆ
+L , where
+ˆ
+L (x, y) = µ(y)
+µ(x)L (y, x).
+A short calculation now shows that, for each bounded g : E → R and T > 0, the process (PT −sg( ˆXs))0≤s≤T
+is a (( ˆFt)0≤t≤T , P)-martingale, where ˆFt = σ(XT −s : 0 ≤ s ≤ t). Indeed, we can use the chain rule to
+calculate
+∂tPT −tg(x) = − ˆ
+L PT −tg(x),
+so the correction term in (1.1) vanishes. Note that it is crucial to use the time-reversed process here,
+since the correction term does not cancel out if one uses the forward process.
+By convexity this directly implies that the time-reversed trajectorial Φ-entropy, i.e., the process defined
+by
+Φ(PT −sg(X(T − s))),
+0 ≤ t ≤ T,
+is a submartingale. This stochastic monotonicity can be seen as a trajectorial version of the classical
+Φ-entropy decay and after taking expectations with respect to P, one obtains the classical results as in
+Proposition 1.1. Therefore, one can interpret these trajectorial results as the probabilistic version of the
+decay of Φ-entropies.
+The main work is now to establish that a similar argument can also be used to treat infinite-dimensional
+systems like the interacting particle systems we consider. To our best knowledge, the first results of this
+kind, in the context of diffusions in Rn, have been achieved in [FJ16].
+More recently, starting with
+[KST20], these results have been extended to more and more classes of Markov processes, including
+continuous time Markov chains on countable state spaces, see [KMS21]. The works [KY22] and [TCY23]
+are also in a similar spirit.
+The setting will be made precise in Section 2, but roughly speaking, we consider continuous-time
+Markov jump processes on general configuration spaces Ω = ΩS
+0 , where S is an arbitrary countable set
+and Ω0 is a compact Polish space. We will refer to the elements of S as sites and call Ω0 the local
+state-space. In most examples considered in the literature, S is the vertex set of some graph like the
+d-dimensional hypercubic lattice Zd, a tree or the Cayley graph of a group. This underlying spatial
+geometry dictates which particles can interact with each other and we are therefore not in the setting
+of mean-field systems but in an infinite-dimensional setting. This of course brings with it its own set of
+technical difficulties which need to be dealt with for making the time-reversal arguments work.
+The main technical difficulties come from making sure that the time-reversal is again a well-defined
+interacting particle system and from obtaining a description of its generator. This is made possible by
+assuming some local regularity of the local conditional distributions of the time-stationary measure µ.
+Namely, by the assumption that µ is actually a Gibbs measure with respect to a quasilocal specification
+that additionaly satisfies a certain smoothness condition. This condition is e.g. satisfied if the specification
+is given in terms of a potential Φ = (ΦB)B⋐S such that
+sup
+x∈S
+�
+B⋐S: B∋x
+|B| ∥ΦB∥∞ < ∞.
+Note that this condition is for example satisfied for any translation-invariant finite-range potential, so
+our theory applies to a fairly large class of models.
+1.3. Organization of the manuscript. The rest of this article is organized as follows. We will first
+collect the necessary notation and formulate our main results in Section 2. Then, as a first step, we
+investigate the time-reversal of interacting particle systems in equilibrium in Section 3 with the main
+goal of obtaining an explicit representation of the (formal) generator of the time-reversed dynamics. In
+Section 4, we will then apply these results to establish pathwise properties of general Φ-entropy functionals
+and derive the classical DeBrujin-like decay property as a corollary.
+2. Setting and main results
+Let (Ω0, B0) be a compact Polish space equipped with its Borel σ-algebra and λ0 a probability measure
+on (Ω0, B0), which will serve as our reference measure. We will consider Markovian dynamics on the
+configuration space Ω = ΩS
+0 , where S is some countable set whose elements we will refer to as sites. In
+most applications this will be the set of vertices of some graph, e.g. Zd or a tree. We equip Ω with the
+product topology and corresponding Borel σ-algebra F. Note that F coincides with the product σ-algebra
+⊗x∈SB0. For ∆ ⊂ S we will also write Ω∆ := Ω∆
+0 for the set of partial configurations. We will also equip
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+4
+Ω∆ with the product σ-algebra and the probability measure λ∆ = ⊗x∈∆λ0. For Λ ⊂ S, let FΛ be the
+sub-σ-algebra of F that is generated by the projections ω �→ ω∆ ∈ Ω∆ for ∆ ⋐ Λ, where we write ⋐ to
+signify that a set is a finite subset of another set. For ∆ ⊂ S and (partial) configurations η∆c ∈ Ω∆c and
+ξ∆ ∈ Ω∆, we will write ξ∆η∆c for the configuration that is defined on all of S and agrees with η∆c on ∆c
+and with ξ∆ on ∆. For a topological space E, we will denote its Borel σ-algebra by B(E) and the space
+of continuous real-valued functions on E by C(E). The space of non-negative measures on E, or more
+precisely on B(E), will be denoted by M(E) and is equipped with the topology of weak-convergence.
+The total variation distance on M(E) will be denoted by ∥·∥TV.
+2.1. Interacting particle systems and Gibbs measures.
+2.1.1. Interacting particle systems. We will consider time-continuous Markovian dynamics on Ω, namely
+interacting particle systems characterized by time-homogeneous generators L with domain dom(L ) ⊂
+C(Ω) and the associated Markovian semigroup (Pt)t≥0 on C(Ω). For interacting particle systems we
+adopt the notation and exposition of the standard reference [Lig05, Chapter I].
+In our setting, the generator L is given by a collection of transition measures (c∆(·, dξ))∆⋐S in finite
+volumes ∆ ⋐ S, i.e., mappings
+Ω ∋ η �→ c∆(η, dξ∆) ∈ M(Ω∆).
+These transition measures can be interpreted as the infinitesimal rates at which the particles inside ∆
+switch from the configuration η∆ to ξ∆, given that the rest of the system is currently in state η∆c. The
+full dynamics of the interacting particle system is then given as the superposition of these local dynamics,
+L f(η) =
+�
+∆⋐S
+�
+Ω∆
+[f(ξ∆η∆c) − f(η)] c∆(η, dξ∆).
+(2.1)
+In [Lig05, Chapter I] it is shown that the following conditions are sufficient to guarantee well-definedness.
+(L1) For each ∆ ⋐ Ω the mapping
+Ω ∋ η �→ c∆(η, dξ∆) ∈ M(Ω∆)
+is continuous.
+(L2) The total rate at which a single particle switches its state is uniformly bounded, i.e.,
+sup
+x∈S
+�
+∆∋x
+sup
+η∈Ω
+c∆(η, Ω∆) < ∞.
+(L3) The total influence of all other particles on the transition rates of a single particle is uniformly
+bounded, i.e.,
+M := sup
+x∈S
+�
+∆∋x
+�
+y̸=x
+δyc∆ < ∞,
+where
+δyc∆ := sup {∥c∆(η, ·) − c∆(ξ, ·)∥TV : ηyc = ξyc} .
+Under these conditions, the core of the operator L is given by
+D(Ω) :=
+�
+f ∈ C(Ω) : |||f||| :=
+�
+x∈S
+δx(f) < ∞
+�
+,
+where for x ∈ S
+δx(f) :=
+sup
+η,ξ: ηxc=ξxc
+|f(η) − f(ξ)|
+is the oscillation of a function f : Ω → R at the site x. In addition, one can show the following estimates
+for L and the action of the semigroup (Pt)t≥0 generated by L . We will need these later on.
+Lemma 2.1. Assume that the generator L satisfies (L1) − (L3) and denote by (Pt)t≥0 the semigroup
+generated by L .
+i. For f ∈ D(Ω) we have Ptf ∈ D(Ω) for all t ≥ 0 and
+|||Ptf||| ≤ exp ((M − ε)t) |||f|||.
+ii. For all f ∈ D(Ω) it holds that
+∥L f∥∞ ≤
+�
+sup
+x∈S
+�
+∆∋x
+sup
+η∈Ω
+c∆(η, Ω∆)
+�
+|||f|||.
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+5
+The constants are explicitly given by
+M = sup
+x∈S
+�
+∆∋x
+�
+y̸=x
+δyc∆ < ∞,
+ε = inf
+x∈S
+inf
+η,ζ : ηxc=ζxc ,ηx̸=ζx
+�
+∆∋x
+
+
+�
+ξ∆:ξx=ζx
+c∆(η, ξ∆) +
+�
+ξ∆:ξx=ηx
+c∆(ζ, ξ∆)
+
+ .
+Proof. Combine the results from Proposition 3.2(a) and Theorem 3.9.(d) in [Lig05, Chapter I].
+□
+For our purposes, the mere well-definedness of an interacting particle system is not sufficient and we
+need to assume some more regularity. All the additional assumptions we put will be used to make the
+generator of the time-reversal well-defined.
+(R1) For each ∆ and η ∈ Ω the measure c∆(η, dξ∆) is absolutely continuous w.r.t. the reference measure
+λ∆(dξ∆) with density c∆(η, ·).
+(R2) For each ∆ ∈ Ω the map
+Ω × Ω∆ ∋ (η, ξ∆) �→ c∆(η, ξ∆) ∈ R
+is continuous w.r.t. the product topology.
+(R3) The total rate of transition for a single site is uniformly bounded from above
+sup
+x∈S
+�
+∆∋x
+sup
+η∈Ω
+∥c∆(η, ·)∥∞ < ∞.
+(R4) The condition (L3) is satisfied, i.e.,
+sup
+x∈S
+�
+∆∋x
+�
+y̸=x
+δyc∆ < ∞.
+(R5) There exists an R > 0 such that for all ∆ ⋐ Zd with |∆| > R we have
+sup
+η∈Ω,ξ∆∈Ω∆
+c∆(η, ξ∆) = 0.
+We will comment on where and why we need these assumptions and their connection to the classical
+conditions (L1) − (L3) at the end of Section 2.1.2, after we have stated our assumptions on the local
+conditional distribution of the time-stationary measure µ.
+2.1.2. Gibbs measures and the DLR formalism. We will mainly be interested in the situation where the
+process generated by L admits a time-stationary measure µ with a well-behaved local representation,
+namely that µ is a Gibbs measure w.r.t. to a sufficiently nice specification γ. Let us therefore first recall
+the general definition of a specification.
+Definition 2.2. A specification γ = (γΛ)Λ⋐S is a family of probability kernels γΛ from ΩΛc to M1(Ω)
+that additionally satisfies the following properties.
+i. Each γΛ is proper, i.e., for all B ∈ FΛc it holds that
+γΛ(B|·) = 1B(·).
+ii. The probability kernels are consistent in the sense that if ∆ ⊂ Λ ⋐ S, then for all A ∈ F
+γΛγ∆(A|·) = γΛ(A|·),
+where the concatenation of two probability kernels is defined as usual via
+γΛγ∆(A|η) =
+�
+Ω
+γ∆(A|ω)γΛ(dω|η).
+For the existence and further properties of Gibbs measures with specification γ one needs to impose
+some conditions on the specification γ. One sufficient condition for the existence of a Gibbs measure for
+a specification γ is quasilocality, see e.g. [FV17] or [Geo11]. For the following sections we will need to
+assume some more regularity for the specification γ. In particular, these assumptions will guarantee that
+γ is quasilocal.
+(S1) For each ∆ ⋐ S and η ∈ Ω, the probability measure γ∆(dξ∆|η) is absolutely continuous w.r.t. the
+reference measure λ∆(dξ∆) with density γ∆(·|η).
+(S2) For all ∆ ⋐ S, the map
+Ω × Ω∆ ∋ (η, ξ∆) �→ γ∆(ξ∆|η∆c) ∈ [0, ∞)
+is continuous (w.r.t. the product topology).
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+6
+(S3) The conditional densities on the single spin spaces are uniformly bounded away from zero and
+infinity, i.e.,
+0 < δ ≤ inf
+x∈S inf
+η∈Ω γx(ηx|ηxc) ≤ sup
+x∈S
+sup
+η∈Ω
+γx(ηx|ηxc) ≤ δ−1 < ∞.
+(S4) We have
+sup
+x∈S
+�
+∆∋x: c∆>0
+�
+y̸=x
+δyγ∆ < ∞,
+where
+δyγ∆ = sup {∥γ∆(dξ∆|η) − γ∆(dξ∆|ζ)∥TV : ηyc = ζyc} .
+Remark 2.3. Now that we have stated all of the conditions that we need, let us briefly comment on why
+and where we need them.
+i. Assumption (R3) clearly implies (L2) and together with (R4) ensures that the interacting particle
+system is well-defined.
+ii. Assumption (R1) and (S1) allow us to write down the local transition density of the time-reversal
+and (S3) makes sure that we are not performing a division by zero.
+iii. The further regularity assumptions (R3), (R5) (S3), (S4) and the continuity assumptions (R2) and
+(S2) make sure that the local transition densities of the time-reversal also give rise to a well-defined
+interacting particle system.
+iv. The quantity in (S4) is similar to the classical Dobrushin uniqueness condition, see [Geo11]. How-
+ever, we only need it to be finite and not strictly smaller than one.
+Example 2.4. One particular class of models to which our theory can be applied to are spin systems
+for which the specification γ is defined via a potential Φ = (ΦB)B⋐S that satisfies
+sup
+x∈S
+�
+B∋x
+|B| ∥ΦB∥∞ < ∞,
+and where the rates are of the form
+c∆(η, ξ∆) =
+�
+exp
+�
+−β �
+B:B∩∆̸=∅ ΦB(ξ∆η∆c)
+�
+,
+if |∆| = 1,
+0,
+otherwise.
+Instead of these single-site updates one could also consider updates in larger regions with a bounded
+diameter. Then the rates satisfy (R1) − (R5) and the specification satisfies (S1) − (S4), as one can see
+by using similar arguments as in the proof of [FV17, Lemma 6.28].
+2.2. The time-reversal of an interacting particle system. In the notation of above, assume that
+µ ∈ G (γ) is Gibbs measure for a quasilocal specification γ, i.e., assume that µ satisfies the DLR equations
+µ(f) = µ(γΛ(f|·))
+for all Λ ⋐ S and bounded measurable functions f. Further assume that µ is time-stationary with respect
+to the Markovian dynamics with generator L . Denote the semigroup generated by L by (Pt)t≥0 and the
+corresponding process on Ω by (η(t))t≥0. As discussed in Section 1.2, for each fixed T > 0 the process
+(η(T − t))0≤t≤T is again a time-homogeneous Markov process and under some suitable assumptions its
+associated semigroup has a generator
+ˆ
+L . But what does this generator look like? For general Markov
+processes it is not possible to give a closed form expression, but in our case we can use the special
+structure of L as the superposition of local dynamics in finite volumes. In each of these finite volumes,
+it is clear how the time-reversal w.r.t. µ should look and we can hope that we can again write
+ˆ
+L as
+the superposition of finite volume processes. With this Ansatz, the probabilistic intuition dictates the
+educated guess
+ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)
+γ∆(η∆|η∆c)
+(2.2)
+for the transition densities appearing in the generator of the time-reversed interacting particle system.
+However, at this stage, it is not obvious that the generator of the time-reversed system is again of the
+form (2.1) and has precisely these rates. For Markov processes on finite state spaces this is an easy
+calculation but we have to put in some more work, which will be carried out in Section 3. The main
+result there is the following.
+Proposition 2.5 (Time-reversal generator). Let the rates of an interacting particle system with generator
+L satisfy (R1) − (R5) and assume that µ is a time-stationary measure for the corresponding Markov
+semigroup (P(t))t≥0 on C(Ω) that is generated by L such that µ is a Gibbs measure w.r.t. a specification
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+7
+γ that satisfies (S1)−(S4). Then, the time-reversed process has a generator
+ˆ
+L whose transition densities
+(w.r.t. the reference measure λ0) are given by
+ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)
+γ∆(η∆|η∆c).
+The proof of this can be found at the end of Section 3.
+2.3. Trajectorial decay of Φ-entropies. With this auxiliary result at hand, we can then obtain the
+following result, which describes the dissipation of general Φ-entropies on a trajectorial level. Before we
+state the theorem, let us introduce some further notation to express the main equation in a cleaner way.
+The Bregman Φ-divergence associated with Φ : I → R is defined as
+divΦ(p|q) := Φ(p) − Φ(q) − (p − q)Φ′(q),
+p, q ∈ I.
+This is precisely the difference between the value of Φ at the point p and the value of the first-order
+Taylor expansion of Φ around q, evaluated at p and is non-negative, since we assumed that Φ is convex.
+Bregman divergences are sometimes also referred to as Bregman distances, despite not being a metric
+since they are in general not symmetric and do not satisfy the triangle inequality. They are however
+still useful for applying techniques from optimization theory in more general contexts, e.g. in statistical
+learning theory [BMDG05].
+Note that we now have to be careful with the probability space and filtration we are working with,
+since we are talking about results on a trajectorial level.
+Theorem 2.6 (Trajectorial decay of Φ-entropies). Let (Ω, A, P) be a probability space on which the
+interacting particle system (η(s))s≥0 is defined. Denote the generator of the interacting particle system
+by L and assume that its rates satisfy (R1) − (R5) and assume that under P we have η0 ∼ µ, where µ
+is a time-stationary measure for the corresponding Markov semigroup (Pt)t≥0 on C(Ω) that is generated
+by L such that µ is a Gibbs measure w.r.t. a specification γ that satisfies (S1) − (S4). Then, for any
+f ∈ D(Ω) and T > 0, the process defined by
+LΦ,f(s) := Φ(PT −sf(ηT −s)),
+0 ≤ s ≤ T,
+(2.3)
+is a (( ˆGt)0≤t≤T , P)-submartingale, where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t). Its Doob–Meyer decomposition is
+given by
+LΦ,f(t) = M Φ,f(t)
+(2.4)
++
+� t
+0
+�
+∆⋐S
+�
+Ω∆
+ˆc∆(η(T − s), ξ∆)divΦ(PT −s(f(ξ∆η∆c(T − s))|PT −sf(η(T − s)))λ∆(dξ∆)ds.
+The proof of this theorem can be found at the end of Section 4. As mentioned before, by taking
+expectations one recovers the classical DeBrujin-like decay of Φ-entropies as stated in Proposition 1.1.
+For the sake of concreteness, let us write out the result explicitly for one of the simplest cases, namely
+the trajectorial decay of variance, corresponding to Φ : u �→ u2.
+Corollary 2.7 (Trajectorial decay of variance). In the setting of Theorem 2.6, we have that, for any f ∈
+D(Ω) and T > 0, the process defined by (PT −sf(ηT −s))2, 0 ≤ s ≤ T , is a (( ˆGt)0≤t≤T , P)-submartingale,
+where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t). Its Doob–Meyer decomposition is given by
+(PT −sf(ηT −s))2 = M f(t) +
+� t
+0
+�
+∆⋐S
+�
+Ω∆
+ˆc∆(η(T − s), ξ∆) [f(ξ∆η∆c(T − s)) − f(η(T − s))]2 λ∆(dξ∆)ds.
+2.4. Outlook. Even though we were able to show the trajectorial decay for the relative entropy under
+quite general assumptions on the dynamics, these results are not fully satisfactory in the context of
+statistical mechanics. The usually more interesting Lyapunov functional in this setting is the so-called
+relative entropy density, as e.g. considered in [FV17], which is not only defined for measures ν that are
+absolutely continuous w.r.t. µ. Therefore, it would be much more natural to work with this functional
+h(·|µ) : Minv
+1
+(Ω) → R instead and one can show that is also a Lyapunov function for interacting particle
+systems under quite general assumptions, see [JK22], but it is somewhat unclear how to even formulate
+conjectures about the trajectorial properties of this functional, since one cannot naively evaluate it
+pointwise.
+As we already saw in the case of a continuous time Markov chain on a finite state space, the main
+ingredient for this type of result is to obtain an explicit description of the generator of the time-reversed
+process.
+Another class of processes that could be of interest and is not covered by our results are
+systems which evolve continuously on their single spin spaces, as opposed to our pure-jump processes.
+The first example that comes to mind are of course systems of (infinitely-many) interacting diffusions,
+e.g. indexed by Zd. We expect that, if a given system of interacting diffusions admits a Gibbs measure
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+8
+as an invariant probability measure, then a combination of the techniques in [KST20] and this article
+should yield analogous results – of course under some suitable regularity conditions on the coefficients.
+3. The time-reversed interacting particle system and its generator
+The main goal of this section is to prove Proposition 2.5, thereby establishing that the generator of
+the time-reversal is indeed given by
+ˆ
+L . For this we will need to establish some regularity properties for
+the transition densities as defined in (2.2).
+3.1. Upper and lower bounds for the conditional densities. Since we will need to deal with
+quotients involving the conditional densities γ∆ on arbitrary finite subsets ∆ ⋐ S, we will need to lift
+the upper and lower bounds from (S3) to this more general case. This is essentially the content of the
+following lemma.
+Lemma 3.1. Let γ be a specification that satisfies (S1) and (S3). Then, there exists a constant C ∈
+(0, ∞) such that for all ∆ ⋐ S we have the estimate
+e−C|∆| ≤ inf
+η∈Ω γ∆(η∆|η∆c) ≤ sup
+η∈Ω
+γ∆(η∆|η∆c) ≤ eC|∆|.
+This constant is precisely given by C = |log δ|.
+Proof. For this, fix an enumeration i1, . . . , ik of the elements of ∆ and introduce the notation
+[ij, ik] := {ij, ij+1, . . . , ik} ,
+1 ≤ j ≤ k.
+With this notation at hand, we can use the chain rule for conditional probability densities to write
+γ∆(η∆|η∆c) =
+k
+�
+j=1
+γ[i1,ij](ηij|η[ij+1,ik]η∆c),
+(3.1)
+where γ[i1,ij](ηij|η[ij+1,ik]η∆c) is the marginal conditional density of the measure γ[i1,ij](dη[i1,ij]|η[ij+1,ik]η∆c)
+w.r.t. the site ij. But, using consistency of the specification γ, we have
+γ[i1,ij](ηij|η[ij+1,ik]η∆c) =
+�
+γ[i1,ij](dξ[i1,ij]|η[ij+1,ik]η∆c)γij(ηij |ξ[i1,ij−1]η[ij+1,ik]η∆c),
+which is, by assumption, upper bounded by δ−1 and lower bounded by δ.
+In conjunction with the
+representation (3.1) this implies the desired upper and lower bound where the constant C is explicitly
+given by C = |log(δ)|.
+□
+As a corollary we now get the following estimate for the quotients that appear in the definition of
+transition density of the time-reversal (2.2).
+Lemma 3.2. Let ∆ ⋐ S and γ be a specification that satisfies (S1) and (S3). Then, for all ∆ ⋐ S,
+η ∈ Ω and ξ∆ ∈ Ω∆, we have
+0 < e−2C|∆| ≤ γ∆(ξ∆|η∆c)
+γ∆(η∆|η∆c) ≤ e2C|∆|.
+3.2. The switching lemma. Now that we can be sure that the densities as in (2.2) are actually well-
+defined and we are not performing a division by zero, we can start showing that
+ˆ
+L is indeed the generator
+of the time-reversed process. The main technical tool will be the following lemma.
+Lemma 3.3. Let the rates of a well-defined interacting particle system with generator L satisfy (R1)
+and assume that µ is a time-stationary measure for L and µ is a Gibbs measure w.r.t. a specification γ
+that satisfies (S1) and (S3). Then, we have for all bounded and measurable f, g : Ω → R and ∆ ⋐ S that
+�
+Ω∆
+�
+Ω
+c∆(ω, ξ∆)f(ω)g(ξ∆ω∆c)µ(dω)λ∆(dξ∆) =
+�
+Ω∆
+�
+Ω
+ˆc∆(ω, ξ∆)f(ξ∆ω∆c)g(ω)µ(dω)λ∆(dξ∆),
+(3.2)
+where
+ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)
+γ∆(η∆|η∆c).
+To keep the notation for conditional expectations in the upcoming proof simple, we will denote integration
+with respect to µ by E[·].
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+9
+Proof. As a first step, note that for fixed ∆ ⋐ S and ξ∆ ∈ Ω∆ the maps
+Ω ∋ ω �→ g(ξ∆ω∆c) ∈ R,
+Ω ∋ ω �→ f(ξ∆ω∆c) ∈ R,
+are F∆c-measurable. Therefore, we can use that γ is the local conditional distribution of µ and the
+definition of the rates ˆc to obtain the µ-almost-sure identity
+E [c∆(·, ξ∆)f(·)g(ξ∆·∆c)|F∆c] (ω) = g(ξ∆ω∆c)E [c∆(·, ξ∆)f(·)|F∆c] (ω)
+= g(ξ∆ω∆c)
+�
+Ω∆
+γ∆(ζ∆|ω∆c)c∆(ζ∆ω∆c, ξ∆)f(ζ∆ω∆c)λ∆(dζ∆)
+= g(ξ∆ω∆c)
+�
+Ω∆
+γ∆(ξ∆|ω∆c)ˆc∆(ξ∆ω∆c, ζ∆)f(ζ∆ω∆c)λ∆(dζ∆).
+If we now integrate over ξ∆, exchange the order of integration (via Fubini) and apply the same arguments
+in reverse – with f taking the role of g and vice versa – we get
+�
+Ω∆
+E [c∆(·, ξ∆)f(·)g(ξ∆·∆c)|F∆c] (η)λ∆(dξ∆) =
+�
+Ω∆
+E [ˆc∆(·, ζ∆)f(ζ∆·∆c)g(·)|F∆c] (η)λ∆(dζ∆).
+By integrating both sides with respect to µ, exchanging the order of integration, and applying the law of
+total expectation we obtain
+�
+Ω∆
+�
+Ω∆
+(ω, ξ∆)f(ω)g(ξ∆ω∆c)µ(dω)λ∆(dξ∆) =
+�
+Ω∆
+�
+Ω
+ˆc∆(ω, ζ∆)f(ζ∆ω∆c)g(ω)µ(dω)λ∆(dζ∆),
+as desired.
+□
+3.3. Regularity of the time-reversal rates. To make sure that
+ˆ
+L is actually the generator of a well-
+defined interacting particle system we now show that the collection of transition measures (ˆc∆(·, ·))∆⋐S
+satisfies the three conditions (L1) − (L3).
+Proposition 3.4. Let the rates of an interacting particle system with generator L satisfy (R1) − (R5)
+and assume that µ is a time-stationary measure for L and such that µ is a Gibbs measure w.r.t. a
+specification γ that satisfies (S1)−(S4). Then, the transition measures (ˆc∆(·, dξ∆))∆⋐Zd with λ∆-densities
+given by
+ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)
+γ∆(η∆|η∆c)
+satisfy the conditions (L1) − (L3).
+Proof. Ad (L1): This follows from the continuity assumptions (R2) and (S2), together with assumption
+(S3) and Lemma 3.2.
+Ad (L2): Note that for fixed ∆ ⋐ S, ξ∆ ∈ Ω∆ and η ∈ Ω we have by Lemma 3.2 and assumption (R5)
+|c∆(η, ξ∆)| =
+����c∆(ξ∆η∆c, η∆) γ∆(ξ∆|η∆c
+γ∆(η∆|η∆c)
+���� ≤ 1
+δ eRc∆(ξ∆η∆c, η∆).
+So we get
+sup
+η∈Ω
+ˆc∆(η, Ω∆) = sup
+η∈Ω
+�
+Ω∆
+ˆc∆(η, ξ∆)λ∆(dξ∆) ≤ 1
+δ eR sup
+η∈Ω
+�
+Ω∆
+c∆(ξ∆η∆c, η∆)λ∆(dξ∆)
+≤ 1
+δ eR sup
+η∈Ω
+∥c∆(η, ·)∥∞ .
+Therefore, assumption (R3) implies that (L2) is also satisfied.
+Ad (L3): Fix ∆ ⋐ S, y ∈ S and two configurations η1, η2 that only disagree at y. Then, for any ξ∆ ∈ Ω∆
+we have
+��ˆc∆(η1, ξ∆) − ˆc∆(η2, ξ∆)
+�� =
+����c∆(ξ∆η1
+∆c, η1
+∆)γ∆(ξ∆|η1
+∆c)
+γ∆(η1
+∆|η1
+∆c) − c∆(ξ∆η2
+∆c, η2
+∆)γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+����
+≤
+��c∆(ξ∆η1
+∆c, η1
+∆)
+��
+����
+γ∆(ξ∆|η1
+∆c)
+γ∆(η1
+∆|η1
+∆c) − γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+����
++
+����
+γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+����
+��c∆(ξ∆η1
+∆c, η1
+∆) − c∆(ξ∆η2
+∆c, η2
+∆)
+�� .
+To estimate this further, we will have to make a case distinction over whether the site y is contained in
+∆ or not. If y is contained in ∆, then we can naively use Lemma 3.2 and assumption (R5) to obtain the
+rough estimate
+��ˆc∆(η1, ξ∆) − ˆc∆(η2, ξ∆)
+�� ≤ 41
+δ eR
+sup
+η∈Ω,ξ∆∈Ω∆
+|c∆(η, ξ∆)| ≤ 4eRK(c)
+δ
+.
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+10
+In the case where y is not contained in ∆, we can (and have to) be a bit more precise. Via the elementary
+algebraic rule
+ac − bd = 1
+2 [(a − b)(c + d) + (a + b)(c − d)] ,
+and Lemma 3.2 plus assumption (R5) one obtains
+��c∆(ξ∆η1
+∆c, η1
+∆)
+��
+����
+γ∆(ξ∆|η1
+∆c)
+γ∆(η1
+∆|η1
+∆c) − γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+���� +
+����
+γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+����
+��c∆(ξ∆η1
+∆c, η1
+∆) − c∆(ξ∆η2
+∆c, η2
+∆)
+��
+= 1
+2
+��c∆(ξ∆η1
+∆c, η1
+∆)
+��
+����
+1
+γ∆(η1
+∆|η1
+∆c)γ∆(η2
+∆|η2
+∆c)
+����
+��γ∆(ξ∆|η1
+∆c) − γ∆(ξ∆|η2
+∆c)
+�� ��γ∆(η1
+∆|η1
+∆c) + γ∆(η2
+∆|η2
+∆c)
+��
++
+����
+γ∆(ξ∆|η2
+∆c)
+γ∆(η2
+∆|η2
+∆c)
+����
+��c∆(ξ∆η1
+∆c, η1
+∆) − c∆(ξ∆η2
+∆c, η2
+∆)
+��
+≤
+1
+2δ2 e2RK(c)K(γ)
+��γ∆(ξ∆|η1
+∆c) − γ∆(ξ∆|η2
+∆c)
+�� + 1
+δ eR ��c∆(ξ∆η1
+∆c, η1
+∆) − c∆(ξ∆η2
+∆c, η2
+∆)
+�� .
+Now, by integrating this pointwise difference of the densities over ξ∆, we obtain via all of the other
+assumptions that
+sup
+x∈S
+�
+∆∋x
+�
+y̸=x
+δyˆc∆ < ∞.
+But this is precisely (L3) and the proof is finished.
+□
+With these two intermediate results at hand, we can now show that
+ˆ
+L is indeed the generator of the
+time-reversal of (ηt)t≥0 (w.r.t. the time-stationary measure µ).
+Proof of Proposition 2.5. It only remains to show that for all f, g ∈ D(Ω) we have
+�
+Ω
+f(ω)L g(ω)µ(dω) =
+�
+Ω
+�
+ˆ
+L f
+�
+(ω)g(ω)µ(dω),
+since then the claimed time-reversal duality follows from Lemma A.4.
+For this, we first note that it suffices to show that the duality relation for the generators holds for
+all local functions f, g : Ω → R. Indeed, if it holds for all pairs of local functions, we can then extend
+it to functions with bounded total oscillation by using the estimates from Lemma 2.1 and dominated
+convergence. Therefore, let f, g be two local functions and let Λ ⋐ S be sufficiently large such that
+both f and g only depend on coordinates in Λ. By first applying Lemma 3.3 and then using that µ is
+time-stationary with respect to the Markovian dynamics generated by L , we see that
+�
+Ω
+f(ω)L g(ω)µ(dω) −
+�
+Ω
+�
+ˆ
+L f(ω)
+�
+g(ω)µ(dω)
+=
+�
+∆∩Λ̸=∅
+�
+Ω∆
+�
+Ω
+c∆(ω, ξ∆)[f · g(ξ∆ω∆c) − f · g(ω)]µ(dω)λ∆(dξ∆) =
+�
+Ω
+L (f · g)(ω)µ(dω) = 0,
+which finishes the proof.
+□
+4. Trajectorial decay of Φ-entropies
+In this section we use the time-reversed process and a martingale argument to prove Theorem 2.6.
+4.1. The time-dependent martingale property. The main technical tool will be the following lemma
+which can be seen as an extension of [RW00, Lemma IV.20.12] to our setting.
+Lemma 4.1. Let L be the generator of an interacting particle system (η(s))s≥0 such that its transition
+rates satisfy (L1)−(L3) and let µ be a time-stationary measure w.r.t. L . Then, for all f : [0, ∞)×Ω → R
+such that
+i. f(·, η) ∈ C1([0, ∞)) for all η ∈ Ω and
+ii. for all T > 0 it holds that
+sup
+0≤t≤T
+|||f(t, ·)||| < ∞,
+the process defined by
+f(t, η(t)) −
+� t
+0
+(∂s + L )f(s, η(s))ds
+is a martingale w.r.t. the filtration Gt := σ(η(u) : 0 ≤ u ≤ t).
+The proof of this lemma is not difficult but hard to find in the existing literature, we therefore give it
+in some detail.
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+11
+Proof. For functions f as above, we define
+M(s) := f(s, η(s)) −
+� s
+0
+(∂u + L )f(u, η(u))du,
+s ≥ 0
+and denote by (Pt)t≥0 the Markov semigroup generated by (∂s + L ). Then, for s < t, the Markov
+property and the elementary identity
+d
+dtPt = Pt(∂t + L ) = (∂t + L )Pt,
+give us
+E
+�
+f(t, η(t)) −
+� t
+0
+(∂u + L )f(u, η(u))du
+���Gs
+�
+= Pt−sf(s, η(s)) −
+� s
+0
+(∂u + L )f(u, η(u))du −
+� t
+s
+Pu−s(∂u + L )f(s, η(s))du
+= Pt−sf(s, η(s)) −
+� s
+0
+(∂u + L )f(u, η(u))du −
+� t
+s
+d
+duPu−sf(s, η(s))du
+= f(s, η(s)) −
+� s
+0
+(∂u + L )f(u, η(u))du.
+This shows that the process (M(s))s≥0 is indeed a martingale.
+□
+This abstract tool now lets us establish the analogue of the first step in the case of a finite state space
+considered in Section 1.2.
+Proposition 4.2. Let (Ω, A, P) be a probability space on which the interacting particle system (η(s))s≥0
+is defined. Denote the generator of (η(s))s≥0 by L and assume that the rates satisfy (R1) − (R5) and
+assume that under P we have η(0) ∼ µ, where µ is a time-stationary measure for the corresponding Markov
+semigroup (Pt)t≥0 on C(Ω) that is generated by L and that µ is a Gibbs measure w.r.t. a specification γ
+that satisfies (S1) − (S4). Then, for all f ∈ D(Ω) and T > 0, the process defined by
+PT −sf(η(T − s)),
+0 ≤ s ≤ T,
+is a (( ˆGt)0≤t≤T , P)-martingale, where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t).
+Proof. Note that by Lemma 2.1 we can apply Lemma 4.1 to the function
+[0, T ] × Ω ∋ (s, η) �→ PT −sf(η).
+But since we have by the chain rule
+∂sPT −sf = − ˆ
+L PT −sf,
+the correction term cancels out and we obtain the claimed martingale property.
+□
+4.2. Trajectorial decay of Φ-entropies. With this preliminary result in place, we can now come to
+the proof of our main result.
+Proof of Theorem 2.6. Submartingale property: By Jensen’s inequality and Proposition 4.2 we immedi-
+ately see that the process (LΦ,f(t))t≥0, as defined in (2.3), is a submartingale.
+Doob–Meyer decomposition: Here we want to apply Lemma 4.1 to the function
+g : [0, T ] × Ω ∋ (s, η) �→ Φ(PT −sf) ∈ R.
+Via the chain rule we see that
+∂sg(s, ·) = ∂sΦ(PT −sf(·)) = −Φ′(PT −sf(·)) ˆ
+L PT −sf(·).
+Applying the generator
+ˆ
+L for fixed s ∈ [0, T ] yields
+ˆ
+L g(s, η) =
+�
+∆⋐S
+�
+Ω∆
+ˆc∆(η, ξ∆) [Φ(PT −sf(ξ∆η∆c)) − Φ(PT −sf(η))] λ∆(dξ∆).
+By putting these two ingredients together and using the previously introduced notation for the Bregman
+Φ-divergence we obtain
+(∂s +
+ˆ
+L )g(s, η) =
+�
+∆⋐S
+�
+Ω∆
+ˆc∆(η, ξ∆)divΦ(PT −sf(ξ∆η∆c)|PT −sf(η))λ∆(dξ∆).
+The claimed Doob–Meyer decomposition (2.4) of the submartingale LΦ,f now follows from Lemma 4.1.
+□
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+12
+Acknowledgements
+The authors acknowledge the financial support of the Leibniz Association within the Leibniz Junior
+Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz
+Competition.
+Appendix A. The time-reversal of Markov processes in equilibrium
+In this section, we briefly summarize some properties of the time-reversal of a Markov process w.r.t.
+a time-stationary measure. These results are classical but not particularly easy to find in the literature,
+at least in this formulation.
+We start by making precise what we mean by time-reversal of a stochastic process.
+Recall that
+any time-stationary stochastic process (Xt))t≥0 can be extended (in law) to a process (Xt)−∞<t<∞ via
+Kolmogorov’s extension theorem.
+Definition A.1. Let (Xt)t≥0 be a time-stationary stochastic process.
+We call a process (Yt)t≥0 the
+time-reversal of X if
+Law((X−t)t≥0) = Law((Yt)t≥0).
+The intuition behind this definition is that forward in time the process Y looks like the process X
+run backwards. For Markov processes this notion can be characterized in terms of the semigroups and
+generators as follows.
+Proposition A.2. Let X = (Xt)t≥0 and Y = (Yt)t≥0 be Markov processes on a compact topological space
+E with associated semigroups (Tt)t≥0 and (St)t≥0. Assume that X has a time-stationary measure ν and
+for all f, g ∈ C(E) we have
+�
+E
+(Ttf)gdν =
+�
+E
+f(Stg)dν.
+(A.1)
+Then ν is also time-stationary for Y and Y is the time-reversal of X (w.r.t. ν).
+Proof. By the duality relation (A.1), ν is also time-stationary for Y and we can extend it to a process
+(Yt)−∞<t<∞ with Y0 ∼ ν. To show that Y is the time-reversal of X (w.r.t. ν) it suffices to show that for
+arbitrary n ∈ N and times t0 < t2 < · · · < tn < tn+1 and functions f1, . . . , fn ∈ C(E) it holds that
+EY0∼ν [f1(Yt1) · · · fn(Ytn)] = EX0∼ν [f1(X−t1) · · · fn(X−tn)] .
+We will do this as follows. First, we introduce the notation f0 = fn+1 ≡ 1 and define functions g0, . . . , gn+1
+and h0, . . . , hn+1 by g0 = hn+1 ≡ 1,
+gl(x) := Ex
+�l−1
+�
+k=0
+fk(Xtl−tk)
+�
+,
+1 ≤ l ≤ n + 1
+and
+hl(y) := Ey
+� n+1
+�
+k=l+1
+fk(Ytk−tl)
+�
+,
+0 ≤ l ≤ n.
+With this notation it suffices to show that the quantity
+αl :=
+�
+E
+gl(x)fl(x)hl(x)ν(dx),
+0 ≤ l ≤ n + 1,
+does not depend on l. Indeed, by stationarity of X and Y , this will then imply that
+EY0∼ν [f1(Yt1) · · · fn(Ytn)] =
+�
+E
+h0dν = α0 = αn+1 =
+�
+E
+gn+1dν = EX0∼ν [f1(X−t1) · · · fn(X−tn)] ,
+exactly as we wanted. In order to show that αl does not depend on l, we use the duality relation as
+follows. For 0 ≤ l ≤ n it holds that
+αl =
+�
+E
+glflhldν
+=
+�
+E
+gl(x)fl(x)Ttl+1−tl[fl+1hl+1](x)ν(dx)
+=
+�
+E
+Stl+1−tl[glfl](x)fl+1(x)hl+1(x)ν(dx)
+=
+�
+E
+gl+1fl+1hl+1dν = αl+1.
+Therefore, αl does not depend on l and the claim follows.
+□
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+13
+The duality relation (A.1) for the semigroups can also be verified by checking a similiar property on
+the level of generators. The main technical tool for establishing this relation between the generator of a
+Markov process and its semigroup will unsurprisingly be the celebrated Hille–Yosida theorem which we
+recall here.
+Theorem A.3 (Hille–Yosida). There is a one-to-one correspondence between Markov generators on C(E)
+and Markov semigroups on C(E). This correspondence is explicitly given by
+i.
+dom(L ) =
+�
+f ∈ C(E) : lim
+t↓0
+Stf − f
+t
+exists
+�
+,
+and
+L f := lim
+t↓0
+Stf − f
+t
+,
+f ∈ dom(L ).
+ii.
+Stf = lim
+n→∞
+�
+Id − t
+nL
+�−n
+f,
+f ∈ C(E), t ≥ 0.
+Moreover,
+iii. if f ∈ dom(L ), then Stf ∈ dom(L ) for all t ≥ 0 and
+d
+dtStf = L Stf = StL f and
+iv. for g ∈ C(E) and λ ≥ 0, the solution to the resolvent equation f − λL f = g is given by
+f =
+� ∞
+0
+e−tSλtgdt.
+Proof. See Theorem 1.2.6 and Theorem 4.2.2 in [EK86].
+□
+With this at hand, we can now formulate the time-reversal duality for generators.
+Lemma A.4. Let (Tt)t≥0 and (St)t≥0 be two Markov semigroups on C(E) where E is a compact topo-
+logical space with time-stationary measure ν. Let (A , dom(A )) and (B, dom(B)) be their generators. If
+for all f ∈ dom(A ) and g ∈ dom(B) it holds that
+�
+E
+(A f) g dν =
+�
+E
+f (Bg) dν,
+(A.2)
+then (A.1) holds. It suffices if (A.2) holds for f, g in a core of the respective generators.
+Proof. The duality relation implies that for all f ∈ dom(A ), g ∈ dom(B) and λ ≥ 0 it holds that
+�
+E
+f(g − λL g) dν =
+�
+E
+g(f − λ ˆ
+L f) dν.
+Now we can replace f by (Id − λL )−1f and g by (Id − λ ˆ
+L )−1g to see that for all f, g ∈ C(Ω)
+�
+E
+[(Id − λA )−1f]g dν =
+�
+E
+[(Id − λB)−1g]f dν.
+By iteratively applying this equality we obtain that for all n ∈ N and λ ≥ 0 it holds that
+�
+E
+[(Id − λA )−nf]g dν =
+�
+E
+[(Id − λB)−ng]f dν.
+For fixed t ≥ 0 we can replace λ by t/n and use Part ii. of Theorem A.3 to see that
+lim
+n→∞
+�
+E
+�
+(Id − t
+nA )−nf
+�
+g dν =
+�
+E
+gTtf dν
+and
+lim
+n→∞
+�
+E
+�
+(Id − t
+nB)−ng
+�
+f dν =
+�
+E
+fStg dν,
+as desired.
+□
+To sum this up, in order to show that a stationary Markov process Y is the time-reversal of a
+stationary Markov process X, it suffices to check that their generators satisfy the duality relation
+(A.2).
+References
+[BMDG05] Arindam Banerjee, Srujana Merugu, Inderjit S. Dhillon, and Joydeep Ghosh. Clustering with Bregman Diver-
+gences. Journal of Machine Learning Research, 6(58):1705–1749, 2005. (Cited on 7)
+[Cha04]
+Djalil Chafa¨ı. Entropies, convexity, and functional inequalities, On Φ-entropies and Φ-Sobolev inequalities.
+Kyoto Journal of Mathematics, 44(2), January 2004. (Cited on 2)
+[EK86]
+Stewart N. Ethier and Thomas G. Kurtz, editors. Markov Processes. Wiley Series in Probability and Statistics.
+John Wiley & Sons, Inc., Hoboken, NJ, USA, March 1986. (Cited on 13)
+[FJ16]
+Joaquin Fontbona and Benjamin Jourdain. A trajectorial interpretation of the dissipations of entropy and Fisher
+information for stochastic differential equations. The Annals of Probability, 44(1), January 2016. (Cited on 3)
+
+TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS
+14
+[FV17]
+Sacha Friedli and Yvan Velenik. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduc-
+tion. Cambridge University Press, 1 edition, November 2017. (Cited on 5, 6, 7)
+[Geo11]
+Hans-Otto Georgii. Gibbs measures and phase transitions. Number 9 in De Gruyter studies in mathematics. De
+Gruyter, Berlin ; New York, 2nd ed edition, 2011. OCLC: ocn706965527. (Cited on 5, 6)
+[HS76]
+Richard A. Holley and Daniel W. Stroock. L2 theory for the stochastic Ising model. Zeitschrift f¨ur Wahrschein-
+lichkeitstheorie und Verwandte Gebiete, 35(2):87–101, 1976. (Cited on 2)
+[JK22]
+Benedikt Jahnel and Jonas K¨oppl. Dynamical Gibbs Variational Principles for Irreversible Interacting Particle
+Systems with Applications to Attractor Properties, May 2022. arXiv:2205.02738 [math]. (Cited on 7)
+[KMS21]
+Ioannis Karatzas, Jan Maas, and Walter Schachermayer. Trajectorial dissipation and gradient flow for the
+relative entropy in Markov chains. Communications in Information and Systems, 21(4):481–536, 2021. (Cited
+on 3)
+[KST20]
+Ioannis Karatzas, Walter Schachermayer, and Bertram Tschiderer. Trajectorial Otto calculus, March 2020.
+arXiv:1811.08686 [math]. (Cited on 3, 8)
+[KY22]
+Donghan Kim and Lane Chun Yeung. A trajectorial approach to entropy dissipation for degenerate parabolic
+equations. arXiv:2210.16158, October 2022. (Cited on 3)
+[Lig05]
+Thomas M. Liggett. Interacting Particle Systems. Classics in Mathematics. Springer Berlin Heidelberg, Berlin,
+Heidelberg, 2005. (Cited on 1, 4, 5)
+[RW00]
+L. C. G. Rogers and David Williams. Diffusions, Markov Processes, and Martingales. Cambridge University
+Press, 2 edition, April 2000. (Cited on 2, 10)
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+Bertram Tschiderer and Lane Chun Yeung. A trajectorial approach to relative entropy dissipation of McK-
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+Institut f¨ur Mathematische Stochastik, Technische Universit¨at Braunschweig, Universit¨atsplatz 2, 38106
+Braunschweig, Germany & Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39,
+10117 Berlin, Germany
+Email address: benedikt.jahnel@tu-braunschweig.de
+Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
+Email address: jonas.koeppl@wias-berlin.de
+
diff --git a/r9E2T4oBgHgl3EQffAdt/content/tmp_files/load_file.txt b/r9E2T4oBgHgl3EQffAdt/content/tmp_files/load_file.txt
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@@ -0,0 +1,644 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf,len=643
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='03922v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='PR]  10 Jan 2023  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING  PARTICLE SYSTEMS  BENEDIKT JAHNEL AND JONAS K ¨OPPL  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' A classical approach for the analysis of the longtime behavior of Markov processes is to  consider suitable Lyapunov functionals like the variance or more generally Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Via purely  analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite  general assumptions on the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In this paper, we complement these classical results via a more  probabilistic approach and show that dissipation is already present on the level of individual trajectories  for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry  and compact local spin spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This extends previous results from the setting of finite-state Markov  chains or diffusions in Rn to an infinite-dimensional setting with weak assumptions on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Introduction  There are many different techniques to study the long-time behavior of Markov processes that excel  in different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' One very common and powerful technique is the use of Lyapunov functionals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=',  functionals that are monotone in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' An example of such a functional is the variance  Varµ(f) := Eµ[f 2] − Eµ[f]2,  f ∈ L2(µ),  where µ is an invariant measure for some Markov process with semigroup (Pt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' If we now fix an  observable f and consider the function  [0, ∞) ∋ t �→ Varµ(Ptf) ∈ [0, ∞),  then it is easy to see that this is non-increasing and under some further assumptions one can even  show that it is strictly decreasing for all non-constant observables f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This whole viewpoint, however, is  purely based on functional analytic arguments and one does not even need to speak about the underlying  Markov process itself to carry out the corresponding calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' From a probabilistic point of view, this  is somewhat unsatisfactory and we therefore want to specify this coarse and non-probabilistic approach  with a finer, more probabilistic technique that allows us to obtain trajectorial versions of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Here, by trajectorial we mean results on the level of single realizations of a stochastic process, as opposed  to averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Thereby our goal is to uncover more of the underlying probabilistic mechanisms  behind the decay of variance, or more generally the decay of Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For this, we will first briefly  recall the notion of Φ-entropies and then explain our main results and ideas with the help of the simple  example of a continuous-time Markov chain on a finite state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The rest of the article is then devoted  to extending these ideas to the setting of spatially extended systems of infinitely many interacting particles  as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' considered in [Lig05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Φ-entropies and their decay under Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let Φ : I → R be a smooth and  convex function defined on a not necessarily bounded interval I ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (E, B(E)) be a Polish space  equipped with its Borel σ-algebra and assume that µ is a probability measure on (E, B(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The Φ-  entropy functional is then defined on the set of µ-integrable functions f : E → I by  EntΦ  µ(f) :=  �  E  Φ(f)dµ − Φ  ��  E  fdµ  �  = Eµ [Φ(f)] − Φ (Eµ [f]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By Jensen’s inequality one can immediately deduce that the Φ-entropy functional takes its values in  R+ ∪ {+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Moreover, EntΦ  µ(f) vanishes if its argument is constant and if Φ is strictly convex, then the  converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For special choices of Φ one can recover the classical variance and relative entropy  functionals since we have  Entu�→u2  µ  = Varµ,  Entu�→u log u  µ  = h(·|µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Now let (X(t))t≥0 be a Markov process on our Polish space E with associated semigroup (Pt)t≥0 acting  on Cb(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' R), the space of continuous and bounded real-valued functions on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let us assume that there  Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Primary 82C20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Secondary 60K35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Interacting particle systems, phi-entropy, time-reversal, martingale representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  1  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  2  exists an invariant probability measure µ and denote by L the generator of the semigroup (Pt)t≥0 with  domain dom(L ) ⊂ Cb(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By invariance of µ and Jensen’s inequality one can now deduce that for all f ∈ Cb(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' R)  EntΦ  µ(Ptf) = Eµ [Φ(Ptf)] − Φ (Eµ [Ptf]) ≤ Eµ [PtΦ(f)] − Φ (Eµ [f]) = EntΦ  µ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  This tells us that the Φ-entropy is non-increasing as a function of t and can be used as a Lyapunov  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' More precisely, with purely analytic arguments, one can even deduce the following general  result about the decay of Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1 (DeBruijn like property for Markov semigroups, [Cha04]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (X(t))t≥0 be a Markov  process on a Polish space E equipped with its Borel σ-algebra B(E) and let (Pt)t≥0 be the associated  Markov semigroup with generator L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Assume that µ is an invariant probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for any  continuous and bounded function f : E → I and any t > 0, it holds that  ∂t EntΦ  µ(Ptf) = Eµ [Φ′(Ptf)L (Ptf)] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  This result is classical, but we nevertheless recall its short analytic proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will later also provide a  more probabilistic argument to obtain the same result in the context of interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The chain rule and the definition of the generator L directly imply that  ∂t EntΦ  µ(Ptf) = Eµ  �  Φ′(Ptf) d  dt(Ptf)  �  = Eµ [Φ′(Ptf)L (Ptf)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  To see that the left-hand side is actually non-positive, it suffices to observe that the convexity of Φ implies  via Jensen’s inequality for conditional expectations  Φ(Pt+sg) ≤ Pt(Φ(Psg))  for any s, t ≥ 0 and hence for all f we have  EntΦ  µ(Pt+sf) ≤ EntΦ  µ(Psf),  by invariance of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  By integrating one obtains the following classical corollary, which links exponential decay of Φ-entropies  and functional inequalities involving Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In the setting of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1, the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' There exists a constant c > 0 such that for all f ∈ dom(L )  EntΦ  µ(f) ≤ −cEµ [Φ′(f)L f] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' There exists a constant c > 0 such that for all continuous and bounded f : E → I  EntΦ  µ(Ptf) ≤ e− t  c EntΦ  µ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Note that, in the special case Φ : u �→ u2, one recovers the Poincar´e inequality  Varµ(f) ≤ − c  2 ⟨f, L f⟩L2(µ),  which is well-known to be equivalent to exponential L2 ergodicity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' [HS76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For a more detailed  review of Φ-entropies and further results we refer the interested reader to [Cha04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' A finite state-space example for the trajectorial approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' As one can see, the results above  can be obtained without even mentioning the underlying stochastic process and just dealing with the  semigroup and its generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We want to supplement this viewpoint with a more probabilistic approach  that allows us to obtain a somewhat finer result on a trajectorial level, which also implies the classical  results by taking expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  For simplicity, we will discuss the main ideas for the example of a continuous-time Markov chain on  a finite state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' More precisely, let (Xt)t≥0 be a Markov chain on a finite set E with irreducible  generator L and strictly positive invariant measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Hence, the corresponding Markov semigroup is  given by the matrix exponential (etL )t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Denote the underlying probability space by (Ω, A, P) and  assume that X0 ∼ µ under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  It is easy to check that for all bounded f : [0, ∞)×E → R such that for all x ∈ E the partial derivatives  ∂tf(·, x) are continuous and bounded, the process defined by  f(t, Xt) −  � t  0  (∂s + L )f(s, Xs)ds,  t ≥ 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1)  is a martingale w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the canonical filtration generated by (Xt)t≥0, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' [RW00, Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  3  If we now fix a finite time horizon T > 0 and consider the time-reversal ( ˆXt)0≤t≤T of (Xt)t≥0, where  ˆXt = XT −t, then under P the time-reversed process is again a time-homogeneous Markov process with  generator  ˆ  L , where  ˆ  L (x, y) = µ(y)  µ(x)L (y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  A short calculation now shows that, for each bounded g : E → R and T > 0, the process (PT −sg( ˆXs))0≤s≤T  is a (( ˆFt)0≤t≤T , P)-martingale, where ˆFt = σ(XT −s : 0 ≤ s ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Indeed, we can use the chain rule to  calculate  ∂tPT −tg(x) = − ˆ  L PT −tg(x),  so the correction term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Note that it is crucial to use the time-reversed process here,  since the correction term does not cancel out if one uses the forward process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By convexity this directly implies that the time-reversed trajectorial Φ-entropy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=', the process defined  by  Φ(PT −sg(X(T − s))),  0 ≤ t ≤ T,  is a submartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This stochastic monotonicity can be seen as a trajectorial version of the classical  Φ-entropy decay and after taking expectations with respect to P, one obtains the classical results as in  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Therefore, one can interpret these trajectorial results as the probabilistic version of the  decay of Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The main work is now to establish that a similar argument can also be used to treat infinite-dimensional  systems like the interacting particle systems we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' To our best knowledge, the first results of this  kind, in the context of diffusions in Rn, have been achieved in [FJ16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  More recently, starting with  [KST20], these results have been extended to more and more classes of Markov processes, including  continuous time Markov chains on countable state spaces, see [KMS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The works [KY22] and [TCY23]  are also in a similar spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The setting will be made precise in Section 2, but roughly speaking, we consider continuous-time  Markov jump processes on general configuration spaces Ω = ΩS  0 , where S is an arbitrary countable set  and Ω0 is a compact Polish space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will refer to the elements of S as sites and call Ω0 the local  state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In most examples considered in the literature, S is the vertex set of some graph like the  d-dimensional hypercubic lattice Zd, a tree or the Cayley graph of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This underlying spatial  geometry dictates which particles can interact with each other and we are therefore not in the setting  of mean-field systems but in an infinite-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This of course brings with it its own set of  technical difficulties which need to be dealt with for making the time-reversal arguments work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The main technical difficulties come from making sure that the time-reversal is again a well-defined  interacting particle system and from obtaining a description of its generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This is made possible by  assuming some local regularity of the local conditional distributions of the time-stationary measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Namely, by the assumption that µ is actually a Gibbs measure with respect to a quasilocal specification  that additionaly satisfies a certain smoothness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This condition is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' satisfied if the specification  is given in terms of a potential Φ = (ΦB)B⋐S such that  sup  x∈S  �  B⋐S: B∋x  |B| ∥ΦB∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Note that this condition is for example satisfied for any translation-invariant finite-range potential, so  our theory applies to a fairly large class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Organization of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will first  collect the necessary notation and formulate our main results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, as a first step, we  investigate the time-reversal of interacting particle systems in equilibrium in Section 3 with the main  goal of obtaining an explicit representation of the (formal) generator of the time-reversed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In  Section 4, we will then apply these results to establish pathwise properties of general Φ-entropy functionals  and derive the classical DeBrujin-like decay property as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Setting and main results  Let (Ω0, B0) be a compact Polish space equipped with its Borel σ-algebra and λ0 a probability measure  on (Ω0, B0), which will serve as our reference measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will consider Markovian dynamics on the  configuration space Ω = ΩS  0 , where S is some countable set whose elements we will refer to as sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In  most applications this will be the set of vertices of some graph, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Zd or a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We equip Ω with the  product topology and corresponding Borel σ-algebra F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Note that F coincides with the product σ-algebra  ⊗x∈SB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For ∆ ⊂ S we will also write Ω∆ := Ω∆  0 for the set of partial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will also equip  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  4  Ω∆ with the product σ-algebra and the probability measure λ∆ = ⊗x∈∆λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For Λ ⊂ S, let FΛ be the  sub-σ-algebra of F that is generated by the projections ω �→ ω∆ ∈ Ω∆ for ∆ ⋐ Λ, where we write ⋐ to  signify that a set is a finite subset of another set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For ∆ ⊂ S and (partial) configurations η∆c ∈ Ω∆c and  ξ∆ ∈ Ω∆, we will write ξ∆η∆c for the configuration that is defined on all of S and agrees with η∆c on ∆c  and with ξ∆ on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For a topological space E, we will denote its Borel σ-algebra by B(E) and the space  of continuous real-valued functions on E by C(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The space of non-negative measures on E, or more  precisely on B(E), will be denoted by M(E) and is equipped with the topology of weak-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The total variation distance on M(E) will be denoted by ∥·∥TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Interacting particle systems and Gibbs measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will consider time-continuous Markovian dynamics on Ω, namely  interacting particle systems characterized by time-homogeneous generators L with domain dom(L ) ⊂  C(Ω) and the associated Markovian semigroup (Pt)t≥0 on C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For interacting particle systems we  adopt the notation and exposition of the standard reference [Lig05, Chapter I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  In our setting, the generator L is given by a collection of transition measures (c∆(·, dξ))∆⋐S in finite  volumes ∆ ⋐ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=', mappings  Ω ∋ η �→ c∆(η, dξ∆) ∈ M(Ω∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  These transition measures can be interpreted as the infinitesimal rates at which the particles inside ∆  switch from the configuration η∆ to ξ∆, given that the rest of the system is currently in state η∆c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The  full dynamics of the interacting particle system is then given as the superposition of these local dynamics,  L f(η) =  �  ∆⋐S  �  Ω∆  [f(ξ∆η∆c) − f(η)] c∆(η, dξ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1)  In [Lig05, Chapter I] it is shown that the following conditions are sufficient to guarantee well-definedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (L1) For each ∆ ⋐ Ω the mapping  Ω ∋ η �→ c∆(η, dξ∆) ∈ M(Ω∆)  is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (L2) The total rate at which a single particle switches its state is uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=',  sup  x∈S  �  ∆∋x  sup  η∈Ω  c∆(η, Ω∆) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (L3) The total influence of all other particles on the transition rates of a single particle is uniformly  bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=',  M := sup  x∈S  �  ∆∋x  �  y̸=x  δyc∆ < ∞,  where  δyc∆ := sup {∥c∆(η, ·) − c∆(ξ, ·)∥TV : ηyc = ξyc} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Under these conditions, the core of the operator L is given by  D(Ω) :=  �  f ∈ C(Ω) : |||f||| :=  �  x∈S  δx(f) < ∞  �  ,  where for x ∈ S  δx(f) :=  sup  η,ξ: ηxc=ξxc  |f(η) − f(ξ)|  is the oscillation of a function f : Ω → R at the site x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In addition, one can show the following estimates  for L and the action of the semigroup (Pt)t≥0 generated by L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will need these later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Assume that the generator L satisfies (L1) − (L3) and denote by (Pt)t≥0 the semigroup  generated by L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For f ∈ D(Ω) we have Ptf ∈ D(Ω) for all t ≥ 0 and  |||Ptf||| ≤ exp ((M − ε)t) |||f|||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For all f ∈ D(Ω) it holds that  ∥L f∥∞ ≤  �  sup  x∈S  �  ∆∋x  sup  η∈Ω  c∆(η, Ω∆)  �  |||f|||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  5  The constants are explicitly given by  M = sup  x∈S  �  ∆∋x  �  y̸=x  δyc∆ < ∞,  ε = inf  x∈S  inf  η,ζ : ηxc=ζxc ,ηx̸=ζx  �  ∆∋x  \uf8eb  \uf8ed  �  ξ∆:ξx=ζx  c∆(η, ξ∆) +  �  ξ∆:ξx=ηx  c∆(ζ, ξ∆)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Combine the results from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2(a) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' (d) in [Lig05, Chapter I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  For our purposes, the mere well-definedness of an interacting particle system is not sufficient and we  need to assume some more regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' All the additional assumptions we put will be used to make the  generator of the time-reversal well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (R1) For each ∆ and η ∈ Ω the measure c∆(η, dξ∆) is absolutely continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the reference measure  λ∆(dξ∆) with density c∆(η, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (R2) For each ∆ ∈ Ω the map  Ω × Ω∆ ∋ (η, ξ∆) �→ c∆(η, ξ∆) ∈ R  is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (R3) The total rate of transition for a single site is uniformly bounded from above  sup  x∈S  �  ∆∋x  sup  η∈Ω  ∥c∆(η, ·)∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (R4) The condition (L3) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=',  sup  x∈S  �  ∆∋x  �  y̸=x  δyc∆ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (R5) There exists an R > 0 such that for all ∆ ⋐ Zd with |∆| > R we have  sup  η∈Ω,ξ∆∈Ω∆  c∆(η, ξ∆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  We will comment on where and why we need these assumptions and their connection to the classical  conditions (L1) − (L3) at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2, after we have stated our assumptions on the local  conditional distribution of the time-stationary measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Gibbs measures and the DLR formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We will mainly be interested in the situation where the  process generated by L admits a time-stationary measure µ with a well-behaved local representation,  namely that µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' to a sufficiently nice specification γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let us therefore first recall  the general definition of a specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' A specification γ = (γΛ)Λ⋐S is a family of probability kernels γΛ from ΩΛc to M1(Ω)  that additionally satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Each γΛ is proper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=', for all B ∈ FΛc it holds that  γΛ(B|·) = 1B(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The probability kernels are consistent in the sense that if ∆ ⊂ Λ ⋐ S, then for all A ∈ F  γΛγ∆(A|·) = γΛ(A|·),  where the concatenation of two probability kernels is defined as usual via  γΛγ∆(A|η) =  �  Ω  γ∆(A|ω)γΛ(dω|η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  For the existence and further properties of Gibbs measures with specification γ one needs to impose  some conditions on the specification γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' One sufficient condition for the existence of a Gibbs measure for  a specification γ is quasilocality, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' [FV17] or [Geo11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For the following sections we will need to  assume some more regularity for the specification γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In particular, these assumptions will guarantee that  γ is quasilocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (S1) For each ∆ ⋐ S and η ∈ Ω, the probability measure γ∆(dξ∆|η) is absolutely continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the  reference measure λ∆(dξ∆) with density γ∆(·|η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (S2) For all ∆ ⋐ S, the map  Ω × Ω∆ ∋ (η, ξ∆) �→ γ∆(ξ∆|η∆c) ∈ [0, ∞)  is continuous (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the product topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  6  (S3) The conditional densities on the single spin spaces are uniformly bounded away from zero and  infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=',  0 < δ ≤ inf  x∈S inf  η∈Ω γx(ηx|ηxc) ≤ sup  x∈S  sup  η∈Ω  γx(ηx|ηxc) ≤ δ−1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (S4) We have  sup  x∈S  �  ∆∋x: c∆>0  �  y̸=x  δyγ∆ < ∞,  where  δyγ∆ = sup {∥γ∆(dξ∆|η) − γ∆(dξ∆|ζ)∥TV : ηyc = ζyc} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Now that we have stated all of the conditions that we need, let us briefly comment on why  and where we need them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Assumption (R3) clearly implies (L2) and together with (R4) ensures that the interacting particle  system is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Assumption (R1) and (S1) allow us to write down the local transition density of the time-reversal  and (S3) makes sure that we are not performing a division by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The further regularity assumptions (R3), (R5) (S3), (S4) and the continuity assumptions (R2) and  (S2) make sure that the local transition densities of the time-reversal also give rise to a well-defined  interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The quantity in (S4) is similar to the classical Dobrushin uniqueness condition, see [Geo11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' How-  ever, we only need it to be finite and not strictly smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' One particular class of models to which our theory can be applied to are spin systems  for which the specification γ is defined via a potential Φ = (ΦB)B⋐S that satisfies  sup  x∈S  �  B∋x  |B| ∥ΦB∥∞ < ∞,  and where the rates are of the form  c∆(η, ξ∆) =  �  exp  �  −β �  B:B∩∆̸=∅ ΦB(ξ∆η∆c)  �  ,  if |∆| = 1,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Instead of these single-site updates one could also consider updates in larger regions with a bounded  diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then the rates satisfy (R1) − (R5) and the specification satisfies (S1) − (S4), as one can see  by using similar arguments as in the proof of [FV17, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The time-reversal of an interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In the notation of above, assume that  µ ∈ G (γ) is Gibbs measure for a quasilocal specification γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=', assume that µ satisfies the DLR equations  µ(f) = µ(γΛ(f|·))  for all Λ ⋐ S and bounded measurable functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Further assume that µ is time-stationary with respect  to the Markovian dynamics with generator L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Denote the semigroup generated by L by (Pt)t≥0 and the  corresponding process on Ω by (η(t))t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' As discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2, for each fixed T > 0 the process  (η(T − t))0≤t≤T is again a time-homogeneous Markov process and under some suitable assumptions its  associated semigroup has a generator  ˆ  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' But what does this generator look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For general Markov  processes it is not possible to give a closed form expression, but in our case we can use the special  structure of L as the superposition of local dynamics in finite volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In each of these finite volumes,  it is clear how the time-reversal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' µ should look and we can hope that we can again write  ˆ  L as  the superposition of finite volume processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' With this Ansatz, the probabilistic intuition dictates the  educated guess  ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)  γ∆(η∆|η∆c)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2)  for the transition densities appearing in the generator of the time-reversed interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  However, at this stage, it is not obvious that the generator of the time-reversed system is again of the  form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1) and has precisely these rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For Markov processes on finite state spaces this is an easy  calculation but we have to put in some more work, which will be carried out in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The main  result there is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='5 (Time-reversal generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let the rates of an interacting particle system with generator  L satisfy (R1) − (R5) and assume that µ is a time-stationary measure for the corresponding Markov  semigroup (P(t))t≥0 on C(Ω) that is generated by L such that µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' a specification  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  7  γ that satisfies (S1)−(S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, the time-reversed process has a generator  ˆ  L whose transition densities  (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the reference measure λ0) are given by  ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)  γ∆(η∆|η∆c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The proof of this can be found at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Trajectorial decay of Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' With this auxiliary result at hand, we can then obtain the  following result, which describes the dissipation of general Φ-entropies on a trajectorial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Before we  state the theorem, let us introduce some further notation to express the main equation in a cleaner way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The Bregman Φ-divergence associated with Φ : I → R is defined as  divΦ(p|q) := Φ(p) − Φ(q) − (p − q)Φ′(q),  p, q ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  This is precisely the difference between the value of Φ at the point p and the value of the first-order  Taylor expansion of Φ around q, evaluated at p and is non-negative, since we assumed that Φ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Bregman divergences are sometimes also referred to as Bregman distances, despite not being a metric  since they are in general not symmetric and do not satisfy the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' They are however  still useful for applying techniques from optimization theory in more general contexts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' in statistical  learning theory [BMDG05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Note that we now have to be careful with the probability space and filtration we are working with,  since we are talking about results on a trajectorial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='6 (Trajectorial decay of Φ-entropies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (Ω, A, P) be a probability space on which the  interacting particle system (η(s))s≥0 is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Denote the generator of the interacting particle system  by L and assume that its rates satisfy (R1) − (R5) and assume that under P we have η0 ∼ µ, where µ  is a time-stationary measure for the corresponding Markov semigroup (Pt)t≥0 on C(Ω) that is generated  by L such that µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' a specification γ that satisfies (S1) − (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for any  f ∈ D(Ω) and T > 0, the process defined by  LΦ,f(s) := Φ(PT −sf(ηT −s)),  0 ≤ s ≤ T,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3)  is a (( ˆGt)0≤t≤T , P)-submartingale, where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Its Doob–Meyer decomposition is  given by  LΦ,f(t) = M Φ,f(t)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4)  +  � t  0  �  ∆⋐S  �  Ω∆  ˆc∆(η(T − s), ξ∆)divΦ(PT −s(f(ξ∆η∆c(T − s))|PT −sf(η(T − s)))λ∆(dξ∆)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The proof of this theorem can be found at the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' As mentioned before, by taking  expectations one recovers the classical DeBrujin-like decay of Φ-entropies as stated in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  For the sake of concreteness, let us write out the result explicitly for one of the simplest cases, namely  the trajectorial decay of variance, corresponding to Φ : u �→ u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='7 (Trajectorial decay of variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='6, we have that, for any f ∈  D(Ω) and T > 0, the process defined by (PT −sf(ηT −s))2, 0 ≤ s ≤ T , is a (( ˆGt)0≤t≤T , P)-submartingale,  where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Its Doob–Meyer decomposition is given by  (PT −sf(ηT −s))2 = M f(t) +  � t  0  �  ∆⋐S  �  Ω∆  ˆc∆(η(T − s), ξ∆) [f(ξ∆η∆c(T − s)) − f(η(T − s))]2 λ∆(dξ∆)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Even though we were able to show the trajectorial decay for the relative entropy under  quite general assumptions on the dynamics, these results are not fully satisfactory in the context of  statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The usually more interesting Lyapunov functional in this setting is the so-called  relative entropy density, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' considered in [FV17], which is not only defined for measures ν that are  absolutely continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Therefore, it would be much more natural to work with this functional  h(·|µ) : Minv  1  (Ω) → R instead and one can show that is also a Lyapunov function for interacting particle  systems under quite general assumptions, see [JK22], but it is somewhat unclear how to even formulate  conjectures about the trajectorial properties of this functional, since one cannot naively evaluate it  pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  As we already saw in the case of a continuous time Markov chain on a finite state space, the main  ingredient for this type of result is to obtain an explicit description of the generator of the time-reversed  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Another class of processes that could be of interest and is not covered by our results are  systems which evolve continuously on their single spin spaces, as opposed to our pure-jump processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The first example that comes to mind are of course systems of (infinitely-many) interacting diffusions,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' indexed by Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' We expect that, if a given system of interacting diffusions admits a Gibbs measure  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  8  as an invariant probability measure, then a combination of the techniques in [KST20] and this article  should yield analogous results – of course under some suitable regularity conditions on the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The time-reversed interacting particle system and its generator  The main goal of this section is to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='5, thereby establishing that the generator of  the time-reversal is indeed given by  ˆ  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For this we will need to establish some regularity properties for  the transition densities as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Upper and lower bounds for the conditional densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Since we will need to deal with  quotients involving the conditional densities γ∆ on arbitrary finite subsets ∆ ⋐ S, we will need to lift  the upper and lower bounds from (S3) to this more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This is essentially the content of the  following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let γ be a specification that satisfies (S1) and (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, there exists a constant C ∈  (0, ∞) such that for all ∆ ⋐ S we have the estimate  e−C|∆| ≤ inf  η∈Ω γ∆(η∆|η∆c) ≤ sup  η∈Ω  γ∆(η∆|η∆c) ≤ eC|∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  This constant is precisely given by C = |log δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For this, fix an enumeration i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' , ik of the elements of ∆ and introduce the notation  [ij, ik] := {ij, ij+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' , ik} ,  1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  With this notation at hand, we can use the chain rule for conditional probability densities to write  γ∆(η∆|η∆c) =  k  �  j=1  γ[i1,ij](ηij|η[ij+1,ik]η∆c),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1)  where γ[i1,ij](ηij|η[ij+1,ik]η∆c) is the marginal conditional density of the measure γ[i1,ij](dη[i1,ij]|η[ij+1,ik]η∆c)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the site ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' But, using consistency of the specification γ, we have  γ[i1,ij](ηij|η[ij+1,ik]η∆c) =  �  γ[i1,ij](dξ[i1,ij]|η[ij+1,ik]η∆c)γij(ηij |ξ[i1,ij−1]η[ij+1,ik]η∆c),  which is, by assumption, upper bounded by δ−1 and lower bounded by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  In conjunction with the  representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1) this implies the desired upper and lower bound where the constant C is explicitly  given by C = |log(δ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  As a corollary we now get the following estimate for the quotients that appear in the definition of  transition density of the time-reversal (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let ∆ ⋐ S and γ be a specification that satisfies (S1) and (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for all ∆ ⋐ S,  η ∈ Ω and ξ∆ ∈ Ω∆, we have  0 < e−2C|∆| ≤ γ∆(ξ∆|η∆c)  γ∆(η∆|η∆c) ≤ e2C|∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The switching lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Now that we can be sure that the densities as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2) are actually well-  defined and we are not performing a division by zero, we can start showing that  ˆ  L is indeed the generator  of the time-reversed process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The main technical tool will be the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let the rates of a well-defined interacting particle system with generator L satisfy (R1)  and assume that µ is a time-stationary measure for L and µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' a specification γ  that satisfies (S1) and (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, we have for all bounded and measurable f, g : Ω → R and ∆ ⋐ S that  �  Ω∆  �  Ω  c∆(ω, ξ∆)f(ω)g(ξ∆ω∆c)µ(dω)λ∆(dξ∆) =  �  Ω∆  �  Ω  ˆc∆(ω, ξ∆)f(ξ∆ω∆c)g(ω)µ(dω)λ∆(dξ∆),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2)  where  ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)  γ∆(η∆|η∆c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  To keep the notation for conditional expectations in the upcoming proof simple, we will denote integration  with respect to µ by E[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' As a first step, note that for fixed ∆ ⋐ S and ξ∆ ∈ Ω∆ the maps  Ω ∋ ω �→ g(ξ∆ω∆c) ∈ R,  Ω ∋ ω �→ f(ξ∆ω∆c) ∈ R,  are F∆c-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Therefore, we can use that γ is the local conditional distribution of µ and the  definition of the rates ˆc to obtain the µ-almost-sure identity  E [c∆(·, ξ∆)f(·)g(ξ∆·∆c)|F∆c] (ω) = g(ξ∆ω∆c)E [c∆(·, ξ∆)f(·)|F∆c] (ω)  = g(ξ∆ω∆c)  �  Ω∆  γ∆(ζ∆|ω∆c)c∆(ζ∆ω∆c, ξ∆)f(ζ∆ω∆c)λ∆(dζ∆)  = g(ξ∆ω∆c)  �  Ω∆  γ∆(ξ∆|ω∆c)ˆc∆(ξ∆ω∆c, ζ∆)f(ζ∆ω∆c)λ∆(dζ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  If we now integrate over ξ∆, exchange the order of integration (via Fubini) and apply the same arguments  in reverse – with f taking the role of g and vice versa – we get  �  Ω∆  E [c∆(·, ξ∆)f(·)g(ξ∆·∆c)|F∆c] (η)λ∆(dξ∆) =  �  Ω∆  E [ˆc∆(·, ζ∆)f(ζ∆·∆c)g(·)|F∆c] (η)λ∆(dζ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By integrating both sides with respect to µ, exchanging the order of integration, and applying the law of  total expectation we obtain  �  Ω∆  �  Ω∆  (ω, ξ∆)f(ω)g(ξ∆ω∆c)µ(dω)λ∆(dξ∆) =  �  Ω∆  �  Ω  ˆc∆(ω, ζ∆)f(ζ∆ω∆c)g(ω)µ(dω)λ∆(dζ∆),  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Regularity of the time-reversal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' To make sure that  ˆ  L is actually the generator of a well-  defined interacting particle system we now show that the collection of transition measures (ˆc∆(·, ·))∆⋐S  satisfies the three conditions (L1) − (L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let the rates of an interacting particle system with generator L satisfy (R1) − (R5)  and assume that µ is a time-stationary measure for L and such that µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' a  specification γ that satisfies (S1)−(S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, the transition measures (ˆc∆(·, dξ∆))∆⋐Zd with λ∆-densities  given by  ˆc∆(η, ξ∆) = c∆(ξ∆η∆c, η∆)γ∆(ξ∆|η∆c)  γ∆(η∆|η∆c)  satisfy the conditions (L1) − (L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Ad (L1): This follows from the continuity assumptions (R2) and (S2), together with assumption  (S3) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Ad (L2): Note that for fixed ∆ ⋐ S, ξ∆ ∈ Ω∆ and η ∈ Ω we have by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2 and assumption (R5)  |c∆(η, ξ∆)| =  ����c∆(ξ∆η∆c, η∆) γ∆(ξ∆|η∆c  γ∆(η∆|η∆c)  ���� ≤ 1  δ eRc∆(ξ∆η∆c, η∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  So we get  sup  η∈Ω  ˆc∆(η, Ω∆) = sup  η∈Ω  �  Ω∆  ˆc∆(η, ξ∆)λ∆(dξ∆) ≤ 1  δ eR sup  η∈Ω  �  Ω∆  c∆(ξ∆η∆c, η∆)λ∆(dξ∆)  ≤ 1  δ eR sup  η∈Ω  ∥c∆(η, ·)∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Therefore, assumption (R3) implies that (L2) is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Ad (L3): Fix ∆ ⋐ S, y ∈ S and two configurations η1, η2 that only disagree at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for any ξ∆ ∈ Ω∆  we have  ��ˆc∆(η1, ξ∆) − ˆc∆(η2, ξ∆)  �� =  ����c∆(ξ∆η1  ∆c, η1  ∆)γ∆(ξ∆|η1  ∆c)  γ∆(η1  ∆|η1  ∆c) − c∆(ξ∆η2  ∆c, η2  ∆)γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ����  ≤  ��c∆(ξ∆η1  ∆c, η1  ∆)  ��  ����  γ∆(ξ∆|η1  ∆c)  γ∆(η1  ∆|η1  ∆c) − γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ����  +  ����  γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ����  ��c∆(ξ∆η1  ∆c, η1  ∆) − c∆(ξ∆η2  ∆c, η2  ∆)  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  To estimate this further, we will have to make a case distinction over whether the site y is contained in  ∆ or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' If y is contained in ∆, then we can naively use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2 and assumption (R5) to obtain the  rough estimate  ��ˆc∆(η1, ξ∆) − ˆc∆(η2, ξ∆)  �� ≤ 41  δ eR  sup  η∈Ω,ξ∆∈Ω∆  |c∆(η, ξ∆)| ≤ 4eRK(c)  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  10  In the case where y is not contained in ∆, we can (and have to) be a bit more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Via the elementary  algebraic rule  ac − bd = 1  2 [(a − b)(c + d) + (a + b)(c − d)] ,  and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2 plus assumption (R5) one obtains  ��c∆(ξ∆η1  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η1  ∆)  ��  ����  γ∆(ξ∆|η1  ∆c)  γ∆(η1  ∆|η1  ∆c) − γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ���� +  ����  γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ����  ��c∆(ξ∆η1  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η1  ∆) − c∆(ξ∆η2  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η2  ∆)  ��  = 1  2  ��c∆(ξ∆η1  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η1  ∆)  ��  ����  1  γ∆(η1  ∆|η1  ∆c)γ∆(η2  ∆|η2  ∆c)  ����  ��γ∆(ξ∆|η1  ∆c) − γ∆(ξ∆|η2  ∆c)  �� ��γ∆(η1  ∆|η1  ∆c) + γ∆(η2  ∆|η2  ∆c)  ��  +  ����  γ∆(ξ∆|η2  ∆c)  γ∆(η2  ∆|η2  ∆c)  ����  ��c∆(ξ∆η1  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η1  ∆) − c∆(ξ∆η2  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η2  ∆)  ��  ≤  1  2δ2 e2RK(c)K(γ)  ��γ∆(ξ∆|η1  ∆c) − γ∆(ξ∆|η2  ∆c)  �� + 1  δ eR ��c∆(ξ∆η1  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η1  ∆) − c∆(ξ∆η2  ∆c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' η2  ∆)  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Now, by integrating this pointwise difference of the densities over ξ∆, we obtain via all of the other  assumptions that  sup  x∈S  �  ∆∋x  �  y̸=x  δyˆc∆ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  But this is precisely (L3) and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  With these two intermediate results at hand, we can now show that  ˆ  L is indeed the generator of the  time-reversal of (ηt)t≥0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the time-stationary measure µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' It only remains to show that for all f, g ∈ D(Ω) we have  �  Ω  f(ω)L g(ω)µ(dω) =  �  Ω  �  ˆ  L f  �  (ω)g(ω)µ(dω),  since then the claimed time-reversal duality follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  For this, we first note that it suffices to show that the duality relation for the generators holds for  all local functions f, g : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Indeed, if it holds for all pairs of local functions, we can then extend  it to functions with bounded total oscillation by using the estimates from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1 and dominated  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Therefore, let f, g be two local functions and let Λ ⋐ S be sufficiently large such that  both f and g only depend on coordinates in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' By first applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3 and then using that µ is  time-stationary with respect to the Markovian dynamics generated by L , we see that  �  Ω  f(ω)L g(ω)µ(dω) −  �  Ω  �  ˆ  L f(ω)  �  g(ω)µ(dω)  =  �  ∆∩Λ̸=∅  �  Ω∆  �  Ω  c∆(ω, ξ∆)[f · g(ξ∆ω∆c) − f · g(ω)]µ(dω)λ∆(dξ∆) =  �  Ω  L (f · g)(ω)µ(dω) = 0,  which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Trajectorial decay of Φ-entropies  In this section we use the time-reversed process and a martingale argument to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The time-dependent martingale property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The main technical tool will be the following lemma  which can be seen as an extension of [RW00, Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='12] to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let L be the generator of an interacting particle system (η(s))s≥0 such that its transition  rates satisfy (L1)−(L3) and let µ be a time-stationary measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for all f : [0, ∞)×Ω → R  such that  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' f(·, η) ∈ C1([0, ∞)) for all η ∈ Ω and  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' for all T > 0 it holds that  sup  0≤t≤T  |||f(t, ·)||| < ∞,  the process defined by  f(t, η(t)) −  � t  0  (∂s + L )f(s, η(s))ds  is a martingale w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' the filtration Gt := σ(η(u) : 0 ≤ u ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The proof of this lemma is not difficult but hard to find in the existing literature, we therefore give it  in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For functions f as above, we define  M(s) := f(s, η(s)) −  � s  0  (∂u + L )f(u, η(u))du,  s ≥ 0  and denote by (Pt)t≥0 the Markov semigroup generated by (∂s + L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for s < t, the Markov  property and the elementary identity  d  dtPt = Pt(∂t + L ) = (∂t + L )Pt,  give us  E  �  f(t, η(t)) −  � t  0  (∂u + L )f(u, η(u))du  ���Gs  �  = Pt−sf(s, η(s)) −  � s  0  (∂u + L )f(u, η(u))du −  � t  s  Pu−s(∂u + L )f(s, η(s))du  = Pt−sf(s, η(s)) −  � s  0  (∂u + L )f(u, η(u))du −  � t  s  d  duPu−sf(s, η(s))du  = f(s, η(s)) −  � s  0  (∂u + L )f(u, η(u))du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  This shows that the process (M(s))s≥0 is indeed a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  This abstract tool now lets us establish the analogue of the first step in the case of a finite state space  considered in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (Ω, A, P) be a probability space on which the interacting particle system (η(s))s≥0  is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Denote the generator of (η(s))s≥0 by L and assume that the rates satisfy (R1) − (R5) and  assume that under P we have η(0) ∼ µ, where µ is a time-stationary measure for the corresponding Markov  semigroup (Pt)t≥0 on C(Ω) that is generated by L and that µ is a Gibbs measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' a specification γ  that satisfies (S1) − (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Then, for all f ∈ D(Ω) and T > 0, the process defined by  PT −sf(η(T − s)),  0 ≤ s ≤ T,  is a (( ˆGt)0≤t≤T , P)-martingale, where ˆGt = σ(η(T − s) : 0 ≤ s ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Note that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1 we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1 to the function  [0, T ] × Ω ∋ (s, η) �→ PT −sf(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  But since we have by the chain rule  ∂sPT −sf = − ˆ  L PT −sf,  the correction term cancels out and we obtain the claimed martingale property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Trajectorial decay of Φ-entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' With this preliminary result in place, we can now come to  the proof of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Submartingale property: By Jensen’s inequality and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2 we immedi-  ately see that the process (LΦ,f(t))t≥0, as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3), is a submartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Doob–Meyer decomposition: Here we want to apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1 to the function  g : [0, T ] × Ω ∋ (s, η) �→ Φ(PT −sf) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Via the chain rule we see that  ∂sg(s, ·) = ∂sΦ(PT −sf(·)) = −Φ′(PT −sf(·)) ˆ  L PT −sf(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Applying the generator  ˆ  L for fixed s ∈ [0, T ] yields  ˆ  L g(s, η) =  �  ∆⋐S  �  Ω∆  ˆc∆(η, ξ∆) [Φ(PT −sf(ξ∆η∆c)) − Φ(PT −sf(η))] λ∆(dξ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By putting these two ingredients together and using the previously introduced notation for the Bregman  Φ-divergence we obtain  (∂s +  ˆ  L )g(s, η) =  �  ∆⋐S  �  Ω∆  ˆc∆(η, ξ∆)divΦ(PT −sf(ξ∆η∆c)|PT −sf(η))λ∆(dξ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The claimed Doob–Meyer decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4) of the submartingale LΦ,f now follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  12  Acknowledgements  The authors acknowledge the financial support of the Leibniz Association within the Leibniz Junior  Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz  Competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The time-reversal of Markov processes in equilibrium  In this section, we briefly summarize some properties of the time-reversal of a Markov process w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  a time-stationary measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' These results are classical but not particularly easy to find in the literature,  at least in this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  We start by making precise what we mean by time-reversal of a stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Recall that  any time-stationary stochastic process (Xt))t≥0 can be extended (in law) to a process (Xt)−∞<t<∞ via  Kolmogorov’s extension theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (Xt)t≥0 be a time-stationary stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  We call a process (Yt)t≥0 the  time-reversal of X if  Law((X−t)t≥0) = Law((Yt)t≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  The intuition behind this definition is that forward in time the process Y looks like the process X  run backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For Markov processes this notion can be characterized in terms of the semigroups and  generators as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let X = (Xt)t≥0 and Y = (Yt)t≥0 be Markov processes on a compact topological space  E with associated semigroups (Tt)t≥0 and (St)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Assume that X has a time-stationary measure ν and  for all f, g ∈ C(E) we have  �  E  (Ttf)gdν =  �  E  f(Stg)dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1)  Then ν is also time-stationary for Y and Y is the time-reversal of X (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' By the duality relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1), ν is also time-stationary for Y and we can extend it to a process  (Yt)−∞<t<∞ with Y0 ∼ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' To show that Y is the time-reversal of X (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' ν) it suffices to show that for  arbitrary n ∈ N and times t0 < t2 < · · · < tn < tn+1 and functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' , fn ∈ C(E) it holds that  EY0∼ν [f1(Yt1) · · · fn(Ytn)] = EX0∼ν [f1(X−t1) · · · fn(X−tn)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  We will do this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' First, we introduce the notation f0 = fn+1 ≡ 1 and define functions g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' , gn+1  and h0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' , hn+1 by g0 = hn+1 ≡ 1,  gl(x) := Ex  �l−1  �  k=0  fk(Xtl−tk)  �  ,  1 ≤ l ≤ n + 1  and  hl(y) := Ey  � n+1  �  k=l+1  fk(Ytk−tl)  �  ,  0 ≤ l ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  With this notation it suffices to show that the quantity  αl :=  �  E  gl(x)fl(x)hl(x)ν(dx),  0 ≤ l ≤ n + 1,  does not depend on l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Indeed, by stationarity of X and Y , this will then imply that  EY0∼ν [f1(Yt1) · · · fn(Ytn)] =  �  E  h0dν = α0 = αn+1 =  �  E  gn+1dν = EX0∼ν [f1(X−t1) · · · fn(X−tn)] ,  exactly as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' In order to show that αl does not depend on l, we use the duality relation as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' For 0 ≤ l ≤ n it holds that  αl =  �  E  glflhldν  =  �  E  gl(x)fl(x)Ttl+1−tl[fl+1hl+1](x)ν(dx)  =  �  E  Stl+1−tl[glfl](x)fl+1(x)hl+1(x)ν(dx)  =  �  E  gl+1fl+1hl+1dν = αl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Therefore, αl does not depend on l and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  TRAJECTORIAL DISSIPATION OF Φ-ENTROPIES FOR INTERACTING PARTICLE SYSTEMS  13  The duality relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1) for the semigroups can also be verified by checking a similiar property on  the level of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The main technical tool for establishing this relation between the generator of a  Markov process and its semigroup will unsurprisingly be the celebrated Hille–Yosida theorem which we  recall here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3 (Hille–Yosida).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' There is a one-to-one correspondence between Markov generators on C(E)  and Markov semigroups on C(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' This correspondence is explicitly given by  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  dom(L ) =  �  f ∈ C(E) : lim  t↓0  Stf − f  t  exists  �  ,  and  L f := lim  t↓0  Stf − f  t  ,  f ∈ dom(L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Stf = lim  n→∞  �  Id − t  nL  �−n  f,  f ∈ C(E), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Moreover,  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' if f ∈ dom(L ), then Stf ∈ dom(L ) for all t ≥ 0 and  d  dtStf = L Stf = StL f and  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' for g ∈ C(E) and λ ≥ 0, the solution to the resolvent equation f − λL f = g is given by  f =  � ∞  0  e−tSλtgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' See Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='6 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2 in [EK86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  With this at hand, we can now formulate the time-reversal duality for generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (Tt)t≥0 and (St)t≥0 be two Markov semigroups on C(E) where E is a compact topo-  logical space with time-stationary measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Let (A , dom(A )) and (B, dom(B)) be their generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' If  for all f ∈ dom(A ) and g ∈ dom(B) it holds that  �  E  (A f) g dν =  �  E  f (Bg) dν,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2)  then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' It suffices if (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2) holds for f, g in a core of the respective generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' The duality relation implies that for all f ∈ dom(A ), g ∈ dom(B) and λ ≥ 0 it holds that  �  E  f(g − λL g) dν =  �  E  g(f − λ ˆ  L f) dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  Now we can replace f by (Id − λL )−1f and g by (Id − λ ˆ  L )−1g to see that for all f, g ∈ C(Ω)  �  E  [(Id − λA )−1f]g dν =  �  E  [(Id − λB)−1g]f dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  By iteratively applying this equality we obtain that for all n ∈ N and λ ≥ 0 it holds that  �  E  [(Id − λA )−nf]g dν =  �  E  [(Id − λB)−ng]f dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  For fixed t ≥ 0 we can replace λ by t/n and use Part ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='3 to see that  lim  n→∞  �  E  �  (Id − t  nA )−nf  �  g dν =  �  E  gTtf dν  and  lim  n→∞  �  E  �  (Id − t  nB)−ng  �  f dν =  �  E  fStg dν,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  □  To sum this up, in order to show that a stationary Markov process Y is the time-reversal of a  stationary Markov process X, it suffices to check that their generators satisfy the duality relation  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='  References  [BMDG05] Arindam Banerjee, Srujana Merugu, Inderjit S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content=' Dhillon, and Joydeep Ghosh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
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+page_content=' (Cited on 3)  Institut f¨ur Mathematische Stochastik, Technische Universit¨at Braunschweig, Universit¨atsplatz 2, 38106  Braunschweig, Germany & Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39,  10117 Berlin, Germany  Email address: benedikt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='jahnel@tu-braunschweig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='de  Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany  Email address: jonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='koeppl@wias-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
+page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQffAdt/content/2301.03922v1.pdf'}
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+Oscillations in the magnetothermal conductivity of α-RuCl3: Evidence of transition anomalies
+Étienne Lefrançois,1, ∗ Jordan Baglo,1, † Q. Barthélemy,1 S. Kim,2 Young-June Kim,2 and Louis Taillefer1, 3, ‡
+1Institut Quantique, Département de physique & RQMP,
+Université de Sherbrooke, Sherbrooke, Québec, Canada
+2Department of Physics, University of Toronto, Toronto, Ontario, Canada
+3Canadian Institute for Advanced Research, Toronto, Ontario, Canada
+(Dated: January 16, 2023)
+The 2D layered insulator α-RuCl3 is a candidate material for a quantum spin-liquid state, which may be re-
+alized when a magnetic field suppresses the antiferromagnetic order present at low temperature. Oscillations in
+the field dependence of the thermal conductivity, observed for an in-plane magnetic field B up to a critical field
+B⋆, have been attributed to exotic charge-neutral fermions, viewed as evidence of a quantum spin-liquid state
+between the critical field Bc ≃ 7 T at which the antiferromagnetic phase ends and B⋆. Here we report measure-
+ments of the thermal conductivity of α-RuCl3 as a function of magnetic field up to 15 T applied in two distinct
+in-plane directions: parallel and perpendicular to the Ru-Ru bond. We find that the number of oscillations be-
+tween Bc and B⋆ is the same for the two field directions even though the field interval between Bc and B⋆ is
+different. In other words, the period of the oscillations is controlled by the transition fields Bc and B⋆. We con-
+clude that these are not true oscillations—coming from putative fermions in a spin-liquid state—but anomalies
+associated with a sequence of magnetic transitions.
+I.
+INTRODUCTION
+A major objective in the field of quantum materials is to
+confirm experimentally, in a real material, the existence of a
+quantum spin-liquid (QSL) state predicted theoretically. In
+this respect, the magnetic insulator α-RuCl3 is a promising
+material [1–3], with its Ru atoms lying on weakly coupled 2D
+honeycomb layers whose interactions are such that they nearly
+satisfy the Kitaev model [4], a model whose exact solution is a
+QSL with Majorana fermions as emergent excitations. In real-
+ity, the ground state of α-RuCl3 is not a QSL, but a state with
+long-range antiferromagnetic order, setting in below a critical
+temperature TN ≃ 7 K (Fig. 1). However, this ordered state
+can be suppressed by applying a magnetic field parallel to the
+honeycomb layers, until it ends at a critical field Bc ≃ 7 T
+(Fig. 1). The question then is this: What is the nature of the
+state just above Bc, at low temperature?
+To address that question experimentally, thermal transport
+has emerged as a fruitful probe. Early measurements of the
+thermal Hall conductivity, κxy, revealed the existence of a
+non-zero κxy, which was attributed to the Majorana fermions
+expected from the Kitaev model, given indications of a plateau
+in κxy vs field at a half-quantized value [5]. Although some
+later studies again find indications of half-quantization [6–8],
+others do not, and instead attribute the measured κxy to chiral
+magnons [9] or phonons [10, 11]. It is fair to say that based on
+κxy data the case for a QSL in α-RuCl3 above Bc is currently
+not compelling.
+In parallel with thermal Hall studies, measurements of the
+longitudinal thermal conductivity, κxx (or κ), have revealed
+the existence of oscillations as a function of in-plane mag-
+netic field B [12–14] (see Fig. 2). These oscillations have
+∗ etienne.lefrancois@usherbrooke.ca
+† jordan.baglo@usherbrooke.ca; É.L. and J.B. contributed equally.
+‡ louis.taillefer@usherbrooke.ca
+been interpreted as quantum oscillations akin to those pro-
+duced by Landau quantization of electron states in a metal
+when a magnetic field is applied, but this time coming from
+putative charge-neutral fermions, e.g. gapless spinons with a
+Fermi surface. Emergent neutral fermions would be a clear
+signature of a QSL state, albeit a different one from that ex-
+pected from the Kitaev model [4, 15]. While other groups con-
+firm the existence of these oscillations in κ vs B (Fig. 2), they
+attribute them to a sequence of magnetic transitions [13, 14].
+In this paper, we revisit these oscillations, with a focus on
+their anisotropy as the field direction within the honeycomb
+plane is changed from being perpendicular to the Ru-Ru bond
+(B ∥ a) to being parallel (B ∥ b) (Fig. 1).
+T
+B
+a
+b
+AF zigzag
+Partially spin 
+polarized
+QSL ?
+Paramagnet
+TN
+Bc
+B0
+FIG. 1.
+Phase diagram of α-RuCl3 as a function of temperature T
+and in-plane magnetic field B. The phase of long-range antiferro-
+magnetic (AF) order is shown in orange, with a transition tempera-
+ture TN ≃ 7 K and a critical field Bc ≃ 7 T. Spins on the Ru sites
+are arranged in a zigzag pattern of ferromagnetic chains (see sketch),
+but that pattern changes at B0 ≃ 6 T, just before reaching Bc. The
+question is whether the state just above the AF phase is a quantum
+spin liquid (QSL).
+arXiv:2301.05254v1  [cond-mat.str-el]  12 Jan 2023
+
+0.0
+0.5
+1.0
+1.5
+2.0
+B/Bc
+0
+4
+8
+12
+κ (mW/K⋅cm)
+Czajka et al.
+Bruin et al.
+Suetsugu et al.
+This work
+α-RuCl3
+B ∥ a
+T = 1 K
+B1
+B2
+FIG. 2. Thermal conductivity of α-RuCl3 as a function of in-plane
+magnetic field B, at T ≃ 1 K, plotted as κ vs B/Bc, where Bc is
+the critical field where the antiferromagnetic phase ends (Fig. 1).
+Data from four different studies are compared: Czajka et al. (pink;
+T = 0.96 K and Bc = 7.21 T) [12]; Bruin et al. (orange; T = 1.0 K
+and Bc = 7.20 T) [13]; Suetsugu et al. (green; T = 1 K and
+Bc = 7.15 T) [14]; this work (black, sample S1; T = 1.16 K and
+Bc = 7.69 T). In all cases, the field is applied along the a axis and
+the current is parallel to the field (J ∥ B). The vertical dashed lines
+mark the location of three minima, at Bc, B1, and B2. These minima
+in the oscillations of κ vs B are seen to be in the same locations for
+all four studies. The grey shaded region marks the regime at high
+field where oscillations are no longer observed.
+II.
+METHODS
+Single crystals were grown via the chemical vapor transport
+(CVT) method, using RuCl3 powder from Sigma-Aldrich.
+The powder, composed of 45–55% ruthenium, was sealed in
+a quartz ampoule under vacuum and the ampoule was then
+placed inside a two-zone tube furnace. The powder was an-
+nealed for two days at ∼ 800 ◦C in a temperature gradient of
+70 ◦C (warmest side was 850 ◦C), followed by a cooldown
+at 4 ◦C/hour while maintaining the temperature gradient. For
+more details, see Refs. 16 and 17. Here we report data on two
+as-grown (uncut) crystals, labeled S1 and S2, handled very
+carefully to minimize any strain induced when installing the
+contacts and fixing them on the experimental mount. Samples
+S1 and S2 are rectangular platelets with planar surface area
+1 mm × 1 mm and thicknesses of 130 and 110 µm, respec-
+tively. The contacts on the samples were made by attaching
+25 µm-diameter silver wires with DuPont 4929N silver paste.
+The heater was attached to the sample with a 100 µm-diameter
+silver wire with silver paste as well.
+Measurements were performed by a steady-state method
+using a standard four-terminal technique, with the thermal
+current applied along the length of the sample within the hon-
+eycomb layers: perpendicular to the Ru-Ru bonds for S1 (J ∥
+a) and parallel to the Ru-Ru bonds for S2 (J ∥ b). The ther-
+mal conductivity κ was measured by employing a standard
+one-heater–two-thermometers method, using a 10 kΩ resistor
+and two RuOx chip sensors whose magnetoresistances have
+been carefully taken into account: for interpolation of temper-
+ature from measured resistance and field, detailed R(T, B)
+calibration surfaces were collected for each individual ther-
+mometer in situ. All thermometers were measured using Lake
+Shore model 370 temperature controllers. A main calibrated
+RuOx sensor placed in the field-compensated mixing chamber
+region of the dilution refrigerator was used for the reference
+temperature T0, with typical control stability within ±6 µK
+at 100 mK and better than 0.03% over the full measurement
+range. A constant heat current J was injected at one end of
+the sample, while the other end of the sample is heat sunk
+to a copper block held at a temperature T0. The heat cur-
+rent was generated by sending an electric current through the
+10 kΩ strain gauge, whose resistance was measured to be es-
+sentially independent of temperature and magnetic field. The
+longitudinal temperature gradient ∆T = T + − T − is mea-
+sured at two points along the length of the sample, separated
+by a distance l. The longitudinal thermal conductivity is given
+by κ = J/ (∆Tα), where α = wt/l is the geometric factor of
+the sample (width w, thickness t).
+The two temperatures T + and T − were measured as the
+magnetic field, applied parallel to the honeycomb layers, was
+slowly swept from 0 to 15 T, while keeping the temperature
+T0 constant. For each sample, a series of field sweeps, taken
+at various temperatures T0 from 0.1 to 5 K, were obtained for
+two field directions: along the a axis and along the b axis of
+the crystal structure (Fig. 1). The magnetic field was swept
+sufficiently slowly (0.04 T/min) to avoid any magnetocaloric
+effect within the sample during the measurement, verified to
+be minimal by comparison with sweeps at different rates. The
+samples were firmly attached to their respective heat sinks
+in order to withstand the torque that could result from any
+slight misalignment of the field away from the intended high-
+symmetry direction, and thus avoid any bending of the sam-
+ples. The error bars on the absolute values of κ come mostly
+from the uncertainty in estimating the sample dimensions (l,
+w, and t), amounting approximately to ±20%. The applied
+current was chosen such that ∆T/T ≃ 3–5 % and the result-
+ing κ was verified to be independent of applied current.
+III.
+RESULTS
+In Fig. 3a, we display three isotherms of κ(B) taken on
+sample S1 (J ∥ a) with B ∥ a, plotted as κ(B)/κ(15 T)
+vs B. We observe four minima. The deepest minimum, lo-
+cated at Bc ≃ 7.6 T, corresponds to the end of the AF phase,
+as determined by thermodynamic measurements such as spe-
+cific heat [14, 18], magnetostriction [19, 20] and magnetiza-
+tion [13]. The minimum located at a field B0 ≃ 6.5 T, just
+below Bc, marks a transition internal to the AF phase (Fig. 1)
+where the AF ordering changes from one spin pattern to an-
+other [21]. The upper two minima, at B1 and B2, are both
+2
+
+0.0
+0.5
+1.0
+κ(B)/κ(15 T)
+0
+5
+10
+15
+B (T)
+0.0
+0.5
+1.0
+κ(B)/κ(15 T)
+0.22
+0.90
+1.84
+α-RuCl3
+B ∥ a
+S1
+J ∥ a
+(a)
+α-RuCl3
+J ∥ a
+B ∥ b
+(b)
+T =
+K
+B1 B2
+B *
+B1B2
+B *
+FIG. 3. Normalized thermal conductivity of sample S1 at T = 0.22,
+0.88, and 1.84 K, plotted as κ(B)/κ(15 T) vs B, for a thermal cur-
+rent J applied along the a axis. The magnetic field was applied par-
+allel to the a axis (a) and perpendicular to the a axis (b).
+above Bc, and so outside the phase of bulk AF order. The lo-
+cation of B1 and B2 is nicely consistent with prior studies, as
+seen in Fig. 2. It is possible that weak signatures of structure
+in κ(B) are visible further below B0 and in the AF state, in
+all four studies; this may be related to similar small anomalies
+in dielectric constant observed by Mi et al. [22]. The main
+purpose of our study was to investigate the anisotropy of this
+oscillatory pattern. In Fig. 3b, we display the corresponding
+isotherms taken on the same sample (S1) but for a field di-
+rection in the other high-symmetry in-plane direction, namely
+B ∥ b. We again observe only two minima above Bc, but this
+time they are much closer to each other (and to B⋆). Indeed,
+the separation between B1 and B2, which one might view as
+the “period” of the oscillations (∆B ≡ B2 − B1), is roughly
+twice as large for B ∥ a compared to B ∥ b.
+In order to confirm this anisotropy in the period of the os-
+cillations as the in-plane field changes direction, we measured
+a second sample (S2), this time with the heat current flowing
+along the b axis (J ∥ b). These data are displayed in Fig. 4,
+where we see that two minima are present above Bc and their
+separation ∆B is again significantly larger for B ∥ a com-
+0.0
+0.5
+1.0
+κ(B)/κ(15 T)
+0
+5
+10
+15
+B (T)
+0.0
+0.5
+1.0
+κ(B)/κ(15 T)
+0.25
+0.93
+2.02
+α-RuCl3
+B ∥ a
+S2
+J ∥ b
+(a)
+α-RuCl3
+J ∥ b
+B ∥ b
+(b)
+T =
+K
+B1
+B2
+B *
+B1
+B2
+B *
+FIG. 4. Normalized thermal conductivity of sample S2 at T = 0.24,
+0.90, and 2.02 K, plotted as κ(B)/κ(15 T) vs B, for a thermal cur-
+rent J applied along the b axis. The magnetic field was applied per-
+pendicular to the b axis (a) and parallel to the b axis (b).
+pared to B ∥ b. The current direction does not seem to make
+a significant difference.
+An important feature in our data is the existence of a thresh-
+old field, B⋆, above which there are no oscillations (or min-
+ima) anymore. This is especially clear in the isotherms at the
+lowest temperature (T ≃ 0.2 K), as in Fig. 3b for B ∥ b,
+where B⋆ ≃ 10 T, and in Fig. 4a for B ∥ a, where B⋆ ≃ 12 T.
+Note the anisotropy of B⋆.
+In Fig. 5, we directly compare isotherms for B ∥ a and
+B ∥ b, both at T ≃ 0.2 K. We see that the two oscillations
+observed for both field directions, with minima at B1 and B2,
+fit neatly in the interval between the two transitions, at Bc (end
+of AF phase) and B⋆ (end of oscillatory pattern), even though
+that interval shrinks by a factor of 2 in going from B ∥ a to
+B ∥ b. This reveals a connection between oscillations and
+transitions, the main finding of our anisotropy study.
+In Fig. 6, we plot the derivative ∂κ/∂B, obtained from our
+full sets of isotherms taken on our two samples (Figs. 7 and
+8), displayed in two contour maps as a function of tempera-
+ture and field, one panel for each field direction: B ∥ a and
+B ∥ b. These contour maps clearly show how the four char-
+3
+
+0.0
+0.1
+0.2
+0.3
+1.0
+1.5
+2.0
+B/Bc
+0.00
+0.05
+0.10
+0.15
+κ (mW/K⋅cm)
+α-RuCl3
+T = 0.2 K
+B ∥ b
+B ∥ a
+B1 B2
+B*
+B*
+B2
+B1
+FIG. 5. Anisotropy of characteristic fields. Comparison of isotherms
+at T ≃ 0.2 K, plotted as κ vs B/Bc, for the two field directions: B ∥
+a (red; sample S2, T = 0.20 K, Bc = 7.2 T) vs B ∥ b (blue; sample
+S1, T = 0.22 K, Bc = 7.7 T). Arrows mark the location of the
+two minima at B1 and B2, and the threshold field B⋆ above which
+oscillations are no longer observed. The period of the oscillations
+(|B1 − B2|) is seen to scale with the width of the interval between
+the field-induced transitions (|Bc − B⋆|), both shrinking roughly by
+half when the field is redirected from B ∥ a to B ∥ b. This suggests
+that the oscillations are related to the transitions.
+acteristic fields Bc, B1, B2, and B⋆ remain equally spaced in
+going from B ∥ a to B ∥ b even though the spacing shrinks by
+a factor of 2 or so. This again highlights the intimate link be-
+tween the oscillations (B1 and B2) and the transitions (Bc and
+B⋆).
+IV.
+DISCUSSION
+A.
+Nature of the heat carriers
+Before we discuss the origin of the oscillations in κ vs B
+observed in α-RuCl3, let us consider the nature of the heat
+carriers responsible for κ. The dependence of κ in α-RuCl3 on
+both temperature, over the full range up to 100 K, and in-
+plane magnetic field, up to 18 T, is well described by assuming
+that the only carriers of heat are phonons and that these are
+scattered by spin fluctuations [23]. Such a model, involving
+a magnetic excitation spectrum with a gap that grows with
+field, captures remarkably well the dramatic changes observed
+in κ as the in-plane field is increased, for temperatures above
+7 K, the regime where no order or oscillations are observed
+(Fig. 1). Phonons clearly dominate the thermal conduction in
+α-RuCl3.
+What about the regime below 4 K and between Bc and
+B⋆ where oscillations are observed? Is there evidence here
+for heat carriers beyond phonons? A recent study reports a
+(a)
+(b)
+4
+6
+8
+10
+12
+14
+B (T)
+0
+1
+2
+3
+4
+T (K)
+0.00
+0.01
+0.02
+∂κ/∂B
+α-RuCl3
+4
+6
+8
+10
+12
+14
+B (T)
+0
+1
+2
+3
+4
+T (K)
+−0.01
+0.00
+0.01
+0.02
+0.03
+∂κ/∂B
+α-RuCl3
+(a)
+(b)
+(mW/K·cm·T)
+(mW/K·cm·T)
+J || a
+B || a
+B || b
+J || b
+Bc
+B*
+B1
+B2
+Bc
+B*
+B1 B2
+FIG. 6. Contour plots of the derivative ∂κ/∂B, mapped as a func-
+tion of temperature and in-plane field, for: (a) B ∥ a, J ∥ a; (b)
+B ∥ b, J ∥ b. The vertical black lines mark the location of the four
+characteristic fields Bc, B1, B2, and B⋆.
+slight anisotropy in the specific heat C of α-RuCl3 whereby
+C(B ∥ b) > C(B ∥ a) below T
+≃ 2 K [24].
+This
+anisotropy is attributed to fermions that are gapless for B ∥ b
+and gapped for B ∥ a. Suetsugu et al. propose that a simi-
+lar anisotropy is reflected in the thermal conductivity, as they
+observe κ(B ∥ b) > κ(B ∥ a) below T ≃ 0.5 K [14], but
+this is not convincing as the reverse is true at high tempera-
+ture, i.e., κ(B ∥ a) > κ(B ∥ b) at T = 1.0 K [14]. We also
+observe the same reversal of the anisotropy with increasing
+temperature (Figs. 7,8). These small anisotropic effects of the
+magnetic field on κ are most likely coming from an anisotropy
+in the magnetic scattering of phonons.
+Suetsugu et al. also emphasize the sharp jump at B⋆ for
+B ∥ a, and attribute it to a (first-order) topological transition,
+because they observe it only for B ∥ a and not for B ∥ b. In
+our own data, however, the sharp jump at B⋆ is present for
+both field directions, as seen in Fig. 5.
+4
+
+10
+−1
+10
+0
+10
+1
+κ/T3 (mW/K4⋅cm)
+0
+5
+10
+15
+B (T)
+10
+−1
+10
+0
+10
+1
+κ/T3 (mW/K4⋅cm)
+0.17
+0.24
+0.35
+0.49
+0.81
+1.16
+1.21
+1.45
+1.91
+1.99
+2.27
+2.82
+3.40
+3.98
+4.54
+5.07
+T =
+K
+0.17
+0.22
+0.33
+0.45
+0.68
+0.90
+1.06
+1.33
+1.59
+1.84
+2.10
+2.60
+3.12
+3.65
+4.17
+4.68
+T =
+K
+α-RuCl3
+B ∥ a
+J ∥ a
+(a)
+α-RuCl3
+B ∥ b
+J ∥ a
+(b)
+FIG. 7.
+Full set of isotherms, measured on sample S1, plotted as
+κ/T 3 vs B in a semilog plot, for J ∥ a: (a) B ∥ a; (b) B ∥ b.
+In summary, there is nothing in the thermal conductivity
+of α-RuCl3 that indicates clearly the presence of fermionic
+excitations, or indeed of any excitations other than phonons.
+(In the partially spin polarized state beyond the AF phase,
+magnons should contribute to heat transport, but not at low
+temperature since they are gapped.)
+B.
+Origin of the oscillations
+Some theoretical studies find that a QSL state is plau-
+sible just above Bc, and its excitations could be gapless
+spinons [15].
+Such spinons have a Fermi surface and this
+could give rise to quantum oscillations. However, two facts
+about the oscillations in κ vs B observed in α-RuCl3 (Fig. 1)
+immediately argue against quantum oscillations. The first is
+the field direction being parallel to the layers. Indeed, in a 2D
+system the Fermi surface is expected to be a cylinder with its
+axis normal to the layers—as in the quasi-2D metal Sr2RuO4,
+for example [25]—which would not produce quantum oscil-
+lations for B ∥ a (or B ∥ b). Assigning the oscillations in
+α-RuCl3 to be quantum oscillations implies that the spinon
+Fermi surface is a 3D object like a sphere—difficult to imag-
+10
+−1
+10
+0
+10
+1
+κ/T3 (mW/K4⋅cm)
+0
+5
+10
+15
+B (T)
+10
+−1
+10
+0
+10
+1
+κ/T3 (mW/K4⋅cm)
+0.20
+0.25
+0.40
+0.52
+0.81
+1.17
+1.21
+1.50
+1.80
+2.08
+2.36
+2.91
+3.44
+3.97
+4.49
+5.00
+T =
+K
+0.20
+0.25
+0.40
+0.52
+0.93
+1.21
+1.43
+1.50
+2.02
+2.08
+2.37
+2.91
+3.45
+3.97
+4.50
+5.04
+T =
+K
+α-RuCl3
+B ∥ a
+J ∥ b
+(a)
+α-RuCl3
+B ∥ b
+J ∥ b
+(b)
+FIG. 8.
+Full set of isotherms, measured on sample S2, plotted as
+κ/T 3 vs B in a semilog plot, for J ∥ b: (a) B ∥ a; (b) B ∥ b.
+ine for a 2D system. The second fact is the sheer magnitude
+of the oscillations. In Fig. 2, we see that the oscillatory part
+of the conductivity can account for as much as 50% (peak-to-
+peak) of the background κ at T ≃ 1 K and B ≃ 10 T [12].
+This is enormous if attributed to quantum oscillations due to
+fermions. Indeed, in a metal like the cuprate YBa2Cu3Oy, for
+example, quantum oscillations in κ vs B coming from elec-
+trons have a peak-to-peak amplitude of 0.2% at T = 1.8 K
+and B = 45 T [26]. Note also the comparison in Fig. 2 of
+our data on sample S1 (black curve) and the data of Czajka
+et al. [12] (pink curve). Our sample has a higher conductiv-
+ity, which reflects a lower level of disorder. Yet its oscilla-
+tions are smaller, whereas quantum oscillations grow expo-
+nentially with decreasing disorder (increasing mean free path
+of fermions).
+Our anisotropy study provides a third argument against
+quantum oscillations from fermions. If the minima at B1 and
+B2 are those of quantum oscillations intrinsic to a QSL state,
+as proposed by Czajka et al. [12], why are there only two such
+minima (above Bc)? The authors postulate that the QSL phase
+ends at B⋆, in accordance with the absence of further oscilla-
+tions at higher field. Now the period of quantum oscillations
+is an intrinsic property of the QSL, imposed by the volume
+of the Fermi surface of those neutral fermions (e.g., gapless
+5
+
+spinons). The fact that two periods happen to fit in the inter-
+val created by the two phase transitions at Bc and B⋆, say for
+B ∥ a, is necessarily an accident. If we then change the field
+direction to B ∥ b, the period of quantum oscillations will
+change according to the topology of the spinon Fermi surface,
+which has nothing to do with the critical fields Bc and B⋆.
+Yet what we observe, as illustrated in Fig. 5, is that the pe-
+riod of the oscillations in α-RuCl3 changes in proportion to
+the change in the separation between the two critical fields
+Bc and B⋆. There are only two oscillations for both field di-
+rections, even though the interval between Bc and B⋆ is twice
+as large for B ∥ a. We conclude that the oscillations are inti-
+mately related to B⋆. More specifically, B⋆ marks the end of
+a transition centered at B2. In other words, the minima at B1
+and B2 mark two additional magnetic transitions above those
+at B0 and Bc, as proposed by Bruin et al. [13] on the basis that
+additional anomalies are also detected in the magnetization, at
+the same field values, and these can be tracked to transitions
+vs temperature in zero field, lying above TN.
+The nature of the transitions at B1 and B2 remains to be
+elucidated. The fact that the values of B1 and B2 are the same
+in different samples (Fig. 2), from different growth methods,
+suggests that they are generic features of α-RuCl3. However,
+the weakness of the associated specific heat anomalies sug-
+gests that the magnetic structures involved may not occupy
+the full volume of the sample and could be nucleated by local
+defects in the crystal structure, such as stacking faults.
+V.
+SUMMARY
+We investigated the proposal that oscillations detected in
+the thermal conductivity κ of the Kitaev material α-RuCl3 as
+a function of magnetic field B are quantum oscillations from
+neutral fermions that are the emergent excitations of a quan-
+tum spin liquid (QSL) state, which would exist in a regime
+of in-plane magnetic fields just above its antiferromagnetic
+(AF) phase, and below some critical field B⋆, as proposed by
+Czajka and coworkers [12]. We have measured the thermal
+conductivity κ of α-RuCl3 as a function of field up to 15 T
+at several temperatures between T = 0.2 K and T = 5 K
+for the two in-plane field directions B ∥ a and B ∥ b. For
+both field directions, we observe two oscillations in κ vs B
+contained between Bc, the critical field where the AF phase
+ends, and B⋆, the threshold field above which no further
+oscillations are seen (the putative critical field where the
+QSL phase ends). We find that as the field is changed from
+B ∥ a to B ∥ b, the interval between the two transition fields
+Bc and B⋆ shrinks by a factor of 2, and so does the period
+of the two oscillations contained in that interval.
+Because
+there is no reason for quantum oscillations—dictated by the
+Fermi surface of fermions—to be related to the critical field
+of the QSL phase (B⋆), we argue that the correlation between
+oscillation period and field interval (B⋆ − Bc) is instead
+evidence that the oscillations are produced by secondary
+magnetic transitions similar to the main transition at Bc.
+Our conclusion is consistent with that of Bruin et al. [13]
+and with theoretical works that report the absence of an inter-
+mediate field-induced region, in the phase diagram as a func-
+tion of in-plane field (Fig. 1), that could potentially harbor a
+QSL state [27, 28].
+ACKNOWLEDGMENTS
+We thank S. Fortier for his assistance with the experiments.
+L. T. acknowledges support from the Canadian Institute for
+Advanced Research (CIFAR) as a CIFAR Fellow and funding
+from the Institut Quantique, the Natural Sciences and Engi-
+neering Research Council of Canada (NSERC; PIN:123817),
+the Fonds de Recherche du Québec - Nature et Technologies
+(FRQNT), the Canada Foundation for Innovation (CFI), and a
+Canada Research Chair. This research was undertaken thanks
+in part to funding from the Canada First Research Excel-
+lence Fund. Work at the University of Toronto was supported
+by NSERC (RGPIN-2019-06449 and RTI-2019-00809), CFI,
+and the Ontario Ministry of Research and Innovation.
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+taev magnet α-RuCl3, Nature Materials (2022).
+[10] E. Lefrançois, G. Grissonnanche, J. Baglo, P. Lampen-Kelley,
+J.-Q. Yan, C. Balz, D. Mandrus, S. Nagler, S. Kim, Y.-J. Kim,
+N. Doiron-Leyraud, and L. Taillefer, Evidence of a Phonon Hall
+Effect in the Kitaev Spin Liquid Candidate α-RuCl3, Physical
+Review X 12, 021025 (2022).
+[11] R. Hentrich, M. Roslova, A. Isaeva, T. Doert, W. Brenig,
+B. Büchner, and C. Hess, Large thermal Hall effect in α-
+RuCl3: Evidence for heat transport by Kitaev-Heisenberg para-
+magnons, Physical Review B 99, 085136 (2019).
+[12] P. Czajka, T. Gao, M. Hirschberger, P. Lampen-Kelley,
+A. Banerjee, J. Yan, D. G. Mandrus, S. E. Nagler, and N. Ong,
+Oscillations of the thermal conductivity in the spin-liquid state
+of α-RuCl3, Nature Physics 17, 915 (2021).
+[13] J. Bruin, R. Claus, Y. Matsumoto, J. Nuss, S. Laha, B. V.
+Lotsch, N. Kurita, H. Tanaka, and H. Takagi, Origin of oscilla-
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+tive Kitaev magnet α-RuCl3, APL Materials 10, 090703 (2022).
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+the Quantum Spin Liquid State of a Kitaev Candidate α-RuCl3,
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+of α-RuCl3, Physical Review Letters 125, 097203 (2020).
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+Y. Takano, E. L. Brosha, P. F. S. Rosa, S. E. Nagler, K. Kindo,
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+Review Letters 120, 117204 (2018).
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+a field-angle-dependent Majorana gap in a Kitaev spin liquid,
+Nature Physics 18, 429 (2022).
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+and Y. Maeno, Detailed Topography of the Fermi Surface of
+Sr2RuO4, Physical Review Letters 84, 2662 (2000).
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+W. N. Hardy, R. Liang, S. Adachi, N. E. Hussey, B. Vig-
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+Theory of the field-revealed Kitaev spin liquid, Nature Com-
+munications 10, 2470 (2019).
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+Probing α-RuCl3 Beyond Magnetic Order: Effects of Temper-
+ature and Magnetic Field, Physical Review Letters 120, 077203
+(2018).
+7
+
diff --git a/s9E4T4oBgHgl3EQfxA03/content/tmp_files/load_file.txt b/s9E4T4oBgHgl3EQfxA03/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a960ef6a2ef37266f7707c7de802eabdac9e6b7b
--- /dev/null
+++ b/s9E4T4oBgHgl3EQfxA03/content/tmp_files/load_file.txt
@@ -0,0 +1,722 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf,len=721
+page_content='Oscillations in the magnetothermal conductivity of α-RuCl3: Evidence of transition anomalies  Étienne Lefrançois,1, ∗ Jordan Baglo,1, † Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Barthélemy,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Kim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 Young-June Kim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 and Louis Taillefer1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' ‡  1Institut Quantique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Département de physique & RQMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Université de Sherbrooke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Sherbrooke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Québec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Canada  2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Canada  3Canadian Institute for Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Canada  (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2023)  The 2D layered insulator α-RuCl3 is a candidate material for a quantum spin-liquid state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' which may be re-  alized when a magnetic field suppresses the antiferromagnetic order present at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Oscillations in  the field dependence of the thermal conductivity, observed for an in-plane magnetic field B up to a critical field  B⋆, have been attributed to exotic charge-neutral fermions, viewed as evidence of a quantum spin-liquid state  between the critical field Bc ≃ 7 T at which the antiferromagnetic phase ends and B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Here we report measure-  ments of the thermal conductivity of α-RuCl3 as a function of magnetic field up to 15 T applied in two distinct  in-plane directions: parallel and perpendicular to the Ru-Ru bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We find that the number of oscillations be-  tween Bc and B⋆ is the same for the two field directions even though the field interval between Bc and B⋆ is  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In other words, the period of the oscillations is controlled by the transition fields Bc and B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We con-  clude that these are not true oscillations—coming from putative fermions in a spin-liquid state—but anomalies  associated with a sequence of magnetic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  INTRODUCTION  A major objective in the field of quantum materials is to  confirm experimentally, in a real material, the existence of a  quantum spin-liquid (QSL) state predicted theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In  this respect, the magnetic insulator α-RuCl3 is a promising  material [1–3], with its Ru atoms lying on weakly coupled 2D  honeycomb layers whose interactions are such that they nearly  satisfy the Kitaev model [4], a model whose exact solution is a  QSL with Majorana fermions as emergent excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In real-  ity, the ground state of α-RuCl3 is not a QSL, but a state with  long-range antiferromagnetic order, setting in below a critical  temperature TN ≃ 7 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' However, this ordered state  can be suppressed by applying a magnetic field parallel to the  honeycomb layers, until it ends at a critical field Bc ≃ 7 T  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The question then is this: What is the nature of the  state just above Bc, at low temperature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  To address that question experimentally, thermal transport  has emerged as a fruitful probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Early measurements of the  thermal Hall conductivity, κxy, revealed the existence of a  non-zero κxy, which was attributed to the Majorana fermions  expected from the Kitaev model, given indications of a plateau  in κxy vs field at a half-quantized value [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Although some  later studies again find indications of half-quantization [6–8],  others do not, and instead attribute the measured κxy to chiral  magnons [9] or phonons [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' It is fair to say that based on  κxy data the case for a QSL in α-RuCl3 above Bc is currently  not compelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In parallel with thermal Hall studies, measurements of the  longitudinal thermal conductivity, κxx (or κ), have revealed  the existence of oscillations as a function of in-plane mag-  netic field B [12–14] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' These oscillations have  ∗ etienne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='lefrancois@usherbrooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='ca  † jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='baglo@usherbrooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  ‡ louis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='taillefer@usherbrooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='ca  been interpreted as quantum oscillations akin to those pro-  duced by Landau quantization of electron states in a metal  when a magnetic field is applied, but this time coming from  putative charge-neutral fermions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' gapless spinons with a  Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Emergent neutral fermions would be a clear  signature of a QSL state, albeit a different one from that ex-  pected from the Kitaev model [4, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' While other groups con-  firm the existence of these oscillations in κ vs B (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2), they  attribute them to a sequence of magnetic transitions [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In this paper, we revisit these oscillations, with a focus on  their anisotropy as the field direction within the honeycomb  plane is changed from being perpendicular to the Ru-Ru bond  (B ∥ a) to being parallel (B ∥ b) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  T  B  a  b  AF zigzag  Partially spin  polarized  QSL ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Paramagnet  TN  Bc  B0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Phase diagram of α-RuCl3 as a function of temperature T  and in-plane magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The phase of long-range antiferro-  magnetic (AF) order is shown in orange, with a transition tempera-  ture TN ≃ 7 K and a critical field Bc ≃ 7 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Spins on the Ru sites  are arranged in a zigzag pattern of ferromagnetic chains (see sketch),  but that pattern changes at B0 ≃ 6 T, just before reaching Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The  question is whether the state just above the AF phase is a quantum  spin liquid (QSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='05254v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='str-el]  12 Jan 2023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  B/Bc  0  4  8  12  κ (mW/K⋅cm)  Czajka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Bruin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Suetsugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  This work  α-RuCl3  B ∥ a  T = 1 K  B1  B2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Thermal conductivity of α-RuCl3 as a function of in-plane  magnetic field B, at T ≃ 1 K, plotted as κ vs B/Bc, where Bc is  the critical field where the antiferromagnetic phase ends (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Data from four different studies are compared: Czajka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (pink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='96 K and Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='21 T) [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Bruin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0 K  and Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='20 T) [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Suetsugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' T = 1 K and  Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='15 T) [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' this work (black, sample S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='16 K and  Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='69 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In all cases, the field is applied along the a axis and  the current is parallel to the field (J ∥ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The vertical dashed lines  mark the location of three minima, at Bc, B1, and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' These minima  in the oscillations of κ vs B are seen to be in the same locations for  all four studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The grey shaded region marks the regime at high  field where oscillations are no longer observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  METHODS  Single crystals were grown via the chemical vapor transport  (CVT) method, using RuCl3 powder from Sigma-Aldrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  The powder, composed of 45–55% ruthenium, was sealed in  a quartz ampoule under vacuum and the ampoule was then  placed inside a two-zone tube furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The powder was an-  nealed for two days at ∼ 800 ◦C in a temperature gradient of  70 ◦C (warmest side was 850 ◦C), followed by a cooldown  at 4 ◦C/hour while maintaining the temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' For  more details, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Here we report data on two  as-grown (uncut) crystals, labeled S1 and S2, handled very  carefully to minimize any strain induced when installing the  contacts and fixing them on the experimental mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Samples  S1 and S2 are rectangular platelets with planar surface area  1 mm × 1 mm and thicknesses of 130 and 110 µm, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The contacts on the samples were made by attaching  25 µm-diameter silver wires with DuPont 4929N silver paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  The heater was attached to the sample with a 100 µm-diameter  silver wire with silver paste as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Measurements were performed by a steady-state method  using a standard four-terminal technique, with the thermal  current applied along the length of the sample within the hon-  eycomb layers: perpendicular to the Ru-Ru bonds for S1 (J ∥  a) and parallel to the Ru-Ru bonds for S2 (J ∥ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The ther-  mal conductivity κ was measured by employing a standard  one-heater–two-thermometers method, using a 10 kΩ resistor  and two RuOx chip sensors whose magnetoresistances have  been carefully taken into account: for interpolation of temper-  ature from measured resistance and field, detailed R(T, B)  calibration surfaces were collected for each individual ther-  mometer in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' All thermometers were measured using Lake  Shore model 370 temperature controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' A main calibrated  RuOx sensor placed in the field-compensated mixing chamber  region of the dilution refrigerator was used for the reference  temperature T0, with typical control stability within ±6 µK  at 100 mK and better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='03% over the full measurement  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' A constant heat current J was injected at one end of  the sample, while the other end of the sample is heat sunk  to a copper block held at a temperature T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The heat cur-  rent was generated by sending an electric current through the  10 kΩ strain gauge, whose resistance was measured to be es-  sentially independent of temperature and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The  longitudinal temperature gradient ∆T = T + − T − is mea-  sured at two points along the length of the sample, separated  by a distance l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The longitudinal thermal conductivity is given  by κ = J/ (∆Tα), where α = wt/l is the geometric factor of  the sample (width w, thickness t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  The two temperatures T + and T − were measured as the  magnetic field, applied parallel to the honeycomb layers, was  slowly swept from 0 to 15 T, while keeping the temperature  T0 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' For each sample, a series of field sweeps, taken  at various temperatures T0 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='1 to 5 K, were obtained for  two field directions: along the a axis and along the b axis of  the crystal structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The magnetic field was swept  sufficiently slowly (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='04 T/min) to avoid any magnetocaloric  effect within the sample during the measurement, verified to  be minimal by comparison with sweeps at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The  samples were firmly attached to their respective heat sinks  in order to withstand the torque that could result from any  slight misalignment of the field away from the intended high-  symmetry direction, and thus avoid any bending of the sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The error bars on the absolute values of κ come mostly  from the uncertainty in estimating the sample dimensions (l,  w, and t), amounting approximately to ±20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The applied  current was chosen such that ∆T/T ≃ 3–5 % and the result-  ing κ was verified to be independent of applied current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  RESULTS  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 3a, we display three isotherms of κ(B) taken on  sample S1 (J ∥ a) with B ∥ a, plotted as κ(B)/κ(15 T)  vs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We observe four minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The deepest minimum, lo-  cated at Bc ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='6 T, corresponds to the end of the AF phase,  as determined by thermodynamic measurements such as spe-  cific heat [14, 18], magnetostriction [19, 20] and magnetiza-  tion [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The minimum located at a field B0 ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5 T, just  below Bc, marks a transition internal to the AF phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1)  where the AF ordering changes from one spin pattern to an-  other [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The upper two minima, at B1 and B2, are both  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  κ(B)/κ(15 T)  0  5  10  15  B (T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  κ(B)/κ(15 T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='84  α-RuCl3  B ∥ a  S1  J ∥ a  (a)  α-RuCl3  J ∥ a  B ∥ b  (b)  T =  K  B1 B2  B *  B1B2  B *  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Normalized thermal conductivity of sample S1 at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='22,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='88, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='84 K, plotted as κ(B)/κ(15 T) vs B, for a thermal cur-  rent J applied along the a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The magnetic field was applied par-  allel to the a axis (a) and perpendicular to the a axis (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  above Bc, and so outside the phase of bulk AF order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The lo-  cation of B1 and B2 is nicely consistent with prior studies, as  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' It is possible that weak signatures of structure  in κ(B) are visible further below B0 and in the AF state, in  all four studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' this may be related to similar small anomalies  in dielectric constant observed by Mi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The main  purpose of our study was to investigate the anisotropy of this  oscillatory pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 3b, we display the corresponding  isotherms taken on the same sample (S1) but for a field di-  rection in the other high-symmetry in-plane direction, namely  B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We again observe only two minima above Bc, but this  time they are much closer to each other (and to B⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Indeed,  the separation between B1 and B2, which one might view as  the “period” of the oscillations (∆B ≡ B2 − B1), is roughly  twice as large for B ∥ a compared to B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In order to confirm this anisotropy in the period of the os-  cillations as the in-plane field changes direction, we measured  a second sample (S2), this time with the heat current flowing  along the b axis (J ∥ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' These data are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 4,  where we see that two minima are present above Bc and their  separation ∆B is again significantly larger for B ∥ a com-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  κ(B)/κ(15 T)  0  5  10  15  B (T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  κ(B)/κ(15 T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='93  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='02  α-RuCl3  B ∥ a  S2  J ∥ b  (a)  α-RuCl3  J ∥ b  B ∥ b  (b)  T =  K  B1  B2  B *  B1  B2  B *  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Normalized thermal conductivity of sample S2 at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='24,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='90, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='02 K, plotted as κ(B)/κ(15 T) vs B, for a thermal cur-  rent J applied along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The magnetic field was applied per-  pendicular to the b axis (a) and parallel to the b axis (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  pared to B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The current direction does not seem to make  a significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  An important feature in our data is the existence of a thresh-  old field, B⋆, above which there are no oscillations (or min-  ima) anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' This is especially clear in the isotherms at the  lowest temperature (T ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 K), as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 3b for B ∥ b,  where B⋆ ≃ 10 T, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 4a for B ∥ a, where B⋆ ≃ 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Note the anisotropy of B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 5, we directly compare isotherms for B ∥ a and  B ∥ b, both at T ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We see that the two oscillations  observed for both field directions, with minima at B1 and B2,  fit neatly in the interval between the two transitions, at Bc (end  of AF phase) and B⋆ (end of oscillatory pattern), even though  that interval shrinks by a factor of 2 in going from B ∥ a to  B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' This reveals a connection between oscillations and  transitions, the main finding of our anisotropy study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 6, we plot the derivative ∂κ/∂B, obtained from our  full sets of isotherms taken on our two samples (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 7 and  8), displayed in two contour maps as a function of tempera-  ture and field, one panel for each field direction: B ∥ a and  B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' These contour maps clearly show how the four char-  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0  B/Bc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='15  κ (mW/K⋅cm)  α-RuCl3  T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 K  B ∥ b  B ∥ a  B1 B2  B*  B*  B2  B1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Anisotropy of characteristic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Comparison of isotherms  at T ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 K, plotted as κ vs B/Bc, for the two field directions: B ∥  a (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' sample S2, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='20 K, Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 T) vs B ∥ b (blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' sample  S1, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='22 K, Bc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='7 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Arrows mark the location of the  two minima at B1 and B2, and the threshold field B⋆ above which  oscillations are no longer observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The period of the oscillations  (|B1 − B2|) is seen to scale with the width of the interval between  the field-induced transitions (|Bc − B⋆|), both shrinking roughly by  half when the field is redirected from B ∥ a to B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' This suggests  that the oscillations are related to the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  acteristic fields Bc, B1, B2, and B⋆ remain equally spaced in  going from B ∥ a to B ∥ b even though the spacing shrinks by  a factor of 2 or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' This again highlights the intimate link be-  tween the oscillations (B1 and B2) and the transitions (Bc and  B⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Nature of the heat carriers  Before we discuss the origin of the oscillations in κ vs B  observed in α-RuCl3, let us consider the nature of the heat  carriers responsible for κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The dependence of κ in α-RuCl3 on  both temperature, over the full range up to 100 K, and in-  plane magnetic field, up to 18 T, is well described by assuming  that the only carriers of heat are phonons and that these are  scattered by spin fluctuations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Such a model, involving  a magnetic excitation spectrum with a gap that grows with  field, captures remarkably well the dramatic changes observed  in κ as the in-plane field is increased, for temperatures above  7 K, the regime where no order or oscillations are observed  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Phonons clearly dominate the thermal conduction in  α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  What about the regime below 4 K and between Bc and  B⋆ where oscillations are observed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Is there evidence here  for heat carriers beyond phonons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' A recent study reports a  (a)  (b)  4  6  8  10  12  14  B (T)  0  1  2  3  4  T (K)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='02  ∂κ/∂B  α-RuCl3  4  6  8  10  12  14  B (T)  0  1  2  3  4  T (K)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='03  ∂κ/∂B  α-RuCl3  (a)  (b)  (mW/K·cm·T)  (mW/K·cm·T)  J || a  B || a  B || b  J || b  Bc  B*  B1  B2  Bc  B*  B1 B2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Contour plots of the derivative ∂κ/∂B, mapped as a func-  tion of temperature and in-plane field, for: (a) B ∥ a, J ∥ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (b)  B ∥ b, J ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The vertical black lines mark the location of the four  characteristic fields Bc, B1, B2, and B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  slight anisotropy in the specific heat C of α-RuCl3 whereby  C(B ∥ b) > C(B ∥ a) below T  ≃ 2 K [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  This  anisotropy is attributed to fermions that are gapless for B ∥ b  and gapped for B ∥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Suetsugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' propose that a simi-  lar anisotropy is reflected in the thermal conductivity, as they  observe κ(B ∥ b) > κ(B ∥ a) below T ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='5 K [14], but  this is not convincing as the reverse is true at high tempera-  ture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=', κ(B ∥ a) > κ(B ∥ b) at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='0 K [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We also  observe the same reversal of the anisotropy with increasing  temperature (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 7,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' These small anisotropic effects of the  magnetic field on κ are most likely coming from an anisotropy  in the magnetic scattering of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Suetsugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' also emphasize the sharp jump at B⋆ for  B ∥ a, and attribute it to a (first-order) topological transition,  because they observe it only for B ∥ a and not for B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In  our own data, however, the sharp jump at B⋆ is present for  both field directions, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  4  10  −1  10  0  10  1  κ/T3 (mW/K4⋅cm)  0  5  10  15  B (T)  10  −1  10  0  10  1  κ/T3 (mW/K4⋅cm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='27  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='82  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='98  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='54  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='07  T =  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='60  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='65  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='68  T =  K  α-RuCl3  B ∥ a  J ∥ a  (a)  α-RuCl3  B ∥ b  J ∥ a  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Full set of isotherms, measured on sample S1, plotted as  κ/T 3 vs B in a semilog plot, for J ∥ a: (a) B ∥ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (b) B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  In summary, there is nothing in the thermal conductivity  of α-RuCl3 that indicates clearly the presence of fermionic  excitations, or indeed of any excitations other than phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  (In the partially spin polarized state beyond the AF phase,  magnons should contribute to heat transport, but not at low  temperature since they are gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=')  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Origin of the oscillations  Some theoretical studies find that a QSL state is plau-  sible just above Bc, and its excitations could be gapless  spinons [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Such spinons have a Fermi surface and this  could give rise to quantum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' However, two facts  about the oscillations in κ vs B observed in α-RuCl3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1)  immediately argue against quantum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The first is  the field direction being parallel to the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Indeed, in a 2D  system the Fermi surface is expected to be a cylinder with its  axis normal to the layers—as in the quasi-2D metal Sr2RuO4,  for example [25]—which would not produce quantum oscil-  lations for B ∥ a (or B ∥ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Assigning the oscillations in  α-RuCl3 to be quantum oscillations implies that the spinon  Fermi surface is a 3D object like a sphere—difficult to imag-  10  −1  10  0  10  1  κ/T3 (mW/K4⋅cm)  0  5  10  15  B (T)  10  −1  10  0  10  1  κ/T3 (mW/K4⋅cm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='80  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='91  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='44  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='97  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='49  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='00  T =  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='91  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='45  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='97  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='50  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='04  T =  K  α-RuCl3  B ∥ a  J ∥ b  (a)  α-RuCl3  B ∥ b  J ∥ b  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Full set of isotherms, measured on sample S2, plotted as  κ/T 3 vs B in a semilog plot, for J ∥ b: (a) B ∥ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' (b) B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  ine for a 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The second fact is the sheer magnitude  of the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2, we see that the oscillatory part  of the conductivity can account for as much as 50% (peak-to-  peak) of the background κ at T ≃ 1 K and B ≃ 10 T [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  This is enormous if attributed to quantum oscillations due to  fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Indeed, in a metal like the cuprate YBa2Cu3Oy, for  example, quantum oscillations in κ vs B coming from elec-  trons have a peak-to-peak amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2% at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='8 K  and B = 45 T [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Note also the comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2 of  our data on sample S1 (black curve) and the data of Czajka  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' [12] (pink curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Our sample has a higher conductiv-  ity, which reflects a lower level of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Yet its oscilla-  tions are smaller, whereas quantum oscillations grow expo-  nentially with decreasing disorder (increasing mean free path  of fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Our anisotropy study provides a third argument against  quantum oscillations from fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' If the minima at B1 and  B2 are those of quantum oscillations intrinsic to a QSL state,  as proposed by Czajka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' [12], why are there only two such  minima (above Bc)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The authors postulate that the QSL phase  ends at B⋆, in accordance with the absence of further oscilla-  tions at higher field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Now the period of quantum oscillations  is an intrinsic property of the QSL, imposed by the volume  of the Fermi surface of those neutral fermions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=', gapless  5  spinons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The fact that two periods happen to fit in the inter-  val created by the two phase transitions at Bc and B⋆, say for  B ∥ a, is necessarily an accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' If we then change the field  direction to B ∥ b, the period of quantum oscillations will  change according to the topology of the spinon Fermi surface,  which has nothing to do with the critical fields Bc and B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Yet what we observe, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 5, is that the pe-  riod of the oscillations in α-RuCl3 changes in proportion to  the change in the separation between the two critical fields  Bc and B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' There are only two oscillations for both field di-  rections, even though the interval between Bc and B⋆ is twice  as large for B ∥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We conclude that the oscillations are inti-  mately related to B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' More specifically, B⋆ marks the end of  a transition centered at B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' In other words, the minima at B1  and B2 mark two additional magnetic transitions above those  at B0 and Bc, as proposed by Bruin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' [13] on the basis that  additional anomalies are also detected in the magnetization, at  the same field values, and these can be tracked to transitions  vs temperature in zero field, lying above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  The nature of the transitions at B1 and B2 remains to be  elucidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' The fact that the values of B1 and B2 are the same  in different samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 2), from different growth methods,  suggests that they are generic features of α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' However,  the weakness of the associated specific heat anomalies sug-  gests that the magnetic structures involved may not occupy  the full volume of the sample and could be nucleated by local  defects in the crystal structure, such as stacking faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  SUMMARY  We investigated the proposal that oscillations detected in  the thermal conductivity κ of the Kitaev material α-RuCl3 as  a function of magnetic field B are quantum oscillations from  neutral fermions that are the emergent excitations of a quan-  tum spin liquid (QSL) state, which would exist in a regime  of in-plane magnetic fields just above its antiferromagnetic  (AF) phase, and below some critical field B⋆, as proposed by  Czajka and coworkers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We have measured the thermal  conductivity κ of α-RuCl3 as a function of field up to 15 T  at several temperatures between T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='2 K and T = 5 K  for the two in-plane field directions B ∥ a and B ∥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' For  both field directions, we observe two oscillations in κ vs B  contained between Bc, the critical field where the AF phase  ends, and B⋆, the threshold field above which no further  oscillations are seen (the putative critical field where the  QSL phase ends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' We find that as the field is changed from  B ∥ a to B ∥ b, the interval between the two transition fields  Bc and B⋆ shrinks by a factor of 2, and so does the period  of the two oscillations contained in that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Because  there is no reason for quantum oscillations—dictated by the  Fermi surface of fermions—to be related to the critical field  of the QSL phase (B⋆), we argue that the correlation between  oscillation period and field interval (B⋆ − Bc) is instead  evidence that the oscillations are produced by secondary  magnetic transitions similar to the main transition at Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  Our conclusion is consistent with that of Bruin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' [13]  and with theoretical works that report the absence of an inter-  mediate field-induced region, in the phase diagram as a func-  tion of in-plane field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' 1), that could potentially harbor a  QSL state [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Fortier for his assistance with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' acknowledges support from the Canadian Institute for  Advanced Research (CIFAR) as a CIFAR Fellow and funding  from the Institut Quantique, the Natural Sciences and Engi-  neering Research Council of Canada (NSERC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' PIN:123817),  the Fonds de Recherche du Québec - Nature et Technologies  (FRQNT), the Canada Foundation for Innovation (CFI), and a  Canada Research Chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' This research was undertaken thanks  in part to funding from the Canada First Research Excel-  lence Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
+page_content=' Work at the University of Toronto was supported  by NSERC (RGPIN-2019-06449 and RTI-2019-00809), CFI,  and the Ontario Ministry of Research and Innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E4T4oBgHgl3EQfxA03/content/2301.05254v1.pdf'}
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+111
+DiRaC-I: Identifying Diverse and Rare Training Classes for
+Zero-Shot Learning
+SANDIPAN SARMA and ARIJIT SUR, Indian Institute of Technology Guwahati, India
+Inspired by strategies like Active Learning, it is intuitive that intelligently selecting the training classes from a
+dataset for Zero-Shot Learning (ZSL) can improve the performance of existing ZSL methods. In this work, we
+propose a framework called Diverse and Rare Class Identifier (DiRaC-I) which, given an attribute-based dataset,
+can intelligently yield the most suitable “seen classes” for training ZSL models. DiRaC-I has two main goals –
+constructing a diversified set of seed classes, followed by a visual-semantic mining algorithm initialized by
+these seed classes that acquires the classes capturing both diversity and rarity in the object domain adequately.
+These classes can then be used as “seen classes” to train ZSL models for image classification. We adopt a
+real-world scenario where novel object classes are available to neither DiRaC-I nor the ZSL models during
+training and conducted extensive experiments on two benchmark data sets for zero-shot image classification
+— CUB and SUN. Our results demonstrate DiRaC-I helps ZSL models to achieve significant classification
+accuracy improvements.
+CCS Concepts: • Computing methodologies → Transfer learning.
+Additional Key Words and Phrases: Zero-shot learning, deep learning, object recognition, image classification
+ACM Reference Format:
+Sandipan Sarma and Arijit Sur. 2018. DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot
+Learning. J. ACM 37, 4, Article 111 (August 2018), 22 pages. https://doi.org/XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+Object recognition has witnessed a significant improvement in the recent past using deep learning
+methods [15–17, 26, 45, 47] trained on large, annotated data sets in a supervised fashion. However,
+such methods fail when novel concepts are encountered. For example, an underwater robot exploring
+deep-sea biodiversity should trigger an alert if it encounters a novel or rare species — like a
+Manocherian’s Catshark (Fig. 2b) — but would probably fail as its recognition model is not trained
+on visual images of that species. A human, on the contrary, can recognize it if he/she has a visual
+perception about sharks and is given additional information that it looks like a small shark with
+some characteristic attributes — a whitish, porcelain-colored body with a white spot on the tail tip.
+The idea of zero-shot learning (ZSL) [25, 55] stems from this ability of humans to recognize unseen
+objects by learning a mapping function associating the visual samples from the seen classes with
+their semantics (or attributes). This function is then used to recognize both seen and unseen objects.
+The ZSL community can be divided into three groups (Fig. 1) based on their contributions towards
+ZSL – (1) Dataset constructors, who collect labeled data and semantics for a fixed number (say 𝑘) of
+object classes. Before releasing the dataset for ZSL research, they define a disjoint seen-unseen
+split of the 𝑘 classes manually; (2) ZSL researchers, who use these predetermined seen classes for
+Authors’ address: Sandipan Sarma, sandipan.sarma@iitg.ac.in; Arijit Sur, arijit@iitg.ac.in, Indian Institute of Technology
+Guwahati, Multimedia Lab, Department of Computer Science and Engineering, Guwahati, Assam, India, 781039.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee
+provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and
+the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored.
+Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires
+prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2018 Association for Computing Machinery.
+0004-5411/2018/8-ART111 $15.00
+https://doi.org/XXXXXXX.XXXXXXX
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+arXiv:2301.00236v1  [cs.CV]  31 Dec 2022
+
+111:2
+Sandipan Sarma and Arijit Sur
+Total claasses = 200 
+Attributes per class = 312 
+Class names = { c1, c2,
+c3, .... c200}
+Manual 
+split 
+of classes
+Dataset constructor
+Collected image
+database
+Unseen
+set
+Seen set
+{c1, c3, c4,...c198}
+{c2, c5, c6,...c200}
+Annotated image collection
+150 classes
+50 classes
+ZSL researcher
+Proposes  
+new  
+ZSL  
+model
+Training
+Evaluate  
+trained  
+model
+Real-time clases encountered  
+Seen (from dataset) = {c1, c3,
+c4,...c198} 
+Unseen (from dataset) = {c2, c5,
+c6,...c200} 
+Novel (from the wild) = {c201,
+c202,....}
+Experimental 
+ input
+Real-time  
+deployment
+Trained ZSL model 
+Domain experts
+Sem (S)
+Sem (U)
+Sem (N)
+Real-time  
+image
+Predicted class 
+Fig. 1. ZSL community and the contributors. Dataset constructors manually define split of seen-unseen
+classes which is received by ZSL researchers and used for training and evaluating ZSL models. High-
+performance models would be used for real-world deployment for image classification. DiRaC-I targets the
+work of the constructors (green dotted box) and aims to replace manual splits by intelligent splits
+automatically.
+training the ZSL models they propose, and the predetermined unseen classes to evaluate these
+models and simulate their ability to classify unseen classes of the wild when deployed in future; (3)
+AI-aided industries, which deploy the state-of-the-art models to solve real-world problems using
+ZSL. For ZSL-based classification models to be widely accepted in the future by industries, the
+need for high-performance ZSL models trained with classes that capture rarity and diversity of the
+object domain (defined in Sec. 4) is paramount. However, ZSL models proposed by researchers today
+only passively learn from the predetermined set of seen classes provided by dataset constructors,
+like Xian et al. [55]. They claim that class diversity is maintained while manually defining the
+seen-unseen sets; however, such splits are not designed for best zero-shot performance [25]. Hence,
+an intelligent seen-unseen split of the 𝑘 classes of the collected dataset should be designed such
+that the designated seen classes automatically capture the diversity and rarity of the object domain.
+Only a few studies have addressed this issue in ZSL, mostly using Active Learning (AL) approaches.
+[58, 59] experiment with textual datasets only. Recently, in the image classification area, [52]
+proposed an GCN-based AL framework for selecting the most crucial classes as seen classes for
+training. However, all these works initialize the AL algorithm with a randomly selected set of classes
+with labeled examples, called the seed set. Additionally, they do not consider the rare attributes for
+enriching the training set.
+In this work, we propose a two-stage framework named Diverse and Rare Class Identifier (DiRaC-I)
+inspired by AL which targets the attribute-based dataset constructors. From the 𝑘-class dataset
+provided by the constructors, DiRaC-I aims to select the most suitable seen classes for training
+ZSL models while trying to capture visual diversity and semantic rarity. The first stage is seed-set
+construction, where the 𝑘 classes are clustered based on semantic similarity. A single representative
+is picked from each cluster to ensure diversity while jointly prioritizing semantic rarity, forming a
+seed set. Intuitively, doing so would incorporate a generalized initial understanding of the object
+domain within the seed set, which is used as input for the next stage — Visual-Semantic Mining
+(VSM). Here, we seek to maximize the diversity between visual samples of the seed and those of the
+other classes by estimating the distribution of related classes, based on the work by [4], and select a
+few candidate non-seed classes. We define a semantic score to be computed for each of them, and a
+few classes having the highest semantic scores are added to the seed set. This process continues
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:3
+iteratively till we get a fixed number of seed classes, which become our seen classes to be provided
+as an output to researchers for training ZSL models.
+We ensure a fair comparison of the knowledge gained while training using our seen classes (from
+Proposed seen-unseen Splits (PS)) and the predetermined (from Existing seen-unseen Splits (ES)) by
+evaluating the performance of existing ZSL models on a common set of unseen classes. For a given
+data set, this set is derived by randomly picking 50% classes from all the unseen classes (for reasons
+given in Sec. 3.2) used in ES. These classes are not used during the entire operation of DiRaC-I or
+ZSL model training but only during the evaluation of ZSL models. Extensive experiments conducted
+on two challenging benchmark data sets — CUB and SUN — demonstrate that zero-shot accuracy
+of most models are enhanced when trained with seen classes acquired by DiRaC-I. In real-world
+situations, our framework should be able to select seen classes from a given attribute-based dataset
+to help improve the training of the ZSL models to be deployed for image classification.
+We summarize our contributions as follows. (1) We design a framework, DiRaC-I, that intelligently
+captures the diversity and rarity of the object domain within a set of seen classes on which ZSL
+models can be trained to have a comprehensive idea of the domain. (2) For initializing the VSM
+algorithm, a few diverse seed classes are selected as per some attribute-based scores, instead of
+just selecting them randomly. The rare attributes play a key role in computing these scores. (3)
+ZSL model performance is evaluated on a set of unseen classes common to both existing splits (ES)
+and proposed splits (PS) that are unavailable to DiRaC-I and the model during training (simulating
+practical scenarios of encountering novel classes in the wild). Hence, a fair comparison between the
+knowledge gained by the model trained with predetermined seen classes of ES and the acquired seen
+classes of PS is ensured. (4) Unlike [52], DiRaC-I can be used by ZSL researchers as a predecessor to
+select the seen classes from a given attribute-based dataset, and therefore can adapt to real-world
+scenarios.
+2
+RELATED WORK
+2.1
+Zero-shot learning (ZSL)
+Motivated by the problems faced in supervised learning, several new learning paradigms have been
+proposed in the last decade or so, such as few-shot [5, 19] and one-shot learning [10, 27]. However,
+methods under these paradigms are still not able to cope with scenarios where we have “zero"
+training samples of certain classes – e.g. a rare fish species. Therefore, in the recent years there has
+been an increasing amount of interest in zero-shot learning, which defines a setting where visual
+features for unseen classes are unavailable during model training. However, seen and unseen classes
+can be linked through their semantics. ZSL has been applied to a wide array of computer vision
+tasks such as object detection [3], action recognition [8] and cross-modal retrieval [60] to name a
+few. We, however, focus on the zero-shot image classification task in the following discussion.
+The early works on ZSL [20, 25, 40, 41] tried to learn intermediate attribute classifiers to transfer
+knowledge from the seen to unseen. Several other approaches that followed [1, 2, 13, 23, 42, 46, 54]
+directly set up bi-linear and non-linear compatibility functions between the visual and semantic
+spaces. At test time, unseen visual features are projected to the semantic space using the learned
+functions, and the predicted class is the one achieving maximum compatibility score. The approach
+of learning a mixture model of seen classes to represent the images and semantic embeddings
+is taken up in [6, 33, 61]. While these approaches work well in the conventional setting (CZSL)
+of unseen test classes only, in practice, a model should be able to classify samples from both
+seen and unseen classes when deployed. Generalized zero-shot learning (GZSL) is a setting that
+considers such a scenario. Most existing works that show improvements in GZSL have incorporated
+generative models [11, 12, 31, 32, 48, 50, 56, 57], where the aim is to synthesize high-quality unseen
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:4
+Sandipan Sarma and Arijit Sur
+Non-overlapping  
+unseen classes (given by ES) 
+ 
+Rhinoceros Auklet
+Tree Sparrow
+California Gull
+Orchard Oriole
+Black-billed Cuckoo
+Pileated Woodpecker
+American Pipit
+Mockingbird
+Caspian Tern
+Kentucky Warbler
+.
+.
+ 
+UE
+Seen classes (given by ES) 
+Blue-headed Vireo 
+Chipping Sparrow
+Laysan Albatross
+Painted Bunting
+Anna Hummingbird
+Blue Jay
+Geococcyx
+Black Tern
+Ringed Kingfisher
+.
+.
+.
+.
+SE
+Collected categories
+C
+Common unseen classes 
+Rhinoceros Auklet
+California Gull
+Orchard Oriole
+Pileated Woodpecker
+Mockingbird
+Usable classes  
+in framework 
+ 
+Tree Sparrow
+Black-billed Cuckoo
+American Pipit
+Caspian Tern
+Kentucky Warbler
+Ucom
+Categories available to
+framework
+random splitting
+(a) Splitting the available set of classes
+in a collected dataset to obtain an ob-
+ject domain to work with and the novel
+classes for model evaluation
+Available categories in database 
+1
+3
+2
+4
+4
+2
+3
+1
+Semantic space
+Visual diversity
+Semantic rarity
+Red fins
+Gill raker
+Dermal outgrowths
+Dorsal spine venom
+Z 
+S 
+L 
+ 
+M 
+O 
+D 
+E 
+L
+Classes for training
+Recognize unseen classes
+Deploy in the wild
+Test model
+(b) “Seen classes” designated by DiRaC-I for train-
+ing ZSL models – an application
+Fig. 2. The target of DiRaC-I and its application. (Left) In a real-world scenario, data for a certain number
+of object categories are collected by dataset constructors. For example, Existing Split (ES) proposed by Xian
+et al. [55] defines some fixed classes to be used as “seen” (SE) and “unseen” (UE) while performing ZSL. We
+randomly split the set UE and obtain a set of common unseen classes (U𝑐𝑜𝑚). The proposed framework has
+access to only the remaining classes belonging to set ˜𝐶, which constitute the known object domain for the
+framework. After training several existing ZSL models with seen classes both from ES and PS (our Proposed
+Splits), model performance is evaluated only on classes from U𝑐𝑜𝑚 for fair comparison, because these classes
+are common to both ES and PS. We can consider classes from U𝑐𝑜𝑚 to act as novel classes encountered in the
+wild; (Right) Leveraging the DiRaC-I framework for selecting suitable seen classes from the object domain of
+fish – a real-world application
+class samples or visual features, converting the ZSL problem into a simple supervised classification
+task. A benchmark providing standard evaluation protocols and seen-unseen splits for some of the
+widely-used data sets in ZSL is given by [55]. However, manually created seen-unseen splits might
+not capture the diversity and rarity well enough for training ZSL models, affecting their knowledge
+about the object domain.
+2.2
+Selecting suitable seen classes for ZSL
+The idea of training ZSL models with seen classes more informative than the predetermined ones is
+relatively new. Recently, Active Learning (AL) [14] strategies have been employed in this direction.
+However, contrary to the traditional way of acquiring the most informative instances from a data set
+for training classifiers, in the zero-shot setting, the objective of AL changes to acquiring informative
+classes. [59] proposes a probabilistic method that focuses on two properties — informativeness of
+the seen classes and their connectivity to the unseen. An extension of this work [58] demonstrates
+the impact of AL on ZSL for extreme multi-label classification. However, it experiments with textual
+data sets only. [52] adopts an AL approach for GCN-based zero-shot image classification. Their
+work extends the k-center algorithm with a Laplacian energy-based strategy for selecting the most
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:5
+crucial classes as seen classes. However, it is limited to GCN frameworks for ZSL and initializes the
+algorithm with a randomly selected seed set, like the other works on AL-based ZSL.
+Research in traditional AL has shown that instead of selecting the seed set randomly, an intelligent
+selection can propel AL in better directions. In an effort to justify this, [49] proposes to manually
+prepare the seed set, artificially enriched with rare class examples. [9] gives an automatic approach
+that follows the same principle. Nevertheless, to the best of our knowledge, no work has shown the
+combined benefits of intelligently acquiring seed classes and using them to obtain seen classes that
+capture the diversity and rarity from the object domain. Our proposed framework (DiRaC-I) first
+constructs a seed set with a diverse initial representation of the object domain. It then initializes an
+AL-inspired algorithm with this seed set and iteratively acquires a fixed number of seen classes
+using which ZSL models are to be trained. The most recent work with a similar objective picks the
+seed set randomly and is compatible only with GCN-based zero-shot frameworks [52]. Moreover,
+their experiments are on a single dataset, and evaluation metrics are not comparable to the standard
+ones [55]. On the other hand, DiRaC-I can work with any attribute-based data set in practical
+scenarios. We also evaluate the prediction accuracy of several existing ZSL models trained with
+seen classes from Existing and Proposed Splits using the standard metrics and obtain encouraging
+results.
+3
+PROBLEM SETTING AND NOTATIONS
+3.1
+Object recognition using zero-shot learning
+In a typical zero-shot setting, we have sets S and U of 𝑁𝑠 and 𝑁𝑢 number of seen and unseen classes
+respectively, such that S ∩ U = ∅. Let C = S ∪ U denote the set of all classes for a given data
+set. The associated semantic embeddings for these sets can be represented by P(S) ∈ R𝑁𝑠×𝑑 and
+P(U) ∈ R𝑁𝑢×𝑑 respectively, where attributes of a class 𝑐 are represented by a𝑑−dimensional vector
+⟨𝑎1
+𝑐,𝑎2
+𝑐, ...𝑎𝑑
+𝑐 ⟩. These embeddings are available in several forms like human-annotated attributes [55],
+word embeddings like Word2Vec [29] and GloVe [35], or hierarchical embeddings like WordNet [30].
+X𝑠 ∈ R𝑚×𝑘 and X𝑢 ∈ R𝑛×𝑘 represent the visual data for seen and unseen samples respectively,
+usually available in the form of visual features extracted from a CNN like ResNet-101 [15, 55],
+pretrained on a large-scale visual dataset like ImageNet [44]. 𝑚 and 𝑛 denote the number of seen
+and unseen class samples respectively, with each image being represented by a 𝑘−dimensional
+feature vector. Then, given training data D = {(𝑥𝑠
+𝑗,𝑦𝑠
+𝑗) ∈ X𝑠 × S} along with P(S) and P(U),
+the task in CZSL is to learn a classifier 𝑓𝑐𝑧𝑠𝑙 : X𝑢 → U. In GZSL, a small subset of X𝑠 (X𝑠
+𝑠𝑢𝑏) is
+used as the set of seen samples at test time. Then, the objective changes to learning a classifier
+𝑓𝑔𝑧𝑠𝑙 : X𝑠
+𝑠𝑢𝑏 ∪ X𝑢 → S ∪ U to classify both seen and unseen objects.
+3.2
+Practical insights into seen-unseen splits
+Scarcity of labeled data and dealing with unseen concepts are two potential areas where ZSL can
+contribute significantly in the future when deployed in practical applications. To name a few, with
+ZSL models: (1) autonomous vehicles [18, 37, 38] should be able to recognize unseen concept cars
+while driving; (2) previously unseen diseases like COVID-19 could be diagnosed based on their novel
+characteristics combined with the similarity to other known diseases like asthma [7, 36, 39]; (3)
+Autonomous Underwater Vehicles (AUVs) deployed in underwater explorations [22, 24] should be
+able to recognize new fish or coral species if encountered (Fig. 2b). However, for a target application,
+data for only a fixed number of available categories (comprising a set ˜C) can be collected by the
+dataset constructors. Although they have access to labeled examples of all the | ˜C| classes and can
+provide them to ZSL researchers, the researchers cannot train their models with all | ˜C| classes
+as they would always need a disjoint set of unseen classes to evaluate ZSL models, as per ZSL
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:6
+Sandipan Sarma and Arijit Sur
+criteria. Consequently, training ZSL models requires a subset S ⊂ ˜C (i.e. the set of seen classes),
+which DiRaC-I helps the constructors to obtain. Classes from the other subset of the collected
+dataset(U = ˜C \ S) can be considered unseen classes by researchers to evaluate their model
+performance. Finally, the trained model can be deployed in the future to recognize novel classes
+(with known attributes) in the wild (Fig. 2b).
+For zero-shot classification, current researchers use seen-unseen splits predetermined by [55].
+However, unlike these Existing Splits (ES), we try to emulate the real-world scenario via our
+Proposed Splits (PS), where the seen classes exhibiting diversity and rarity can be automatically
+acquired from ˜C itself. For fair comparison of the knowledge gained by existing ZSL models when
+trained with seen classes from ES and PS, they should be evaluated on the same set of unseen classes.
+Since we do not have data from classes that are completely unknown to us during experimentation,
+we extract a few classes from the unseen set originally given by ES, and make them unavailable to
+both DiRaC-I and the ZSL models during their training. We first dissociate the set U of ES (UE)
+into two halves randomly — U𝑐𝑜𝑚 becomes the set of 𝑁𝑢𝑐𝑜𝑚 unseen classes of the wild and
+˜
+UE
+the other half, having 𝑁 ˜𝑢 classes. Then, the proposed framework acquires seen classes from the
+set ˜C = SE ∪
+˜
+UE and ZSL model can train on the acquired classes (SP). Figure 2a gives a better
+understanding of this process. Finally, let Φ𝐸 = {Φ𝑀1
+𝐸 , Φ𝑀2
+𝐸 , ...Φ𝑀𝑛
+𝐸 } and Φ𝑃 = {Φ𝑀1
+𝑃 , Φ𝑀2
+𝑃 , ...Φ𝑀𝑛
+𝑃 }
+denote the sets of models 𝑀1, 𝑀2, ...𝑀𝑛 trained using seen classes from ES and PS respectively. We
+compare the performance of the models Φ𝑥
+𝐸 and Φ𝑥
+𝑃 on the test set U𝑐𝑜𝑚 that is unseen to both
+Φ𝑥
+𝐸 and Φ𝑥
+𝑃 (𝑥 = 𝑀1, 𝑀2, ...𝑀𝑛). Note that in our framework, classes in U𝑐𝑜𝑚 do not overlap with
+the ImageNet 1K classes used for pretraining ResNet-101, following the ZSL assumption provided
+by [55]. Moreover, PS is not fixed – since we induce randomness while splitting UE, we repeat
+the entire process (from initializing DiRaC-I to evaluating ZSL models trained with the acquired
+seen classes) three times so that three different sets of classes are available to our framework at its
+inception. We show our results in each case, demonstrating the robustness of our framework to the
+available object domain.
+4
+DIRAC-I: DIVERSE AND RARE CLASS IDENTIFIER
+In this work, we focus on data sets having homogeneous categories only — e.g. having all bird
+categories. For such a data set, we say that its object domain is birds. Heuristically, training a ZSL
+model with seen classes that capture both the diversity in the visual space and rarity in the semantic
+space would provide it with a more generalized idea of the object domain. Hence, the key to our
+approach is exploring the entire available object domain (defined by classes from ˜C) for diversity
+and rarity using a method inspired by Active Learning. Adopting such a principle enhances the
+capability of ZSL models for knowledge transfer from the seen to unseen classes during evaluation.
+Moreover, the novel classes exhibiting rare attributes have a better chance of being recognized
+accurately, as suggested by the results of our experiments on two benchmark data sets (Tab. 2).
+DiRaC-I consists of two stages, which are discussed in the following sections.
+4.1
+Stage 1: Seed-set construction
+Let ˜Ψ𝑖 = {Ψ1
+𝑖 , Ψ2
+𝑖 ...Ψ𝑖
+𝑖 } denote the set of 𝑖 clusters, where Ψ𝑗
+𝑖 denotes the 𝑗𝑡ℎ cluster of classes
+represented by their semantic vectors (𝑗 ≤ 𝑖) when 𝑖 clusters are obtained. We run hierarchical ag-
+glomerative clustering (HAC) multiple times to decide the optimal number of clusters by evaluating
+the goodness of clusters in each ˜Ψ𝑖 as:
+𝑁𝑧 =
+𝑎𝑟𝑔𝑚𝑎𝑥
+2≤ 𝑖 ≤(𝑁𝑠+𝑁 ˜𝑢−1)
+𝑀𝑆𝐶( ˜Ψ𝑖)
+(1)
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:7
+C1
+C2
+C3
+Semantic space
+ 
+C3
+C2
+C1
+shape: swallow-like, 
+eye color: grey, 
+wing-color: purple........
+bill color: red, 
+under tail color: blue, 
+size: large,.............
+head pattern: crested, 
+shape: owl-like, 
+throat color: yellow......
+bill shape: curved, 
+primary color: yellow, 
+shape: duck-like.....
+bill shape: hooked, 
+size: very large, 
+shape: hawk-like...
+wing color: green, 
+eye color: pink, 
+bill shape: needle.........
+C1
+C3
+C2
+Filter Irrelevant 
+Attributes (Eq. 3)
+Neglect Unremarkable 
+Attributes  (Eq. 4)
+Seed
+classes
+Feature extractor
+Train
+Unlabeled pool
+MAVs for seed 
+classes 
+AVs for unlabeled samples 
+Visual diversity
+Unique candidates =
+{Ci , Cj , Ck}
+PZ
+a1
+a2
+...
+ad
+c1
+a11
+a12
+...
+a1d
+c2
+a2
+1
+a2
+2
+...
+a2
+d
+c3
+a3
+1
+a3
+2
+...
+a3
+d
+Compute  
+attribute  
+weights 
+(Eqs. 6 and 11) 
+Semantic score calculator 
+Semantic  
+vectors
+Semantic 
+weights
+Cluster representative designator 
+Acquired classes
+Add to seed set
+Semantic 
+rarity
+STAGE 1
+STAGE 2
+Fig. 3. DiRaC-I workflow. In stage 1, clustering in the semantic space selects a representative from each
+cluster while filtering out irrelevant and unremarkable attributes, and this forms the seed set. In stage 2, the
+seed set expands iteratively while considering visual diversity and semantic rarity, until it contains a fixed
+number of classes. The resulting set can act as seen set for training ZSL models
+where 𝑀𝑆𝐶(.) is the mean silhouette coefficient [21, 43]:
+𝑀𝑆𝐶( ˜Ψ𝑖) =
+1
+(𝑁𝑠 + 𝑁 ˜𝑢)
+∑︁
+𝑘 ∈ ˜C
+𝑏𝑖
+𝑘 − 𝑎𝑖
+𝑘
+𝑚𝑎𝑥{𝑎𝑖
+𝑘,𝑏𝑖
+𝑘}
+(2)
+where 𝑎𝑖
+𝑘 and 𝑏𝑖
+𝑘 are the mean intra-cluster distance and mean nearest-cluster distance for semantic
+vector of class 𝑘 when 𝑖 clusters are formed by HAC. From the optimal set of clusters( ˜Ψ𝑁𝑧), a single
+representative is selected from each Ψ𝑗
+𝑁𝑧 based on information from the cluster-specific semantic
+space. However, for a cluster, some attributes might not be present at all (irrelevant) or may occur
+in minimal amounts (unremarkable) and hence can be ignored while searching for its suitable
+representative. Therefore, for a cluster Ψ𝑗
+𝑁𝑧, we formally recognize these two groups of attributes
+respectively from the semantic space (P(Ψ𝑗
+𝑁𝑧)) spanned by its member classes:
+𝐼𝐴(Ψ𝑗
+𝑁𝑧) = {𝑎𝑙 ∈ R | 𝑎𝑙
+𝑐 = 0, ∀ 𝑐 ∈ Ψ𝑗
+𝑁𝑧 }
+(3)
+𝑈𝐴(Ψ𝑗
+𝑁𝑧) = {𝑎𝑙 ∈ R | B𝑙
+𝑐 (Ψ𝑗
+𝑁𝑧) = 0, ∀ 𝑐 ∈ Ψ𝑗
+𝑁𝑧 }
+(4)
+where for an attribute 𝑎𝑙:
+B𝑙
+𝑐 (Ψ𝑗
+𝑁𝑧) =
+
+
+0,
+if 𝑎𝑙
+𝑐 ≤
+1
+|{𝑐 ∈Ψ𝑗
+𝑁𝑧 | 𝑎𝑙𝑐≠0}|
+�
+𝑐 ∈Ψ𝑗
+𝑁𝑧
+𝑎𝑙
+𝑐
+1,
+otherwise
+(5)
+Here, for cluster Ψ𝑗
+𝑁𝑧, 𝐼𝐴(.) and 𝑈𝐴(.) denote the sets of irrelevant and unremarkable attributes
+respectively, and B(.) is a binary class-attribute matrix procured from P(.) after ignoring the
+irrelevant attributes. Unremarkable attributes are also discarded from both B(.) and P(.). To
+account for the rarity in semantic space P(Ψ𝑗
+𝑁𝑧), we calculate per-attribute frequencies with the
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+EricBeginJMPAll rights reserved @ Emila Yusof111:8
+Sandipan Sarma and Arijit Sur
+help of the corresponding B(Ψ𝑗
+𝑁𝑧) — rarer the attribute, more the importance given to it by sampling
+weights from the function:
+𝑓 (𝜃𝑎𝑙 ) = − log(𝜃𝑎𝑙 )
+(6)
+where we obtain attribute frequencies from the diagonal values (𝑑𝑙
+𝑙 ) of matrix (B(Ψ𝑗
+𝑁𝑧))𝑇 · B(Ψ𝑗
+𝑁𝑧)
+as:
+𝜃𝑎𝑙 =
+𝑑𝑙
+𝑙
+|Ψ𝑗
+𝑁𝑧 |
+(7)
+According to Eq. 7, 𝜃 ∈ (0, 1] and 𝑓 (𝜃𝑎𝑙 ) ∈ [0, ∞). We use − log(𝜃𝑎𝑙 ) to sample attribute-weights as
+it is strictly decreasing on the interval (0, 1], providing a higher weight if 𝑎𝑙 is rare (i.e., 𝜃𝑎𝑙 is low),
+and a lower weight otherwise. Finally, we get the seed set as Z = {𝜅(Ψ1
+𝑁𝑧),𝜅(Ψ2
+𝑁𝑧), ...𝜅(Ψ𝑁𝑧
+𝑁𝑧 )} in
+which a representative class from each cluster is selected as:
+𝜅(Ψ𝑗
+𝑁𝑧) = 𝑎𝑟𝑔𝑚𝑎𝑥
+𝑐 ∈Ψ𝑗
+𝑁𝑧
+(B(Ψ𝑗
+𝑁𝑧) ⊙ P(Ψ𝑗
+𝑁𝑧)) · W
+(8)
+where ⊙ denotes element-wise matrix multiplication and W is a vector of weights for attributes
+present in P(.). Such representatives from different clusters boost diversity while promoting
+semantic rarity via Eqs. 6 and 8. We consider an outlier class (not a member of any cluster) to be
+diverse enough from the other classes and take it directly into the seed set.
+4.2
+Stage 2: Visual-Semantic Mining (VSM)
+The samples belonging to classes from Z act as a labeled data set used to initialize our VSM
+algorithm. VSM is inspired by Active Learning (AL), where inputs from an Oracle (the source of
+ground truth labels, e.g. a human expert) are used to label some of the most informative samples from
+an unlabeled pool (A) to train machine learning models. For our framework, this pool corresponds
+to the samples not belonging to classes from the seed set (for a given VSM iteration only; dataset
+constructors actually provide labels for all samples available to DiRaC-I). We aim to adopt a similar
+strategy to iteratively acquire 𝑁𝑠 classes exhibiting diversity and rarity for training ZSL models.
+In each iteration of VSM, we retrain a ResNet-101 (M) [15] pretrained on ImageNet [44] to behave
+as a feature extractor for the seed class samples using a transfer learning approach. We capitalize on
+the work done by [4] and use the scores from the penultimate layer of a CNN (Activation Vectors
+or AVs) to estimate the distribution of the related classes, establishing a relationship between the
+unlabeled and labeled samples in the AV space. Each class 𝑐 ∈ Z is represented by its Mean
+Activation Vector (MAV) computed using the AVs of the training samples classified correctly
+by M, obtaining the MAV set V = {𝜇1, 𝜇2, ...𝜇|Z|}. For the unlabeled samples, we extract the AVs
+using the trained M to obtain F = {𝑓1, 𝑓2, ...𝑓|A |}. We intend to capture the visually most diverse
+samples leveraging the AV space by first obtaining the set:
+Π = {𝑘 ∈ R | 𝑘 = 𝑚𝑖𝑛
+𝜇𝑐 ∈V 𝛿(𝜇𝑐, 𝑓𝑗), ∀𝑓𝑗 ∈ F }
+(9)
+and then selecting 𝑡 samples from A corresponding to the largest values in Π, where 𝛿 denotes
+the Euclidean-cosine distance [4]. A set of unique candidate classes (H) is formed by querying the
+ground truths of these 𝑡 samples. VSM then explores the rarity in the semantic space spanned
+by these candidate classes (P(H)). A class-wise estimate of the number of images from the seed
+classes exhibiting each attribute can be obtained in a matrix I, where:
+I𝑙
+𝑐 = 𝑎𝑙
+𝑐 · 𝐼𝐶𝑐, 𝑐 ∈ Z
+(10)
+Here, 𝑎𝑙
+𝑐 is an element from the semantic space of the seed classes (P(Z)) and 𝐼𝐶𝑐 gives the number
+of images for seed class 𝑐. New attribute-weights are calculated based on the proportion of each
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:9
+attribute within the currently “known object domain (Z)” for VSM using Eq. 6, except that now:
+𝜃𝑎𝑙 =
+�
+𝑐 ∈Z
+I𝑙
+𝑐
+�
+𝑐 ∈Z
+𝐼𝐶𝑐
+(11)
+Finally, we calculate semantic scores for each candidate class as follows:
+𝛼ℎ = P(H) · W
+(12)
+where W denotes the vector of obtained attribute-weights. Top-𝑞 candidate classes having the
+highest semantic scores are added to Z. Ground truth of the samples from the added classes are
+also queried at the end of an iteration so that samples belonging to classes in Z always remain
+labeled. We repeat this entire process (Fig. 3) until |Z| = 𝑁𝑠 (kept as the same value as in Existing
+Split for a fair comparison).
+It is important to note that VSM is only inspired by Active Learning (AL). We do acknowledge
+the structural similarities with AL, such as seeds and acquisition functions. However, the problem
+setup and goal of VSM are quite different from AL. AL theoretically aims to select the most
+informative samples from a huge unlabeled pool of data, whereas that is not the case for DiRaC-I’s
+target group (dataset constructors). DiRaC-I can query the labels once a class is added to the seed
+set since labeled data for all the collected classes are available with the constructors. Hence, for this
+task, the AL assumption (model should not have access to labels) does not hold, and its absence
+does not make VSM impractical to use.
+5
+EXPERIMENTS
+5.1
+Datasets and seen-unseen splits
+CUB [51] and SUN [34] are two challenging fine-grained data sets, both with several classes but
+limited data per class. CUB contains 11,788 images from 200 bird categories, each of which is defined
+using 312 human-annotated attributes. SUN contains 14,340 images from 717 scene categories
+annotated with 102 attributes. 𝑁𝑠 is 150 for CUB and 645 for SUN for both ES and PS. Moreover,
+before initiating DiRaC-I, we obtain 𝑁𝑢𝑐𝑜𝑚 as 25 and 36 for CUB and SUN. The random split of
+UE is done three times and model evaluation is done on three different U𝑐𝑜𝑚 sets — U1
+𝑐𝑜𝑚, U2
+𝑐𝑜𝑚
+and U3
+𝑐𝑜𝑚 — where 𝑋 in U𝑋
+𝑐𝑜𝑚 denotes the split number. Consequently, DiRaC-I runs three times
+with different object domains ( ˜C) at its inception. We report the image count of classes belonging
+to sets SE, SP and U𝑐𝑜𝑚 in Tab. 1, where the slight difference in image count for ES and PS can
+be attributed to the different seen classes considered in ES and PS. For visual features, we follow
+previous work [55] and use CNN features extracted from pretrained ResNet-101 [15].
+5.2
+Implementation details
+During stage 1, HAC uses Ward’s method [53] to calculate cluster similarity. Obtaining too few
+clusters (and hence, seed classes) using HAC would initialize the deep model (M) in VSM with too
+few training samples. Additionally, some data sets have very few images per class — e.g. 20 for SUN.
+Therefore, we set the lower bound of number of clusters to be formed as 5 to achieve effective model
+training. While retraining M, weights of all the layers are frozen except the last fully-connected
+layer. The learning rates for optimizing M are set to 0.01 and 0.001 for CUB and SUN, respectively.
+𝑞 is set to 2 for CUB and 4 for SUN. We need 𝑡 to be low so that in a practical scenario, only a small
+percentage of the unlabeled images from A need to be queried for their class labels while inferring
+the candidate classes during the entire process of VSM. In our experiments, 𝑡 = max(5, ⌈(3 log𝑎)⌉),
+where 𝑎 = average number of images per class for a given data set, according to which 𝑡 = 13
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:10
+Sandipan Sarma and Arijit Sur
+Table 1. Image count in seen and common unseen classes for ES and PS for different random splits of UE
+Dataset
+Split
+ES
+PS
+SE
+U𝒄𝒐𝒎
+SP
+U𝒄𝒐𝒎
+CUB
+1
+7057
+1489
+7068
+1489
+2
+7057
+1488
+7068
+1488
+3
+7057
+1471
+7075
+1471
+SUN
+1
+10320
+720
+10320
+720
+2
+10320
+720
+10320
+720
+3
+10320
+720
+10320
+720
+for CUB and 𝑡 = 9 for SUN. Across the three runs of DiRaC-I, VSM runs for 58 iterations on an
+average for CUB and queries the labels for 754 samples, i.e., 6.39% of the total samples in CUB. For
+SUN, labels are queried for 10.04% of the total samples (1440) over an average of 160 iterations.
+Furthermore, keeping in mind the real-life scenarios, we prioritize a candidate class to be included
+in Z if it is an overlapping class (Sec. 3.2), so that classes from the set ˜𝐶 \ SP can also serve as test
+set if required without violating zero-shot assumptions [55]. We observe that these classes mostly
+have the highest semantic scores too, and hence conclude that the inclusion is fair.
+5.3
+Performance comparison with Existing Splits
+We report the average per-class top-1 accuracy [55] in the CZSL setting to evaluate ZSL methods
+based on different approaches like compatibility learning and generative frameworks and compare
+the performance of Φ𝐸 and Φ𝑃 (Sec. 3.2). All the methods are implemented in PyTorch. Among
+them, official codes in PyTorch are available for LsrGAN [50] and TF-VAEGAN [32], and the rest
+are re-implemented versions based on the original publications. The hyperparameters for the
+official codes are directly used, whereas they are set on the validation sets for the rest. Note that
+model accuracy on ES reported in Tab. 2 is not comparable with that of the original papers because
+the test set is different in our framework. Table 2 shows that in the CZSL setting, models in Φ𝑃
+show significant improvements on CUB over Φ𝐸 across all three splits. For SUN, we notice a mix
+of improved and similar results with models in Φ𝐸. This might be because DiRaC-I leverages
+information from the semantic space, which lacks attributes with discriminative strength in the
+case of SUN, as explained in Sec. 6.1. Comparing results for ES and PS in the GZSL setting would not
+be fair in the current work because the seen classes in ES and PS might be different. Consequently,
+they might have different influences on the unseen predictions and the harmonic mean of seen
+and unseen accuracy (GZSL evaluation metric). This is true for most models because of their bias
+towards the seen classes.
+6
+FRAMEWORK ANALYSIS
+In this section, we demonstrate some qualitative and quantitative results and analyze the perfor-
+mance of the two stages of our framework, as well as the impact of incorporating diversity and
+rarity of the object domain in the training of ZSL models. In some of the subsections that follow,
+we show some qualitative results on the CUB [51] data set when ˜C = SE ∪ (UE \ U2
+𝑐𝑜𝑚) (U2
+𝑐𝑜𝑚
+denotes the set U𝑐𝑜𝑚 after the randomly splitting UE for the second time). For brevity, we denote
+this set as ˜C2. All the results are shown for the object domain ˜C2 (unless stated otherwise) to
+maintain a correlation between results obtained from various stages of the framework. We choose
+the CUB dataset for qualitative results because the attributes that characterize each class in CUB
+are visually interpretable and hence can be easily verified with the visual results we provide here.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:11
+Table 2. Comparative results (top-1 accuracy in %) of Conventional ZSL with models from sets Φ𝐸 and Φ𝑃
+(defined in Sec. 3.2) on the CUB and SUN data sets. Results on test classes having at least one common
+attribute corresponds to results on all classes (left), since all classes in CUB and SUN exhibit at least one
+common attribute (see Tab. 5). Enhanced results achieved with PS are in BOLD
+Method
+Test set
+ZSL for all
+ZSL for classes having
+test classes
+at least 1 rare attribute
+CUB
+SUN
+CUB
+SUN
+ES
+PS
+ES
+PS
+ESR
+PSR
+ESR
+PSR
+ALE [1]
+U1
+𝑐𝑜𝑚
+47.85
+50.76
+61.25
+59.17
+46.58
+49.89
+63.63
+60.45
+U2
+𝑐𝑜𝑚
+43.76
+47.62
+63.19
+59.86
+41.32
+48.36
+61.84
+58.68
+U3
+𝑐𝑜𝑚
+48.06
+56.10
+60.28
+57.92
+47.99
+54.24
+61.66
+63.33
+SAE [23]
+U1
+𝑐𝑜𝑚
+40.69
+44.07
+47.78
+53.19
+39.26
+42.85
+52.5
+59.54
+U2
+𝑐𝑜𝑚
+32.07
+39.64
+54.17
+55.69
+30.30
+40.12
+53.68
+55.26
+U3
+𝑐𝑜𝑚
+40.41
+43.64
+50.69
+52.92
+40.42
+44.11
+50.00
+55.66
+SJE [2]
+U1
+𝑐𝑜𝑚
+48.01
+50.74
+53.19
+53.06
+47.16
+49.37
+54.31
+55.90
+U2
+𝑐𝑜𝑚
+40.26
+47.98
+53.19
+55.42
+38.32
+47.32
+51.57
+56.57
+U3
+𝑐𝑜𝑚
+54.05
+54.81
+51.67
+50.28
+48.20
+48.72
+55.00
+53.66
+DeViSE [13]
+U1
+𝑐𝑜𝑚
+47.18
+50.27
+53.89
+54.31
+45.46
+48.96
+57.04
+60.22
+U2
+𝑐𝑜𝑚
+42.55
+45.63
+58.47
+55.00
+38.88
+45.56
+56.31
+56.31
+U3
+𝑐𝑜𝑚
+44.58
+45.09
+55.14
+54.17
+41.32
+43.15
+62.00
+66.00
+ESZSL [42]
+U1
+𝑐𝑜𝑚
+53.62
+52.37
+53.89
+49.31
+52.79
+51.01
+59.54
+56.81
+U2
+𝑐𝑜𝑚
+44.46
+50.80
+55.42
+52.78
+44.38
+52.27
+57.36
+53.15
+U3
+𝑐𝑜𝑚
+56.81
+59.84
+57.78
+50.83
+51.63
+55.96
+62.66
+54.00
+LsrGAN [50]
+U1
+𝑐𝑜𝑚
+57.91
+60.19
+59.44
+64.58
+57.61
+59.64
+62.72
+65.90
+U2
+𝑐𝑜𝑚
+56.03
+59.40
+61.67
+64.17
+56.01
+61.28
+61.57
+63.94
+U3
+𝑐𝑜𝑚
+64.43
+61.46
+60.69
+61.81
+60.54
+57.84
+63.66
+66.00
+TF-VAEGAN [32]
+U1
+𝑐𝑜𝑚
+63.48
+66.32
+64.58
+65.83
+63.68
+65.89
+65.00
+67.04
+U2
+𝑐𝑜𝑚
+61.82
+64.98
+69.03
+66.11
+60.55
+65.13
+68.15
+67.10
+U3
+𝑐𝑜𝑚
+67.56
+68.35
+65.83
+65.00
+64.44
+65.06
+67.66
+70.33
+6.1
+Seed-set construction
+We aim to achieve a good quality of clusters during this stage and acquire seed classes that
+provide a comprehensive initial idea of the object domain to the next stage of DiRaC-I. For CUB,
+obtained clusters are visually more interpretable since the semantic space consists of several groups
+of discriminative properties like wing color, bill shape, head pattern, etc. Hence, hummingbirds,
+kingfishers, and gulls get clustered separately, and picking a representative from each cluster
+captures the object domain diversity well enough. However, the SUN attributes come from a
+variety of contexts [34], many of which are applicable to several classes with the same attribute
+strength — e.g. P(SE ∪ (UE \ U1
+𝑐𝑜𝑚)) shows attributes warm and eating have a non-zero value
+for 669 and 346 classes but have only 50 and 49 unique values. This results in a large number of
+classes clustering together in the semantic space (Fig. 8). Hence, our experiments suggest that
+data sets which characterize classes using more discriminative attribute strengths would help in
+selecting better seed classes. Figures 8a and 8e show the seed classes (numbered in ‘black’) for CUB
+and SUN respectively. Randomly selecting such classes could pick all of them from a particular
+region (when every class is semantically very similar) or from very different regions. However, our
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:12
+Sandipan Sarma and Arijit Sur
+Arctic Tern
+Caspian Tern
+Common Tern
+Elegant Tern
+Forster Tern
+Least Tern
+Brewer Blackbird
+Red-winged 
+Blackbird
+Brandt Cormorant
+Red-faced 
+Cormorant
+Pelagic Cormorant
+Bronzed Cowbird
+Shiny Cowbird
+American Crow
+Fish Crow
+Frigatebird
+Common Raven
+White-necked 
+Raven
+Chuck-will Widow
+Gray-crowned 
+Rosy Finch
+Nighthawk
+Whip-poor Will
+Fox Sparrow
+Seaside Sparrow
+Song Sparrow
+Brown Thrasher
+Sage Thrasher
+Black throat with pointed tail and rounded wings 
+Buff colored with bill length shorter than head
+Dagger-shaped bill with striped wings
+Fig. 4. Results of clustering during seed-set construction (for object domain ˜C2). Each row corresponds to
+the members of a specific cluster obtained. The attribute descriptions for each cluster written below each
+row are formed by combining some of the most frequent attributes, procured after discarding the attributes
+adjudged as irrelevant and unremarkable for the cluster (such discarded attributes can be found in Tab. 3)
+Fig. 5. An example from every seed class at the end of stage 1 for CUB with object domain ˜C2. Notice the
+birds exhibiting rare attributes of the domain ˜C2 (see Tab. 6) such as needle-shaped bill (pink box) and orange
+eye (green box)
+approach leverages the semantic relationships between classes and ensures that it picks diverse
+representatives, as evident from Fig. 8.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:13
+Table 3. Sets of irrelevant attributes (𝐼𝐴) and unremarkable attributes (𝑈𝐴) obtained for the clusters corre-
+sponding to the top, middle and bottom rows of Fig. 4, respectively. Ψ𝑐
+25 denotes cluster number 𝑐 out of the
+25 clusters obtained using HAC. (+𝑝) indicates set contains 𝑝 more attributes
+Cluster (c)
+𝑰𝑨(𝚿𝒄
+25)
+𝑼𝑨(𝚿𝒄
+25)
+1
+Purple bill
+Yellow back
+Olive bill
+Purple eye
+Green bill
+Pink forehead
+Rufous crown
+Orange nape
+Green leg (+14)
+2
+Spatulate-shaped bill
+Purple wing
+Green wing
+Green throat
+Purple leg
+Blue under-tail
+Olive crown
+Red nape
+Pink eye (+19)
+Blue belly (+24)
+3
+Purple back
+Orange wing
+Pink underparts
+Red upper-tail
+Green upper-tail
+Brown eye
+Red forehead
+Blue nape
+Olive breast (+57)
+Pink bill (+53)
+1
+2
+3
+4
+5
+6
+7
+8
+Classes queried per VSM iteration (q)
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy (in %) for TF-VAEGAN
+CUB
+SUN
+Fig. 6. Sensitivity to 𝑞 for TF-VAEGAN [32]
+6.1.1
+Clustering in the semantic space. Hierarchical Agglomerative Clustering (HAC) in the se-
+mantic space spanned by classes from set ˜C2 provides 25 clusters, and a single representative is
+designated as the seed class from each cluster. Fig. 4 elucidates the clustering quality by presenting
+the cluster members of three clusters found by HAC. It can be seen that all the cluster members of
+the bottom row belong to the family of terns — hence picking a seed class from this cluster ensures
+that the final seed set at the end of seed-set construction has a member from this family of birds. For
+the other rows, although the cluster members come from several families, they share common visual
+properties. For example, crows, cormorants, blackbirds and others have been clustered together in
+the top row, whereas the middle row consists of small-sized birds like sparrows, finches, etc. This
+suggests that selecting a member from each cluster would provide a good visual representation of
+that cluster to the next stage (VSM). Fig. 5 shows a sample image from each of the seed classes of
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:14
+Sandipan Sarma and Arijit Sur
+(a) After iteration 1
+(b) After iteration 2
+(c) After iteration 3
+(d) After iteration 4
+Fig. 7. Top 𝑡 samples visually most diverse from the seed classes at the start of iteration 𝑖, based on a
+Euclidean-cosine distance [4]
+CUB dataset corresponding to Fig. 8a with object domain ˜C2, providing a visual idea of the domain
+diversity captured within the seed classes.
+6.1.2
+Designating cluster representatives as seeds. Once the clusters are obtained, seed classes are
+selected based on the cluster-specific semantic space using Eqs. 6, 7, and 8. However, to ensure that
+the computations are devoid of the effects of irrelevant and unremarkable attributes of a cluster, we
+defined sets 𝐼𝐴(.) and 𝑈𝐴(.) for every cluster. Table 3 shows these two sets obtained corresponding
+to the three clusters (out of 25) exhibited in Fig. 4. Combining the information from Fig. 4 and
+Tab. 3, we can see that the attributes belonging to the obtained sets 𝐼𝐴(.) and 𝑈𝐴(.) are indeed
+not descriptive enough of the cluster members. However, associating some of the most frequently
+occurring attributes in a given cluster, we construct some cluster descriptions (Fig. 4) and find them
+to be consistent with the visual images of the cluster members, providing a general description of
+the cluster.
+6.2
+Visual-Semantic Mining (VSM)
+The idea of capturing diversity and rarity in the object domain via an iterative VSM algorithm is
+pivotal to our work and has been shown in action in Fig. 8. We notice that classes are captured
+from several regions of the semantic space, maximizing the distance from the existing seed classes
+in most cases. In a few cases, the acquired classes are closer to quite a few existing seed classes,
+like classes labeled as 29 in Fig. 8d and 8 in Fig. 8g. These cases arise when the generated semantic
+scores exceed the visual diversity factor (Eq. 9) by virtue of the rarity of attributes.
+6.2.1
+Qualitative analysis: VSM. Figure 7 shows the 𝑡 samples for the first four VSM iterations
+considered visually most diverse from the existing seed classes at the start of every iteration. For
+CUB data set, 𝑡 = 13 (Sec. 5.2). New classes added to the initial seed set at every iteration have
+been shown in Fig. 9, where classes at the end of iteration 𝑖 serve as the seed classes at the start of
+iteration 𝑖 + 1 (for 𝑖 = 1, 2, 3). The initial 25 seed classes (Fig. 5) are used for acquiring new classes
+in iteration 1 of VSM.
+For iteration 1, Fig. 7a presents different kinds of swallows, albatrosses and sparrows which are
+different from the birds captured in the seed set. At the start of iteration 1, the top-5 rare and
+common attributes captured from the existing seed classes are shown in Tab. 4. These lists are
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:15
+(a) CUB, 𝑖1
+(b) CUB, 𝑖2
+(c) CUB, 𝑖3
+(d) CUB, 𝑖4
+(e) SUN, 𝑖1
+(f) SUN, 𝑖2
+(g) SUN, 𝑖3
+(h) SUN, 𝑖4
+Fig. 8. Visualization of the classes in the semantic space acquired during different iterations of VSM for both
+CUB and SUN by t-SNE method [28] (best viewed in color). Each class is represented by its attribute vector in
+the semantic space, and classes in the same cluster are shown in the same color. The top 𝑞 classes acquired in
+the 𝑘𝑡ℎ iteration are marked with ‘red’ numbers, and the rest of the numbered classes are the seed classes
+before starting iteration named 𝑖𝑘
+obtained according to the fraction of seed class images that the attributes appear in, and hence
+can approximately be verified from the visual images of the seed classes (Fig. 5). The added classes
+after iteration 1 are Sooty albatross and Dark-eyed junco. Referring to the semantic vectors for these
+two classes, we find that both of them marginally exhibit the top-5 rare attributes, except green leg.
+Moreover, as expected, both the added classes exhibit huge amounts of some of the top-5 common
+attributes like black eye and solid belly pattern. After adding these new classes to the previous seed
+set, the list of top-5 rare attributes changes in the second iteration of VSM, indicating that the seed
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+10
+12
+5
+19
+0
+26
+15
+-5
+24
+20
+21
+1.8
+-10
+-15
+-10
+-5
+0
+10
+1510
+12
+5
+19
+0
+-5
+24
+20
+.8
+-10
+-15
+-10
+-5
+0
+5
+10
+1510
+5
+19
+30
+0
+17
+22
+15
+-5
+24
+20
+5
+8
+-10
+-15
+-10
+-5
+0
+5
+10
+1510
+19
+0 -
+8
+27
+26
+5
+-5-
+24
+20
+21
+18
+-10
+10
+-15
+-10
+-5
+0
+5
+10
+1515
+10
+5
+0
+-5
+-10
+-15
+-20
+-40
+-20
+0
+20
+4015
+10
+5
+0
+-5
+-10
+-15
+-20
+-40
+-20
+0
+20
+4016
+15
+10
+5
+0
+-5
+-10
+-15
+-20
+-40
+-20
+0
+20
+40111:16
+Sandipan Sarma and Arijit Sur
+(a) After iteration 1
+(b) After iteration 2
+(c) After iteration 3
+(d) After iteration 4
+Fig. 9. New classes (in red boxes) added to the initial seed set (Fig. 5) after the first four iterations of VSM.
+Corresponding results in the semantic space can be found in Fig. 8. Visual diversity can be observed as
+representatives of various families like sparrows, albatrosses, cormorants, kingfishers and others have been
+acquired by VSM. Rare attributes like purple underparts, needle-shaped bill and others have also been captured
+within these classes
+set has now been enriched with rare attributes. The list of top-5 common attributes remains mostly
+the same, as these attributes are already the most abundant ones in the object domain.
+6.3
+Parameter sensitivity
+During VSM, attribute-weights are computed based on the semantics of classes in Z only (Eq. 11).
+Diversity and rarity expressed by such a small portion of the object domain should not dictate
+the selection of too many classes at a time. Figure 6 suggests that 𝑞 is not very sensitive to ZSL
+model performance, so we set low values of 𝑞 for VSM to steadily explore the object domain while
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:17
+Table 4. The top five rare and common attributes captured by analyzing the semantic space of the classes in
+the seed set at the start of the first four iterations of VSM. For each iteration, the attributes are shown in
+descending order of weights assigned to them, computed using Eqs. 7, 10, and 11
+Iteration Top Rare
+Top Common
+1
+Purple under-tail Small size
+Primarily purple
+Solid belly pattern
+Purple nape
+Bill shorter than head
+Green leg
+Solid breast pattern
+Purple underparts Black eye
+2
+Pink eye
+Small size
+Purple breast
+Bill shorter than head
+Green leg
+Solid breast pattern
+Purple under-tail Solid belly pattern
+Green bill
+Black eye
+3
+Pink eye
+Small size
+Purple eye
+Bill shorter than head
+Green leg
+Solid breast pattern
+Purple under-tail Solid belly pattern
+Pink under-tail
+Black eye
+4
+Owl-like shape
+Small size
+Pink eye
+Bill shorter than head
+Purple eye
+Solid breast pattern
+Purple under-tail Solid belly pattern
+Green leg
+Black eye
+Table 5. Distribution of rare and common attributes for different random splits of UE. 𝐴 = total number
+of attributes; 𝑁 ˜𝐶 = 𝑁𝑠 + 𝑁 ˜𝑢; AR and AC are the number of rare and common attributes; YR and YC are the
+number of common unseen classes having at least one rare and one common attribute respectively
+Dataset A / N ˜𝐶 / Nucom Split AR / AC YR / YC
+CUB
+312 / 175 / 25
+1
+24 / 9
+24 / 25
+2
+22 / 9
+22 / 25
+3
+22 / 10
+19 / 25
+SUN
+102 / 681 / 36
+1
+7 / 2
+22 / 36
+2
+7 / 2
+19 / 36
+3
+6 / 2
+15 / 36
+expanding the set Z. Since the average image count per class for SUN is relatively lower than CUB,
+we set 𝑞 to be higher for SUN to train the feature extractor effectively.
+6.4
+Acknowledging rarity in the object domain
+For attribute-based data, an object class is uniquely characterized by its attributes, so it can be
+reasoned that the more rare attributes a class exhibits, the higher its probability of being a rare class.
+To test the semantic knowledge gained by our framework about the object domain, we develop a
+notion for designating attributes as either rare or common using semantic information from P( ˜C).
+Sets 𝐼𝐴( ˜C) and 𝑈𝐴( ˜C) are developed using Eqs. 3 and 4 and their member attributes are discarded.
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:18
+Sandipan Sarma and Arijit Sur
+Mallard
+Scarlet Tanager
+Orange crowned Warbler
+Barn Swallow
+White crowned Sparrow
+Chestnut sided Warbler
+White eyed Vireo
+Bronzed Cowbird
+Pileated Woodpecker
+Green Violetear
+Scott Oriole
+Mockingbird
+Boat tailed Grackle
+Cape May Warbler
+Henslow Sparrow
+Tree Swallow
+Yellow bellied Flycatcher
+Blue winged Warbler
+Evening Grosbeak
+Red legged Kittiwake
+Brown Creeper
+Field Sparrow
+Pied billed Grebe
+Magnolia Warbler
+Common unseen class
+0
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(a) CUB, DeViSE
+Mallard
+Scarlet Tanager
+Orange crowned Warbler
+Barn Swallow
+White crowned Sparrow
+Chestnut sided Warbler
+White eyed Vireo
+Bronzed Cowbird
+Pileated Woodpecker
+Green Violetear
+Scott Oriole
+Mockingbird
+Boat tailed Grackle
+Cape May Warbler
+Henslow Sparrow
+Tree Swallow
+Yellow bellied Flycatcher
+Blue winged Warbler
+Evening Grosbeak
+Red legged Kittiwake
+Brown Creeper
+Field Sparrow
+Pied billed Grebe
+Magnolia Warbler
+Common unseen class
+20
+30
+40
+50
+60
+70
+80
+90
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(b) CUB, TF-VAEGAN
+geodesic_dome_indoor
+landing_deck
+jacuzzi_indoor
+bazaar_indoor
+volleyball_court_outdoor
+trading_floor
+sandbox
+playground
+parking_lot
+car_interior_frontseat
+casino_outdoor
+firing_range_indoor
+auditorium
+ticket_booth
+elevator_interior
+racecourse
+arena_basketball
+bus_depot_outdoor
+excavation
+betting_shop
+tundra
+rectory
+Common unseen class
+0
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(c) SUN, DeViSE
+geodesic_dome_indoor
+landing_deck
+jacuzzi_indoor
+bazaar_indoor
+volleyball_court_outdoor
+trading_floor
+sandbox
+playground
+parking_lot
+car_interior_frontseat
+casino_outdoor
+firing_range_indoor
+auditorium
+ticket_booth
+elevator_interior
+racecourse
+arena_basketball
+bus_depot_outdoor
+excavation
+betting_shop
+tundra
+rectory
+Common unseen class
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(d) SUN, TF-VAEGAN
+Mallard
+Scarlet Tanager
+Orange crowned Warbler
+Barn Swallow
+White crowned Sparrow
+Chestnut sided Warbler
+White eyed Vireo
+Savannah Sparrow
+Bronzed Cowbird
+Pileated Woodpecker
+Green Violetear
+Scott Oriole
+Mockingbird
+Boat tailed Grackle
+Cape May Warbler
+Henslow Sparrow
+Tree Swallow
+Yellow bellied Flycatcher
+Blue winged Warbler
+Evening Grosbeak
+Red legged Kittiwake
+Brown Creeper
+Field Sparrow
+Pied billed Grebe
+Magnolia Warbler
+Common unseen class
+0
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(e) CUB, DeViSE
+Mallard
+Scarlet Tanager
+Orange crowned Warbler
+Barn Swallow
+White crowned Sparrow
+Chestnut sided Warbler
+White eyed Vireo
+Savannah Sparrow
+Bronzed Cowbird
+Pileated Woodpecker
+Green Violetear
+Scott Oriole
+Mockingbird
+Boat tailed Grackle
+Cape May Warbler
+Henslow Sparrow
+Tree Swallow
+Yellow bellied Flycatcher
+Blue winged Warbler
+Evening Grosbeak
+Red legged Kittiwake
+Brown Creeper
+Field Sparrow
+Pied billed Grebe
+Magnolia Warbler
+Common unseen class
+20
+30
+40
+50
+60
+70
+80
+90
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(f) CUB, TF-VAEGAN
+geodesic_dome_indoor
+landing_deck
+jacuzzi_indoor
+bazaar_indoor
+volleyball_court_outdoor
+trading_floor
+sandbox
+playground
+hotel_room
+artists_loft
+archive
+motel
+parking_lot
+car_interior_frontseat
+casino_outdoor
+firing_range_indoor
+auditorium
+piano_store
+vineyard
+wrestling_ring_indoor
+ticket_booth
+canal_natural
+elevator_interior
+bank_vault
+temple_south_asia
+racecourse
+doorway_indoor
+arena_basketball
+theater_indoor_seats
+bus_depot_outdoor
+excavation
+workshop
+betting_shop
+vestry
+tundra
+rectory
+Common unseen class
+0
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(g) SUN, TF-VAEGAN
+geodesic_dome_indoor
+landing_deck
+jacuzzi_indoor
+bazaar_indoor
+volleyball_court_outdoor
+trading_floor
+sandbox
+playground
+hotel_room
+artists_loft
+archive
+motel
+parking_lot
+car_interior_frontseat
+casino_outdoor
+firing_range_indoor
+auditorium
+piano_store
+vineyard
+wrestling_ring_indoor
+ticket_booth
+canal_natural
+elevator_interior
+bank_vault
+temple_south_asia
+racecourse
+doorway_indoor
+arena_basketball
+theater_indoor_seats
+bus_depot_outdoor
+excavation
+workshop
+betting_shop
+vestry
+tundra
+rectory
+Common unseen class
+20
+40
+60
+80
+100
+CZSL accuracy (in %)
+Existing Split (ES)
+Proposed Split (PS)
+(h) SUN, TF-VAEGAN
+Fig. 10. Class-wise accuracy (in %) in the CZSL setting for test classes from U1𝑐𝑜𝑚 (obtained after randomly
+splitting set U of Existing Split (ES)), containing at least one rare attribute ((a)–(d)) or one common attribute
+((e)–(h)). The curves are obtained by evaluating the performance of two trained models – DeViSE [13] and
+TF-VAEGAN [32]. Common unseen classes are the test classes on which we evaluate models trained with
+seen classes from Existing Splits (ES) and Proposed Splits (PS)
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning
+111:19
+Table 6. Understanding object domain via attributes. Five rare and common attributes (designated as described
+in Sec. 6.4) are listed for the three different object domains for which we show our results.
+˜
+C𝑋 denotes the
+classes in the object domain acquired as
+˜
+C𝑋 = SE ∪ (UE \ U𝑋
+𝑐𝑜𝑚). Here, U𝑋
+𝑐𝑜𝑚 denotes the set of common
+unseen classes for both ES and PS separated out for fair evaluation, created by 𝑋𝑡ℎ random split of set UE.
+(+𝑝) indicates set contains 𝑝 more attributes
+Domain Rare
+Common
+˜C1
+Needle-shaped bill
+Black bill
+Pink throat
+Notched tail
+Red back
+Small size
+Purple eye
+Rounded wings
+Owl-like shape (+19) Black eye (+4)
+˜C2
+Needle-shaped bill
+Rounded wing
+Purple underparts
+Solid breast pattern
+Pink forehead
+Notched tail
+Green leg
+Bill shorter than head
+Orange eye (+17)
+Solid belly pattern (+4)
+˜C3
+Red upper-tail
+Rounded wing
+Pink crown
+Small size
+Green crown
+Solid back pattern
+Purple breast
+Black eye
+Red underparts (+17) Black bill (+5)
+Then, we analyze B( ˜𝐶) (obtained using Eq. 5) and designate the attributes which appear in less
+than 5% of all classes in ˜C as rare, and those appearing in more than 50% of the classes as common
+attributes. Table 5 indicates that there are fewer rare attributes in SUN as compared to CUB. This
+was expected as the attributes in SUN are observed in many different contexts [34] and hence
+appear for many classes. On the other hand, several attributes in CUB are visual variants of a single,
+broader attribute [51]. Hence, several of these attributes are exhibited by a few classes only.
+We report a few rare and common attributes for each object domain (i.e. ˜C1, ˜C2 and ˜C3) in Tab. 6.
+Thereafter, looking back at the seed classes acquired from ˜C2 at the end of stage 1 (Fig. 5), we find
+that our seed-set construction process indeed picks an initial seed set which is not only diverse
+enough, but also captures the rarity from the semantic space of the object domain. For example,
+Fig. 5 shows birds exhibiting needle-shaped bill and orange eye (in colored boxes), which are rare
+attributes considering the domain ˜C2 (see Fig. 6).
+Table 2 conveys that training ZSL models with seen classes that capture rarity in the object
+domain well enough enhance the models’ capability to recognize novel classes exhibiting rare
+attributes. The class-wise top-1 accuracy after training with two ZSL models — DeViSE [13] (a
+compatibility learning framework) and TF-VAEGAN [32] (a generative model-based framework) —
+is depicted in Fig. 10 for test classes from U1
+𝑐𝑜𝑚 exhibiting at least one rare or common attribute. It
+is evident that models in Φ𝑃 (those trained by seen classes selected by DiRaC-I) recognize novel
+classes more accurately.
+7
+CONCLUSION
+In this paper, we propose a novel framework called DiRaC-I for identifying the most suitable classes
+from the available database that can be used to train zero-shot models. Specifically, we emphasize
+J. ACM, Vol. 37, No. 4, Article 111. Publication date: August 2018.
+
+111:20
+Sandipan Sarma and Arijit Sur
+capturing both visual diversity and semantic rarity of an object domain through our framework,
+inspired by Active Learning. Extensive experiments on two challenging fine-grained data sets
+verified that zero-shot models trained with classes acquired by DiRaC-I perform better than models
+trained with predetermined classes. We limit our work to these data sets for fair comparison as
+they have a balanced image count across all the classes, unlike certain others like AwA2 [55]. This
+ensures that even if seen classes for ES and PS are different, it does not adversely affect ZSL model
+performance just due to a huge difference in number of training images. Additionally, we work only
+with human-annotated attributes to account for rarity as they are more semantically descriptive
+and interpretable than word vector representations of classes. Such an attribute space is consistent
+with our real-life goal of zero-shot methods working in a specific object domain. However, manually
+defining attribute ontology is expensive. Hence, an extension of DiRaC-I that can work with word
+vector spaces is worth investigating.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf,len=1600
+page_content='111  DiRaC-I: Identifying Diverse and Rare Training Classes for  Zero-Shot Learning  SANDIPAN SARMA and ARIJIT SUR, Indian Institute of Technology Guwahati, India  Inspired by strategies like Active Learning, it is intuitive that intelligently selecting the training classes from a  dataset for Zero-Shot Learning (ZSL) can improve the performance of existing ZSL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In this work, we  propose a framework called Diverse and Rare Class Identifier (DiRaC-I) which, given an attribute-based dataset,  can intelligently yield the most suitable “seen classes” for training ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DiRaC-I has two main goals –  constructing a diversified set of seed classes, followed by a visual-semantic mining algorithm initialized by  these seed classes that acquires the classes capturing both diversity and rarity in the object domain adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  These classes can then be used as “seen classes” to train ZSL models for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We adopt a  real-world scenario where novel object classes are available to neither DiRaC-I nor the ZSL models during  training and conducted extensive experiments on two benchmark data sets for zero-shot image classification  — CUB and SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Our results demonstrate DiRaC-I helps ZSL models to achieve significant classification  accuracy improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  CCS Concepts: • Computing methodologies → Transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Additional Key Words and Phrases: Zero-shot learning, deep learning, object recognition, image classification  ACM Reference Format:  Sandipan Sarma and Arijit Sur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot  Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM 37, 4, Article 111 (August 2018), 22 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='XXXXXXX  1  INTRODUCTION  Object recognition has witnessed a significant improvement in the recent past using deep learning  methods [15–17, 26, 45, 47] trained on large, annotated data sets in a supervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However,  such methods fail when novel concepts are encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For example, an underwater robot exploring  deep-sea biodiversity should trigger an alert if it encounters a novel or rare species — like a  Manocherian’s Catshark (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2b) — but would probably fail as its recognition model is not trained  on visual images of that species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' A human, on the contrary, can recognize it if he/she has a visual  perception about sharks and is given additional information that it looks like a small shark with  some characteristic attributes — a whitish, porcelain-colored body with a white spot on the tail tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  The idea of zero-shot learning (ZSL) [25, 55] stems from this ability of humans to recognize unseen  objects by learning a mapping function associating the visual samples from the seen classes with  their semantics (or attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This function is then used to recognize both seen and unseen objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  The ZSL community can be divided into three groups (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 1) based on their contributions towards  ZSL – (1) Dataset constructors, who collect labeled data and semantics for a fixed number (say 𝑘) of  object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Before releasing the dataset for ZSL research, they define a disjoint seen-unseen  split of the 𝑘 classes manually;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (2) ZSL researchers, who use these predetermined seen classes for  Authors’ address: Sandipan Sarma, sandipan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='sarma@iitg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Arijit Sur, arijit@iitg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='in, Indian Institute of Technology  Guwahati, Multimedia Lab, Department of Computer Science and Engineering, Guwahati, Assam, India, 781039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='  © 2018 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='XXXXXXX  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='00236v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CV]  31 Dec 2022  111:2  Sandipan Sarma and Arijit Sur  Total claasses = 200  Attributes per class = 312  Class names = { c1, c2,  c3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='. c200}  Manual  split  of classes  Dataset constructor  Collected image  database  Unseen  set  Seen set  {c1, c3, c4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='c198}  {c2, c5, c6,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='c200}  Annotated image collection  150 classes  50 classes  ZSL researcher  Proposes  new  ZSL  model  Training  Evaluate  trained  model  Real-time clases encountered  Seen (from dataset) = {c1, c3,  c4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='c198}  Unseen (from dataset) = {c2, c5,  c6,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='c200}  Novel (from the wild) = {c201,  c202,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='.}  Experimental  input  Real-time  deployment  Trained ZSL model  Domain experts  Sem (S)  Sem (U)  Sem (N)  Real-time  image  Predicted class  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ZSL community and the contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Dataset constructors manually define split of seen-unseen  classes which is received by ZSL researchers and used for training and evaluating ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' High-  performance models would be used for real-world deployment for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DiRaC-I targets the  work of the constructors (green dotted box) and aims to replace manual splits by intelligent splits  automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  training the ZSL models they propose, and the predetermined unseen classes to evaluate these  models and simulate their ability to classify unseen classes of the wild when deployed in future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (3)  AI-aided industries, which deploy the state-of-the-art models to solve real-world problems using  ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For ZSL-based classification models to be widely accepted in the future by industries, the  need for high-performance ZSL models trained with classes that capture rarity and diversity of the  object domain (defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4) is paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, ZSL models proposed by researchers today  only passively learn from the predetermined set of seen classes provided by dataset constructors,  like Xian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' They claim that class diversity is maintained while manually defining the  seen-unseen sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' however, such splits are not designed for best zero-shot performance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence,  an intelligent seen-unseen split of the 𝑘 classes of the collected dataset should be designed such  that the designated seen classes automatically capture the diversity and rarity of the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Only a few studies have addressed this issue in ZSL, mostly using Active Learning (AL) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  [58, 59] experiment with textual datasets only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Recently, in the image classification area, [52]  proposed an GCN-based AL framework for selecting the most crucial classes as seen classes for  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, all these works initialize the AL algorithm with a randomly selected set of classes  with labeled examples, called the seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Additionally, they do not consider the rare attributes for  enriching the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  In this work, we propose a two-stage framework named Diverse and Rare Class Identifier (DiRaC-I)  inspired by AL which targets the attribute-based dataset constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' From the 𝑘-class dataset  provided by the constructors, DiRaC-I aims to select the most suitable seen classes for training  ZSL models while trying to capture visual diversity and semantic rarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The first stage is seed-set  construction, where the 𝑘 classes are clustered based on semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' A single representative  is picked from each cluster to ensure diversity while jointly prioritizing semantic rarity, forming a  seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Intuitively, doing so would incorporate a generalized initial understanding of the object  domain within the seed set, which is used as input for the next stage — Visual-Semantic Mining  (VSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Here, we seek to maximize the diversity between visual samples of the seed and those of the  other classes by estimating the distribution of related classes, based on the work by [4], and select a  few candidate non-seed classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We define a semantic score to be computed for each of them, and a  few classes having the highest semantic scores are added to the seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This process continues  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:3  iteratively till we get a fixed number of seed classes, which become our seen classes to be provided  as an output to researchers for training ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  We ensure a fair comparison of the knowledge gained while training using our seen classes (from  Proposed seen-unseen Splits (PS)) and the predetermined (from Existing seen-unseen Splits (ES)) by  evaluating the performance of existing ZSL models on a common set of unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For a given  data set, this set is derived by randomly picking 50% classes from all the unseen classes (for reasons  given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2) used in ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' These classes are not used during the entire operation of DiRaC-I or  ZSL model training but only during the evaluation of ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Extensive experiments conducted  on two challenging benchmark data sets — CUB and SUN — demonstrate that zero-shot accuracy  of most models are enhanced when trained with seen classes acquired by DiRaC-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In real-world  situations, our framework should be able to select seen classes from a given attribute-based dataset  to help improve the training of the ZSL models to be deployed for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  We summarize our contributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (1) We design a framework, DiRaC-I, that intelligently  captures the diversity and rarity of the object domain within a set of seen classes on which ZSL  models can be trained to have a comprehensive idea of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (2) For initializing the VSM  algorithm, a few diverse seed classes are selected as per some attribute-based scores, instead of  just selecting them randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The rare attributes play a key role in computing these scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (3)  ZSL model performance is evaluated on a set of unseen classes common to both existing splits (ES)  and proposed splits (PS) that are unavailable to DiRaC-I and the model during training (simulating  practical scenarios of encountering novel classes in the wild).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, a fair comparison between the  knowledge gained by the model trained with predetermined seen classes of ES and the acquired seen  classes of PS is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (4) Unlike [52], DiRaC-I can be used by ZSL researchers as a predecessor to  select the seen classes from a given attribute-based dataset, and therefore can adapt to real-world  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  2  RELATED WORK  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Zero-shot learning (ZSL)  Motivated by the problems faced in supervised learning, several new learning paradigms have been  proposed in the last decade or so, such as few-shot [5, 19] and one-shot learning [10, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However,  methods under these paradigms are still not able to cope with scenarios where we have “zero"  training samples of certain classes – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' a rare fish species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Therefore, in the recent years there has  been an increasing amount of interest in zero-shot learning, which defines a setting where visual  features for unseen classes are unavailable during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, seen and unseen classes  can be linked through their semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ZSL has been applied to a wide array of computer vision  tasks such as object detection [3], action recognition [8] and cross-modal retrieval [60] to name a  few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We, however, focus on the zero-shot image classification task in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  The early works on ZSL [20, 25, 40, 41] tried to learn intermediate attribute classifiers to transfer  knowledge from the seen to unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Several other approaches that followed [1, 2, 13, 23, 42, 46, 54]  directly set up bi-linear and non-linear compatibility functions between the visual and semantic  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' At test time, unseen visual features are projected to the semantic space using the learned  functions, and the predicted class is the one achieving maximum compatibility score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The approach  of learning a mixture model of seen classes to represent the images and semantic embeddings  is taken up in [6, 33, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' While these approaches work well in the conventional setting (CZSL)  of unseen test classes only, in practice, a model should be able to classify samples from both  seen and unseen classes when deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Generalized zero-shot learning (GZSL) is a setting that  considers such a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Most existing works that show improvements in GZSL have incorporated  generative models [11, 12, 31, 32, 48, 50, 56, 57], where the aim is to synthesize high-quality unseen  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  111:4  Sandipan Sarma and Arijit Sur  Non-overlapping  unseen classes (given by ES)  Rhinoceros Auklet  Tree Sparrow  California Gull  Orchard Oriole  Black-billed Cuckoo  Pileated Woodpecker  American Pipit  Mockingbird  Caspian Tern  Kentucky Warbler  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  UE  Seen classes (given by ES)  Blue-headed Vireo  Chipping Sparrow  Laysan Albatross  Painted Bunting  Anna Hummingbird  Blue Jay  Geococcyx  Black Tern  Ringed Kingfisher  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='SE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Collected categories  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Rhinoceros Auklet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='California Gull  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orchard Oriole  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pileated Woodpecker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mockingbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Usable classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='in framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Tree Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black-billed Cuckoo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='American Pipit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Caspian Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Kentucky Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Ucom  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Categories available to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='random splitting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(a) Splitting the available set of classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='in a collected dataset to obtain an ob-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='ject domain to work with and the novel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='classes for model evaluation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Available categories in database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Semantic space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Visual diversity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Semantic rarity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red fins  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Gill raker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Dermal outgrowths  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Dorsal spine venom  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='M  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Classes for training  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Recognize unseen classes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Deploy in the wild  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Test model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(b) “Seen classes” designated by DiRaC-I for train-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='ing ZSL models – an application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The target of DiRaC-I and its application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (Left) In a real-world scenario, data for a certain number  of object categories are collected by dataset constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For example, Existing Split (ES) proposed by Xian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' [55] defines some fixed classes to be used as “seen” (SE) and “unseen” (UE) while performing ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We  randomly split the set UE and obtain a set of common unseen classes (U𝑐𝑜𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The proposed framework has  access to only the remaining classes belonging to set ˜𝐶, which constitute the known object domain for the  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' After training several existing ZSL models with seen classes both from ES and PS (our Proposed  Splits), model performance is evaluated only on classes from U𝑐𝑜𝑚 for fair comparison, because these classes  are common to both ES and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We can consider classes from U𝑐𝑜𝑚 to act as novel classes encountered in the  wild;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (Right) Leveraging the DiRaC-I framework for selecting suitable seen classes from the object domain of  fish – a real-world application  class samples or visual features, converting the ZSL problem into a simple supervised classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' A benchmark providing standard evaluation protocols and seen-unseen splits for some of the  widely-used data sets in ZSL is given by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, manually created seen-unseen splits might  not capture the diversity and rarity well enough for training ZSL models, affecting their knowledge  about the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Selecting suitable seen classes for ZSL  The idea of training ZSL models with seen classes more informative than the predetermined ones is  relatively new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Recently, Active Learning (AL) [14] strategies have been employed in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  However, contrary to the traditional way of acquiring the most informative instances from a data set  for training classifiers, in the zero-shot setting, the objective of AL changes to acquiring informative  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' [59] proposes a probabilistic method that focuses on two properties — informativeness of  the seen classes and their connectivity to the unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' An extension of this work [58] demonstrates  the impact of AL on ZSL for extreme multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, it experiments with textual  data sets only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' [52] adopts an AL approach for GCN-based zero-shot image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Their  work extends the k-center algorithm with a Laplacian energy-based strategy for selecting the most  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:5  crucial classes as seen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, it is limited to GCN frameworks for ZSL and initializes the  algorithm with a randomly selected seed set, like the other works on AL-based ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Research in traditional AL has shown that instead of selecting the seed set randomly, an intelligent  selection can propel AL in better directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In an effort to justify this, [49] proposes to manually  prepare the seed set, artificially enriched with rare class examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' [9] gives an automatic approach  that follows the same principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Nevertheless, to the best of our knowledge, no work has shown the  combined benefits of intelligently acquiring seed classes and using them to obtain seen classes that  capture the diversity and rarity from the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Our proposed framework (DiRaC-I) first  constructs a seed set with a diverse initial representation of the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' It then initializes an  AL-inspired algorithm with this seed set and iteratively acquires a fixed number of seen classes  using which ZSL models are to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The most recent work with a similar objective picks the  seed set randomly and is compatible only with GCN-based zero-shot frameworks [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Moreover,  their experiments are on a single dataset, and evaluation metrics are not comparable to the standard  ones [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' On the other hand, DiRaC-I can work with any attribute-based data set in practical  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We also evaluate the prediction accuracy of several existing ZSL models trained with  seen classes from Existing and Proposed Splits using the standard metrics and obtain encouraging  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  3  PROBLEM SETTING AND NOTATIONS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Object recognition using zero-shot learning  In a typical zero-shot setting, we have sets S and U of 𝑁𝑠 and 𝑁𝑢 number of seen and unseen classes  respectively, such that S ∩ U = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Let C = S ∪ U denote the set of all classes for a given data  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The associated semantic embeddings for these sets can be represented by P(S) ∈ R𝑁𝑠×𝑑 and  P(U) ∈ R𝑁𝑢×𝑑 respectively, where attributes of a class 𝑐 are represented by a𝑑−dimensional vector  ⟨𝑎1  𝑐,𝑎2  𝑐, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑎𝑑  𝑐 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' These embeddings are available in several forms like human-annotated attributes [55],  word embeddings like Word2Vec [29] and GloVe [35], or hierarchical embeddings like WordNet [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  X𝑠 ∈ R𝑚×𝑘 and X𝑢 ∈ R𝑛×𝑘 represent the visual data for seen and unseen samples respectively,  usually available in the form of visual features extracted from a CNN like ResNet-101 [15, 55],  pretrained on a large-scale visual dataset like ImageNet [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 𝑚 and 𝑛 denote the number of seen  and unseen class samples respectively, with each image being represented by a 𝑘−dimensional  feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Then, given training data D = {(𝑥𝑠  𝑗,𝑦𝑠  𝑗) ∈ X𝑠 × S} along with P(S) and P(U),  the task in CZSL is to learn a classifier 𝑓𝑐𝑧𝑠𝑙 : X𝑢 → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In GZSL, a small subset of X𝑠 (X𝑠  𝑠𝑢𝑏) is  used as the set of seen samples at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Then, the objective changes to learning a classifier  𝑓𝑔𝑧𝑠𝑙 : X𝑠  𝑠𝑢𝑏 ∪ X𝑢 → S ∪ U to classify both seen and unseen objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Practical insights into seen-unseen splits  Scarcity of labeled data and dealing with unseen concepts are two potential areas where ZSL can  contribute significantly in the future when deployed in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' To name a few, with  ZSL models: (1) autonomous vehicles [18, 37, 38] should be able to recognize unseen concept cars  while driving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (2) previously unseen diseases like COVID-19 could be diagnosed based on their novel  characteristics combined with the similarity to other known diseases like asthma [7, 36, 39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (3)  Autonomous Underwater Vehicles (AUVs) deployed in underwater explorations [22, 24] should be  able to recognize new fish or coral species if encountered (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, for a target application,  data for only a fixed number of available categories (comprising a set ˜C) can be collected by the  dataset constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Although they have access to labeled examples of all the | ˜C| classes and can  provide them to ZSL researchers, the researchers cannot train their models with all | ˜C| classes  as they would always need a disjoint set of unseen classes to evaluate ZSL models, as per ZSL  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  111:6  Sandipan Sarma and Arijit Sur  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Consequently, training ZSL models requires a subset S ⊂ ˜C (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' the set of seen classes),  which DiRaC-I helps the constructors to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Classes from the other subset of the collected  dataset(U = ˜C \\ S) can be considered unseen classes by researchers to evaluate their model  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Finally, the trained model can be deployed in the future to recognize novel classes  (with known attributes) in the wild (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  For zero-shot classification, current researchers use seen-unseen splits predetermined by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  However, unlike these Existing Splits (ES), we try to emulate the real-world scenario via our  Proposed Splits (PS), where the seen classes exhibiting diversity and rarity can be automatically  acquired from ˜C itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For fair comparison of the knowledge gained by existing ZSL models when  trained with seen classes from ES and PS, they should be evaluated on the same set of unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Since we do not have data from classes that are completely unknown to us during experimentation,  we extract a few classes from the unseen set originally given by ES, and make them unavailable to  both DiRaC-I and the ZSL models during their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We first dissociate the set U of ES (UE)  into two halves randomly — U𝑐𝑜𝑚 becomes the set of 𝑁𝑢𝑐𝑜𝑚 unseen classes of the wild and  ˜  UE  the other half, having 𝑁 ˜𝑢 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Then, the proposed framework acquires seen classes from the  set ˜C = SE ∪  ˜  UE and ZSL model can train on the acquired classes (SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Figure 2a gives a better  understanding of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Finally, let Φ𝐸 = {Φ𝑀1  𝐸 , Φ𝑀2  𝐸 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Φ𝑀𝑛  𝐸 } and Φ𝑃 = {Φ𝑀1  𝑃 , Φ𝑀2  𝑃 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Φ𝑀𝑛  𝑃 }  denote the sets of models 𝑀1, 𝑀2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑀𝑛 trained using seen classes from ES and PS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We  compare the performance of the models Φ𝑥  𝐸 and Φ𝑥  𝑃 on the test set U𝑐𝑜𝑚 that is unseen to both  Φ𝑥  𝐸 and Φ𝑥  𝑃 (𝑥 = 𝑀1, 𝑀2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑀𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Note that in our framework, classes in U𝑐𝑜𝑚 do not overlap with  the ImageNet 1K classes used for pretraining ResNet-101, following the ZSL assumption provided  by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Moreover, PS is not fixed – since we induce randomness while splitting UE, we repeat  the entire process (from initializing DiRaC-I to evaluating ZSL models trained with the acquired  seen classes) three times so that three different sets of classes are available to our framework at its  inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We show our results in each case, demonstrating the robustness of our framework to the  available object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  4  DIRAC-I: DIVERSE AND RARE CLASS IDENTIFIER  In this work, we focus on data sets having homogeneous categories only — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' having all bird  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For such a data set, we say that its object domain is birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Heuristically, training a ZSL  model with seen classes that capture both the diversity in the visual space and rarity in the semantic  space would provide it with a more generalized idea of the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, the key to our  approach is exploring the entire available object domain (defined by classes from ˜C) for diversity  and rarity using a method inspired by Active Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Adopting such a principle enhances the  capability of ZSL models for knowledge transfer from the seen to unseen classes during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Moreover, the novel classes exhibiting rare attributes have a better chance of being recognized  accurately, as suggested by the results of our experiments on two benchmark data sets (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I consists of two stages, which are discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Stage 1: Seed-set construction  Let ˜Ψ𝑖 = {Ψ1  𝑖 , Ψ2  𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Ψ𝑖  𝑖 } denote the set of 𝑖 clusters, where Ψ𝑗  𝑖 denotes the 𝑗𝑡ℎ cluster of classes  represented by their semantic vectors (𝑗 ≤ 𝑖) when 𝑖 clusters are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We run hierarchical ag-  glomerative clustering (HAC) multiple times to decide the optimal number of clusters by evaluating  the goodness of clusters in each ˜Ψ𝑖 as:  𝑁𝑧 =  𝑎𝑟𝑔𝑚𝑎𝑥  2≤ 𝑖 ≤(𝑁𝑠+𝑁 ˜𝑢−1)  𝑀𝑆𝐶( ˜Ψ𝑖)  (1)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:7  C1  C2  C3  Semantic space  C3  C2  C1  shape: swallow-like,  eye color: grey,  wing-color: purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='.  bill color: red,  under tail color: blue,  size: large,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  head pattern: crested,  shape: owl-like,  throat color: yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='.  bill shape: curved,  primary color: yellow,  shape: duck-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  bill shape: hooked,  size: very large,  shape: hawk-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  wing color: green,  eye color: pink,  bill shape: needle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  C1  C3  C2  Filter Irrelevant  Attributes (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3)  Neglect Unremarkable  Attributes  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4)  Seed  classes  Feature extractor  Train  Unlabeled pool  MAVs for seed  classes  AVs for unlabeled samples  Visual diversity  Unique candidates =  {Ci , Cj , Ck}  PZ  a1  a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ad  c1  a11  a12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  a1d  c2  a2  1  a2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  a2  d  c3  a3  1  a3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  a3  d  Compute  attribute  weights  (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6 and 11)  Semantic score calculator  Semantic  vectors  Semantic  weights  Cluster representative designator  Acquired classes  Add to seed set  Semantic  rarity  STAGE 1  STAGE 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DiRaC-I workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In stage 1, clustering in the semantic space selects a representative from each  cluster while filtering out irrelevant and unremarkable attributes, and this forms the seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In stage 2, the  seed set expands iteratively while considering visual diversity and semantic rarity, until it contains a fixed  number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The resulting set can act as seen set for training ZSL models  where 𝑀𝑆𝐶(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') is the mean silhouette coefficient [21, 43]:  𝑀𝑆𝐶( ˜Ψ𝑖) =  1  (𝑁𝑠 + 𝑁 ˜𝑢)  ∑︁  𝑘 ∈ ˜C  𝑏𝑖  𝑘 − 𝑎𝑖  𝑘  𝑚𝑎𝑥{𝑎𝑖  𝑘,𝑏𝑖  𝑘}  (2)  where 𝑎𝑖  𝑘 and 𝑏𝑖  𝑘 are the mean intra-cluster distance and mean nearest-cluster distance for semantic  vector of class 𝑘 when 𝑖 clusters are formed by HAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' From the optimal set of clusters( ˜Ψ𝑁𝑧), a single  representative is selected from each Ψ𝑗  𝑁𝑧 based on information from the cluster-specific semantic  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, for a cluster, some attributes might not be present at all (irrelevant) or may occur  in minimal amounts (unremarkable) and hence can be ignored while searching for its suitable  representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Therefore, for a cluster Ψ𝑗  𝑁𝑧, we formally recognize these two groups of attributes  respectively from the semantic space (P(Ψ𝑗  𝑁𝑧)) spanned by its member classes:  𝐼𝐴(Ψ𝑗  𝑁𝑧) = {𝑎𝑙 ∈ R | 𝑎𝑙  𝑐 = 0, ∀ 𝑐 ∈ Ψ𝑗  𝑁𝑧 }  (3)  𝑈𝐴(Ψ𝑗  𝑁𝑧) = {𝑎𝑙 ∈ R | B𝑙  𝑐 (Ψ𝑗  𝑁𝑧) = 0, ∀ 𝑐 ∈ Ψ𝑗  𝑁𝑧 }  (4)  where for an attribute 𝑎𝑙:  B𝑙  𝑐 (Ψ𝑗  𝑁𝑧) =  \uf8f1\uf8f4\uf8f4\uf8f2  \uf8f4\uf8f4\uf8f3  0,  if 𝑎𝑙  𝑐 ≤  1  |{𝑐 ∈Ψ𝑗  𝑁𝑧 | 𝑎𝑙𝑐≠0}|  �  𝑐 ∈Ψ𝑗  𝑁𝑧  𝑎𝑙  𝑐  1,  otherwise  (5)  Here, for cluster Ψ𝑗  𝑁𝑧, 𝐼𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') and 𝑈𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') denote the sets of irrelevant and unremarkable attributes  respectively, and B(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') is a binary class-attribute matrix procured from P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') after ignoring the  irrelevant attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Unremarkable attributes are also discarded from both B(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') and P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' To  account for the rarity in semantic space P(Ψ𝑗  𝑁𝑧), we calculate per-attribute frequencies with the  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  EricBeginJMPAll rights reserved @ Emila Yusof111:8  Sandipan Sarma and Arijit Sur  help of the corresponding B(Ψ𝑗  𝑁𝑧) — rarer the attribute, more the importance given to it by sampling  weights from the function:  𝑓 (𝜃𝑎𝑙 ) = − log(𝜃𝑎𝑙 )  (6)  where we obtain attribute frequencies from the diagonal values (𝑑𝑙  𝑙 ) of matrix (B(Ψ𝑗  𝑁𝑧))𝑇 · B(Ψ𝑗  𝑁𝑧)  as:  𝜃𝑎𝑙 =  𝑑𝑙  𝑙  |Ψ𝑗  𝑁𝑧 |  (7)  According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 7, 𝜃 ∈ (0, 1] and 𝑓 (𝜃𝑎𝑙 ) ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We use − log(𝜃𝑎𝑙 ) to sample attribute-weights as  it is strictly decreasing on the interval (0, 1], providing a higher weight if 𝑎𝑙 is rare (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=', 𝜃𝑎𝑙 is low),  and a lower weight otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Finally, we get the seed set as Z = {𝜅(Ψ1  𝑁𝑧),𝜅(Ψ2  𝑁𝑧), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝜅(Ψ𝑁𝑧  𝑁𝑧 )} in  which a representative class from each cluster is selected as:  𝜅(Ψ𝑗  𝑁𝑧) = 𝑎𝑟𝑔𝑚𝑎𝑥  𝑐 ∈Ψ𝑗  𝑁𝑧  (B(Ψ𝑗  𝑁𝑧) ⊙ P(Ψ𝑗  𝑁𝑧)) · W  (8)  where ⊙ denotes element-wise matrix multiplication and W is a vector of weights for attributes  present in P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Such representatives from different clusters boost diversity while promoting  semantic rarity via Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We consider an outlier class (not a member of any cluster) to be  diverse enough from the other classes and take it directly into the seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Stage 2: Visual-Semantic Mining (VSM)  The samples belonging to classes from Z act as a labeled data set used to initialize our VSM  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' VSM is inspired by Active Learning (AL), where inputs from an Oracle (the source of  ground truth labels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' a human expert) are used to label some of the most informative samples from  an unlabeled pool (A) to train machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For our framework, this pool corresponds  to the samples not belonging to classes from the seed set (for a given VSM iteration only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' dataset  constructors actually provide labels for all samples available to DiRaC-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We aim to adopt a similar  strategy to iteratively acquire 𝑁𝑠 classes exhibiting diversity and rarity for training ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  In each iteration of VSM, we retrain a ResNet-101 (M) [15] pretrained on ImageNet [44] to behave  as a feature extractor for the seed class samples using a transfer learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We capitalize on  the work done by [4] and use the scores from the penultimate layer of a CNN (Activation Vectors  or AVs) to estimate the distribution of the related classes, establishing a relationship between the  unlabeled and labeled samples in the AV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Each class 𝑐 ∈ Z is represented by its Mean  Activation Vector (MAV) computed using the AVs of the training samples classified correctly  by M, obtaining the MAV set V = {𝜇1, 𝜇2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝜇|Z|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For the unlabeled samples, we extract the AVs  using the trained M to obtain F = {𝑓1, 𝑓2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑓|A |}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We intend to capture the visually most diverse  samples leveraging the AV space by first obtaining the set:  Π = {𝑘 ∈ R | 𝑘 = 𝑚𝑖𝑛  𝜇𝑐 ∈V 𝛿(𝜇𝑐, 𝑓𝑗), ∀𝑓𝑗 ∈ F }  (9)  and then selecting 𝑡 samples from A corresponding to the largest values in Π, where 𝛿 denotes  the Euclidean-cosine distance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' A set of unique candidate classes (H) is formed by querying the  ground truths of these 𝑡 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' VSM then explores the rarity in the semantic space spanned  by these candidate classes (P(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' A class-wise estimate of the number of images from the seed  classes exhibiting each attribute can be obtained in a matrix I, where:  I𝑙  𝑐 = 𝑎𝑙  𝑐 · 𝐼𝐶𝑐, 𝑐 ∈ Z  (10)  Here, 𝑎𝑙  𝑐 is an element from the semantic space of the seed classes (P(Z)) and 𝐼𝐶𝑐 gives the number  of images for seed class 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' New attribute-weights are calculated based on the proportion of each  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:9  attribute within the currently “known object domain (Z)” for VSM using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6, except that now:  𝜃𝑎𝑙 =  �  𝑐 ∈Z  I𝑙  𝑐  �  𝑐 ∈Z  𝐼𝐶𝑐  (11)  Finally, we calculate semantic scores for each candidate class as follows:  𝛼ℎ = P(H) · W  (12)  where W denotes the vector of obtained attribute-weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Top-𝑞 candidate classes having the  highest semantic scores are added to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Ground truth of the samples from the added classes are  also queried at the end of an iteration so that samples belonging to classes in Z always remain  labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We repeat this entire process (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3) until |Z| = 𝑁𝑠 (kept as the same value as in Existing  Split for a fair comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  It is important to note that VSM is only inspired by Active Learning (AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We do acknowledge  the structural similarities with AL, such as seeds and acquisition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, the problem  setup and goal of VSM are quite different from AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' AL theoretically aims to select the most  informative samples from a huge unlabeled pool of data, whereas that is not the case for DiRaC-I’s  target group (dataset constructors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DiRaC-I can query the labels once a class is added to the seed  set since labeled data for all the collected classes are available with the constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, for this  task, the AL assumption (model should not have access to labels) does not hold, and its absence  does not make VSM impractical to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  5  EXPERIMENTS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Datasets and seen-unseen splits  CUB [51] and SUN [34] are two challenging fine-grained data sets, both with several classes but  limited data per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' CUB contains 11,788 images from 200 bird categories, each of which is defined  using 312 human-annotated attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' SUN contains 14,340 images from 717 scene categories  annotated with 102 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 𝑁𝑠 is 150 for CUB and 645 for SUN for both ES and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Moreover,  before initiating DiRaC-I, we obtain 𝑁𝑢𝑐𝑜𝑚 as 25 and 36 for CUB and SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The random split of  UE is done three times and model evaluation is done on three different U𝑐𝑜𝑚 sets — U1  𝑐𝑜𝑚, U2  𝑐𝑜𝑚  and U3  𝑐𝑜𝑚 — where 𝑋 in U𝑋  𝑐𝑜𝑚 denotes the split number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Consequently, DiRaC-I runs three times  with different object domains ( ˜C) at its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We report the image count of classes belonging  to sets SE, SP and U𝑐𝑜𝑚 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 1, where the slight difference in image count for ES and PS can  be attributed to the different seen classes considered in ES and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For visual features, we follow  previous work [55] and use CNN features extracted from pretrained ResNet-101 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Implementation details  During stage 1, HAC uses Ward’s method [53] to calculate cluster similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Obtaining too few  clusters (and hence, seed classes) using HAC would initialize the deep model (M) in VSM with too  few training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Additionally, some data sets have very few images per class — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 20 for SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Therefore, we set the lower bound of number of clusters to be formed as 5 to achieve effective model  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' While retraining M, weights of all the layers are frozen except the last fully-connected  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The learning rates for optimizing M are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='001 for CUB and SUN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  𝑞 is set to 2 for CUB and 4 for SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We need 𝑡 to be low so that in a practical scenario, only a small  percentage of the unlabeled images from A need to be queried for their class labels while inferring  the candidate classes during the entire process of VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In our experiments, 𝑡 = max(5, ⌈(3 log𝑎)⌉),  where 𝑎 = average number of images per class for a given data set, according to which 𝑡 = 13  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  111:10  Sandipan Sarma and Arijit Sur  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Image count in seen and common unseen classes for ES and PS for different random splits of UE  Dataset  Split  ES  PS  SE  U𝒄𝒐𝒎  SP  U𝒄𝒐𝒎  CUB  1  7057  1489  7068  1489  2  7057  1488  7068  1488  3  7057  1471  7075  1471  SUN  1  10320  720  10320  720  2  10320  720  10320  720  3  10320  720  10320  720  for CUB and 𝑡 = 9 for SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Across the three runs of DiRaC-I, VSM runs for 58 iterations on an  average for CUB and queries the labels for 754 samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=', 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='39% of the total samples in CUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For  SUN, labels are queried for 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='04% of the total samples (1440) over an average of 160 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Furthermore, keeping in mind the real-life scenarios, we prioritize a candidate class to be included  in Z if it is an overlapping class (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2), so that classes from the set ˜𝐶 \\ SP can also serve as test  set if required without violating zero-shot assumptions [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We observe that these classes mostly  have the highest semantic scores too, and hence conclude that the inclusion is fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  Performance comparison with Existing Splits  We report the average per-class top-1 accuracy [55] in the CZSL setting to evaluate ZSL methods  based on different approaches like compatibility learning and generative frameworks and compare  the performance of Φ𝐸 and Φ𝑃 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' All the methods are implemented in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Among  them, official codes in PyTorch are available for LsrGAN [50] and TF-VAEGAN [32], and the rest  are re-implemented versions based on the original publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The hyperparameters for the  official codes are directly used, whereas they are set on the validation sets for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Note that  model accuracy on ES reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2 is not comparable with that of the original papers because  the test set is different in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Table 2 shows that in the CZSL setting, models in Φ𝑃  show significant improvements on CUB over Φ𝐸 across all three splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For SUN, we notice a mix  of improved and similar results with models in Φ𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This might be because DiRaC-I leverages  information from the semantic space, which lacks attributes with discriminative strength in the  case of SUN, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Comparing results for ES and PS in the GZSL setting would not  be fair in the current work because the seen classes in ES and PS might be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Consequently,  they might have different influences on the unseen predictions and the harmonic mean of seen  and unseen accuracy (GZSL evaluation metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This is true for most models because of their bias  towards the seen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6  FRAMEWORK ANALYSIS  In this section, we demonstrate some qualitative and quantitative results and analyze the perfor-  mance of the two stages of our framework, as well as the impact of incorporating diversity and  rarity of the object domain in the training of ZSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In some of the subsections that follow,  we show some qualitative results on the CUB [51] data set when ˜C = SE ∪ (UE \\ U2  𝑐𝑜𝑚) (U2  𝑐𝑜𝑚  denotes the set U𝑐𝑜𝑚 after the randomly splitting UE for the second time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For brevity, we denote  this set as ˜C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' All the results are shown for the object domain ˜C2 (unless stated otherwise) to  maintain a correlation between results obtained from various stages of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We choose  the CUB dataset for qualitative results because the attributes that characterize each class in CUB  are visually interpretable and hence can be easily verified with the visual results we provide here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:11  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Comparative results (top-1 accuracy in %) of Conventional ZSL with models from sets Φ𝐸 and Φ𝑃  (defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2) on the CUB and SUN data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Results on test classes having at least one common  attribute corresponds to results on all classes (left), since all classes in CUB and SUN exhibit at least one  common attribute (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Enhanced results achieved with PS are in BOLD  Method  Test set  ZSL for all  ZSL for classes having  test classes  at least 1 rare attribute  CUB  SUN  CUB  SUN  ES  PS  ES  PS  ESR  PSR  ESR  PSR  ALE [1]  U1  𝑐𝑜𝑚  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='45  U2  𝑐𝑜𝑚  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='1  Seed-set construction  We aim to achieve a good quality of clusters during this stage and acquire seed classes that  provide a comprehensive initial idea of the object domain to the next stage of DiRaC-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For CUB,  obtained clusters are visually more interpretable since the semantic space consists of several groups  of discriminative properties like wing color, bill shape, head pattern, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, hummingbirds,  kingfishers, and gulls get clustered separately, and picking a representative from each cluster  captures the object domain diversity well enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, the SUN attributes come from a  variety of contexts [34], many of which are applicable to several classes with the same attribute  strength — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' P(SE ∪ (UE \\ U1  𝑐𝑜𝑚)) shows attributes warm and eating have a non-zero value  for 669 and 346 classes but have only 50 and 49 unique values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This results in a large number of  classes clustering together in the semantic space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, our experiments suggest that  data sets which characterize classes using more discriminative attribute strengths would help in  selecting better seed classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Figures 8a and 8e show the seed classes (numbered in ‘black’) for CUB  and SUN respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Randomly selecting such classes could pick all of them from a particular  region (when every class is semantically very similar) or from very different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, our  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='111:12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Sandipan Sarma and Arijit Sur  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Arctic Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Caspian Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Elegant Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Forster Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Least Tern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brewer Blackbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red-winged  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blackbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brandt Cormorant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red-faced  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cormorant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pelagic Cormorant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bronzed Cowbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Shiny Cowbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='American Crow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Fish Crow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Frigatebird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common Raven  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White-necked  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Chuck-will Widow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Gray-crowned  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Whip-poor Will  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Fox Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Seaside Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Sage Thrasher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black throat with pointed tail and rounded wings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Buff colored with bill length shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Dagger-shaped bill with striped wings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Results of clustering during seed-set construction (for object domain ˜C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Each row corresponds to  the members of a specific cluster obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The attribute descriptions for each cluster written below each  row are formed by combining some of the most frequent attributes, procured after discarding the attributes  adjudged as irrelevant and unremarkable for the cluster (such discarded attributes can be found in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' An example from every seed class at the end of stage 1 for CUB with object domain ˜C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Notice the  birds exhibiting rare attributes of the domain ˜C2 (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6) such as needle-shaped bill (pink box) and orange  eye (green box)  approach leverages the semantic relationships between classes and ensures that it picks diverse  representatives, as evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:13  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Sets of irrelevant attributes (𝐼𝐴) and unremarkable attributes (𝑈𝐴) obtained for the clusters corre-  sponding to the top, middle and bottom rows of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Ψ𝑐  25 denotes cluster number 𝑐 out of the  25 clusters obtained using HAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' (+𝑝) indicates set contains 𝑝 more attributes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cluster (c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑰𝑨(𝚿𝒄  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='25)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='𝑼𝑨(𝚿𝒄  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='25)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Yellow back  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Olive bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink forehead  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Rufous crown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange nape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg (+14)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Spatulate-shaped bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Green wing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green throat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Blue under-tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Red nape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink eye (+19)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue belly (+24)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Orange wing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Red upper-tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green upper-tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brown eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red forehead  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue nape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Olive breast (+57)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Classes queried per VSM iteration (q)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Accuracy (in %) for TF-VAEGAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CUB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='SUN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Sensitivity to 𝑞 for TF-VAEGAN [32]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Clustering in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hierarchical Agglomerative Clustering (HAC) in the se-  mantic space spanned by classes from set ˜C2 provides 25 clusters, and a single representative is  designated as the seed class from each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4 elucidates the clustering quality by presenting  the cluster members of three clusters found by HAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' It can be seen that all the cluster members of  the bottom row belong to the family of terns — hence picking a seed class from this cluster ensures  that the final seed set at the end of seed-set construction has a member from this family of birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For  the other rows, although the cluster members come from several families, they share common visual  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For example, crows, cormorants, blackbirds and others have been clustered together in  the top row, whereas the middle row consists of small-sized birds like sparrows, finches, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This  suggests that selecting a member from each cluster would provide a good visual representation of  that cluster to the next stage (VSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5 shows a sample image from each of the seed classes of  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  111:14  Sandipan Sarma and Arijit Sur  (a) After iteration 1  (b) After iteration 2  (c) After iteration 3  (d) After iteration 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Top 𝑡 samples visually most diverse from the seed classes at the start of iteration 𝑖, based on a  Euclidean-cosine distance [4]  CUB dataset corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8a with object domain ˜C2, providing a visual idea of the domain  diversity captured within the seed classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Designating cluster representatives as seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Once the clusters are obtained, seed classes are  selected based on the cluster-specific semantic space using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, to ensure that  the computations are devoid of the effects of irrelevant and unremarkable attributes of a cluster, we  defined sets 𝐼𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') and 𝑈𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') for every cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Table 3 shows these two sets obtained corresponding  to the three clusters (out of 25) exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Combining the information from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4 and  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3, we can see that the attributes belonging to the obtained sets 𝐼𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') and 𝑈𝐴(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=') are indeed  not descriptive enough of the cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, associating some of the most frequently  occurring attributes in a given cluster, we construct some cluster descriptions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4) and find them  to be consistent with the visual images of the cluster members, providing a general description of  the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  Visual-Semantic Mining (VSM)  The idea of capturing diversity and rarity in the object domain via an iterative VSM algorithm is  pivotal to our work and has been shown in action in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We notice that classes are captured  from several regions of the semantic space, maximizing the distance from the existing seed classes  in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In a few cases, the acquired classes are closer to quite a few existing seed classes,  like classes labeled as 29 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8d and 8 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' These cases arise when the generated semantic  scores exceed the visual diversity factor (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 9) by virtue of the rarity of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  Qualitative analysis: VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Figure 7 shows the 𝑡 samples for the first four VSM iterations  considered visually most diverse from the existing seed classes at the start of every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For  CUB data set, 𝑡 = 13 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' New classes added to the initial seed set at every iteration have  been shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 9, where classes at the end of iteration 𝑖 serve as the seed classes at the start of  iteration 𝑖 + 1 (for 𝑖 = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The initial 25 seed classes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5) are used for acquiring new classes  in iteration 1 of VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  For iteration 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 7a presents different kinds of swallows, albatrosses and sparrows which are  different from the birds captured in the seed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' At the start of iteration 1, the top-5 rare and  common attributes captured from the existing seed classes are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' These lists are  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:15  (a) CUB, 𝑖1  (b) CUB, 𝑖2  (c) CUB, 𝑖3  (d) CUB, 𝑖4  (e) SUN, 𝑖1  (f) SUN, 𝑖2  (g) SUN, 𝑖3  (h) SUN, 𝑖4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Visualization of the classes in the semantic space acquired during different iterations of VSM for both  CUB and SUN by t-SNE method [28] (best viewed in color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Each class is represented by its attribute vector in  the semantic space, and classes in the same cluster are shown in the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The top 𝑞 classes acquired in  the 𝑘𝑡ℎ iteration are marked with ‘red’ numbers, and the rest of the numbered classes are the seed classes  before starting iteration named 𝑖𝑘  obtained according to the fraction of seed class images that the attributes appear in, and hence  can approximately be verified from the visual images of the seed classes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The added classes  after iteration 1 are Sooty albatross and Dark-eyed junco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Referring to the semantic vectors for these  two classes, we find that both of them marginally exhibit the top-5 rare attributes, except green leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Moreover, as expected, both the added classes exhibit huge amounts of some of the top-5 common  attributes like black eye and solid belly pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' After adding these new classes to the previous seed  set, the list of top-5 rare attributes changes in the second iteration of VSM, indicating that the seed  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  10  12  5  19  0  26  15  5  24  20  21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='8  10  15  10  5  0  10  1510  12  5  19  0  5  24  20  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='8  10  15  10  5  0  5  10  1510  5  19  30  0  17  22  15  5  24  20  5  8  10  15  10  5  0  5  10  1510  19  0 -  8  27  26  5  5-  24  20  21  18  10  10  15  10  5  0  5  10  1515  10  5  0  5  10  15  20  40  20  0  20  4015  10  5  0  5  10  15  20  40  20  0  20  4016  15  10  5  0  5  10  15  20  40  20  0  20  40111:16  Sandipan Sarma and Arijit Sur  (a) After iteration 1  (b) After iteration 2  (c) After iteration 3  (d) After iteration 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' New classes (in red boxes) added to the initial seed set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5) after the first four iterations of VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Corresponding results in the semantic space can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Visual diversity can be observed as  representatives of various families like sparrows, albatrosses, cormorants, kingfishers and others have been  acquired by VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Rare attributes like purple underparts, needle-shaped bill and others have also been captured  within these classes  set has now been enriched with rare attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The list of top-5 common attributes remains mostly  the same, as these attributes are already the most abundant ones in the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  Parameter sensitivity  During VSM, attribute-weights are computed based on the semantics of classes in Z only (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Diversity and rarity expressed by such a small portion of the object domain should not dictate  the selection of too many classes at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Figure 6 suggests that 𝑞 is not very sensitive to ZSL  model performance, so we set low values of 𝑞 for VSM to steadily explore the object domain while  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:17  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The top five rare and common attributes captured by analyzing the semantic space of the classes in  the seed set at the start of the first four iterations of VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For each iteration, the attributes are shown in  descending order of weights assigned to them, computed using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' and 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Iteration Top Rare  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Top Common  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple under-tail Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Primarily purple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid belly pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple nape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bill shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid breast pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple underparts Black eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple breast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bill shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid breast pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple under-tail Solid belly pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bill shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid breast pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple under-tail Solid belly pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink under-tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Owl-like shape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bill shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid breast pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple under-tail Solid belly pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Distribution of rare and common attributes for different random splits of UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 𝐴 = total number  of attributes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 𝑁 ˜𝐶 = 𝑁𝑠 + 𝑁 ˜𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' AR and AC are the number of rare and common attributes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' YR and YC are the  number of common unseen classes having at least one rare and one common attribute respectively  Dataset A / N ˜𝐶 / Nucom Split AR / AC YR / YC  CUB  312 / 175 / 25  1  24 / 9  24 / 25  2  22 / 9  22 / 25  3  22 / 10  19 / 25  SUN  102 / 681 / 36  1  7 / 2  22 / 36  2  7 / 2  19 / 36  3  6 / 2  15 / 36  expanding the set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Since the average image count per class for SUN is relatively lower than CUB,  we set 𝑞 to be higher for SUN to train the feature extractor effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4  Acknowledging rarity in the object domain  For attribute-based data, an object class is uniquely characterized by its attributes, so it can be  reasoned that the more rare attributes a class exhibits, the higher its probability of being a rare class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  To test the semantic knowledge gained by our framework about the object domain, we develop a  notion for designating attributes as either rare or common using semantic information from P( ˜C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Sets 𝐼𝐴( ˜C) and 𝑈𝐴( ˜C) are developed using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 3 and 4 and their member attributes are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='111:18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Sandipan Sarma and Arijit Sur  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mallard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scarlet Tanager  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange crowned Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Barn Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White crowned Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Chestnut sided Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White eyed Vireo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bronzed Cowbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pileated Woodpecker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green Violetear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scott Oriole  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mockingbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Boat tailed Grackle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cape May Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Henslow Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Tree Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Yellow bellied Flycatcher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue winged Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Evening Grosbeak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red legged Kittiwake  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brown Creeper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Field Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pied billed Grebe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Magnolia Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(a) CUB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DeViSE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mallard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scarlet Tanager  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange crowned Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Barn Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White crowned Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Chestnut sided Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White eyed Vireo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bronzed Cowbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pileated Woodpecker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green Violetear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scott Oriole  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mockingbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Boat tailed Grackle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cape May Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Henslow Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Tree Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Yellow bellied Flycatcher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue winged Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Evening Grosbeak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red legged Kittiwake  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brown Creeper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Field Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pied billed Grebe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Magnolia Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content=' TF-VAEGAN  geodesic_dome_indoor  landing_deck  jacuzzi_indoor  bazaar_indoor  volleyball_court_outdoor  trading_floor  sandbox  playground  parking_lot  car_interior_frontseat  casino_outdoor  firing_range_indoor  auditorium  ticket_booth  elevator_interior  racecourse  arena_basketball  bus_depot_outdoor  excavation  betting_shop  tundra  rectory  Common unseen class  0  20  40  60  80  100  CZSL accuracy (in %)  Existing Split (ES)  Proposed Split (PS)  (c) SUN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DeViSE  geodesic_dome_indoor  landing_deck  jacuzzi_indoor  bazaar_indoor  volleyball_court_outdoor  trading_floor  sandbox  playground  parking_lot  car_interior_frontseat  casino_outdoor  firing_range_indoor  auditorium  ticket_booth  elevator_interior  racecourse  arena_basketball  bus_depot_outdoor  excavation  betting_shop  tundra  rectory  Common unseen class  20  40  60  80  100  CZSL accuracy (in %)  Existing Split (ES)  Proposed Split (PS)  (d) SUN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' TF-VAEGAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mallard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scarlet Tanager  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange crowned Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Barn Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Bronzed Cowbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pileated Woodpecker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green Violetear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scott Oriole  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mockingbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Boat tailed Grackle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cape May Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Henslow Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Tree Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Yellow bellied Flycatcher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue winged Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Evening Grosbeak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red legged Kittiwake  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brown Creeper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Field Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pied billed Grebe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Magnolia Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(e) CUB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' DeViSE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mallard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Scarlet Tanager  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange crowned Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Barn Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='White crowned Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Scott Oriole  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Mockingbird  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Boat tailed Grackle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Cape May Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Henslow Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Tree Swallow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Yellow bellied Flycatcher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Blue winged Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Evening Grosbeak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red legged Kittiwake  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Brown Creeper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Field Sparrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pied billed Grebe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Magnolia Warbler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(g) SUN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' TF-VAEGAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='rectory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common unseen class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='CZSL accuracy (in %)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Existing Split (ES)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Proposed Split (PS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(h) SUN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' TF-VAEGAN  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Class-wise accuracy (in %) in the CZSL setting for test classes from U1𝑐𝑜𝑚 (obtained after randomly  splitting set U of Existing Split (ES)), containing at least one rare attribute ((a)–(d)) or one common attribute  ((e)–(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The curves are obtained by evaluating the performance of two trained models – DeViSE [13] and  TF-VAEGAN [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Common unseen classes are the test classes on which we evaluate models trained with  seen classes from Existing Splits (ES) and Proposed Splits (PS)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  DiRaC-I: Identifying Diverse and Rare Training Classes for Zero-Shot Learning  111:19  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Understanding object domain via attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Five rare and common attributes (designated as described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='4) are listed for the three different object domains for which we show our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ˜  C𝑋 denotes the  classes in the object domain acquired as  ˜  C𝑋 = SE ∪ (UE \\ U𝑋  𝑐𝑜𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Here, U𝑋  𝑐𝑜𝑚 denotes the set of common  unseen classes for both ES and PS separated out for fair evaluation, created by 𝑋𝑡ℎ random split of set UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='(+𝑝) indicates set contains 𝑝 more attributes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Domain Rare  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Common  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='˜C1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Needle-shaped bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink throat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Notched tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red back  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Rounded wings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Owl-like shape (+19) Black eye (+4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='˜C2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Needle-shaped bill  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Rounded wing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple underparts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid breast pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink forehead  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Notched tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green leg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Bill shorter than head  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Orange eye (+17)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid belly pattern (+4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='˜C3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red upper-tail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Rounded wing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Pink crown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Small size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Green crown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Solid back pattern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Purple breast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Black eye  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Red underparts (+17) Black bill (+5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' we analyze B( ˜𝐶) (obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5) and designate the attributes which appear in less  than 5% of all classes in ˜C as rare, and those appearing in more than 50% of the classes as common  attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Table 5 indicates that there are fewer rare attributes in SUN as compared to CUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This  was expected as the attributes in SUN are observed in many different contexts [34] and hence  appear for many classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' On the other hand, several attributes in CUB are visual variants of a single,  broader attribute [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, several of these attributes are exhibited by a few classes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  We report a few rare and common attributes for each object domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ˜C1, ˜C2 and ˜C3) in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Thereafter, looking back at the seed classes acquired from ˜C2 at the end of stage 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5), we find  that our seed-set construction process indeed picks an initial seed set which is not only diverse  enough, but also captures the rarity from the semantic space of the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' For example,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 5 shows birds exhibiting needle-shaped bill and orange eye (in colored boxes), which are rare  attributes considering the domain ˜C2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  Table 2 conveys that training ZSL models with seen classes that capture rarity in the object  domain well enough enhance the models’ capability to recognize novel classes exhibiting rare  attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' The class-wise top-1 accuracy after training with two ZSL models — DeViSE [13] (a  compatibility learning framework) and TF-VAEGAN [32] (a generative model-based framework) —  is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 10 for test classes from U1  𝑐𝑜𝑚 exhibiting at least one rare or common attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' It  is evident that models in Φ𝑃 (those trained by seen classes selected by DiRaC-I) recognize novel  classes more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  7  CONCLUSION  In this paper, we propose a novel framework called DiRaC-I for identifying the most suitable classes  from the available database that can be used to train zero-shot models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Specifically, we emphasize  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  111:20  Sandipan Sarma and Arijit Sur  capturing both visual diversity and semantic rarity of an object domain through our framework,  inspired by Active Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Extensive experiments on two challenging fine-grained data sets  verified that zero-shot models trained with classes acquired by DiRaC-I perform better than models  trained with predetermined classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' We limit our work to these data sets for fair comparison as  they have a balanced image count across all the classes, unlike certain others like AwA2 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' This  ensures that even if seen classes for ES and PS are different, it does not adversely affect ZSL model  performance just due to a huge difference in number of training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Additionally, we work only  with human-annotated attributes to account for rarity as they are more semantically descriptive  and interpretable than word vector representations of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Such an attribute space is consistent  with our real-life goal of zero-shot methods working in a specific object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' However, manually  defining attribute ontology is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Hence, an extension of DiRaC-I that can work with word  vector spaces is worth investigating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  REFERENCES  [1] Zeynep Akata, Florent Perronnin, Zaid Harchaoui, and Cordelia Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content=' IEEE transactions on pattern  analysis and machine intelligence 28, 4 (2006), 594–611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content=' IEEE Transactions on  Neural Networks and Learning Systems 32, 6 (2021), 2506–2520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content=' Corrado, Jon Shlens, Samy Bengio, Jeff Dean, Marc’Aurelio Ranzato, and Tomas Mikolov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content=' arXiv  preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+page_content='  [60] Xing Xu, Jialin Tian, Kaiyi Lin, Huimin Lu, Jie Shao, and Heng Tao Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Zero-shot cross-modal retrieval  by assembling AutoEncoder and generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM Transactions on Multimedia Computing,  Communications, and Applications (TOMM) 17, 1s (2021), 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  [61] Ziming Zhang and Venkatesh Saligrama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Zero-shot learning via semantic similarity embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  4166–4174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' ACM, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' 4, Article 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
+page_content=' Publication date: August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfaPcK/content/2301.00236v1.pdf'}
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+arXiv:2301.13037v1  [econ.TH]  30 Jan 2023
+Royal Processions: Incentives, Efficiency and Fairness in
+Two-sided Matching
+Sophie Bade∗
+Joseph Root†
+January 2023
+Abstract
+We study the set of incentive compatible and efficient two-sided matching mechanisms. We
+classify all such mechanisms under an additional assumption – “gender-neutrality” – which guar-
+antees that the two sides be treated symmetrically. All group strategy-proof, efficient, and gender-
+neutral mechanisms are recursive and the outcome is decided in a sequence of rounds. In each
+round two agents are selected, one from each side. These agents are either “matched-by-default”
+or “unmatched-by-default.” In the former case either of the selected agents can unilaterally force
+the other to match with them while in the latter case, they may only match together if both agree.
+In either case, if this pair of agents is not matched together, each gets their top choices among
+the set of remaining agents. As an important step in the characterization, we first show that in
+one-sided matching all group strategy-proof and efficient mechanisms are sequential dictatorships.
+An immediate corollary is that there are no individually rational, group strategy-proof and efficient
+one-sided matching mechanisms.
+1
+Introduction
+Gale and Shapley (1962) initiated the study of stability in two-sided matching problems using the
+example of a marriage market where each of n men is matched to one of n women. A matching is
+considered stable if no unmarried pair prefers each other to their partners from the matching. Gale
+and Shapley (1962) proved that stable matchings always exist. However, Roth (1982) showed that
+stability and incentive compatibility clash: any stable mechanism is manipulable.
+Faced with this dilemma, the vast literature on two-sided matching typically sacrifices incentive
+compatibility in favor of stability.1 In this paper, we do the opposite: we ignore stability and instead
+study incentive compatible mechanisms.
+Stability is an especially important criterion for settings
+where agents can easily rearrange matchings among themselves (Abdulkadiroglu and S¨onmez 2013).
+However there are many settings where coalitional deviations may be hard to organize. For example,
+people who sign up for a dating app do so since they find it difficult to find dates in real life. In other
+∗Royal Holloway, University of London and Max Planck Institut for Research on Collective Goods, Email:
+so-
+phie.bade@rhul.ac.uk
+†Department of Economics, University of Chicago, Email: jroot@uchicago.edu
+1See for instance Roth and Sotomayor (1992), Roth (2008) and Abdulkadiroglu and S¨onmez (2013)
+1
+
+settings the central planner may have the authority to enforce the outcome of a mechanism. Take, for
+example, the British “homes for Ukraine” scheme that matches refugees with host families. The UK
+home office could condition residency permits on the matches found by an algorithm. So even if some
+families and refugees would like to deviate from the matching, they might find it impossible to do so.
+Finding mechanisms with strong incentive properties so that they are simple to use would, in both
+these contexts, seem more important.
+By abandoning stability, we open the door to the study of incentive compatible and efficient
+mechanisms.
+In particular, we look for mechanisms with robust incentive properties – those that
+are group strategy-proof. A mechanism is group strategy-proof if there is no profile of preferences,
+group of agents and deviation for that group, such that all group members are weakly better-off after
+the deviation (and some strictly).
+A mechanism is efficient if it never chooses a matching where
+an alternative could make all agents better-off. Group strategy-proofness and efficiency are easy to
+attain in two-sided matching problems. Simply declare one side of the market to be “objects” and
+use one of the well-known group strategy-proof and efficient mechanisms for allocation problems.2
+Indeed sometimes “objectification” of one side of the market may be appropriate. Abdulkadiro˘glu
+and S¨onmez (2003), for example, make a convincing case to treat schools as objects in school choice
+problems. However in settings where the two sides are symmetric a priori, one may be hesitant to
+assign all agency to one side of the market. In fact, we may want to require that the two sides of the
+market are treated equally. There is, for example, no reason to treat the men and women on a dating
+app any differently. Likewise one may wish to give an equal amount of say to the refugees and the
+host families in the “Homes for Ukraine” program.
+Our axiom of “gender-neutrality” captures the idea that the two sides should be treated equally.
+Roughly, we require that if the men and women’s preferences are swapped, the outcome of the mecha-
+nism is swapped as well. We not only impose such symmetry between the sides in the overall mechanism
+but on every “continuation submechanism” as well. The notion of “continuation submechanism” cap-
+tures the idea that a mechanism can be recursive. If a mechanism makes initial matches using only the
+preferences of the initially matched agents, ignoring all other agents preferences, then the remaining
+agents face such a continuation submechanism and we ask that they be treated symmetrically as well.
+We characterize the set of all group strategy-proof, efficient and gender-neutral mechanisms as
+the set of “royalty mechanisms.” These work as follows: each round one agent is chosen from either
+side of the market. These “royals” are either “matched-by-default” or “unmatched-by-default.” In the
+former case the royals are matched together if at least one of them top-ranks the other; in the latter
+case, the royals are only matched if both top-rank one another. If the royals are not matched with
+each other, they get their top choices among the set of remaining agents. As long as at least 6 agents
+remain unmatched, a procession of choices by such royal couples continues. Once only four agents
+remain, there are only two possible matches for these agents, however a plethora of (sub)mechanisms
+for these remaining agents satisfy our desiderata.
+Our proof begins with the observation that any gender-neutral two-sided mechanism nests a
+2The set of all group strategy-proof and efficient mechanism for the house allocation problem was characterized
+by Pycia and ¨Unver (2017) and Bade (2020).
+The characterization extends Gale’s top trading cycles (Shapley and
+Scarf 1974) in three ways: agents may “own” multiple houses (P´apai 2000), the agents may “broker” houses (Pycia and
+¨Unver 2017) and when there are exactly three agents and houses there may be “braids” (Bade 2020).
+2
+
+one-sided matching mechanism, where any agent is free to match with any other agent. To get an
+intuition for this result, index the men and women such that mi is the man who mirrors women wi
+under the desired symmetry. If we restrict attention to preference profiles where mi and wi announce
+symmetric preferences, by gender-neutrality they must get symmetric outcomes.
+So for each i, a
+symmetric mechanism either matches wi and mi with each other or with a different symmetric pair
+mj and wj. We can therefore treat the pairs (mi, wi) as the agents in a one-sided matching problem.
+Any pair either matches with themselves, which is analogous to the pair remaining single, or they swap
+partners with another symmetric pair, which can be viewed as a match between these two pairs. The
+embedded mechanism inherits the group strategy-proofness and efficiency of the original mechanism.
+This is helpful since the group strategy-proof and efficient mechanisms admit a neat characterization
+for the one-sided matching problem. We show that all such mechanisms are “sequential dictatorships”
+where sequences of agents or “dictators” choose their most preferred remaining agent as their partner.
+This characterization complements that of Root and Ahn (2020) which finds a similar result when
+agents are not allowed to remain single. Despite the similarities in the result, the proofs are very
+different.
+Our proof stems from our analysis of the three-agent case which is ruled out in Root
+and Ahn (2020) since all agents are required to be matched. Having established that the embedded
+one-sided mechanism is a sequential dictatorship, we see that any gender-neutral, efficient and group
+strategy-proof mechanism agrees with a royalty mechanism on all symmetric preference profiles. Our
+next task is to show that the same holds for asymmetric profiles. The proof is involved, but boils down
+to finding a path from any asymmetric preference profile to a symmetric profile, over which we can
+show the mechanism agrees with a royalty mechanism.
+Gender-neutrality can be seen as a fairness requirement. A common fairness criterion in social
+choice, full symmetry, requires that all agents are treated identically, in the sense that the outcome
+of a mechanism is invariant under any bijective relabeling of all agents. In many environments full
+symmetry conflicts with other desiderata. Bartholdi, Hann-Caruthers, Josyula, Tamuz, and Yariv
+(2021) therefore weaken full symmetry to require that the outcome of a mechanism must only remain
+unchanged for a particular set of bijective relabelings of agents. From this point of view, our notion
+of (weak) gender-neutrality only requires that the mechanism be symmetric with respect to one such
+relabeling. In fact, in two-sided matching, no efficient and group strategy-proof mechanism is invariant
+under more than two relabelings.
+Faced with these limits on achieving symmetry we consider randomization. We first show that
+randomization is no panacea. In our environment, just as in house allocation where random mecha-
+nisms are well-studied, randomization can introduce inefficiency. A typical approach is to “symmetrize”
+a mechanism by choosing agents’ roles in the mechanism uniformly at random. For example, “serial
+dictatorship”, where agents choose their match sequentially in a fixed order, can be symmetrized to
+yield “random serial dictatorship” by selecting the picking order uniformly at random. Despite the
+efficiency of serial dictatorship, random serial dictatorship can be inefficient. Nevertheless, the sym-
+metrized mechanism retains the incentive properties of the deterministic mechanism from which it is
+derived. In house allocation, no mechanism can simultaneously achieve efficiency, incentive compati-
+bility and symmetry (Bogomolnaia and Moulin 2001). On the other hand, random serial dictatorship
+admits a certain form of efficiency: it selects a lottery over deterministic outcomes each of which is
+3
+
+efficient. We say that random serial dictatorship is therefore ex-post efficient. A major open question
+in house allocation is whether any other mechanism satisfies ex-post efficiency, incentive compatibility
+and symmetry. We show that two-sided matching admits multiple mechanisms of this type.
+The rest of the paper proceeds as follows. We first establish the preliminary definitions then
+describe our results. Finally we discuss our axioms. We conclude with a discussion of randomized
+mechanisms.
+2
+Preliminaries
+Let N be a finite set of agents. We will be interested in both “one-sided matching”, where any agent
+can match with any other and “two-sided matching”, where matching is bipartite, so that agents can
+only match with partners from the other side of the market. In two-sided matching, we will assume
+that N is the disjoint union of two sets, M = {m1, . . . , mn} and W = {w1, . . . , wn}. In keeping with
+Gale and Shapley (1962), we refer to the agents in M as “men” and the agents in W as “women.”
+A submatching is a (possibly empty) list of mutually exclusive pairs and singletons. The pairs are
+unordered so that (m, w) and (w, m) both refer to the same pair. In the case of two-sided matching
+singletons are not allowed and each pair has to be made up of one man and one woman. Let Σ1
+0 and
+Σ2
+0 denote the set of one- and two-sided submatchings respectively. For any submatching ν let N(ν)
+denote the set of agents matched in the submatching. A matching is a submatching that lists every
+agent. A proper submatching is a submatching which is not a matching. We alternatively represent
+matchings as bijections µ : N → N of order 2 (i.e. µ ◦ µ = id) so that agent i is matched with j if and
+only if j is matched with i. The requirement that pairs must be made up of one man and one woman
+each then translates to µ(M) = W. We denote the sets of one- and two-sided matchings by Σ1 and
+Σ2 respectively.
+Agents are assumed to have strict preferences (total orders) over their possible partners. In
+one-sided matching, each agent i has a strict preference ≿i over N, where i stands for the option
+to stay unmatched. We write x ≻i y to mean that i strictly prefers being matched with x to being
+matched with y. We write x ≿i y to indicate that either x ≻i y or x = y. In two-sided matching,
+an agents’ preferences range over the other side of the market. That is, each m ∈ M has a strict
+preference ≿m over W and each w ∈ W has a strict preference ≿w over M. A preference profile
+is a list all agents’ preferences (≿i)i∈N. Let Ω1 and Ω2 denote the set of preference profiles for the
+one-sided and two-sided matching problems respectively.
+For any preference ≿i in either context, define top(≿i) as agent i’s most preferred partner and
+bottom(≿i) as agent i’s least preferred partner. For any agent i, we use the notation ≿i: j1, j2, · · · to
+mean an arbitrary preference which top-ranks j1, second-ranks j2 and so on. Given a subset of agents
+S, we write ≿S as a preference profile for just the agents in S. Given a preference profile ≿ and some
+≿′
+S, we write (≿′
+S, ≿−S) for the preference profile in which each agent i from S announces ≿′
+i and each
+agent j from N \ S announces ≿j.
+A (matching) mechanism is a function f : Ωk → Σk for k = 1, 2 that maps each profile of
+preferences to a matching. Such a mechanism f is group strategy-proof if there is no preference
+profile ≿, group of agents S ⊂ N and ≿′
+S such that f(≿′
+S, ≿−S)(i) ≿i f(≿)(i) for all i ∈ S, and for some
+4
+
+j ∈ S, f(≿′
+S, ≿−S)(j) ≻j f(≿)(j). That is, if agents’ true preferences were captured by the profile ≿
+there would be no group of agents who could jointly misreport, making all agents in the group weakly
+better-off, with at least one strictly better-off.3 Given a preference profile ≿, a matching ν Pareto
+dominates a matching η if for all agents i, ν(i) ≿i η(i) and for at least one agent j, ν(j) ≻j η(j).
+A matching η is efficient at the preference profile ≿ if there is no ν which Pareto dominates it. A
+mechanism f is efficient if for every preference profile ≿, f(≿) is efficient.
+We now turn to our fairness requirement. Informally, a mechanism is weakly gender-neutral if
+there is a way to reflect preferences across the sides so that, after reflection, the mechanism chooses
+the same outcome, reflected. To formally define weak gender-neutrality fix a σ ∈ Σ2. While σ is
+a matching, we use it here to denote the reflection across which the mechanism exhibits the desired
+symmetry. Thus σ(wj) = mi is interpreted to mean that agent wj is symmetric to agent mi. We use σ
+both to transform preferences and matchings as follows. Given a matching µ ∈ Σ2, the reflection, σ∗µ,
+of µ under σ is the matching such that (m, w) ∈ σ ∗ µ ⇐⇒ (σ−1(m), σ−1(w)) = (σ(m), σ(w)) ∈ µ.4.
+Equivalently, for any pair, (m, w) matched in µ, (σ(m), σ(w)) are matched in σ ∗ µ. For a preference
+profile ≿∈ Ω2 we define the reflection ≿′= σ(≿) so that j ≿′
+i j′ ⇔ σ(j) ≿σ(i) σ(j′). So if man m
+prefers woman w to woman w′ in the profile ≿, then woman σ(m) prefers man σ(w) to man σ(w′)
+according to the reflected profile σ(≿). Gender-neutrality will ensure that the symmetric agents i and
+σ(i) are is treated equally: a mechanism f is weakly gender-neutral if f(σ(≿)) = σ ∗ f(≿) holds
+for all ≿∈ Σ2.5
+For a fixed symmetry σ we say that a matching µ and respectively a profile of preferences ≿
+are symmetric if they equal to their reflections, so µ = σ ∗ µ and ≿= σ(≿). For notational simplicity,
+when considering a fixed weakly gender-neutral mechanism f we will index the set of men and women
+M = {m1, . . . , mn} and W = {w1, . . . , wn} so that σ(wj) = mj for all j, where σ is the symmetry
+with respect to which f is weakly gender-neutral.
+Example 1. Suppose M = {Ad, Bob, Carl} and W = {Ann, Beth, Connie}. Let
+σ = {(Ann, Ad), (Beth, Bob), (Connie, Carl)}. The reflection of
+µ = {(Ann, Bob), (Beth, Carl), (Connie, Ad)}
+under σ is the matching
+σ ∗ µ = {(Beth, Ad), (Connie, Bob), (Ann, Carl)}
+For instance (Ann, Bob) ∈ µ implies (σ(Ann), σ(Bob)) = (Ad, Beth) ∈ σ∗µ. Fix a profile of preferences
+≿ such that ≿Ad: Connie, Beth, Ann and such that all agents other than Ad rank partners according
+to their alphabetical order. Then the reflected profile ≿′= σ(≿) is such that ≿′
+Ann: Carl, Bob, Ad
+3It turns out that in this context a mechanism is group strategy-proof if and only if no group of agents of size one or
+two can find a jointly profitable misreport. See Alva (2017) and Root and Ahn (2020) for further discussion.
+4The reflection σ ∗ µ is the matching given by the function σ ◦ µ ◦ σ.
+5Note that we require σ to be of order two. This is without loss of generality in the following sense. One could
+have instead asked for an arbitrary bijection ρ : N → N such that ρ(M) = W and required f(ρ(≿)) = ρ ∗ f(≿) for
+all ≿. In this case, since ρ is a permutation, it is a member of the symmetric group. It therefore has a finite order k
+so that ρk is the identity. Since ρ matches all men to women and vice-versa, k is even. Hence ρk/2 is of order 2. If
+f(ρ(≿)) = ρ ∗ f(≿) for all ≿ then f(ρm(≿)) = ρm ∗ f(≿) for any m and any ≿. Thus f(ρk/2(≿)) = ρk/2 ∗ f(≿), and f
+is weakly gender-neutral with respect to ρk/2.
+5
+
+while all other agents rank all possible partners in alphabetical order.
+Example 2. The following is an example of a mechanism which is weakly gender-neutral under
+σ(wj) = mj. For any preference profile, match m1 and w1 with their top choices unless they conflict
+– i.e. if exactly one top ranks the other. In this case, match both with their top choices excluding
+one another. If m1 and w1 are matched together, repeat this with m2 and w2 choosing from the
+remaining agents. Continue until there is a pair mk and wk who do not choose one another. Suppose
+mk is matched with wj and wk is matched with ml. If j < l match the remaining agents using serial
+dictatorship with the men as dictators picking in order of their index. If j > l match the remaining
+agents using serial dictatorship with the women as dictators picking in order of their index.
+While f is weakly gender-neutral in this example, men and women are treated highly unequally
+after most initial choices by m1 and w1: after most initial choices one side of the market retains all
+agency while the other is turned into objects. An agent who is uncertain about the royal’s preferences
+might reason that they have an equal subjective likelihood to be dictator and object. For this agent,
+weak gender-neutrality might be sufficient.
+However, an agent with more information about the
+preferences of the royals might evaluate this mechanism as highly unfair.
+To avoid such unequal
+treatment we impose weak gender-neutrality on “continuation” submechanisms such as the mechanisms
+following on the choices by m1 and w1 in the present example.
+Definition 1. A matching mechanism g for the agents in S is a continuation submechanism of
+the matching mechanism f if there is a profile of preferences ≿−S for N \ S such that g(≿∗
+S)(j) =
+f(≿S, ≿−S)(j) ∈ S for all ≿S and j ∈ S, where ≿∗
+S is the restriction of ≿S to S.
+Definition 2. A bilateral matching mechanism f : Ω2 → Σ2 is gender-neutral if for every continu-
+ation submechanism g of f is weakly gender-neutral.
+Since any mechanism is a continuation submechanism of itself, and gender-neutral mechanism
+is weakly gender-neutral.
+Example 3 (Example 2 continued). To see that the mechanism f defined above is not gender-neutral
+note that upon m1 and w1 respectively choosing w3 and m2 as their partners a continuation sub-
+mechanism for all other agents arises. Since w1’s partner (m2) has a lower index than m1’s (w3) the
+remaining matches are determined by a serial dictatorship of all women. Since serial dictatorships are
+not not weakly gender-neutral, the overall mechanism is not gender-neutral.
+In the preceding example the continuation submechanism following the symmetric match be-
+tween m1 and w1 is weakly gender-neutral under the same symmetry as the original mechanism. This
+observation easily generalizes:
+Lemma 1. Suppose f is a weakly gender-neutral mechanism with symmetry σ. Suppose g is a con-
+tinuation submechanism of f.
+Say g follows on a gender-neutral submatching.
+Then g is weakly
+gender-neutral with the symmetry σ′ that is the restriction of σ to the agents in g.
+6
+
+3
+Results and Approach
+We characterize the class of group strategy-proof, gender-neutral and efficient two-sided matching
+mechanisms. As a key lemma we characterize all group strategy-proof and efficient one-sided matching
+mechanisms, allowing agents to remain unmatched.
+For simplicity we ignore related special cases
+in both results, which we leave to the appendix. These correspond to when there are exactly four
+agents in two-sided matching and when there are exactly two agents in one-sided matching. These are
+special cases for similar reasons: both reduce to social choice problems with exactly two outcomes. As
+described in the introduction, our characterization results in the class of “Royalty mechanisms.” These
+mechanisms sequentially select two agents, one from either side, to choose their matches according to
+one of two regimes: matched-by-default(D) or unmatched-by-default(U). The precise order of these
+agents can vary.
+Royalty mechanisms are therefore parameterized by this order, which we call a
+“succession order”.
+Definition 3. A succession order is a function ϕ : A → (M × W) × {D, U} where A is a subset of
+Σ2
+0. Let ϕ1 : A → (M × W) and ϕ2 : A → {D, U} correspond to the first and second components of
+ϕ. A succession order ϕ must satisfy
+1. ∅ ∈ A
+2. If ν ∈ A and ϕ1(ν) = (m, w) then m, w /∈ N(ν)
+3. If ν ∈ A and ϕ1(ν) = (m, w) then for any m′.w′ /∈ N(ν), ν ∪ {(m, w′), (w, m′)} matches all but
+four or fewer agents in A.
+In addition to the succession order, we need to specify a terminal condition for the algorithm.
+Let χ be the set of group strategy-proof, efficient, and gender-neutral mechanisms with exactly four
+agents6 and let ΣT
+2 be the set of submatchings in Σ2
+0 where exactly four agents are unmatched.
+Definition 4. A terminal condition is a map ϕT : ΣT
+2 → χ.
+Given a succession order ϕ and a terminal condition ϕT , the royalty mechanism R(ϕ,ϕT )
+chooses a matching using the following algorithm.
+Definition 5. The royalty algorithm given a succession order ϕ, a terminal condition ϕT and a
+preference profile ≿ proceeds in a number of steps:
+Initialize: Set ν0 = ∅ and if there are three or more couples in the mechanism go to Step
+1, otherwise go to Step T
+Step k: The agents ϕ1(νk−1) are declared royals. If ϕ2(νk−1) = D then if either royal
+top-ranks the other among N − N(νk−1), they are matched. If ϕ2(νk−1) = U the royals
+are matched only if they both top-rank one another among N − N(νk−1). If the royals
+are not matched, each gets matched with their favorite partner excluding one another in
+N − N(νk−1). Let νk denote the resulting submatching. If νk leaves 6 or more couples
+6These are characterized in appendix section A.
+7
+
+unmatched, go to step k + 1. If νk leaves 4 or fewer agents unmatched go to Step T .
+Step T: If there are exactly two remaining agents, match them together to result in νT .
+Otherwise use the mechanism ϕT (νk) to find a submatching ν for the remaining four agents.
+Set νT = νk ∪ ν. Return νT .
+The last step of royalty mechanisms are a special case which we discuss in the Appendix A.
+There are a number of possible group strategy-proof, efficient and gender-neutral mechanisms. As a
+lead example consider a unanimity rule which sets one of the matchings as a default and chooses the
+other matching only if all four agents prefer it to the default.
+Theorem 1. A two-sided matching mechanism f : Ω2 → Σ2 is group strategy-proof, efficient and
+gender-neutral only if it is a royalty mechanism.
+Theorem 1 does not characterize the set of all group strategy-proof, efficient and gender-neutral
+mechanisms as some royalty mechanisms are not gender-neutral. To see this, consider a royalty mech-
+anism f for 5 or more couples that starts with m1, w1 as the royal couple. Consider the cases that the
+royal couple either choose w2 and m3 as their partners or w3 and m2. Our definition of royalty mech-
+anisms only imposes that these choices lead to either a matched-by-default or a unmatched-by-default
+step. However gender neutrality requires more than that: If ml and wk become the royal couple after
+the choice of w2 and m3 then wl = σ(ml) and mk = σ(wk) become must become the royal couple after
+the choice of w3 and m2.
+To define neutral royalty mechanisms, we proceed inductively over the number of couples in a
+mechanism. To start say that the set of all royalty mechanisms for two couples consists of the gender
+neutral, efficient and group strategy-proof mechanisms characterized in appendix A. Now suppose
+that neutral royalty mechanisms for up to n couples have been defined. To define a neutral royalty
+mechanism for n + 1 couples, fix a symmetry σ on N, with the standard assumption that σ(mi) = wi
+for all i and such that (m1, w1) are chosen as the royal couple at the start of the mechanism.
+Say (m1, w1) choose mk and wl as their partners. If these choices are symmetric with respect to
+σ, so if k = l, then a neutral royalty mechanism that is symmetric with respect to the restriction
+of σ to the unmatched agents must be chosen as the continuation submechanism. If k < l we may
+freely choose any neutral royalty mechanism f k,l to follow on the submatching {(m1, wl), (mk, w1)}.
+If k > l we are bound by the choices for the preceding case. In that case we must use a permutation
+of the royalty mechanism σ ∗ f k,l that is identical to f k,l except that wherever a man m appears in
+f k,l the woman σ(m) should take up his place in σ ∗ f k,l and wherever a woman w appears in f k,l the
+woman σ(w) should take up her place in σ ∗ f k,l.
+Corollary 1. A two-sided matching mechanism f : Ω2 → Σ2 is group strategy-proof, efficient and
+gender-neutral if only if it is a neutral royalty mechanism.
+The proof is involved so we defer the details to the appendix. We sketch the argument here.
+The first key insight is to notice that within any gender-neutral two-sided mechanism is a one-sided
+mechanism. Suppose as usual that σ(mi) = wi for all i. Let Ω2
+symm be the set of symmetric preference
+profiles (i.e. the ≿ such that σ(≿) =≿). Since σ(≿) =≿, gender-neutrality implies that f(≿) = σ∗f(≿
+). Then if (mi, wj) ∈ f(≿) so is (σ(mi), σ(wj)) = (wi, mj). From this we see that given any symmetric
+8
+
+preference profile ≿, for any pair (mi, wi), either mi and wi are matched in f(≿) or they swap partners
+with another symmetric pair (mj, wj). By treating each symmetric pair (mi, wi) as a single agent,
+ci we can extract a one-sided matching mechanism. We interpret the swap of partners between say
+(mi, wi) and (mj, wj) as a match between ci and cj and a match between mi and wi as the agent ci
+being unmatched. Root and Ahn (2020) showed that all group strategy-proof and efficient one-sided
+mechanisms are sequential dictatorships. However, they did not allow agents to remain unmatched,
+a key requirement for us. We complement their characterization by showing the same holds even if
+agents are allowed to remain unmatched.
+Definition 6. let N ′ = {1, 2, . . ., n} and N = {m1, . . . , mn} ∪ {w1, . . . , wn}. Given a gender-neutral
+mechanism f for N, we define a one-sided matching mechanism g for N ′ which we call the induced
+one-sided mechanism for f. For any ≿ in the one-sided matching market, let ≿∗ be the preference
+profile in the two-sided market where mk ≿∗
+wj ml and wk ≿∗
+mj wl if and only if k ≿j l for all j, k, l.
+Then for any j, g(≿)(j) is the agent i such that f(≿∗)(mj) = wi and f(≿∗)(wj) = mi.
+Lemma 2. For any gender-neutral two-sided matching mechanism f, the induced one-sided mechanism
+g is group strategy-proof and efficient if f is.
+Proof. Assume that f is group strategy-proof and efficient.
+We will first show that g is efficient.
+Consider any preference profile ≿ in the one-sided matching market and the matching g(≿).
+For
+any other one-sided matching µ ̸= g(≿) we can define the symmetric two-sided matching ν so that
+ν(mj) = wi and ν(wj) = mi if and only if µ(j) = i. Since f is efficient, there must be some j such that
+f(≿∗)(mj) ≻∗
+mj ν(mj). However by definition we then have that g(≿)(j) ≻j ν(j). Hence ν cannot
+Pareto dominate g(≿).
+To see that g is group strategy-proof, fix any preference profile ≿, a group of agents S and some
+≿′
+S such that g(≿) ̸= g(≿′
+S, ≿−S). Since f is group strategy-proof, there is some j among the set of men
+and women who have the same index as an agent in S such that f(≿∗)(mj) ≻mj f((≿′
+S, ≿−S)∗)(mj)
+or f(≿∗)(wj) ≻wj f((≿′
+S, ≿−S)∗)(wj). By definition then g(≿) ≻j g(≿′
+S, ≿−S)(j), so ≿′
+S is not a
+profitable deviation for the coalition S. Since the coalition, deviation and original preference profile
+were arbitrary, g is group strategy-proof.
+Before stating our characterization of one-sided matching mechanisms, let’s recall the notion of
+sequential dictatorship.
+Definition 7. A picking order is a function φ : A → N where A is a subset of Σ1
+0 and
+1. ∅ ∈ A
+2. If ν ∈ A and φ(ν) = i then i /∈ N(ν)
+3. If ν ∈ A, φ(ν) = i and j /∈ N(ν) then ν ∪ {(i)} and ν ∪ {(i, j)} are either in A or are matchings.
+Definition 8. A mechanism f is a sequential dictatorship with respect to φ if for any ≿, f(≿) is
+the matching resulting from the following algorithm:
+9
+
+Step 1: Agent φ(∅) is matched with her favorite partner (including herself). Let ν1 be this
+submatching.
+Step k ≥ 2: Agent φ(νk−1) is matched with her favorite remaining partner (including
+herself). Let νk be the matching νk−1 ∪{(φ(νk−1)} if φ(νk−1) prefers to remain unmatched
+or νk−1 ∪{(φ(νk−1, j)} if j is φ(νk−1)s favorite remaining partner. If νk is a matching, stop
+and return νk.
+Theorem 2. A one-sided matching mechanism f : Ω1 → Σ1 is group strategy-proof and efficient if
+and only if it is a sequential dictatorship.
+We prove Theorem 2 by induction over the number of agents. In a sequence of Lemmas we show
+that the result holds for 3 agents. For 4 agents we use the fact that restricted to the case where each
+agent bottom ranks being single, we face a classical social choice problem over 3 alternatives. This case
+must by the Gibbard-Satterthwaite Theorem be a dictatorship (Gibbard 1973)(Satterthwaite 1975).
+We show that the dictator in this constrained problem must also be the dictator in the four agent
+problem when agent may prefer to be single to some matches. The cases of five or more agents then
+do not require any special treatment. This result agrees with Root and Ahn (2020) who find a similar
+characterization in the setting where agents are not allowed to remain unmatched. Since agents are
+not able to veto the dictators’ choices in a sequential dictatorship, Theorem 2 admits an immediate
+corollary:
+Corollary 2. There are no individually rational, group strategy-proof and efficient one-sided matching
+mechanisms
+Having observed that a gender-neutral mechanism is equivalent to a one-sided matching mech-
+anism on symmetric profiles, we see that on these profiles a group strategy-proof, gender-neutral and
+efficient mechanism must agree with some royalty mechanism. At a high level, the rest of the proof
+can be summarized by the following procedure. For any preference profile ≿ we find a sequence of
+profiles ≿0, . . . , ≿m where ≿m=≿ and ≿0 is symmetric. We then argue that for every pair of adjacent
+profiles ≿k, ≿k+1 in this sequence if a group strategy-proof, efficient and gender-neutral mechanism
+agrees with a royalty mechanism at ≿k it must also do so at ≿k+1. The difficulty is in finding exactly
+the right sequence.
+4
+The axioms
+No axiom can be dropped from the characterizations in Theorems 1 and 2.
+Considering first the
+characterization of one-sided mechanisms in Theorem 2 note that any constant mechanism is group
+strategy-proof but not efficient.
+Most efficient mechanisms are not group strategy-proof.
+For an
+example define a one-to-one function R from the set of matchings to the natural numbers. Then map
+any profile ≿ to the matching µ that minimizes R(µ) over the set of all efficient matchings.
+To see that this mechanism is not group strategy-proof fix a setting with just three agents
+{1, 2, 3}. Say R assigns values 1,2,3 and 4 to the matchings {(1, 2), (3)}, {(1), (2, 3)}, {(1, 3), (2)} and
+{(1), (2), (3)} respectively.
+Fix the profile ≿∗ where each agent holds the same ranking ≿∗
+i which
+ranks 3 above 1 above 2. Since each matching is efficient at ≿∗, the mechanism chooses {(1, 2), (3)}.
+10
+
+Now modify agent 2’s preference to be ≿′
+2: 3, 2, 1. Since now agents 1 and 2 would rather be single
+than be together the matching {(1, 2), (3)} is not efficient. Since {(1), (2, 3)} is efficient and because
+R({(1), (2, 3)}) = 2 the mechanism maps (≿′
+2, ≿∗
+−2) to {(1), (2, 3)}. However, agent 2 strictly prefers
+to be matched with agent 3 than to agent 1 so the mechanism is not strategy-proof.
+Things get more interesting with Theorem 1. The constant mechanism which maps each profile
+of preferences to the matching σ with respect to which f is gender-neutral, is not only group strategy-
+proof but also gender-neutral. It is clearly not efficient. For a gender-neutral and efficient mechanism
+f that is not group strategy-proof we fix, as above, a one-to-one mapping R from the set of gender-
+neutral matchings to the natural numbers. To define f(≿) first check whether there there exist any
+symmetric efficient matchings at ≿. If so choose the matching that minimizes R(µ) over the set of
+all efficient symmetric matchings. If not, use some fixed neutral royalty mechanism to calculate f(≿).
+To see that the mechanism is gender-neutral fix a profile ≿. Note that the set of efficient symmetric
+matchings at ≿ and σ(≿) coincide and that σ ∗ µ holds for any symmetric matching.
+Therefore
+σ ∗ f(≿) = f(≿) = f(σ(≿)) both equal the minimal R(µ) over the set of all matchings µ that are
+efficient at σ (and therefore also at σ(≿)).
+Since neutral royalty mechanisms are gender neutral,
+f(≿) = f(σ(≿)) also holds for the alternative case where no gender neutral matchings are efficient at
+≿. The fact that f is not group strategy-proof follows from the fact that the analog mechanisms for
+roommate problems is not group strategy-proof.7
+If we drop gender-neutrality and modify efficiency to only include the preferences of one side of
+the market, we obtain the set of efficient and group strategy-proof one-sided matching mechanisms as
+characterized by (Pycia and ¨Unver 2017) and (Bade 2020). When we replace gender-neutrality with
+weak gender-neutrality, our proof continues to hold until the point where we show that a royal couple
+must be matched or unmatched-by-default. However if this royal couple chooses partners m and w
+that are not symmetric, the continuation submechanism need not be symmetric. This is illustrated by
+example 2.
+4.1
+The incompatibility of gender-neutrality and stability
+The tension between stability and incentive compatibility was first described in Roth (1982). Here
+we show that stability also clashes with gender-neutrality. Stability therefore forces the designer to
+introduce asymmetry where none exists.
+Theorem 3. No stable mechanism f for at least three couples is gender-neutral and stable.
+Proof. Fix a matching problem with three couples {(m1, w1), (m2, w2), (m3, w3)}. Suppose f was an
+stable mechanism that is gender-neutral with respect to σ. Fix the following preferences
+≿m1
+w3
+w2
+w1
+≿m2
+w1
+w3
+w2
+≿m3
+w2
+w1
+w3
+≿w1
+m3
+m2
+m1
+≿w2
+m1
+m3
+m2
+≿w3
+m2
+m1
+m3
+7If f was group strategy-proof it would have to be group strategy-proof on the subdomain of symmetric preferences.
+Now consider a problem with three couples. If the couples preferences and the function R are given as in the discussion
+of roommate problems, we see that couple two has an incentive to lie about their preferences.
+11
+
+Notice that there are two stable matchings
+M − optimal = {(m1, w3), (m2, w1), (m3, w2)}
+W − optimal = {(m1, w2), (m2, w3), (m3, w1)}
+To see that there is no other stable matching, notice that in any other matching at least one pair of
+agents with the same index (mi, wi) are matched. In this case, there is an agent mj and an agent wj
+who top-rank wi and mi respectively. These form blocking pairs.
+Since σ(≿) =≿ and since f is gender-neutral with respect to σ, f(≿) must be symmetric.
+However neither stable matching is symmetric: (m1, w3) are matched in the M-optimal stable match,
+but (σ(m1), σ(w3)) = (w1, m3) are not. Likewise (m1, w2) are matched in the W-optimal stable match,
+but (σ(m1), σ(w2)) = (w1, m2) are not.
+5
+Two-sided Mechanisms with Randomization
+A typical approach to achieving fairness in social choice is to employ randomization. A deterministic
+mechanism is symmetrized by randomizing the role agents play in the mechanism. For example, one
+could symmetrize serial dictatorship, where agents are called in a fixed order to choose their preferred
+matches, by selecting a picking order uniformly at random. This gives a mechanism analogous to the
+mechanism known as random serial dictatorship (RSD) in house allocation problems. The symmetrized
+mechanism retains some of the features of the original mechanism. If the original mechanism is strategy-
+proof, so is the symmetrized version8. On the other hand, there is no guarantee that the symmetrized
+mechanism will be efficient, even if the original mechanism is. For example, consider the following
+preferences:
+≿m1
+w3
+w2
+w1
+≿m2
+w1
+w3
+w2
+≿m3
+w2
+w1
+w3
+≿w1
+m3
+m2
+m1
+≿w2
+m1
+m3
+m2
+≿w3
+m2
+m1
+m3
+If we run serial dictatorship, selecting the picking order uniformly at random we get the following
+random allocation.
+w1
+w2
+w3
+m1
+1/12
+11/24
+11/24
+m2
+11/24
+1/12
+11/24
+m3
+11/24
+11/24
+1/12
+However, the following random allocation gives a first-order stochastic improvement for all agents.
+w1
+w2
+w3
+m1
+0
+1/2
+1/2
+m2
+1/2
+0
+1/2
+m3
+1/2
+1/2
+0
+8In the sense that truthfully reporting gives a lottery which first-order stochastically dominates any other lottery that
+could be achieved by a misreport.
+12
+
+This is a common issue in randomized mechanisms in a variety of environments (Echenique, Root, and
+Sandomirskiy 2022). In house allocation, Bogomolnaia and Moulin (2001) showed that it is inevitable:
+there is no efficient, strategy-proof and symmetric mechanism. There is a sense in which the mechanism
+is efficient: the outcome can be decomposed as a lottery over efficient deterministic outcomes. This is
+known as ex-post efficiency. Bade (2020) showed that symmetrizing any one of the many efficient and
+group strategy-proof house allocation mechanism gives nothing other than random serial dictatorship.
+This, along with the impossibility result of Bogomolnaia and Moulin (2001) has lead to interest in the
+question of whether RSD is the unique ex-post efficient, strategy-proof and symmetric mechanism.
+Symmetrizing our royalty mechanisms gives a negative answer to the same question in two-sided
+matching. For example when symmetrizing a royalty mechanism where the royals are matched-by-
+default in each round we get the following allocation given the preferences above.
+w1
+w2
+w3
+m1
+1/9
+4/9
+4/9
+m2
+4/9
+1/9
+4/9
+m3
+4/9
+4/9
+1/9
+Notice that these three allocation matrices are Pareto ranked with the second matrix dominating
+RSD which in turn dominates the uniform match-by-default outcome. The ranking of RSD and uniform
+match-by-default is an artifact of this example; there are examples where the Pareto ranking is flipped
+and in general the two random allocations cannot be ranked.
+References
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+review, 93(3), 729–747.
+Abdulkadiroglu, A., and T. S¨onmez (2013): “Matching markets: Theory and practice,” Advances in Economics and
+Econometrics, 1, 3–47.
+Alva, S. (2017): “When is manipulation all about the ones and twos,” Unpublished.
+Bade, S. (2020): “Random serial dictatorship: the one and only,” Mathematics of Operations Research, 45(1), 353–368.
+Bartholdi, L., W. Hann-Caruthers, M. Josyula, O. Tamuz, and L. Yariv (2021):
+“Equitable voting rules,”
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+Bogomolnaia, A., and H. Moulin (2001): “A new solution to the random assignment problem,” Journal of Economic
+theory, 100(2), 295–328.
+Echenique, F., J. Root, and F. Sandomirskiy (2022): “Efficiency in Random Resource Allocation and Social Choice,”
+arXiv preprint arXiv:2203.06353.
+Gale, D., and L. S. Shapley (1962): “College admissions and the stability of marriage,” The American Mathematical
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+Pycia, M., and M. U. ¨Unver (2017): “Incentive compatible allocation and exchange of discrete resources,” Theoretical
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+Root, J., and D. S. Ahn (2020): “Incentives and Efficiency in Constrained Allocation Mechanisms,” arXiv preprint
+arXiv:2006.06776.
+13
+
+Roth, A. E. (1982): “The economics of matching: Stability and incentives,” Mathematics of operations research, 7(4),
+617–628.
+(2008): “Deferred acceptance algorithms: History, theory, practice, and open questions,” international Journal
+of game Theory, 36(3), 537–569.
+Roth, A. E., and M. Sotomayor (1992): “Two-sided matching,” Handbook of game theory with economic applications,
+1, 485–541.
+Satterthwaite, M. A. (1975): “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems
+for voting procedures and social welfare functions,” Journal of economic theory, 10(2), 187–217.
+Shapley, L., and H. Scarf (1974): “On cores and indivisibility,” Journal of mathematical economics, 1(1), 23–37.
+14
+
+A
+The case of four agents in two-sided matching
+In this section we characterize all group strategy-proof, efficient and gender-neutral bilateral matching
+mechanisms for four agents {m1, m2, w1, w2}.
+In this case Σ2 contains only two matchings: ν :=
+{(m1, w1), (m2, w2)} and µ := {(m1, w2), (m2, w1)}. Given that there are only two matchings, each
+agent has exactly two preferences: the agent either prefers ν or µ. A preference profile ≿ can then
+be summarized as the set S or all agents i with ν(i) ≿i µ(i). In the present context it is easier to
+represent preference profiles as such sets S, meaning that mechanisms now map from the collection of
+all subsets of N to {µ, ν}.
+In line with this representation of preference profiles a mechanism partitions the set of all
+subsets of N into Λν and Λµ with the understanding that f(S) = ν iff S ∈ Λν. Our three desiderata
+the translate to the following requirements in the environment with just two couples.
+• The mechanism f is efficient iff N ∈ Λν and ∅ ∈ Λµ.
+• The mechanism f is group strategy-proof9 iff S ∈ Λν and S ⊊ S′ imply that S′ ∈ Λν.
+• The mechanism f is gender-neutral if S ∈ Λν implies that σ(S) ∈ Λν.
+Any group strategy-proof mechanism can therefore be represented by a set Λ
+ν ⊂ Λν of all
+minimal sets S ∈ Λν.
+Lemma 3. The following list of sets Λ
+ν is - up to renaming - an exhaustive list of all efficient group
+strategy-proof and gender-neutral mechanisms.
+a) {S | |S| = x} for x = 1, 2, 3, 4,
+b) {{m1, w1, w2}, {m1, w1, m2}, {m2, w2}},
+c) {{m1, w1, w2}, {m1, w1, m2}},
+d) Any subset of the sets {{m1, w1}}, {{m2, w2}}, {{m1, w2}, {m2, w1}}, {{m1, m2}, {w1, w2}}.
+e) {{m1}, {w1}, {m2, w2}}
+f) {{m1}, {w1}}
+The 4 mechanisms listed in a) are moreover the exhaustive subset of fully symmetric mechanisms.
+For notational convenience Lemma 3 includes a certain number of double counting. The mech-
+anism where ν is chosen if at least two agents prefer it for example falls in groups a) and d) above.
+We could have also economized the definition by noting that the mechanisms listed in b) and c) are
+equivalent to the mechanisms in e) and f) upon exchanging the matchings ν and µ. The mechanism in
+f) is the matched-by-default mechanism with m1 and w1 the royal couple, the mechanism defined by
+{m1, w1}, which appears in d), is the unmatched-by-default mechanism with m1 and w1 as the royals.
+9In fact, group strategy-proofness is equivalent to individual strategy-proofness with two outcomes, so this condition
+is necessary and sufficient for the mechanism to be strategy-proof.
+15
+
+∅
+{m2}
+{m1}
+{w2}
+{w1}
+{m1, w2}
+{m1, m2}
+{w1, m2}
+{w1, w2}
+{m1, w1}
+{m2, w2}
+{m1, w1, w2}
+{m2, w1, w2}
+{m1, m2, w1}
+{m1, m2, w2}
+{m1, m2, w1, w2}
+Figure 1: Four agents: Any gender-neutral, strategy-proof mechanism can be expressed as a set of
+nodes in the lattice above that is closed upwards and which is symmetric with respect to the the
+reflection over the vertical dotted line.
+Proof. For each of the mechanisms group strategy-proofness holds by definition since Λν is in each
+case defined as the set of all supersets in Λ
+ν. Each mechanism is Pareto optimal by the preceding
+argument together with the observation that ∅ /∈ Λ
+ν for any of the lists sets of subsets. Finally each
+mechanism is gender-neutral since each of the generating sets is gender-neutral.
+So we only have to show that the above list is exhaustive. To do so first consider Λ
+ν with
+{m1, w1, w2} ∈ Λ
+ν. By gender-neutrality σ({m1, w1, w2}) = {m1, w1, m2} ∈ Λ
+ν must hold. Since
+{m1, w1, w2}, {m1, w1, m2} ∈ Λ
+ν no subset of either {m1, w1, w2} or {m1, w1, m2} can be in Λ
+ν. So
+the latter cannot contain any of the singleton sets. {m2, w2} is the only two agent set that can be
+contained in Λ
+ν. We in sum found that the sets Λ
+ν which contain a three agent set are exactly the sets
+listed in a), b) and c). Exchanging ν and µ in the agents preferences we see that the sets Λ
+ν which
+contain a singleton are exactly the sets listed in a) e) and f).
+Finally to see that d) is an exhaustive list of mechanisms generated by two agent sets, note that
+d) lists the complete set of gender-neutral sets of two agent sets.
+B
+The case of two agents in one-sided matching
+Suppose that N = {1, 2}. With just two agents, there are exactly two outcomes in one-sided matching:
+the matching {(1, 2)} and the matching {(1), (2)}.
+Definition 9. We say that a mechanism f is a dictatorship if there is an agent k such that f(≿
+)(k) = top(≿k) for all ≿.
+Definition 10. We say that a mechanism f is a unanimity rule one of the two matchings is chosen
+unless both agree top-rank the other.
+16
+
+Lemma 4. If N = {1, 2}, then a one-sided mechanism f : Ω1 → Σ1 is group strategy-proof and
+efficient if and only if it is either a dictatorship or a unanimity rule.
+Proof. Since N = {1, 2} there exist exactly two matchings: one pairs the two agents, the other keeps
+both single. A mechanism is then efficient if and only if it chooses the matching preferred by both
+agents if they agree. The four such mechanisms map the two profiles where the agents disagree to
+the two different matchings. It is easy to check that these four mechanisms correspond to the two
+dictatorships and the two unanimity rules, and that they are strategy-proof.
+C
+The proof of Theorem 2
+C.1
+Sequential Dictatorships are efficient and group strategy-proof.
+Lemma 5. Sequential Dictatorships are efficient and group strategy-proof.
+Proof. Fix a sequential dictatorship SD : Ω1 → Σ1.
+Suppose SD was not efficient or not group
+strategy-proof. So suppose there either exists some profile ≿ and either a matching µ Pareto dominates
+SD(≿) or there is a group S ⊂ N and a deviation ≿′
+S such that all members of S weakly prefer
+SD(≿′
+S, ≿−S) to SD(≿) while some members of the group strictly hold this preference. Define k1 and
+k2 as the first rounds at which the SD(≿)-algorithm finds a match that differs from respectively µ
+and SD(≿′
+S, ≿−S). If only two agents remain unmatched at step k1 or k2 we obtain a contradiction
+via Lemma 4.
+So say that at least three agents remain unmatched at steps k1 and k2. Say i1 and i2 are
+the dictators at these steps, so that SD(≿)(i1) ̸= µ(i1) and SD(≿)(i2) ̸= SD(≿′
+S, ≿−S)(i2). Since
+the dictator ix gets matched with his top-ranked unmatched partner at Step kx, and since µ(i1) and
+SD(≿′
+S, ≿−S)(i2) respectively stay available at Steps k1 and k2, we get the contradictions SD(≿
+)(i1) ≻i µ(i1) and SD(≿)(i2) ≻i SD(≿′
+S, ≿−S)(i2).
+C.2
+Submechanisms
+For any two different agents j, k ∈ N define Ω−j and Ω−j,k as the restriction of Ω1 to all agents but
+agent j and j, k respectively.10 For any group strategy-proof and efficient mechanism f, define two
+mechanisms f −j and f −j,k on Ω−j and Ω−j,k respectively as follows. Set f −j(≿−j)(i) = f(≿)(i) for
+all ≿−j∈ Ω−j and i ∈ N \ {j} where ≿ is any preference profile such that (1) when restricted to
+N − {j} it gives ≿−j and (2) all agents i ̸= j rank j at the bottom while agent j ranks herself at the
+top. Similarly let f −j,k(≿−j,k)(i) = f(≿)(i) for all ≿−j,k∈ Ω−j,k and i ∈ N \ {j, k} where ≿ is any
+preference profile such that (1) when restricted to N − {j, k} it gives ≿−j,k and (2) all agents i ̸= j, k
+rank j and k at the bottom while agent j and k ranks each other at the top. To simplify notation we
+drop the superscripts −j and −j, k from the preference profiles ≿−j and ≿−j,k in the sequel; f −j(≿)
+then stands for the application of f −j to the restriction of ≿ to all agents but j.
+10That is, Ω−j is the set of preference profiles for all agents other than j and such that j is excluded from all other
+agents’ preferences. Ω−j,k is similar except now j and k are excluded.
+17
+
+Lemma 6. Say f is a group strategy-proof and efficient mechanism and j, k ∈ N are two differ-
+ent agents. Then the mechanisms f −j and f −j,k are well defined, group strategy-proof and efficient.
+Moreover if f(≿)(j) = j, then f(≿)(i) = f −j(≿)(i) for all i ∈ N \ {j} and if f(≿)(j) = k, then
+f(≿)(i) = f −j,k(≿)(i) for all i ∈ N \ {j, k}.
+Proof. The arguments are similar for f −j and f −j,k, so we will just prove this for f −j. To see that
+f is well-defined, notice that for any ≿−j and any profiles ≿ and ≿′ where the conditions described
+above hold (namely (1) when restricted to N − {j} the profiles ≿ and ≿′ give ≿−j and (2) all agents
+i ̸= j rank j at the bottom while agent j ranks herself at the top in both profiles) ≿ and ≿′ can only
+possibly differ in j’s ranking. However since f is efficient, in both-cases j will be matched to herself. By
+group strategy-proofness, all other agents matches must also remain the same. Now suppose, by way
+of contradiction, that f −j is not group strategy-proof. Then there is a preference profile ≿−j, some
+coalition of agents S, and an profile ≿′
+S for the agents in S so that f −j(≿′
+S, ≿−j
+−S)(i) ≿i f −j(≿−j)(i)
+for all i in S and for some k in S, f −j(≿′
+S, ≿−j
+−S)(k) ≿k f −j(≿−j)(k). Let ≿∗
+S be the profile for the
+agents in S such that all agents bottom-rank j and their ranking is the same as in ≿S otherwise.
+Likewise, let ≿−j∗ be the profile where all agents bottom-rank j and their preferences are the same
+as ≿−j otherwise. Let ≿j be an arbitrary preference for j where j top-ranks herself. By definition
+f −j(≿−j)(i) = f(≿j, ≿−j∗)(i) and f −j(≿′
+S, ≿−j
+−S)(i) = f(≿∗
+S, ≿j, ≿−j∗
+−S )(i) for all i ̸= j.
+However
+this leads to a violation of group strategy-proofness for f, a contradiction. Finally, efficiency follows
+immediately from the efficiency of f.
+C.3
+The Inductive Structure of the Proof
+Lemma 7. If each efficient and group strategy-proof mechanism f : Ω1 → Σ1 for three or more agents
+has a dictator, then any efficient group strategy-proof mechanism is a sequential dictatorship.
+Proof. Fix a group strategy-proof and efficient mechanism f : Ω1 → Σ1. If there are only two agents
+then f is by Lemma 4 a sequential dictatorship. If there are more than two agents then f has by
+assumption a dictator i, so that f(≿) = top(≿i) for any ≿ . If, upon matching i with top(≿i) only one
+agent remains unmatched, this agent must stay single. If not, then Lemma 6 implies that any choice
+of agent 1 is followed by a group strategy-proof and efficient submechanism f ′. By group strategy-
+proofness this submechanism f ′ only depends on agent i’s choice (and not on agent i’s rankings over
+the options he did not choose or on the preferences of the agent j ̸= i that i did choose). If two agents
+remain unmatched, f ′ must by Lemma 4 be either a dictatorship or a unanimity rule, and we are
+done. If more than two agents remain unmatched f ′ has by the assumption in the Lemma a dictator.
+Proceeding inductively we reach the case where at most two agents remain.
+Following Lemma 7 it suffices to show that each group strategy-proof and efficient mechanism
+with more than two agents has a dictator. The next three section show that each group strategy-proof
+and efficient mechanisms with respectively n = 3, n = 4 and n ≥ 5 agents has a dictator.
+C.4
+The case of three agents
+Throughout this section fix a group strategy-proof and efficient mechanism f : Ω1 → Σ1 for the agents
+N = {1, 2, 3}, so that Σ1 contains only four matchings. If any two agents are matched, the remaining
+18
+
+agent is clearly left single. The proof that f must be a sequential dictatorship revolves around the
+notion of “ownership”. Say agent i owns an agent j if agent i can always choose to be matched with
+agent j. Formally top(≿i) = j implies f(≿i, ≿−i)(j) = i for all ≿−i. After establishing the preliminary
+Lemma 8 on ownership, Lemma 9 shows that each agent in i ∈ {1, 2, 3} must be owned. Lemma 10
+then rules out all ownership structures except then one where one agent owns all agents. Throughout
+this section, it will be useful to have some additional notation. Let ≿1,2
+i
+: 1, 2, 3, ≿1,3
+i : 1, 3, 2 and ≿1
+i
+as an arbitrary preferences with top(≿i) = 1, so ≿1
+i equals either ≿1,2
+i
+or ≿1,3
+i
+. Likewise let ≿1,2, ≿1,3
+and ≿1 denote preference profiles where all agents’ preferences are ≿1,2
+i , ≿1,3
+i
+and ≿1
+i respectively.
+The first lemma describes a sufficient condition to establish ownership.
+Lemma 8. Agent j∗ owns agent 1 if and only if f
+�
+≿1,2 �
+(j∗) = f
+�
+≿1,3 �
+(j∗) = 1.
+Proof. If agent j∗ owns agent 1, then f(≿)(j∗) = 1 holds for any ≿ with top(≿j∗) = 1, in particular
+≿1,2 and ≿1,3.
+So suppose that we have f
+�
+≿1,2 �
+(j∗) = f
+�
+≿1,3 �
+(j∗) = 1 for some agent j∗. For j∗ to
+own 1 it is sufficient to show that f(≿1)(j∗) = 1 holds for all ≿1. To see that latter fix any ≿−j∗.
+Define ≿′
+i for the two agents i ̸= j∗ such that top(≿′
+i) = 1 and 2 ≿′
+i 3 ⇔ 2 ≿i 3. By the assumption
+f(≿1
+j∗, ≿′
+−j∗)(j∗) = 1. Group strategy-proofness then yields f(≿1
+j∗, ≿′
+−j∗) = f(≿1
+j∗, ≿−j∗), so that
+f(≿1
+j∗, ≿−j∗)(j∗) = 1.
+Case 1: j∗ = 1.
+Case 1.1: f
+�
+≿1,2 �
+(2) = 2. Then the group strategy-proofness of f implies f
+�
+≿1,2
+2 , ≿1
+−2
+�
+(1) =
+1. The strategy-proofness of f implies that f
+�
+≿1,3
+2 , ≿1
+−2
+�
+(2) ∈ {2, 3}, and therefore, by either group
+strategy-proofness or feasibility respectively, f
+�
+≿1,3
+2 , ≿1
+−2
+�
+(1) = 1. We in sum get that f(≿1)(j∗) = 1
+for any ≿1, which establishes the claim.
+Case 1.2: f
+�
+≿1,3 �
+(3) = 3. Applying the arguments from Case 1.1. mutatis mutandis we get
+f(≿1)(j∗) = 1 for any ≿1.
+Case 1.3: f
+�
+≿1,2 �
+(3) = f
+�
+≿1,3 �
+(3) = 2. So f
+�
+≿1,2 �
+= f
+�
+≿1,3 �
+and by group strategy-
+proofness f must also match 3 with 2 for any profile where agent 1 ranks being single at the top
+and where at least one agent in {2, 3} prefers being matched with the other to being single.
+For
+the last remaining case where agent 1 top ranks being single while the other two prefer being single
+to being matched with each other start with the observation that f(≿1
+1, ≿1,2
+−1) and f(≿1
+1, ≿1,3
+−1) both
+match agents 2 and 3.
+By strategy-proofness we then get f(≿1
+1, ≿1,2
+2 ≿1,3
+3 )(2) ∈ {2, 3} as well as
+f(≿1
+1, ≿1,2
+2 ≿1,3
+3 )(3) ∈ {2, 3}. The efficiency of f then implies that agents 2 and 3 both stay single in
+f(≿1
+1, ≿1,2
+2 ≿1,3
+3 ), so that once again f(≿1
+1, ≿1,2
+2 ≿1,3
+3 )(1) = 1 and in sum f(≿1)(1) = 1.
+Case 2: j∗ ̸= 1.
+W.o.l.g say j∗ = 2 so that f
+�
+≿1,2 �
+(2) = f
+�
+≿1,3 �
+(2) = 1.
+Now say
+f(≿∗)(2) ̸= 1 did hold for some ≿∗ with top(≿∗
+1)(i) = 1 for i = 1, 2, 3. The assumption f
+�
+≿1,2
+�
+(2) = f
+�
+≿1,3 �
+(2) = 1 together with group strategy-proofness implies that f(≿1,2
+1 , ≿1
+−1)(1) = 2 (A)
+as well as f(≿1,3
+3 , ≿1
+−3)(1) = 2 (B) for any ≿1. So for f(≿∗)(1) ̸= 2 to hold we must have ≿∗
+1=≿1,3
+1
+and ≿∗
+3=≿1,2
+3 . Case 1: f(≿∗)(1) = f(≿1,3
+1 , ≿∗
+2, ≿1,2
+3 )(1) = 3. In that case we get by group strategy-
+proofness that f(≿1,3
+1 , ≿∗
+2, ≿1,2
+3 ) = f(≿1,3
+1 , ≿∗
+2, ≿1,3
+3 ) which leads to a contradiction to (B) established
+above. Case 1: f(≿∗)(1) = f(≿1,3
+1 , ≿∗
+2, ≿1,2
+3 )(1) = 1. In that case we get by group strategy-proofness
+that f(≿1,3
+1 , ≿∗
+2, ≿1,2
+3 ) = f(≿1,2
+1 , ≿∗
+2, ≿1,2
+3 ) which leads to a contradiction to (A) established above.
+19
+
+Lemma 9. According to f, each agent i ∈ {1, 2, 3} is owned.
+Proof. By Lemma 8 (and the interchangeability of all agents) it suffices to show f(≿1,2)(1) = f(≿1,3
+)(1). So suppose we had f(≿1,2)(1) ̸= f(≿1,3)(1).
+Case 1: Agent 1 is not alone at either ≿1,3 or ≿1,2, so f(≿1,2)(1) ̸= 1 ̸= f(≿1,3)(1).
+Suppose we had f(≿1,2)(3) = f(≿1,3)(2) = 1. The group strategy-proofness of f implies f(≿1,2
+) = f(≿1,3
+3 , ≿1,2
+−3) as well as f(≿1,3) = f(≿1,2
+2 , ≿1,3
+−2) = f(≿1,3
+3 , ≿1,2
+−3). We in sum get the contradiction
+f(≿1,2) = f(≿1,3) to the assumption that f(≿1,2) and f(≿1,3) respectively match agent 1 with agent
+2 and 3. So we must have f(≿1,2)(2) = f(1, 3)(3) = 1.
+The proof derives a contradiction by showing that f must equal two different matchings at the
+profile ≿ given by
+≿1: 1, 3, 2
+≿2: 3, 1, 2
+≿3: 1, 2, 3
+Starting at f
+�
+≿1,2 �
+(1) = 2 change agent 2’s preference to ≿2: 3, 1, 2. By strategy-proofness
+f
+�
+≿2, ≿1,2
+−2
+�
+(2) ∈ {1, 3}. Since the matching {(1), (2, 3)} Pareto dominates the matching {(1, 2), 3}
+at (≿2, ≿1,2
+−2), f
+�
+≿2, ≿1,2
+−2
+�
+must match agents 2 and 3. The group srategy-proofness then implies
+that f
+�
+≿2, ≿1,2
+−2
+�
+= f(≿). On the other hand, the group strategy-proofness of f, together with the
+observation that ≿1,3
+1 =≿1 implies that f
+�
+≿1,3 �
+= f
+�
+≿
+�
+. A contradiction arises since f
+�
+≿1,3 �
+(1) =
+3 while we have shown above that f
+�
+≿
+�
+(1) = 1.
+Case 2: Agent 1 is alone at ≿1,2 or ≿1,3, not both. W.o.l.g f(≿1,2)(1) = 1.
+Case 2.1. f
+�
+≿1,2 �
+(2) = 2. Group strategy-proofness then implies f
+�
+≿1,2
+2 , ≿1,3
+−2
+�
+(1) = 1.
+Strategy-proofness yields that f
+�
+≿1,3 �
+(2) equals 2 or 3. We then get f
+�
+≿1,3 �
+(1) = 1 (in the first
+case by group strategy-proofness and in the second by feasibility) a contradiction to the assumption
+that agent 1 is not alone at ≿1,3.
+Case 2.2 f
+�
+≿1,2 �
+(2) = 3. Group strategy-proofness then implies f
+�
+≿1,2 �
+= f
+�
+≿1,2
+3 , ≿1,3
+−3
+�
+.
+Since f
+�
+≿1,3 �
+̸= f
+�
+≿1,2 �
+= f
+�
+≿1,2
+3 , ≿1,3
+−3
+�
+and since ≿1,2
+3
+and ≿1,3
+3
+differ only in their ranking of
+agents 2 and 3 in second and third place, f
+�
+≿1,3 �
+(3) = 3. So for f
+�
+≿1,3 �
+(1) ̸= 1 to hold we must
+have f
+�
+≿1,3 �
+(1) = 2.
+Just as above the proof derives a contradiction by showing that f maps the following ≿ to two
+different matchings,
+≿1: 3, 1, 2
+≿2: 1, 2, 3
+≿3: 1, 3, 2.
+20
+
+Strategy-proofness, ≿1,2
+2 =≿2, and f
+�
+≿1,2 �
+(1) = 1 imply f
+�
+≿1,2
+3 , ≿−3
+�
+(1) ∈ {1, 3}.
+If
+f
+�
+≿1,2
+3 , ≿−3
+�
+(1) = 1, then the group strategy-proofness of f implies f
+�
+≿1,2 �
+= f
+�
+≿1,2
+3 , ≿−3
+�
+. A
+contradiction arises, since f
+�
+≿1,2 �
+is at
+�
+≿1,2
+3 , ≿−3
+�
+dominated by µ with µ(1) = 3. So f
+�
+≿1,2
+3 , ≿−3
+�
+(1) = 3 and f
+�
+≿1,2
+3 , ≿−3
+�
+= µ. Since f is strateyproof and since µ(3) = 1 = top(≿3) = top(≿1,2
+3 )
+we get f
+�
+≿
+�
+= µ. Starting with f
+�
+≿1,3 �
+and noting that ≿1,3
+3 =≿3 the group strategy-proofness of
+f yields the contradiction f
+�
+≿
+�
+= f
+�
+≿1,3 �
+̸= µ.
+Lemma 10. The mechanism f must have a dictator.
+Proof. By Lemma 9 each agent is owned. Suppose we had an “ownership chain” in the sense that
+agent i owns j who in turn owns k with {1, 2, 3} = {i, j, k}. For a profile ≿ with top(≿i) = j and
+top(≿j) = k we would then obtain the contradiction (i, j), (j, k) ∈ f(≿). Given that there can be no
+such ownership chains we have to consider only 3 ownership structures (up to renaming): 1. Each
+agent owns herself. 2. Agent 1 owns agent 2 and agent 3 owns herself. 3. One agent owns all three
+agents. To show that the third case must hold we rule out the first two.
+Case 1: Each agent owns herself. The classic example of a roommate problems without a
+stable matching serves to obtain a contradiction. Define ≿ as follows:
+≿1: 2, 3, 1
+≿2: 3, 1, 2
+≿3: 1, 2, 3
+Since f is efficient at least two agents must get matched.
+Assume that f(≿)(1) = 2 and
+consider the deviation to ≿′
+2: 3, 2, 1 and ≿′
+3: 2, 3, 1 for agents 2 and 3. Since each agent owns herself,
+f(≿1, ≿′
+−1)(2) ≿2 2 and f(≿1, ≿′
+−1)(3) ≿3 3. So f(≿1, ≿′
+−1) either keeps all agents single or pairs up
+agents 2 and 3. Since the latter Pareto dominates the former, f(≿1, ≿′
+−1)(2) = 3 must hold. Since
+f(≿1, ≿′
+−1)(2) = 3 ≻2 1 = f(≿)(2) and f(≿1, ≿′
+−1)(3) = 2 ≻3 3 = f(≿)(3) a contradiction to the
+group strategy-proofness of f results. Mutatis mutandis the same arguments rule out the remaining
+two matchings. So it cannot be that each agent owns herself.
+Case 2: Agent 1 owns agent 2 and agent 3 owns herself. Transform the profile ≿ in two
+steps to (≿2, ≿′
+−2) where
+≿1: 3, 2, 1
+≿′
+1: 3, 1, 2
+≿2: 3, 2, 1
+≿3: 2, 1, 3
+≿′
+3: 2, 3, 1.
+Since agent 1 owns agent 2, f(≿)(1) equals 2 or 3. The latter must hold since the matching {(1, 2), ()}
+is (at ≿) Pareto dominated by {(1, 3)(2)}. By group strategy-proofness f(≿) = f(≿′
+1, ≿−1). Now
+swap agent 3’s preference to ≿′
+3: By strategy-proofness and since agent 3 owns herself 2 ≻3 f(≿2, ≿′
+−2
+)(3) ≿′
+3 3, so that f(≿2, ≿′
+−2)(3) = 3. Conditioning on agent 3 staying single, agents 1 and 2 must, by
+21
+
+efficiency, also stay single at f(≿2, ≿′
+−2). A contradiction arises since matching agents 2 and 3 Pareto
+dominates f(≿2, ≿′
+−2) at (≿2, ≿′
+−2).
+C.5
+The case of four agents
+In the present subsection assume that f : Ω1 → Σ1 is a group strategy-proof and efficient mechanism
+for four agents. For each profile of preferences ≿ define ≿ so that each agent i ranks being single at the
+bottom keeping all other rankings identical to ≿. Say Ω
+1 is the subdomain of such profiles where each
+agent ranks being single at the bottom. The set of matchings where no agent is single is Σ
+1. There
+are exactly four such matchings.
+Lemma 11. The restriction f of f to Ω
+1 is a dictatorship.
+Proof. Since f is group strategy-proof and efficient, its restriction to Ω
+1 is so too. Since f is efficient
+and since each agent ranks being single at the bottom no agent stays single according to f(≿) for any
+≿∈ Ω
+1 and we can represent f as a mechanism mapping Ω
+1 to Σ
+1. Since Σ
+1 contains three matchings,
+each of which is fully determined by the match of a single agent, we are facing a classic social choice
+problem with three options where four agents may hold any preferences over these three options. By
+the Gibbard Satterthwaite theorem f : Ω
+1 → Σ
+1 must be a dictatorship.
+For the reminder of the present section say agent 1 is the dictator in f : Ω
+1 → Σ
+1.
+Lemma 12. For each i ∈ {1, 2, 3, 4}, f −i is a sequential dictatorship.
+Proof. By Lemma 6 f −i is efficient and group strategy-proof for three agents. By the preceding section
+f −i is a sequential dictatorship.
+For the remainder of the section assume that agent 2 is the dictator in f −1.
+Lemma 13. An agent i∗ ∈ {1, 2} is the dictator in both f −3 and f −4.
+Proof. First fix a profile ≿ such that
+≿1:
+1, ·, ·, ·
+≿2:
+2, ·, ·, ·
+≿3:
+2, 3, 4, 1
+≿4:
+2, 4, 3, 1
+The efficiency of f implies f(≿)(1) = 1, and Lemma 6 then implies f −1(≿) ⊂ f(≿). Since
+top(≿2) = 2 and since agent 2 is the dictator in f −1, f(≿)(2) = f −1(≿)(2) = 2. By efficiency agents 3
+and 4 also remain single. By Lemma 6 f(≿) must then be consistent with f −3(≿) and f −4(≿). Since
+f(≿)(4) = 4 and since 2 ≿4 4, agent 4 cannot be the dictator in f −3. Mutatis mutandis we see that
+agent 3 cannot be the dictator in f −4.
+22
+
+To see that f −3 and f −4 must have the same dictator fix ≿ such that
+≿1:
+1, 2, ·, ·,
+≿2:
+1, 2, ·, ·,
+≿3:
+3, ·, ·, ·,
+≿4:
+4, ·, ·, ·.
+Since f is efficient, f(≿)(3) = 3 and f(≿)(4) = 4. By Lemma 6, f(≿)(1) = f −3(≿)(1) = f −4(≿)(1).
+Since top(≿1) = top(≿2) = 1, we then get f −3(≿)(1) = f −4(≿)(1) = i∗ so f −3 and f −4 have the same
+dictator.
+Lemma 14. Agent 1 is the dictator in f −j for j = 2, 3, 4.
+Proof. Fix ≿ and the deviations ≿′
+≿1: 3, 1, 2, 4
+≿′
+1: 3, 4, 1, 2
+≿2: 3, 2, 1, 4
+≿3: 2, 1, 3, 4
+≿4: 4, ·, ·, ·
+≿′
+4: 1, 4, ·, ·
+By efficiency f(≿)(4) = 4. By Lemma 6 f −4(≿) ⊂ f(≿). Suppose 1 is not the dictator in f −4. So 2 or 3
+must be the dictator in f −4 and we get f(≿) = {{1}, {2, 3}, {4}}. Changing agent 1 and 4’s preferences
+to ≿′
+1,4, group strategy-proofness implies f(≿′
+1,4, ≿2,3)(1) = 4. By efficiency f(≿′
+1,4, ≿2,3)(2) = 3. Now
+define ≿′′ to be identical to (≿′
+1,4, ≿2,3) except that each agent ranks being single at the bottom. By
+group strategy-proofness f(≿′
+1,4, ≿2,3) = f(≿′′). A contradiction arises since ≿′′∈ Σ
+1 and agent 1 is
+by assumption the dictator for problems where all agents rank being single at the bottom. So agent 1
+must be the dictator in f −4. By Lemma 13 agent 1 is also the dictator in f −3.
+Now define ≿
+≿1:
+1, 3, ·, ·,
+≿2:
+2, ·, ·, ·
+≿3:
+1, 3, ·, ·,
+≿4:
+4, ·, ·, ·.
+By efficiency f(≿)(2) = 2 and f(≿)(4) = 4. By Lemma 6 f(≿) is consistent with f −2(≿) and
+f −4(≿). Since agent 1 is the dictator in f −4, f −4(≿)(1) = 1 = f(≿)(1) = f −2(≿)(1). The latter
+implies that agent 3 cannot be the dictator in f −2. Mutatis mutandis the dictator of f −2 cannot be
+agent 4 either. So the dictator of f −2 must be agent 1.
+Lemma 15. Agent 1 is the dictator in f.
+23
+
+Proof. We have to show that f(≿)(1) = top(≿1) for all ≿. Fix an arbitrary ≿. Assume without loss
+of generality that top(≿1) ∈ {1, 2}. If f(≿)(k) = k for k = 3 or k = 4, then f −k(≿) ⊂ f(≿). Since 1
+is the dictator in f −k for k = 3, 4 we get that f −k(≿)(1) = f(≿)(1) = top(≿1). So for the remainder
+assume that (3, 4) ∈ f(≿).
+Case 1: top(≿1) = 2. Suppose we have f(≿)(1) ̸= 2.
+Since (3, 4) ∈ f(≿), f(≿) must then equal {(1), (2), (3, 4)}. By group strategy-proofness we can
+w.l.o.g assume that
+≿1:
+2, 1, 3, ·,
+≿2:
+2, ·, ·, ·
+≿3:
+4, ·, ·, ·,
+≿4:
+3, 4·, ·, ·.
+By strategy-proofness f(≿′
+1, ≿−1)(1) ∈ {1, 3} for ≿′
+1: 2, 3, 1. If f(≿′
+1, ≿−1)(1) = 1, then group
+strategy-proofness implies f(≿′
+1, ≿−1) = f(≿), and f(≿′
+1, ≿−1) is consistent with f −2(≿′
+1, ≿−1). A
+contradiction arises since agent 1, would as the dictator in f −2 choose agent 3. So we must have
+f(≿′
+1, ≿−1)(1) = 3. Efficiency then implies f(≿′
+1, ≿−1)(4) = 4 so that f(≿′
+1, ≿−1) is consistent with
+f −4(≿′
+1, ≿−1). A contradiction arises since agent 1, would as the dictator in f −4 choose agent 2.
+Case 2: top(≿1) = 1. Since (3, 4) ∈ f(≿), f(≿) must then equal {(1, 2), (3, 4)}. . By group
+strategy-proofness assume w.l.o.g. that ≿ is given by
+≿1:
+1, 2, 3, ·,
+≿2:
+1, 2·, ·,
+≿3:
+4, ·, ·, ·,
+≿4:
+3, 4·, ·, .
+Now swap agents 2 and 3 in agent 1’s ranking, so that ≿′
+1: 1, 3, 2. By strategy-proofness, f(≿′
+1, ≿−1
+)(1) ∈ {2, 3}. Suppose f(≿′
+1, ≿−1) = 2, then we have f(≿′
+1, ≿−1) = f(≿) by group strategy-proofness.
+Now define ≿′′ to be identical to (≿′
+1, ≿−1) except that each agent drops being single to the bottom of
+their ranking. By group strategy-proofness f(≿′′) = f(≿′
+1, ≿−1). A contradiction arises since ≿′′∈ Σ
+1,
+but f(≿′′)(1) = 2 ̸= top(≿′′)(1) = 3. So we must have f(≿′
+1, ≿−1)(1) = 3. By efficiency agents 2 and 4
+must then stay single. A contradiction arises since f(≿′
+1, ≿−1)(1) must then equal f −2(≿′
+1, ≿−1)(1) = 1
+since agent 1 is the dictator in f −2.
+24
+
+C.6
+The case of n ≥ 4 agents
+For this section fix a group strategy-proof and efficient mechanism f for n + 1 agents and assume that
+all such mechanisms for n or fewer agents are sequential dictatorships.
+Lemma 16. If agent 1 is the dictator in f −2, then some agent j ∈ {1, 2} is the dictator in f −k for
+all k /∈ {1, 2}.
+Proof. Suppose agent 4 was the dictator in f −3. Fix ≿ such that each agent i ̸= 3 ranks agents 1
+and 3 respectively at the top and at the bottom, agents i ̸= 1, 3 rank being single in second place and
+top(≿3) = 3.
+≿1:
+1, ·, ·, · · · , 3
+≿2:
+1, 2, ·, · · · , 3
+≿3:
+3, ·, ·, · · ·
+≿4:
+1, 4, ·, · · · , 3
+·
+·
+By efficiency f(≿) is consistent with f −3(≿). Since agent 4 is the dictator in f −3, and since
+top(≿4) = 1, we have f(≿)(4) = 1. By efficiency f(≿)(i) = i for all i /∈ {1, 4}, in particular f(≿
+)(2) = 2, so that f −2(≿) is consistent with f(≿). Since agent 1 is the dictator in f −2, we obtain the
+contradiction that f(≿)(1) = f −2(≿)(1) = 1. Since agents 3 and 4 were chosen arbitrarily in N \{1, 2}
+the dictator in f −k is for each k /∈ {1, 2} is either agent 1 or agent 2.
+To see that f −k for each k /∈ {1, 2} must have the same dictator consider the profile ≿ where
+agents 1 and 2 both rank agents 1 and 2 in first and second place and where any agent i /∈ {1, 2} top
+ranks being single, so
+≿1:
+1, 2, ·, ·,
+≿2:
+1, 2, ·, ·, ·,
+≿3:
+3, ·, ·, ·,
+≿4:
+4, ·, ·, ·,
+·
+·
+By efficiency, f(≿)(i) = i for each i /∈ {1, 2}.
+So f(≿) is consistent with f −i(≿) for each
+i /∈ {1, 2}. By the preceding paragraph some agent j ∈ {1, 2} is the dictator in j−3, so f(≿)(j) =
+f −3(≿)(j) = 1. Now consider any k /∈ {1, 2, 3}. Since f(≿) is consistent with f −k, agent j must also
+be the dictator in f −k
+25
+
+By Lemma 16 one agent j is the dictator in all submechanisms f −k with j ̸= k.
+For the
+remainder assume w.l.o.g that agent 1 is this agent.
+Lemma 17. Agent 1 is the dictator in f −j,k for any two agents j, k such that 1 /∈ {j, k}.
+Proof. Fix ≿ such that two agents other than agent 1, say agents 2 and 3 top rank each other while a
+third agent, say agent 4, top ranks himself. All remaining agents top rank agent 1 and rank agents 2,
+3 and 4 at the bottom.
+≿1:
+1, ·, . . . , 2, 3, 4
+≿2:
+3, ·, . . . ,
+≿3:
+2, ·, . . . ,
+≿4:
+4, ·, . . . ,
+≿5:
+1, ·, . . . , 2, 3, 4
+·
+·
+·
+By efficiency f(≿) matches agent 4 with himself and agents 2 and 3 with each other. So f(≿)
+is consistent with f −4(≿) and f −2,3(≿). Since agent 1 is the dictator in f −4(≿), f(≿)(1) = 1. Since
+f −2,3(≿) is consistent with f(≿), f −2,3(≿)(1) = 1. Since all agents but agent 4 top rank agent 1 in ≿,
+the dictator in f −2,3 is either agent 1 or agent 4. Repeating the same arguments with swapping agent
+4 and 5, we see that either agent 1 or agent 5 is the dictator in f −2,3. In sum, agent 1 must be the
+dictator in f −2,3. Since agents 2 and 3 were chosen arbitrarily, agent 1 is the dictator in any f −j,k for
+1 /∈ {j, k}.
+Lemma 18. Agent 1 is the dictator in f.
+Proof. Fix a profile ≿. Suppose f(≿)(1) ̸= top(≿1). Since n ≥ 5, we can fix an agent j such that
+{j, f(≿)(j)} ∩ {1, f(≿)(1), top(≿1)} = ∅. If j = f(≿)(j), then f(≿) is consistent with f −j(≿). Since
+agent 1 is by Lemma 16 the dictator in f −j and since top(≿1) ̸= j we then obtain f(≿)(1) = f −j(≿
+)(1) = top(≿1). If j ̸= f(≿)(j), define f(≿)(j) = k. In that case f(≿) is consistent with f −j,k(≿).
+Since agent 1 is by Lemma 17 the dictator in f −j,k and since top(≿1) /∈ {j, k} we can conclude as
+above that f(≿)(1) = f j,k(≿)(1) = top(≿1).
+D
+Proof of Theorem 1
+Fix a mechanism f : Ω2 → Σ2. Restricted to the domain Ω2
+symm the mechanism f is by Lemma 2 and
+Theorem 2 a serial dictatorship in which one couple can choose to either stay alone (marry each other)
+or pair up with a different couple (swap partners). Recall that f is assumed to be weakly gender-
+neutral with respect to σ that is defined such that σ(mi) = wi for all i. Without loss of generality say
+the couple (m1, w1) is the dictator in the embedded roommates mechanism. We call this the “royal
+26
+
+couple” and all other agents “commoners.” The rest of this proof is dedicated to showing that the
+royal couples’ powers also apply to symmetric preferences. At the same time we must show that any
+conflict between the royal couples interests (when one wants the royals to marry and the other does
+not) must be mediated using either the matched-by-default or the unmatched-by-default protocol.
+For Lemmas 19, 20, and 21 fix a symmetric profile ≿sym
+−m1,w1 for all commoners.
+Lemma 19. Say Ωl,k
+m1,w1 is the set of all preferences of m1 and w1 that respectively top rank wl and
+mk. Then either
+• f(≿m1,w1, ≿sym
+−m1,w1)(m1) = wl and f(≿m1,w1, ≿sym
+−m1,w1)(w1) = mk for all ≿m1,w1∈ Ωl,k
+m1,w1, or
+• f(≿m1,w1, ≿sym
+−m1,w1)(m1) ̸= wl and f(≿m1,w1, ≿sym
+−m1,w1)(w1) = mk for all ≿m1,w1∈ Ωl.k
+m1,w1 or
+• f(≿m1,w1, ≿sym
+−m1,w1)(m1) = wl and f(≿m1,w1, ≿sym
+−m1,w1)(w1) ̸= mk for all ≿m1,w1∈ Ωl,k
+m1,w1.
+Proof. Pick an arbitrary ≿m1,w1∈ Ωl,k
+m1,w1.
+Case 1: k = l Let ≿′
+m1= σ(≿w1) so that Theorem 2 implies (m1, wk), (w1, mk) ∈ f(≿′
+m1, ≿w1
+, ≿sym
+−m1,w1) Since f(≿′
+m1, ≿w1, ≿sym
+−m1,w1)(m1) = top(≿′
+m1) = top(≿m1), group strategy-proofness then
+implies that (m1, wk), (w1, mk) ∈ f(≿m1,w1, ≿sym
+−m1,w1).
+Case 2: k ̸= l
+Let ≿′
+m1: wl, wk and ≿′
+w1: mk, ml. By group strategy-proofness and Case 1, f(≿′
+m1,w1, ≿sym
+−m1,w1)
+either marries the royal couple with their most preferred partners (Case 2.1), or with ml and wl
+(Case 2.2) or with mk and wk (Case 2.3). In Case 2.1 the group strategy-proofness of f implies
+f(≿m1,w1, ≿sym
+−m1,w1) matches each royal with their top choice.
+For Case 2.2 Strategy-proofness implies f(≿w1, ≿′
+m1, ≿sym
+−m1,m1)(w1) ̸= mk. Now since m1 could
+swap to make an announcement symmetric to ≿w1, strategy-proofness and Case 1 imply that f(≿w1
+, ≿′
+m1, ≿sym
+−m1,m1)(m1) ≿′
+m1 wk. However, if f(≿w1, ≿′
+m1, ≿sym
+−m1,m1)(m1) = wk, we obtain a violation of
+group strategy-proofness since if w1 announces σ(≿m1) she gets the ≿w1-preferred mk. So f(≿w1, ≿′
+m1
+, ≿sym
+−m1,m1)(m1) = wl must hold. Finally, by strategy-proofness for m1 we have f(≿w1, ≿′
+m1, ≿sym
+−m1,m1
+)(m1) = f(≿m1,w1, ≿sym
+−m1,m1)(m1) = wl. Group strategy-proofness then implies f(≿w1, ≿′
+m1, ≿sym
+−m1,m1
+) = f(≿m1,w1, ≿sym
+−m1,m1), so that
+f(≿w1, ≿′
+m1, ≿sym
+−m1,m1 (m1) = f(≿m1,w1, ≿sym
+−m1,m1)(w1) ̸= mk
+as required.
+Mutatis mutandis the arguments of Case 2.2 apply to Case 2.3.
+Lemma 20. Say Ω∗
+m1,w1 is the set of preferences for the royals where exactly one royal top ranks the
+other. Then either a) or b) holds for all ≿m1,w1∈ Ω∗
+m1,w1
+a) (m1, w1) ∈ f(≿m1,w1, ≿sym
+−m1,m1) .
+b) f(≿m1,w1, ≿sym
+−m1,m1) matches the royal who top ranks a commoner with that commoner.
+Proof. Let ≿′
+m1: w1, wk and ≿′
+w1: mk, m1 for k ̸= 1. Since it is not possible to match both royals with
+their top-ranked partners, Lemma 19 yields that one of the royals gets their top partner.
+27
+
+Case 1: (m1, w1) ∈ f(≿′
+m1,w1, ≿sym
+−m1,m1). Then Lemma 19 implies that (m1, w1) ∈ f(≿′′
+m1,w1
+, ≿sym
+−m1,w1) for any ≿′′
+m1,w1 with top(≿′′
+m1) = w1 and top(≿′′
+w1) = mk (including ≿′
+w1 that rank m1
+last). Then group strategy-proofness implies that (m1, w1) ∈ f(≿m1,w1, ≿sym
+−m1,w1) for any ≿m1,w1 with
+top(≿m1) = w1. By gender-neutrality we also get that (m1, w1) ∈ f(≿m1,w1, ≿sym
+−m1,w1) if top(≿w1) =
+m1 and if ≿m1 is any preference.
+Case 2: f(≿′
+m1,w1, ≿sym
+−m1,m1)(w1) = mk. If (m1, w1) ∈ f(≿′
+m1,w1, ≿sym
+−m1,m1) held for some
+≿m1,w1 with top(≿m1) = w1 and top(≿w1) = ml for l ̸= 1 case 1 would imply the contradiction that
+(m1, w1) ∈ f(≿′
+m1,w1, ≿sym
+−m1,m1). Hence by Lemma 19 we have (w1, ml) ∈ f(≿(m1,w1), ≿sym
+−m1,w1)(w1) =
+ml. gender-neutrality then implies the result.
+Lemma 21. Say top(≿m1) = wl and top(≿w1) = mk. If k = 1 = l or if k ̸= 1 ̸= l, f(≿m1,w1, ≿sym
+−m1,w1)
+matches the royal couple with their most preferred partners.
+Proof. If k = 1 = l the claim follows from first part of the proof of Lemma 19. So let k ̸= 1 ̸= l. From
+Lemma 20 there are two cases to consider.
+Case 1: If exactly one royal top-ranks the other, the royals are matched Consider
+≿′
+m1: wl, w1 and ≿′
+w1: mk, m1.
+By Lemma 19 at least one royal must get their top partner.
+By
+strategy-proofness, neither royal can do worse than the other royal. Since m1 is only matched with
+w1 if w1 is matched with m1 we then get (w1, mk), (m1, wl) ∈ f(≿′
+m1,w1, ≿sym
+−m1,w1) and by group
+strategy-proofness f(≿m1,w1, ≿sym
+−m1,w1) = f(≿′
+m1,w1, ≿sym
+−m1,w1).
+Case 2: If exactly one royal top-ranks the other, the royal top-ranking a commoner
+gets their top match Consider the preferences
+≿′
+m1: wl, w1,
+≿′′
+m1: w1, wl
+≿′
+w1: mk, m1,
+≿′′
+w1: m1, mk
+derived from ≿m1 and ≿w1 keeping all else equal.
+Since we are in Case 2, we have f(≿′
+m1, ≿′′
+w1, ≿sym
+−m1,w1)(m1) = wl and f(≿′
+w1, ≿′′
+m1, ≿sym
+−m1,w1
+)(w1) = mk. By group strategy-proofness we then get that (m1, wl), (w1, mk) ∈ f(≿′
+m1, ≿′
+w1, ≿sym
+−m1,w1
+)(m1). Applying group strategy-proofness once again to drop each other in their royals’ rankings we
+get (m1, wl), (w1, mk) ∈ f(≿m1,w1, ≿sym
+−m1,w1) as required.
+The preceding three Lemmas referred to an arbitrarily fixed symmetric profile for all commoners
+≿sym
+−m1,w1.
+We showed that for any such fixed profile, the royals must be matched according to a
+matched-by-default or a unmatched-by-default protocol. The next Lemma shows that for a vast set of
+profiles for the commoners the royals get their top choices, if these top choices do not stand in conflict
+(so if they both want to marry commoners or want to marry each other.)
+Lemma 22. Fix ≿ such that a subset of all pairs of commoners have symmetric preferences, while all
+remaining commoners bottom-rank the royals, but do not necessarily have symmetric preferences. Say
+top(≿w1) = mk and top(≿m1) = wl. If l = k = 1 or l ̸= 1 ̸= k, then (m1, wl), (w1, mk) ∈ f(≿).
+28
+
+Proof. We use induction over the number m of pairs who do not have symmetric preferences.
+Start of the induction: m = 0 so that ≿−m1,w1 is symmetric. In this case the claim holds
+by Lemma 21.
+Induction step: Suppose the claim holds up to some m < n. Fix an arbitrary profile ≿−m1,w1
+such that m + 1 pairs have (potentially) asymmetric preferences, ranking the royals at the bottom,
+and such that the remaining n − m − 1 pairs have symmetric preferences. Without loss, suppose that
+(m2, w2) are a type that do not have symmetric preferences and consider ≿′
+w1: mk, m2.
+Case 1: k ̸= 1 ̸= l. Suppose by way of contradiction that f(≿)(w1) ̸= mk.
+Case 1.1: Couple (mk, wk) does not have symmetric preferences. For ≿′
+mk= σ(≿wk), bottom(≿wk
+) = m1 implies bottom(≿′
+mk) = w1. Since (≿′
+mk, ≿−mk) is covered by the hypothesis of the induc-
+tion f(≿′
+mk, ≿−mk)(w1) = mk. We then obtain a contradiction to strategy-proofness since w1 ̸= f(≿
+)(mk) ≻′
+mk f(≿′
+mk, ≿−mk)(mk) = w1 = bottom(≿′
+mk).
+Case 1.2 Couple (mk, wk) does have symmetric preferences. By strategy-proofness and the as-
+sumption that f(≿)(w1) ̸= mk, we have f(≿′
+w1, ≿−w1)(w1) ̸= mk. By Case 1.1, w1 would get matched
+with m2 if she were to top rank m2 (since we’ve assumed m2 and w2 announce asymmetric prefer-
+ences). So by strategy-proofness f(≿′
+w1, ≿−w1)(w1) = m2. Now consider ≿′
+m2= σ(≿w2). Since couple
+(m2, w2) does not have symmetric preferences bottom(≿m2) = w1. We then obtain a contradiction
+to strategy-proofness since (≿′
+w1,m2, ≿−w1,m2) is covered by the hypothesis of the induction so that
+f(≿′
+w1,m2, ≿−w1,m2)(w1) = mk. The latter implies that m2 can improve his match by switching his
+preference from ≿m2 to ≿′
+m2 to avoid being matched with w1 = bottom(≿m2).
+Cases 1.1 and 1.2 proves that (w1, mk) ∈ f(≿) holds in Case 1. By gender-neutrality (m1, wl) ∈
+f(≿) also holds.
+Case 2: k=l=1. Suppose (m1, w1) /∈ f(≿). As in the proof of Case 1.2 f(≿′
+w1, ≿−w1)(w1) =
+m2. Just as in that proof we obtain a contradiction to strategy-proofness since m2 would be better-off
+if he symmetrized his preference with w2.
+The next two Lemmas pertain to the case that the royals top rank a symmetric pair (mi, wi).
+In this case the royals are matched with their top choices - for any profile of commoners.
+Lemma 23. Fix ≿ so that the royals top rank each other. Then f(≿) marries the royals.
+Proof. Suppose not. Define ≿′
+−m1,w1 as a modification of ≿−m1,w1 where m1 and w1 are moved to
+the top of all agents preferences, but are otherwise unchanged. Sequentially swap each commoner
+i’s preference from ≿i to ≿′
+i. By group strategy-proofness, with each such swap the matching either
+stays constant or commoner i marries a royal. Therefore (m1, w1) /∈ f(≿m1,w1, ≿′
+−m1,w1). Suppose
+(m1, wi), (w1, mj) ∈ f(≿m1,w1, ≿′
+−m1,w1), with i ̸= 1 ̸= j and i = j permitted. Start with ≿′
+−m1,w1,
+to define a new profile ≿′′
+−m1,w1 by dropping the royals to the bottom of all commoners’ rankings all
+commoners other than in wi, mi, wj and mj rankings. Moreover let ≿′
+wi :
+= σ(≿mi) and if i ̸= j
+also ≿′
+mj : = σ(≿wj) so that the couples (mi, wi) as well as (mj, wj) (if different) have symmetric
+preferences where the agents who are matched with the royal couples by f(≿m1,w1, ≿′
+−m1,w1) top
+rank the royal couple. By group strategy-proofness none of these changes affect the match so that
+29
+
+f(≿m1,w1, ≿′
+−m1,w1) = f(≿m1,w1, ≿′′
+−m1,w1). However, a contradiction arises since ≿′′
+−m1,w1 is covered
+by Lemma 22, so (m1, w1) ∈ f(≿m1,w1, ≿′′
+−m1,w1).
+Lemma 24. If top(≿m1) = wl and top(≿w1) = ml, then (m1, wl), (w1, ml) ∈ f(≿).
+Proof. Fix the royal’s preferences such that they rank each other in second place. Suppose f(≿) did
+not match both royals with their top ranked partners.
+Observation 1: One royal must get their most preferred partner.
+By the preceding Lemma and group strategy-proofness the royals cannot both prefer each other
+to their matches f(≿)(m1) and f(≿)(w1). Now suppose we had f(≿)(w1) = m1. By group strategy-
+proofness f(≿) = f(≿m1,w1, ≿′
+−m1,w1) where the commoners’ preferences ≿′
+−m1,w1 are derived from
+≿−m1,w1 by dropping the royals to the bottom of each commoner’s preference keeping all else equal. A
+contradiction arises since ≿′
+−m1,w1 is covered by Lemma 22, so that (m1, w1) ∈ f(≿m1,w1, ≿′
+−m1,w1).
+In the only remaining case one royal gets their most preferred partner.
+Observation 2: We may w.l.o.g assume that bottom(≿ml) = w1 and bottom(≿wl) = m1.
+Suppose this did not hold. For concreteness assume that (m1, wl) ∈ f(≿), noting that the
+following arguments apply mutatis mutandis to the alternative case where (w1, ml) ∈ f(≿). Since
+(w1, ml) /∈ f(≿), we may by group strategy-proofness w.l.o.g assume that bottom(≿ml) = w1. Now
+define ≿′
+wl by dropping m1 to the bottom of wl’s ranking keeping all else equal to ≿wl. If (m1, wl) /∈
+f(≿′
+wl, ≿−wl), then Observation 1 implies that (w1, ml) ∈ f(≿′
+wl, ≿−wl).
+By Observations 1 and 2 and group strategy-proofness we may assume that (m1, wl)(w1, mj) ∈
+f(≿) for some j ̸= l, top(≿mj) = w1, bottom(≿mi) = w1 for all i ̸= j, 1 and bottom(≿wi) = m1 for all
+i ̸= 1
+Now lift m1 in wj’s ranking so that top(≿′
+wj) = m1 keeping all else equal to ≿wj. By group
+strategy-proofness we either have f(≿) = f(≿′
+wj, ≿−wj) or (m1, wj) ∈ f(≿′
+wj, ≿−wj). Now define
+≿′′
+mj,wj as two gender-neutral preferences ≿′′
+mj= σ(≿′′
+wj) so that an agent in {mj, wj} who is matched
+with a commoner keeps their preference.
+By group strategy-proofness f(≿) = f(≿′
+wj, ≿−wj) =
+f(≿′′
+wj,mj, ≿−mj,wj). A contradiction then arises since (≿′′
+wj,mj, ≿−mj,wj) is covered by Lemma 22.
+And we can conclude that (m1, wl), (w1, ml) ∈ f(≿).
+To conclude the proof, note that the group strategy-proofness of f together with (m1, wl), (w1, ml) ∈
+f(≿) imply that the royals keep their most preferred partners if they drop each other to any place in
+their rankings.
+Lemma 25. If top(≿m1) = wl, top(≿w1) = mk, and k ̸= 1 ̸= l, then (m1, wl), (w1, mk) ∈ f(≿).
+Proof. Define ≿′
+w1: mk, ml, ≿′
+m1: wl, wk. Keeping all else equal. By the Lemma 24 and group strategy-
+proofness the royals either get their top choices or they marry mk and wk or ml and wl. So suppose that
+(m1, wk), (w1, mk) ∈ f(≿′
+m1,w1, ≿−m1,w1). Define ≿′
+−m1,w1 for the commoners so that top(≿′
+mk) = w1,
+≿′
+wk= σ(≿′
+mk), bottom(≿′
+mi) = w1 and bottom(≿′
+wi) = m1 for all i ̸= k keeping all else equal. By
+group strategy-proofness we have f(≿′
+m1,w1, ≿−m1,w1) = f(≿′). A contradiction arises since ≿′
+−m1,w1 is
+30
+
+covered by Lemma 22 so that the royals must get their top choices in f(≿′). Mutatis mutandis the same
+arguments rule out the case that (m1, wl), (w1, ml) ∈ f(≿′
+m1,w1, ≿−m1,w1) and f(≿′
+m1,w1, ≿−m1,w1)
+must match the royals with their top choices. Dropping the less liked partners ml and wk in their
+rankings, group strategy-proofness yields f(≿′
+m1,w1, ≿−m1,w1) = f(≿).
+Lemma 26. For any fixed ≿−m1,w1 the royals are either
+• matched-by-default
+• unmatched-by-default
+• choosing sequentially dictatorially with m1 going first
+• choosing sequentially dictatorially with w1 going first
+Proof. Since all four regimens find the same choices when the royals either top rank each other or when
+they both top rank commoners, we focus on the case where exactly one royal wants to marry the other.
+In particular we assume top(≿w1) = mk and ≿m1: w1, wl for k ̸= 1 ̸= l and note that the arguments
+apply by gender-neutrality to the case where w1 wants to marry m1. Suppose that (m1, w1) /∈ f(≿).
+By Lemmas 24 and 25 and group strategy-proofness f(≿)(m1) ≿m1 wl. Since m1 is by assumption
+not matched with the only partner he prefers to wl we have f(≿)(m1) = wl and f(≿)(w1) = mk
+Fix any ≿′
+m1,w1 with ≿′
+w1= mk′, mk and ≿′
+m1: w1, wl′ for l′ ̸= 1 ̸= k′. By group strategy-
+proofness we may in the preceding paragraph assume w.l.o.g that ≿w1: mk, mk′ and ≿m1: w1, wl, wl′.
+By strategy-proofness f(≿′
+m1, ≿−m1)(m1) ∈ {wl′, wl}. By the above arguments, (m1, wl′), (w1, mk) ∈
+f(≿′
+m1, ≿−m1). Now swap mk and mk′ in w1’s ranking. By strategy-proofness f(≿′
+m1,w1, ≿−m1,w1
+)(w1) ∈ {mk, m′
+k}. By Lemmas 24 and 25 and group strategy-proofness (m1, wl′), (w1, mk′) ∈ f(≿′
+m1,w1
+, ≿−m1,w1). Dropping mk in woman w1’s ranking we see that woman w1 always gets her will (given
+≿−m1,w1) to marry a commoner if she once gets her will. By symmetry, she would always get her will
+to marry m1 if she once gets her will. So the only possible regimes at and ≿−m1,w1 are the two serial
+dictatorships as well as matched and unmatched-by-default.
+Lemma 27. The royals are either matched-by-default or unmatched-by-default.
+Proof. Fix an arbitrary ≿−m1,w1. By Lemma 26 the royals are either matched or unmatched-by-default
+or engaged in a serial dictatorship with one of the royals choosing first, the other second. Suppose that
+different profiles for the commoners were governed by different regimes. Fix two such profiles where
+the regimes change with the preference of one agent. Say w.l.o.g that this agent is m2. So the picking
+regime for w1 and m1 is governed by different rules at ≿−m1,w1 and at (≿′
+w2, ≿−m1,w1,m2).
+The outcomes of all regimes are identical when the two royals either top rank each other or
+when they both rank commoners.
+So we fix a profile ≿m1,w1 where exactly one royal top ranks
+a commoner and say that at ≿−m1,w1 the royal who wants to marry the other royal wins, while
+at (≿′
+w2, ≿−m1,w1,m2) the royals marry commoners. For concreteness assume that ≿m1: w1, wl and
+≿w1: mk, so that the regime change from ≿−m1,w1 to (≿′
+w2, ≿−m1,w1,m2) is one from matched-by-
+default or serial dictatorship with m1 going first to unmatched-by-default or serial dictatorship with
+w1 going first. Mutatis mutandis the same arguments apply when ≿m1,w1 is such that w1 wants to
+31
+
+marry m1 who in turn wants to marry a commoner.
+In sum we have that (m1, w1) ∈ f(≿) and
+(w1, mk), (m1, wl) ∈ f(≿′
+m2, ≿−m2).
+Since the regime stays fixed as long as the commoners’ preferences stay fixed we may w.l.o.g
+assume that mk = m2 so that f(≿′
+w2, ≿−m1,w1,m2)(m2) = w1. By group strategy-proofness f(≿) =
+f(≿∗
+m2, ≿−m2) for any ≿∗
+m2 that top ranks f(≿)(m2) and f(≿′′
+m2, ≿−m1,w1,m2) = f(≿′
+w2, ≿−m1,w1,m2)
+for any ≿′′
+m2: w1, f(≿)(m2).
+Now change the royal couples preferences to ≿′
+m1: w1, f(≿)(m2) and
+≿′
+w1: m3. Give that the regimes stay fixed at (≿∗
+m2, ≿−m1,w1,m2) and (≿′′
+m2, ≿−m1,w1,m2) we get
+f(≿′
+m1,w1, ≿∗
+m2, ≿−m1,w1,m2) = f(≿)
+(m1, f(≿)(m2)), (w1, m3) ∈ f(≿′
+m1,w1, ≿′′
+m2, ≿−m1,w1,m2).
+A contradiction to strategy-proofness results since f(≿′
+m1,w1, ≿∗
+m2, ≿−m1,w1,m2) matches m2
+with f(≿)(m2) which is according to ≿′′
+m2, m2’s second favorite wife. Conversely, since
+(m1, f(≿)(m2)), (w1, m3) ∈ f(≿′
+m1,w1, ≿′′
+m2, ≿−m1,w1,m2)
+the latter neither matches m2 with his ≿′′
+m2-favorite wife w1 nor with his second favorite f(≿)(w2).
+We in sum get that the regime with which the royals m1, w1 choose partners stays fixed for
+all ≿−m1,w1. By gender-neutrality the royals must use a symmetric regime when ≿−m1,w1 is gender-
+neutral. So neither of the two serial dictatorships can govern the choices by the royals and we must
+have that the regime is either matched-by-default or unmatched-by-default.
+We have shown that there is a single royal couple who are either matched-by default, or un-
+matched by default. The notion of gender-neutrality implies that, given the matches of the royal couple,
+the remaining agents are engaged in a continuation mechanism which is required to be gender-neutral
+with respect to some symmetry of order two. Applying the same arguments to these submechanisms
+gives a second royal couple. Continuing in this way we get a sequence of royal couples until just four
+agents remain at which point our arguments break down, and any one of the four-agent mechanisms
+described in section A can be used.
+32
+
diff --git a/wdFPT4oBgHgl3EQfPTQG/content/tmp_files/load_file.txt b/wdFPT4oBgHgl3EQfPTQG/content/tmp_files/load_file.txt
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@@ -0,0 +1,1038 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf,len=1037
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='13037v1  [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='TH]  30 Jan 2023  Royal Processions: Incentives, Efficiency and Fairness in  Two-sided Matching  Sophie Bade∗  Joseph Root†  January 2023  Abstract  We study the set of incentive compatible and efficient two-sided matching mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We  classify all such mechanisms under an additional assumption – “gender-neutrality” – which guar-  antees that the two sides be treated symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' All group strategy-proof, efficient, and gender-  neutral mechanisms are recursive and the outcome is decided in a sequence of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In each  round two agents are selected, one from each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' These agents are either “matched-by-default”  or “unmatched-by-default.” In the former case either of the selected agents can unilaterally force  the other to match with them while in the latter case, they may only match together if both agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In either case, if this pair of agents is not matched together, each gets their top choices among  the set of remaining agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As an important step in the characterization, we first show that in  one-sided matching all group strategy-proof and efficient mechanisms are sequential dictatorships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  An immediate corollary is that there are no individually rational, group strategy-proof and efficient  one-sided matching mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  1  Introduction  Gale and Shapley (1962) initiated the study of stability in two-sided matching problems using the  example of a marriage market where each of n men is matched to one of n women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A matching is  considered stable if no unmarried pair prefers each other to their partners from the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Gale  and Shapley (1962) proved that stable matchings always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However, Roth (1982) showed that  stability and incentive compatibility clash: any stable mechanism is manipulable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Faced with this dilemma, the vast literature on two-sided matching typically sacrifices incentive  compatibility in favor of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1 In this paper, we do the opposite: we ignore stability and instead  study incentive compatible mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Stability is an especially important criterion for settings  where agents can easily rearrange matchings among themselves (Abdulkadiroglu and S¨onmez 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  However there are many settings where coalitional deviations may be hard to organize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For example,  people who sign up for a dating app do so since they find it difficult to find dates in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In other  ∗Royal Holloway, University of London and Max Planck Institut for Research on Collective Goods, Email:  so-  phie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='bade@rhul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='uk  †Department of Economics, University of Chicago, Email: jroot@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='edu  1See for instance Roth and Sotomayor (1992), Roth (2008) and Abdulkadiroglu and S¨onmez (2013)  1  settings the central planner may have the authority to enforce the outcome of a mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Take, for  example, the British “homes for Ukraine” scheme that matches refugees with host families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The UK  home office could condition residency permits on the matches found by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So even if some  families and refugees would like to deviate from the matching, they might find it impossible to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Finding mechanisms with strong incentive properties so that they are simple to use would, in both  these contexts, seem more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By abandoning stability, we open the door to the study of incentive compatible and efficient  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In particular, we look for mechanisms with robust incentive properties – those that  are group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A mechanism is group strategy-proof if there is no profile of preferences,  group of agents and deviation for that group, such that all group members are weakly better-off after  the deviation (and some strictly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  A mechanism is efficient if it never chooses a matching where  an alternative could make all agents better-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Group strategy-proofness and efficiency are easy to  attain in two-sided matching problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Simply declare one side of the market to be “objects” and  use one of the well-known group strategy-proof and efficient mechanisms for allocation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2  Indeed sometimes “objectification” of one side of the market may be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Abdulkadiro˘glu  and S¨onmez (2003), for example, make a convincing case to treat schools as objects in school choice  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However in settings where the two sides are symmetric a priori, one may be hesitant to  assign all agency to one side of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In fact, we may want to require that the two sides of the  market are treated equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' There is, for example, no reason to treat the men and women on a dating  app any differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Likewise one may wish to give an equal amount of say to the refugees and the  host families in the “Homes for Ukraine” program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Our axiom of “gender-neutrality” captures the idea that the two sides should be treated equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Roughly, we require that if the men and women’s preferences are swapped, the outcome of the mecha-  nism is swapped as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We not only impose such symmetry between the sides in the overall mechanism  but on every “continuation submechanism” as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The notion of “continuation submechanism” cap-  tures the idea that a mechanism can be recursive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If a mechanism makes initial matches using only the  preferences of the initially matched agents, ignoring all other agents preferences, then the remaining  agents face such a continuation submechanism and we ask that they be treated symmetrically as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We characterize the set of all group strategy-proof, efficient and gender-neutral mechanisms as  the set of “royalty mechanisms.” These work as follows: each round one agent is chosen from either  side of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' These “royals” are either “matched-by-default” or “unmatched-by-default.” In the  former case the royals are matched together if at least one of them top-ranks the other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' in the latter  case, the royals are only matched if both top-rank one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If the royals are not matched with  each other, they get their top choices among the set of remaining agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As long as at least 6 agents  remain unmatched, a procession of choices by such royal couples continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Once only four agents  remain, there are only two possible matches for these agents, however a plethora of (sub)mechanisms  for these remaining agents satisfy our desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Our proof begins with the observation that any gender-neutral two-sided mechanism nests a  2The set of all group strategy-proof and efficient mechanism for the house allocation problem was characterized  by Pycia and ¨Unver (2017) and Bade (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The characterization extends Gale’s top trading cycles (Shapley and  Scarf 1974) in three ways: agents may “own” multiple houses (P´apai 2000), the agents may “broker” houses (Pycia and  ¨Unver 2017) and when there are exactly three agents and houses there may be “braids” (Bade 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  2  one-sided matching mechanism, where any agent is free to match with any other agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To get an  intuition for this result, index the men and women such that mi is the man who mirrors women wi  under the desired symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If we restrict attention to preference profiles where mi and wi announce  symmetric preferences, by gender-neutrality they must get symmetric outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So for each i, a  symmetric mechanism either matches wi and mi with each other or with a different symmetric pair  mj and wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We can therefore treat the pairs (mi, wi) as the agents in a one-sided matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Any pair either matches with themselves, which is analogous to the pair remaining single, or they swap  partners with another symmetric pair, which can be viewed as a match between these two pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The  embedded mechanism inherits the group strategy-proofness and efficiency of the original mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  This is helpful since the group strategy-proof and efficient mechanisms admit a neat characterization  for the one-sided matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We show that all such mechanisms are “sequential dictatorships”  where sequences of agents or “dictators” choose their most preferred remaining agent as their partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  This characterization complements that of Root and Ahn (2020) which finds a similar result when  agents are not allowed to remain single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Despite the similarities in the result, the proofs are very  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Our proof stems from our analysis of the three-agent case which is ruled out in Root  and Ahn (2020) since all agents are required to be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Having established that the embedded  one-sided mechanism is a sequential dictatorship, we see that any gender-neutral, efficient and group  strategy-proof mechanism agrees with a royalty mechanism on all symmetric preference profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Our  next task is to show that the same holds for asymmetric profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The proof is involved, but boils down  to finding a path from any asymmetric preference profile to a symmetric profile, over which we can  show the mechanism agrees with a royalty mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Gender-neutrality can be seen as a fairness requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A common fairness criterion in social  choice, full symmetry, requires that all agents are treated identically, in the sense that the outcome  of a mechanism is invariant under any bijective relabeling of all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In many environments full  symmetry conflicts with other desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Bartholdi, Hann-Caruthers, Josyula, Tamuz, and Yariv  (2021) therefore weaken full symmetry to require that the outcome of a mechanism must only remain  unchanged for a particular set of bijective relabelings of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' From this point of view, our notion  of (weak) gender-neutrality only requires that the mechanism be symmetric with respect to one such  relabeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In fact, in two-sided matching, no efficient and group strategy-proof mechanism is invariant  under more than two relabelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Faced with these limits on achieving symmetry we consider randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We first show that  randomization is no panacea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In our environment, just as in house allocation where random mecha-  nisms are well-studied, randomization can introduce inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A typical approach is to “symmetrize”  a mechanism by choosing agents’ roles in the mechanism uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For example, “serial  dictatorship”, where agents choose their match sequentially in a fixed order, can be symmetrized to  yield “random serial dictatorship” by selecting the picking order uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Despite the  efficiency of serial dictatorship, random serial dictatorship can be inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Nevertheless, the sym-  metrized mechanism retains the incentive properties of the deterministic mechanism from which it is  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In house allocation, no mechanism can simultaneously achieve efficiency, incentive compati-  bility and symmetry (Bogomolnaia and Moulin 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' On the other hand, random serial dictatorship  admits a certain form of efficiency: it selects a lottery over deterministic outcomes each of which is  3  efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We say that random serial dictatorship is therefore ex-post efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A major open question  in house allocation is whether any other mechanism satisfies ex-post efficiency, incentive compatibility  and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We show that two-sided matching admits multiple mechanisms of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The rest of the paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We first establish the preliminary definitions then  describe our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Finally we discuss our axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We conclude with a discussion of randomized  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  2  Preliminaries  Let N be a finite set of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We will be interested in both “one-sided matching”, where any agent  can match with any other and “two-sided matching”, where matching is bipartite, so that agents can  only match with partners from the other side of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In two-sided matching, we will assume  that N is the disjoint union of two sets, M = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , mn} and W = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , wn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In keeping with  Gale and Shapley (1962), we refer to the agents in M as “men” and the agents in W as “women.”  A submatching is a (possibly empty) list of mutually exclusive pairs and singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The pairs are  unordered so that (m, w) and (w, m) both refer to the same pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In the case of two-sided matching  singletons are not allowed and each pair has to be made up of one man and one woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let Σ1  0 and  Σ2  0 denote the set of one- and two-sided submatchings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any submatching ν let N(ν)  denote the set of agents matched in the submatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A matching is a submatching that lists every  agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A proper submatching is a submatching which is not a matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We alternatively represent  matchings as bijections µ : N → N of order 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' µ ◦ µ = id) so that agent i is matched with j if and  only if j is matched with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The requirement that pairs must be made up of one man and one woman  each then translates to µ(M) = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We denote the sets of one- and two-sided matchings by Σ1 and  Σ2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Agents are assumed to have strict preferences (total orders) over their possible partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In  one-sided matching, each agent i has a strict preference ≿i over N, where i stands for the option  to stay unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We write x ≻i y to mean that i strictly prefers being matched with x to being  matched with y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We write x ≿i y to indicate that either x ≻i y or x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In two-sided matching,  an agents’ preferences range over the other side of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' That is, each m ∈ M has a strict  preference ≿m over W and each w ∈ W has a strict preference ≿w over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A preference profile  is a list all agents’ preferences (≿i)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let Ω1 and Ω2 denote the set of preference profiles for the  one-sided and two-sided matching problems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For any preference ≿i in either context, define top(≿i) as agent i’s most preferred partner and  bottom(≿i) as agent i’s least preferred partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any agent i, we use the notation ≿i: j1, j2, · · · to  mean an arbitrary preference which top-ranks j1, second-ranks j2 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given a subset of agents  S, we write ≿S as a preference profile for just the agents in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given a preference profile ≿ and some  ≿′  S, we write (≿′  S, ≿−S) for the preference profile in which each agent i from S announces ≿′  i and each  agent j from N \\ S announces ≿j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  A (matching) mechanism is a function f : Ωk → Σk for k = 1, 2 that maps each profile of  preferences to a matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Such a mechanism f is group strategy-proof if there is no preference  profile ≿, group of agents S ⊂ N and ≿′  S such that f(≿′  S, ≿−S)(i) ≿i f(≿)(i) for all i ∈ S, and for some  4  j ∈ S, f(≿′  S, ≿−S)(j) ≻j f(≿)(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' That is, if agents’ true preferences were captured by the profile ≿  there would be no group of agents who could jointly misreport, making all agents in the group weakly  better-off, with at least one strictly better-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='3 Given a preference profile ≿, a matching ν Pareto  dominates a matching η if for all agents i, ν(i) ≿i η(i) and for at least one agent j, ν(j) ≻j η(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  A matching η is efficient at the preference profile ≿ if there is no ν which Pareto dominates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A  mechanism f is efficient if for every preference profile ≿, f(≿) is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We now turn to our fairness requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Informally, a mechanism is weakly gender-neutral if  there is a way to reflect preferences across the sides so that, after reflection, the mechanism chooses  the same outcome, reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To formally define weak gender-neutrality fix a σ ∈ Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' While σ is  a matching, we use it here to denote the reflection across which the mechanism exhibits the desired  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Thus σ(wj) = mi is interpreted to mean that agent wj is symmetric to agent mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We use σ  both to transform preferences and matchings as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given a matching µ ∈ Σ2, the reflection, σ∗µ,  of µ under σ is the matching such that (m, w) ∈ σ ∗ µ ⇐⇒ (σ−1(m), σ−1(w)) = (σ(m), σ(w)) ∈ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Equivalently, for any pair, (m, w) matched in µ, (σ(m), σ(w)) are matched in σ ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For a preference  profile ≿∈ Ω2 we define the reflection ≿′= σ(≿) so that j ≿′  i j′ ⇔ σ(j) ≿σ(i) σ(j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So if man m  prefers woman w to woman w′ in the profile ≿, then woman σ(m) prefers man σ(w) to man σ(w′)  according to the reflected profile σ(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Gender-neutrality will ensure that the symmetric agents i and  σ(i) are is treated equally: a mechanism f is weakly gender-neutral if f(σ(≿)) = σ ∗ f(≿) holds  for all ≿∈ Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='5  For a fixed symmetry σ we say that a matching µ and respectively a profile of preferences ≿  are symmetric if they equal to their reflections, so µ = σ ∗ µ and ≿= σ(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For notational simplicity,  when considering a fixed weakly gender-neutral mechanism f we will index the set of men and women  M = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , mn} and W = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , wn} so that σ(wj) = mj for all j, where σ is the symmetry  with respect to which f is weakly gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose M = {Ad, Bob, Carl} and W = {Ann, Beth, Connie}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let  σ = {(Ann, Ad), (Beth, Bob), (Connie, Carl)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The reflection of  µ = {(Ann, Bob), (Beth, Carl), (Connie, Ad)}  under σ is the matching  σ ∗ µ = {(Beth, Ad), (Connie, Bob), (Ann, Carl)}  For instance (Ann, Bob) ∈ µ implies (σ(Ann), σ(Bob)) = (Ad, Beth) ∈ σ∗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix a profile of preferences  ≿ such that ≿Ad: Connie, Beth, Ann and such that all agents other than Ad rank partners according  to their alphabetical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then the reflected profile ≿′= σ(≿) is such that ≿′  Ann: Carl, Bob, Ad  3It turns out that in this context a mechanism is group strategy-proof if and only if no group of agents of size one or  two can find a jointly profitable misreport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' See Alva (2017) and Root and Ahn (2020) for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  4The reflection σ ∗ µ is the matching given by the function σ ◦ µ ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  5Note that we require σ to be of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This is without loss of generality in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' One could  have instead asked for an arbitrary bijection ρ : N → N such that ρ(M) = W and required f(ρ(≿)) = ρ ∗ f(≿) for  all ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In this case, since ρ is a permutation, it is a member of the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' It therefore has a finite order k  so that ρk is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since ρ matches all men to women and vice-versa, k is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Hence ρk/2 is of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If  f(ρ(≿)) = ρ ∗ f(≿) for all ≿ then f(ρm(≿)) = ρm ∗ f(≿) for any m and any ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Thus f(ρk/2(≿)) = ρk/2 ∗ f(≿), and f  is weakly gender-neutral with respect to ρk/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  5  while all other agents rank all possible partners in alphabetical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The following is an example of a mechanism which is weakly gender-neutral under  σ(wj) = mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any preference profile, match m1 and w1 with their top choices unless they conflict  – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' if exactly one top ranks the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In this case, match both with their top choices excluding  one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If m1 and w1 are matched together, repeat this with m2 and w2 choosing from the  remaining agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Continue until there is a pair mk and wk who do not choose one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose  mk is matched with wj and wk is matched with ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If j < l match the remaining agents using serial  dictatorship with the men as dictators picking in order of their index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If j > l match the remaining  agents using serial dictatorship with the women as dictators picking in order of their index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  While f is weakly gender-neutral in this example, men and women are treated highly unequally  after most initial choices by m1 and w1: after most initial choices one side of the market retains all  agency while the other is turned into objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' An agent who is uncertain about the royal’s preferences  might reason that they have an equal subjective likelihood to be dictator and object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For this agent,  weak gender-neutrality might be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  However, an agent with more information about the  preferences of the royals might evaluate this mechanism as highly unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To avoid such unequal  treatment we impose weak gender-neutrality on “continuation” submechanisms such as the mechanisms  following on the choices by m1 and w1 in the present example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A matching mechanism g for the agents in S is a continuation submechanism of  the matching mechanism f if there is a profile of preferences ≿−S for N \\ S such that g(≿∗  S)(j) =  f(≿S, ≿−S)(j) ∈ S for all ≿S and j ∈ S, where ≿∗  S is the restriction of ≿S to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A bilateral matching mechanism f : Ω2 → Σ2 is gender-neutral if for every continu-  ation submechanism g of f is weakly gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since any mechanism is a continuation submechanism of itself, and gender-neutral mechanism  is weakly gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Example 3 (Example 2 continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To see that the mechanism f defined above is not gender-neutral  note that upon m1 and w1 respectively choosing w3 and m2 as their partners a continuation sub-  mechanism for all other agents arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since w1’s partner (m2) has a lower index than m1’s (w3) the  remaining matches are determined by a serial dictatorship of all women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since serial dictatorships are  not not weakly gender-neutral, the overall mechanism is not gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In the preceding example the continuation submechanism following the symmetric match be-  tween m1 and w1 is weakly gender-neutral under the same symmetry as the original mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This  observation easily generalizes:  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose f is a weakly gender-neutral mechanism with symmetry σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose g is a con-  tinuation submechanism of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Say g follows on a gender-neutral submatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Then g is weakly  gender-neutral with the symmetry σ′ that is the restriction of σ to the agents in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  6  3  Results and Approach  We characterize the class of group strategy-proof, gender-neutral and efficient two-sided matching  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As a key lemma we characterize all group strategy-proof and efficient one-sided matching  mechanisms, allowing agents to remain unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For simplicity we ignore related special cases  in both results, which we leave to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' These correspond to when there are exactly four  agents in two-sided matching and when there are exactly two agents in one-sided matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' These are  special cases for similar reasons: both reduce to social choice problems with exactly two outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As  described in the introduction, our characterization results in the class of “Royalty mechanisms.” These  mechanisms sequentially select two agents, one from either side, to choose their matches according to  one of two regimes: matched-by-default(D) or unmatched-by-default(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The precise order of these  agents can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Royalty mechanisms are therefore parameterized by this order, which we call a  “succession order”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A succession order is a function ϕ : A → (M × W) × {D, U} where A is a subset of  Σ2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ϕ1 : A → (M × W) and ϕ2 : A → {D, U} correspond to the first and second components of  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A succession order ϕ must satisfy  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ∅ ∈ A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ν ∈ A and ϕ1(ν) = (m, w) then m, w /∈ N(ν)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ν ∈ A and ϕ1(ν) = (m, w) then for any m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='w′ /∈ N(ν), ν ∪ {(m, w′), (w, m′)} matches all but  four or fewer agents in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In addition to the succession order, we need to specify a terminal condition for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Let χ be the set of group strategy-proof, efficient, and gender-neutral mechanisms with exactly four  agents6 and let ΣT  2 be the set of submatchings in Σ2  0 where exactly four agents are unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A terminal condition is a map ϕT : ΣT  2 → χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Given a succession order ϕ and a terminal condition ϕT , the royalty mechanism R(ϕ,ϕT )  chooses a matching using the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The royalty algorithm given a succession order ϕ, a terminal condition ϕT and a  preference profile ≿ proceeds in a number of steps:  Initialize: Set ν0 = ∅ and if there are three or more couples in the mechanism go to Step  1, otherwise go to Step T  Step k: The agents ϕ1(νk−1) are declared royals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ϕ2(νk−1) = D then if either royal  top-ranks the other among N − N(νk−1), they are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ϕ2(νk−1) = U the royals  are matched only if they both top-rank one another among N − N(νk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If the royals  are not matched, each gets matched with their favorite partner excluding one another in  N − N(νk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let νk denote the resulting submatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If νk leaves 6 or more couples  6These are characterized in appendix section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  7  unmatched, go to step k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If νk leaves 4 or fewer agents unmatched go to Step T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Step T: If there are exactly two remaining agents, match them together to result in νT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Otherwise use the mechanism ϕT (νk) to find a submatching ν for the remaining four agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Set νT = νk ∪ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Return νT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The last step of royalty mechanisms are a special case which we discuss in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  There are a number of possible group strategy-proof, efficient and gender-neutral mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As a  lead example consider a unanimity rule which sets one of the matchings as a default and chooses the  other matching only if all four agents prefer it to the default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A two-sided matching mechanism f : Ω2 → Σ2 is group strategy-proof, efficient and  gender-neutral only if it is a royalty mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Theorem 1 does not characterize the set of all group strategy-proof, efficient and gender-neutral  mechanisms as some royalty mechanisms are not gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To see this, consider a royalty mech-  anism f for 5 or more couples that starts with m1, w1 as the royal couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Consider the cases that the  royal couple either choose w2 and m3 as their partners or w3 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Our definition of royalty mech-  anisms only imposes that these choices lead to either a matched-by-default or a unmatched-by-default  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However gender neutrality requires more than that: If ml and wk become the royal couple after  the choice of w2 and m3 then wl = σ(ml) and mk = σ(wk) become must become the royal couple after  the choice of w3 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To define neutral royalty mechanisms, we proceed inductively over the number of couples in a  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To start say that the set of all royalty mechanisms for two couples consists of the gender  neutral, efficient and group strategy-proof mechanisms characterized in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now suppose  that neutral royalty mechanisms for up to n couples have been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To define a neutral royalty  mechanism for n + 1 couples, fix a symmetry σ on N, with the standard assumption that σ(mi) = wi  for all i and such that (m1, w1) are chosen as the royal couple at the start of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Say (m1, w1) choose mk and wl as their partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If these choices are symmetric with respect to  σ, so if k = l, then a neutral royalty mechanism that is symmetric with respect to the restriction  of σ to the unmatched agents must be chosen as the continuation submechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If k < l we may  freely choose any neutral royalty mechanism f k,l to follow on the submatching {(m1, wl), (mk, w1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  If k > l we are bound by the choices for the preceding case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In that case we must use a permutation  of the royalty mechanism σ ∗ f k,l that is identical to f k,l except that wherever a man m appears in  f k,l the woman σ(m) should take up his place in σ ∗ f k,l and wherever a woman w appears in f k,l the  woman σ(w) should take up her place in σ ∗ f k,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A two-sided matching mechanism f : Ω2 → Σ2 is group strategy-proof, efficient and  gender-neutral if only if it is a neutral royalty mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The proof is involved so we defer the details to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We sketch the argument here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The first key insight is to notice that within any gender-neutral two-sided mechanism is a one-sided  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose as usual that σ(mi) = wi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let Ω2  symm be the set of symmetric preference  profiles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' the ≿ such that σ(≿) =≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since σ(≿) =≿, gender-neutrality implies that f(≿) = σ∗f(≿  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then if (mi, wj) ∈ f(≿) so is (σ(mi), σ(wj)) = (wi, mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' From this we see that given any symmetric  8  preference profile ≿, for any pair (mi, wi), either mi and wi are matched in f(≿) or they swap partners  with another symmetric pair (mj, wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By treating each symmetric pair (mi, wi) as a single agent,  ci we can extract a one-sided matching mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We interpret the swap of partners between say  (mi, wi) and (mj, wj) as a match between ci and cj and a match between mi and wi as the agent ci  being unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Root and Ahn (2020) showed that all group strategy-proof and efficient one-sided  mechanisms are sequential dictatorships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However, they did not allow agents to remain unmatched,  a key requirement for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We complement their characterization by showing the same holds even if  agents are allowed to remain unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' let N ′ = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', n} and N = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , mn} ∪ {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , wn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given a gender-neutral  mechanism f for N, we define a one-sided matching mechanism g for N ′ which we call the induced  one-sided mechanism for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any ≿ in the one-sided matching market, let ≿∗ be the preference  profile in the two-sided market where mk ≿∗  wj ml and wk ≿∗  mj wl if and only if k ≿j l for all j, k, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Then for any j, g(≿)(j) is the agent i such that f(≿∗)(mj) = wi and f(≿∗)(wj) = mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any gender-neutral two-sided matching mechanism f, the induced one-sided mechanism  g is group strategy-proof and efficient if f is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Assume that f is group strategy-proof and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We will first show that g is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Consider any preference profile ≿ in the one-sided matching market and the matching g(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For  any other one-sided matching µ ̸= g(≿) we can define the symmetric two-sided matching ν so that  ν(mj) = wi and ν(wj) = mi if and only if µ(j) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f is efficient, there must be some j such that  f(≿∗)(mj) ≻∗  mj ν(mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However by definition we then have that g(≿)(j) ≻j ν(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Hence ν cannot  Pareto dominate g(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To see that g is group strategy-proof, fix any preference profile ≿, a group of agents S and some  ≿′  S such that g(≿) ̸= g(≿′  S, ≿−S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f is group strategy-proof, there is some j among the set of men  and women who have the same index as an agent in S such that f(≿∗)(mj) ≻mj f((≿′  S, ≿−S)∗)(mj)  or f(≿∗)(wj) ≻wj f((≿′  S, ≿−S)∗)(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By definition then g(≿) ≻j g(≿′  S, ≿−S)(j), so ≿′  S is not a  profitable deviation for the coalition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since the coalition, deviation and original preference profile  were arbitrary, g is group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Before stating our characterization of one-sided matching mechanisms, let’s recall the notion of  sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A picking order is a function φ : A → N where A is a subset of Σ1  0 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ∅ ∈ A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ν ∈ A and φ(ν) = i then i /∈ N(ν)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If ν ∈ A, φ(ν) = i and j /∈ N(ν) then ν ∪ {(i)} and ν ∪ {(i, j)} are either in A or are matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A mechanism f is a sequential dictatorship with respect to φ if for any ≿, f(≿) is  the matching resulting from the following algorithm:  9  Step 1: Agent φ(∅) is matched with her favorite partner (including herself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ν1 be this  submatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Step k ≥ 2: Agent φ(νk−1) is matched with her favorite remaining partner (including  herself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let νk be the matching νk−1 ∪{(φ(νk−1)} if φ(νk−1) prefers to remain unmatched  or νk−1 ∪{(φ(νk−1, j)} if j is φ(νk−1)s favorite remaining partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If νk is a matching, stop  and return νk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A one-sided matching mechanism f : Ω1 → Σ1 is group strategy-proof and efficient if  and only if it is a sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We prove Theorem 2 by induction over the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In a sequence of Lemmas we show  that the result holds for 3 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For 4 agents we use the fact that restricted to the case where each  agent bottom ranks being single, we face a classical social choice problem over 3 alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This case  must by the Gibbard-Satterthwaite Theorem be a dictatorship (Gibbard 1973)(Satterthwaite 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We show that the dictator in this constrained problem must also be the dictator in the four agent  problem when agent may prefer to be single to some matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The cases of five or more agents then  do not require any special treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This result agrees with Root and Ahn (2020) who find a similar  characterization in the setting where agents are not allowed to remain unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agents are  not able to veto the dictators’ choices in a sequential dictatorship, Theorem 2 admits an immediate  corollary:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' There are no individually rational, group strategy-proof and efficient one-sided matching  mechanisms  Having observed that a gender-neutral mechanism is equivalent to a one-sided matching mech-  anism on symmetric profiles, we see that on these profiles a group strategy-proof, gender-neutral and  efficient mechanism must agree with some royalty mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' At a high level, the rest of the proof  can be summarized by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any preference profile ≿ we find a sequence of  profiles ≿0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , ≿m where ≿m=≿ and ≿0 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We then argue that for every pair of adjacent  profiles ≿k, ≿k+1 in this sequence if a group strategy-proof, efficient and gender-neutral mechanism  agrees with a royalty mechanism at ≿k it must also do so at ≿k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The difficulty is in finding exactly  the right sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  4  The axioms  No axiom can be dropped from the characterizations in Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Considering first the  characterization of one-sided mechanisms in Theorem 2 note that any constant mechanism is group  strategy-proof but not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Most efficient mechanisms are not group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For an  example define a one-to-one function R from the set of matchings to the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then map  any profile ≿ to the matching µ that minimizes R(µ) over the set of all efficient matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To see that this mechanism is not group strategy-proof fix a setting with just three agents  {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say R assigns values 1,2,3 and 4 to the matchings {(1, 2), (3)}, {(1), (2, 3)}, {(1, 3), (2)} and  {(1), (2), (3)} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Fix the profile ≿∗ where each agent holds the same ranking ≿∗  i which  ranks 3 above 1 above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since each matching is efficient at ≿∗, the mechanism chooses {(1, 2), (3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  10  Now modify agent 2’s preference to be ≿′  2: 3, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since now agents 1 and 2 would rather be single  than be together the matching {(1, 2), (3)} is not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since {(1), (2, 3)} is efficient and because  R({(1), (2, 3)}) = 2 the mechanism maps (≿′  2, ≿∗  −2) to {(1), (2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However, agent 2 strictly prefers  to be matched with agent 3 than to agent 1 so the mechanism is not strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Things get more interesting with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The constant mechanism which maps each profile  of preferences to the matching σ with respect to which f is gender-neutral, is not only group strategy-  proof but also gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' It is clearly not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For a gender-neutral and efficient mechanism  f that is not group strategy-proof we fix, as above, a one-to-one mapping R from the set of gender-  neutral matchings to the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To define f(≿) first check whether there there exist any  symmetric efficient matchings at ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If so choose the matching that minimizes R(µ) over the set of  all efficient symmetric matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If not, use some fixed neutral royalty mechanism to calculate f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To see that the mechanism is gender-neutral fix a profile ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Note that the set of efficient symmetric  matchings at ≿ and σ(≿) coincide and that σ ∗ µ holds for any symmetric matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Therefore  σ ∗ f(≿) = f(≿) = f(σ(≿)) both equal the minimal R(µ) over the set of all matchings µ that are  efficient at σ (and therefore also at σ(≿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since neutral royalty mechanisms are gender neutral,  f(≿) = f(σ(≿)) also holds for the alternative case where no gender neutral matchings are efficient at  ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The fact that f is not group strategy-proof follows from the fact that the analog mechanisms for  roommate problems is not group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='7  If we drop gender-neutrality and modify efficiency to only include the preferences of one side of  the market, we obtain the set of efficient and group strategy-proof one-sided matching mechanisms as  characterized by (Pycia and ¨Unver 2017) and (Bade 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' When we replace gender-neutrality with  weak gender-neutrality, our proof continues to hold until the point where we show that a royal couple  must be matched or unmatched-by-default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However if this royal couple chooses partners m and w  that are not symmetric, the continuation submechanism need not be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This is illustrated by  example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1  The incompatibility of gender-neutrality and stability  The tension between stability and incentive compatibility was first described in Roth (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Here  we show that stability also clashes with gender-neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Stability therefore forces the designer to  introduce asymmetry where none exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' No stable mechanism f for at least three couples is gender-neutral and stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix a matching problem with three couples {(m1, w1), (m2, w2), (m3, w3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose f was an  stable mechanism that is gender-neutral with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix the following preferences  ≿m1  w3  w2  w1  ≿m2  w1  w3  w2  ≿m3  w2  w1  w3  ≿w1  m3  m2  m1  ≿w2  m1  m3  m2  ≿w3  m2  m1  m3  7If f was group strategy-proof it would have to be group strategy-proof on the subdomain of symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now consider a problem with three couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If the couples preferences and the function R are given as in the discussion  of roommate problems, we see that couple two has an incentive to lie about their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  11  Notice that there are two stable matchings  M − optimal = {(m1, w3), (m2, w1), (m3, w2)}  W − optimal = {(m1, w2), (m2, w3), (m3, w1)}  To see that there is no other stable matching, notice that in any other matching at least one pair of  agents with the same index (mi, wi) are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In this case, there is an agent mj and an agent wj  who top-rank wi and mi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' These form blocking pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since σ(≿) =≿ and since f is gender-neutral with respect to σ, f(≿) must be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  However neither stable matching is symmetric: (m1, w3) are matched in the M-optimal stable match,  but (σ(m1), σ(w3)) = (w1, m3) are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Likewise (m1, w2) are matched in the W-optimal stable match,  but (σ(m1), σ(w2)) = (w1, m2) are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  5  Two-sided Mechanisms with Randomization  A typical approach to achieving fairness in social choice is to employ randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A deterministic  mechanism is symmetrized by randomizing the role agents play in the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For example, one  could symmetrize serial dictatorship, where agents are called in a fixed order to choose their preferred  matches, by selecting a picking order uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This gives a mechanism analogous to the  mechanism known as random serial dictatorship (RSD) in house allocation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The symmetrized  mechanism retains some of the features of the original mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If the original mechanism is strategy-  proof, so is the symmetrized version8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' On the other hand, there is no guarantee that the symmetrized  mechanism will be efficient, even if the original mechanism is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For example, consider the following  preferences:  ≿m1  w3  w2  w1  ≿m2  w1  w3  w2  ≿m3  w2  w1  w3  ≿w1  m3  m2  m1  ≿w2  m1  m3  m2  ≿w3  m2  m1  m3  If we run serial dictatorship, selecting the picking order uniformly at random we get the following  random allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  w1  w2  w3  m1  1/12  11/24  11/24  m2  11/24  1/12  11/24  m3  11/24  11/24  1/12  However, the following random allocation gives a first-order stochastic improvement for all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  w1  w2  w3  m1  0  1/2  1/2  m2  1/2  0  1/2  m3  1/2  1/2  0  8In the sense that truthfully reporting gives a lottery which first-order stochastically dominates any other lottery that  could be achieved by a misreport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  12  This is a common issue in randomized mechanisms in a variety of environments (Echenique, Root, and  Sandomirskiy 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In house allocation, Bogomolnaia and Moulin (2001) showed that it is inevitable:  there is no efficient, strategy-proof and symmetric mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' There is a sense in which the mechanism  is efficient: the outcome can be decomposed as a lottery over efficient deterministic outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' This is  known as ex-post efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Bade (2020) showed that symmetrizing any one of the many efficient and  group strategy-proof house allocation mechanism gives nothing other than random serial dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  This, along with the impossibility result of Bogomolnaia and Moulin (2001) has lead to interest in the  question of whether RSD is the unique ex-post efficient, strategy-proof and symmetric mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Symmetrizing our royalty mechanisms gives a negative answer to the same question in two-sided  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For example when symmetrizing a royalty mechanism where the royals are matched-by-  default in each round we get the following allocation given the preferences above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  w1  w2  w3  m1  1/9  4/9  4/9  m2  4/9  1/9  4/9  m3  4/9  4/9  1/9  Notice that these three allocation matrices are Pareto ranked with the second matrix dominating  RSD which in turn dominates the uniform match-by-default outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The ranking of RSD and uniform  match-by-default is an artifact of this example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' there are examples where the Pareto ranking is flipped  and in general the two random allocations cannot be ranked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  References  Abdulkadiro˘glu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' S¨onmez (2003): “School choice: A mechanism design approach,” American economic  review, 93(3), 729–747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Abdulkadiroglu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' S¨onmez (2013): “Matching markets: Theory and practice,” Advances in Economics and  Econometrics, 1, 3–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Alva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (2017): “When is manipulation all about the ones and twos,” Unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Bade, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (2020): “Random serial dictatorship: the one and only,” Mathematics of Operations Research, 45(1), 353–368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Bartholdi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Hann-Caruthers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Josyula, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Tamuz, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Yariv (2021):  “Equitable voting rules,”  Econometrica, 89(2), 563–589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Bogomolnaia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Moulin (2001): “A new solution to the random assignment problem,” Journal of Economic  theory, 100(2), 295–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Echenique, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Root, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Sandomirskiy (2022): “Efficiency in Random Resource Allocation and Social Choice,”  arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='06353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Gale, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
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+page_content=' Shapley (1962): “College admissions and the stability of marriage,” The American Mathematical  Monthly, 69(1), 9–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Gibbard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (1973): “Manipulation of voting schemes: a general result,” Econometrica: journal of the Econometric  Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' 587–601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  P´apai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (2000): “Strategyproof assignment by hierarchical exchange,” Econometrica, 68(6), 1403–1433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Pycia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ¨Unver (2017): “Incentive compatible allocation and exchange of discrete resources,” Theoretical  Economics, 12(1), 287–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Root, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Ahn (2020): “Incentives and Efficiency in Constrained Allocation Mechanisms,” arXiv preprint  arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='06776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  13  Roth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (1982): “The economics of matching: Stability and incentives,” Mathematics of operations research, 7(4),  617–628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  (2008): “Deferred acceptance algorithms: History, theory, practice, and open questions,” international Journal  of game Theory, 36(3), 537–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Roth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Sotomayor (1992): “Two-sided matching,” Handbook of game theory with economic applications,  1, 485–541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Satterthwaite, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' (1975): “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems  for voting procedures and social welfare functions,” Journal of economic theory, 10(2), 187–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Shapley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Scarf (1974): “On cores and indivisibility,” Journal of mathematical economics, 1(1), 23–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  14  A  The case of four agents in two-sided matching  In this section we characterize all group strategy-proof, efficient and gender-neutral bilateral matching  mechanisms for four agents {m1, m2, w1, w2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In this case Σ2 contains only two matchings: ν :=  {(m1, w1), (m2, w2)} and µ := {(m1, w2), (m2, w1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given that there are only two matchings, each  agent has exactly two preferences: the agent either prefers ν or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A preference profile ≿ can then  be summarized as the set S or all agents i with ν(i) ≿i µ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In the present context it is easier to  represent preference profiles as such sets S, meaning that mechanisms now map from the collection of  all subsets of N to {µ, ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In line with this representation of preference profiles a mechanism partitions the set of all  subsets of N into Λν and Λµ with the understanding that f(S) = ν iff S ∈ Λν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Our three desiderata  the translate to the following requirements in the environment with just two couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The mechanism f is efficient iff N ∈ Λν and ∅ ∈ Λµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The mechanism f is group strategy-proof9 iff S ∈ Λν and S ⊊ S′ imply that S′ ∈ Λν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The mechanism f is gender-neutral if S ∈ Λν implies that σ(S) ∈ Λν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Any group strategy-proof mechanism can therefore be represented by a set Λ  ν ⊂ Λν of all  minimal sets S ∈ Λν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The following list of sets Λ  ν is - up to renaming - an exhaustive list of all efficient group  strategy-proof and gender-neutral mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  a) {S | |S| = x} for x = 1, 2, 3, 4,  b) {{m1, w1, w2}, {m1, w1, m2}, {m2, w2}},  c) {{m1, w1, w2}, {m1, w1, m2}},  d) Any subset of the sets {{m1, w1}}, {{m2, w2}}, {{m1, w2}, {m2, w1}}, {{m1, m2}, {w1, w2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  e) {{m1}, {w1}, {m2, w2}}  f) {{m1}, {w1}}  The 4 mechanisms listed in a) are moreover the exhaustive subset of fully symmetric mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For notational convenience Lemma 3 includes a certain number of double counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The mech-  anism where ν is chosen if at least two agents prefer it for example falls in groups a) and d) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We could have also economized the definition by noting that the mechanisms listed in b) and c) are  equivalent to the mechanisms in e) and f) upon exchanging the matchings ν and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The mechanism in  f) is the matched-by-default mechanism with m1 and w1 the royal couple, the mechanism defined by  {m1, w1}, which appears in d), is the unmatched-by-default mechanism with m1 and w1 as the royals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  9In fact, group strategy-proofness is equivalent to individual strategy-proofness with two outcomes, so this condition  is necessary and sufficient for the mechanism to be strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  15  ∅  {m2}  {m1}  {w2}  {w1}  {m1, w2}  {m1, m2}  {w1, m2}  {w1, w2}  {m1, w1}  {m2, w2}  {m1, w1, w2}  {m2, w1, w2}  {m1, m2, w1}  {m1, m2, w2}  {m1, m2, w1, w2}  Figure 1: Four agents: Any gender-neutral, strategy-proof mechanism can be expressed as a set of  nodes in the lattice above that is closed upwards and which is symmetric with respect to the the  reflection over the vertical dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For each of the mechanisms group strategy-proofness holds by definition since Λν is in each  case defined as the set of all supersets in Λ  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Each mechanism is Pareto optimal by the preceding  argument together with the observation that ∅ /∈ Λ  ν for any of the lists sets of subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Finally each  mechanism is gender-neutral since each of the generating sets is gender-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So we only have to show that the above list is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To do so first consider Λ  ν with  {m1, w1, w2} ∈ Λ  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By gender-neutrality σ({m1, w1, w2}) = {m1, w1, m2} ∈ Λ  ν must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  {m1, w1, w2}, {m1, w1, m2} ∈ Λ  ν no subset of either {m1, w1, w2} or {m1, w1, m2} can be in Λ  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So  the latter cannot contain any of the singleton sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' {m2, w2} is the only two agent set that can be  contained in Λ  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We in sum found that the sets Λ  ν which contain a three agent set are exactly the sets  listed in a), b) and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Exchanging ν and µ in the agents preferences we see that the sets Λ  ν which  contain a singleton are exactly the sets listed in a) e) and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Finally to see that d) is an exhaustive list of mechanisms generated by two agent sets, note that  d) lists the complete set of gender-neutral sets of two agent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  B  The case of two agents in one-sided matching  Suppose that N = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' With just two agents, there are exactly two outcomes in one-sided matching:  the matching {(1, 2)} and the matching {(1), (2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We say that a mechanism f is a dictatorship if there is an agent k such that f(≿  )(k) = top(≿k) for all ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We say that a mechanism f is a unanimity rule one of the two matchings is chosen  unless both agree top-rank the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  16  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If N = {1, 2}, then a one-sided mechanism f : Ω1 → Σ1 is group strategy-proof and  efficient if and only if it is either a dictatorship or a unanimity rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since N = {1, 2} there exist exactly two matchings: one pairs the two agents, the other keeps  both single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A mechanism is then efficient if and only if it chooses the matching preferred by both  agents if they agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The four such mechanisms map the two profiles where the agents disagree to  the two different matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' It is easy to check that these four mechanisms correspond to the two  dictatorships and the two unanimity rules, and that they are strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  C  The proof of Theorem 2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1  Sequential Dictatorships are efficient and group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Sequential Dictatorships are efficient and group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix a sequential dictatorship SD : Ω1 → Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Suppose SD was not efficient or not group  strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So suppose there either exists some profile ≿ and either a matching µ Pareto dominates  SD(≿) or there is a group S ⊂ N and a deviation ≿′  S such that all members of S weakly prefer  SD(≿′  S, ≿−S) to SD(≿) while some members of the group strictly hold this preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Define k1 and  k2 as the first rounds at which the SD(≿)-algorithm finds a match that differs from respectively µ  and SD(≿′  S, ≿−S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If only two agents remain unmatched at step k1 or k2 we obtain a contradiction  via Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So say that at least three agents remain unmatched at steps k1 and k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say i1 and i2 are  the dictators at these steps, so that SD(≿)(i1) ̸= µ(i1) and SD(≿)(i2) ̸= SD(≿′  S, ≿−S)(i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  the dictator ix gets matched with his top-ranked unmatched partner at Step kx, and since µ(i1) and  SD(≿′  S, ≿−S)(i2) respectively stay available at Steps k1 and k2, we get the contradictions SD(≿  )(i1) ≻i µ(i1) and SD(≿)(i2) ≻i SD(≿′  S, ≿−S)(i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2  Submechanisms  For any two different agents j, k ∈ N define Ω−j and Ω−j,k as the restriction of Ω1 to all agents but  agent j and j, k respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='10 For any group strategy-proof and efficient mechanism f, define two  mechanisms f −j and f −j,k on Ω−j and Ω−j,k respectively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Set f −j(≿−j)(i) = f(≿)(i) for  all ≿−j∈ Ω−j and i ∈ N \\ {j} where ≿ is any preference profile such that (1) when restricted to  N − {j} it gives ≿−j and (2) all agents i ̸= j rank j at the bottom while agent j ranks herself at the  top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Similarly let f −j,k(≿−j,k)(i) = f(≿)(i) for all ≿−j,k∈ Ω−j,k and i ∈ N \\ {j, k} where ≿ is any  preference profile such that (1) when restricted to N − {j, k} it gives ≿−j,k and (2) all agents i ̸= j, k  rank j and k at the bottom while agent j and k ranks each other at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To simplify notation we  drop the superscripts −j and −j, k from the preference profiles ≿−j and ≿−j,k in the sequel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' f −j(≿)  then stands for the application of f −j to the restriction of ≿ to all agents but j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  10That is, Ω−j is the set of preference profiles for all agents other than j and such that j is excluded from all other  agents’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Ω−j,k is similar except now j and k are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  17  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say f is a group strategy-proof and efficient mechanism and j, k ∈ N are two differ-  ent agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then the mechanisms f −j and f −j,k are well defined, group strategy-proof and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Moreover if f(≿)(j) = j, then f(≿)(i) = f −j(≿)(i) for all i ∈ N \\ {j} and if f(≿)(j) = k, then  f(≿)(i) = f −j,k(≿)(i) for all i ∈ N \\ {j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The arguments are similar for f −j and f −j,k, so we will just prove this for f −j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To see that  f is well-defined, notice that for any ≿−j and any profiles ≿ and ≿′ where the conditions described  above hold (namely (1) when restricted to N − {j} the profiles ≿ and ≿′ give ≿−j and (2) all agents  i ̸= j rank j at the bottom while agent j ranks herself at the top in both profiles) ≿ and ≿′ can only  possibly differ in j’s ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However since f is efficient, in both-cases j will be matched to herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By  group strategy-proofness, all other agents matches must also remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now suppose, by way  of contradiction, that f −j is not group strategy-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then there is a preference profile ≿−j, some  coalition of agents S, and an profile ≿′  S for the agents in S so that f −j(≿′  S, ≿−j  −S)(i) ≿i f −j(≿−j)(i)  for all i in S and for some k in S, f −j(≿′  S, ≿−j  −S)(k) ≿k f −j(≿−j)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ≿∗  S be the profile for the  agents in S such that all agents bottom-rank j and their ranking is the same as in ≿S otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Likewise, let ≿−j∗ be the profile where all agents bottom-rank j and their preferences are the same  as ≿−j otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ≿j be an arbitrary preference for j where j top-ranks herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By definition  f −j(≿−j)(i) = f(≿j, ≿−j∗)(i) and f −j(≿′  S, ≿−j  −S)(i) = f(≿∗  S, ≿j, ≿−j∗  −S )(i) for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  However  this leads to a violation of group strategy-proofness for f, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Finally, efficiency follows  immediately from the efficiency of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='3  The Inductive Structure of the Proof  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If each efficient and group strategy-proof mechanism f : Ω1 → Σ1 for three or more agents  has a dictator, then any efficient group strategy-proof mechanism is a sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix a group strategy-proof and efficient mechanism f : Ω1 → Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If there are only two agents  then f is by Lemma 4 a sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If there are more than two agents then f has by  assumption a dictator i, so that f(≿) = top(≿i) for any ≿ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If, upon matching i with top(≿i) only one  agent remains unmatched, this agent must stay single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If not, then Lemma 6 implies that any choice  of agent 1 is followed by a group strategy-proof and efficient submechanism f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-  proofness this submechanism f ′ only depends on agent i’s choice (and not on agent i’s rankings over  the options he did not choose or on the preferences of the agent j ̸= i that i did choose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If two agents  remain unmatched, f ′ must by Lemma 4 be either a dictatorship or a unanimity rule, and we are  done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If more than two agents remain unmatched f ′ has by the assumption in the Lemma a dictator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proceeding inductively we reach the case where at most two agents remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Following Lemma 7 it suffices to show that each group strategy-proof and efficient mechanism  with more than two agents has a dictator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The next three section show that each group strategy-proof  and efficient mechanisms with respectively n = 3, n = 4 and n ≥ 5 agents has a dictator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='4  The case of three agents  Throughout this section fix a group strategy-proof and efficient mechanism f : Ω1 → Σ1 for the agents  N = {1, 2, 3}, so that Σ1 contains only four matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If any two agents are matched, the remaining  18  agent is clearly left single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The proof that f must be a sequential dictatorship revolves around the  notion of “ownership”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say agent i owns an agent j if agent i can always choose to be matched with  agent j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Formally top(≿i) = j implies f(≿i, ≿−i)(j) = i for all ≿−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' After establishing the preliminary  Lemma 8 on ownership, Lemma 9 shows that each agent in i ∈ {1, 2, 3} must be owned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Lemma 10  then rules out all ownership structures except then one where one agent owns all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Throughout  this section, it will be useful to have some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ≿1,2  i  : 1, 2, 3, ≿1,3  i : 1, 3, 2 and ≿1  i  as an arbitrary preferences with top(≿i) = 1, so ≿1  i equals either ≿1,2  i  or ≿1,3  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Likewise let ≿1,2, ≿1,3  and ≿1 denote preference profiles where all agents’ preferences are ≿1,2  i , ≿1,3  i  and ≿1  i respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The first lemma describes a sufficient condition to establish ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent j∗ owns agent 1 if and only if f  �  ≿1,2 �  (j∗) = f  �  ≿1,3 �  (j∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If agent j∗ owns agent 1, then f(≿)(j∗) = 1 holds for any ≿ with top(≿j∗) = 1, in particular  ≿1,2 and ≿1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So suppose that we have f  �  ≿1,2 �  (j∗) = f  �  ≿1,3 �  (j∗) = 1 for some agent j∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For j∗ to  own 1 it is sufficient to show that f(≿1)(j∗) = 1 holds for all ≿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To see that latter fix any ≿−j∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Define ≿′  i for the two agents i ̸= j∗ such that top(≿′  i) = 1 and 2 ≿′  i 3 ⇔ 2 ≿i 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By the assumption  f(≿1  j∗, ≿′  −j∗)(j∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Group strategy-proofness then yields f(≿1  j∗, ≿′  −j∗) = f(≿1  j∗, ≿−j∗), so that  f(≿1  j∗, ≿−j∗)(j∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: j∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1: f  �  ≿1,2 �  (2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then the group strategy-proofness of f implies f  �  ≿1,2  2 , ≿1  −2  �  (1) =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The strategy-proofness of f implies that f  �  ≿1,3  2 , ≿1  −2  �  (2) ∈ {2, 3}, and therefore, by either group  strategy-proofness or feasibility respectively, f  �  ≿1,3  2 , ≿1  −2  �  (1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We in sum get that f(≿1)(j∗) = 1  for any ≿1, which establishes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2: f  �  ≿1,3 �  (3) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Applying the arguments from Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' mutatis mutandis we get  f(≿1)(j∗) = 1 for any ≿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='3: f  �  ≿1,2 �  (3) = f  �  ≿1,3 �  (3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So f  �  ≿1,2 �  = f  �  ≿1,3 �  and by group strategy-  proofness f must also match 3 with 2 for any profile where agent 1 ranks being single at the top  and where at least one agent in {2, 3} prefers being matched with the other to being single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For  the last remaining case where agent 1 top ranks being single while the other two prefer being single  to being matched with each other start with the observation that f(≿1  1, ≿1,2  −1) and f(≿1  1, ≿1,3  −1) both  match agents 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By strategy-proofness we then get f(≿1  1, ≿1,2  2 ≿1,3  3 )(2) ∈ {2, 3} as well as  f(≿1  1, ≿1,2  2 ≿1,3  3 )(3) ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The efficiency of f then implies that agents 2 and 3 both stay single in  f(≿1  1, ≿1,2  2 ≿1,3  3 ), so that once again f(≿1  1, ≿1,2  2 ≿1,3  3 )(1) = 1 and in sum f(≿1)(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: j∗ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g say j∗ = 2 so that f  �  ≿1,2 �  (2) = f  �  ≿1,3 �  (2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now say  f(≿∗)(2) ̸= 1 did hold for some ≿∗ with top(≿∗  1)(i) = 1 for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The assumption f  �  ≿1,2  �  (2) = f  �  ≿1,3 �  (2) = 1 together with group strategy-proofness implies that f(≿1,2  1 , ≿1  −1)(1) = 2 (A)  as well as f(≿1,3  3 , ≿1  −3)(1) = 2 (B) for any ≿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So for f(≿∗)(1) ̸= 2 to hold we must have ≿∗  1=≿1,3  1  and ≿∗  3=≿1,2  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Case 1: f(≿∗)(1) = f(≿1,3  1 , ≿∗  2, ≿1,2  3 )(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In that case we get by group strategy-  proofness that f(≿1,3  1 , ≿∗  2, ≿1,2  3 ) = f(≿1,3  1 , ≿∗  2, ≿1,3  3 ) which leads to a contradiction to (B) established  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Case 1: f(≿∗)(1) = f(≿1,3  1 , ≿∗  2, ≿1,2  3 )(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In that case we get by group strategy-proofness  that f(≿1,3  1 , ≿∗  2, ≿1,2  3 ) = f(≿1,2  1 , ≿∗  2, ≿1,2  3 ) which leads to a contradiction to (A) established above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  19  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' According to f, each agent i ∈ {1, 2, 3} is owned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 8 (and the interchangeability of all agents) it suffices to show f(≿1,2)(1) = f(≿1,3  )(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So suppose we had f(≿1,2)(1) ̸= f(≿1,3)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: Agent 1 is not alone at either ≿1,3 or ≿1,2, so f(≿1,2)(1) ̸= 1 ̸= f(≿1,3)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Suppose we had f(≿1,2)(3) = f(≿1,3)(2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The group strategy-proofness of f implies f(≿1,2  ) = f(≿1,3  3 , ≿1,2  −3) as well as f(≿1,3) = f(≿1,2  2 , ≿1,3  −2) = f(≿1,3  3 , ≿1,2  −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We in sum get the contradiction  f(≿1,2) = f(≿1,3) to the assumption that f(≿1,2) and f(≿1,3) respectively match agent 1 with agent  2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So we must have f(≿1,2)(2) = f(1, 3)(3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The proof derives a contradiction by showing that f must equal two different matchings at the  profile ≿ given by  ≿1: 1, 3, 2  ≿2: 3, 1, 2  ≿3: 1, 2, 3  Starting at f  �  ≿1,2 �  (1) = 2 change agent 2’s preference to ≿2: 3, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By strategy-proofness  f  �  ≿2, ≿1,2  −2  �  (2) ∈ {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since the matching {(1), (2, 3)} Pareto dominates the matching {(1, 2), 3}  at (≿2, ≿1,2  −2), f  �  ≿2, ≿1,2  −2  �  must match agents 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The group srategy-proofness then implies  that f  �  ≿2, ≿1,2  −2  �  = f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' On the other hand, the group strategy-proofness of f, together with the  observation that ≿1,3  1 =≿1 implies that f  �  ≿1,3 �  = f  �  ≿  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since f  �  ≿1,3 �  (1) =  3 while we have shown above that f  �  ≿  �  (1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: Agent 1 is alone at ≿1,2 or ≿1,3, not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g f(≿1,2)(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' f  �  ≿1,2 �  (2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Group strategy-proofness then implies f  �  ≿1,2  2 , ≿1,3  −2  �  (1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Strategy-proofness yields that f  �  ≿1,3 �  (2) equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We then get f  �  ≿1,3 �  (1) = 1 (in the first  case by group strategy-proofness and in the second by feasibility) a contradiction to the assumption  that agent 1 is not alone at ≿1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 f  �  ≿1,2 �  (2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Group strategy-proofness then implies f  �  ≿1,2 �  = f  �  ≿1,2  3 , ≿1,3  −3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since f  �  ≿1,3 �  ̸= f  �  ≿1,2 �  = f  �  ≿1,2  3 , ≿1,3  −3  �  and since ≿1,2  3  and ≿1,3  3  differ only in their ranking of  agents 2 and 3 in second and third place, f  �  ≿1,3 �  (3) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So for f  �  ≿1,3 �  (1) ̸= 1 to hold we must  have f  �  ≿1,3 �  (1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Just as above the proof derives a contradiction by showing that f maps the following ≿ to two  different matchings,  ≿1: 3, 1, 2  ≿2: 1, 2, 3  ≿3: 1, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  20  Strategy-proofness, ≿1,2  2 =≿2, and f  �  ≿1,2 �  (1) = 1 imply f  �  ≿1,2  3 , ≿−3  �  (1) ∈ {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  If  f  �  ≿1,2  3 , ≿−3  �  (1) = 1, then the group strategy-proofness of f implies f  �  ≿1,2 �  = f  �  ≿1,2  3 , ≿−3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A  contradiction arises, since f  �  ≿1,2 �  is at  �  ≿1,2  3 , ≿−3  �  dominated by µ with µ(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So f  �  ≿1,2  3 , ≿−3  �  (1) = 3 and f  �  ≿1,2  3 , ≿−3  �  = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f is strateyproof and since µ(3) = 1 = top(≿3) = top(≿1,2  3 )  we get f  �  ≿  �  = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Starting with f  �  ≿1,3 �  and noting that ≿1,3  3 =≿3 the group strategy-proofness of  f yields the contradiction f  �  ≿  �  = f  �  ≿1,3 �  ̸= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The mechanism f must have a dictator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 9 each agent is owned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose we had an “ownership chain” in the sense that  agent i owns j who in turn owns k with {1, 2, 3} = {i, j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For a profile ≿ with top(≿i) = j and  top(≿j) = k we would then obtain the contradiction (i, j), (j, k) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Given that there can be no  such ownership chains we have to consider only 3 ownership structures (up to renaming): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Each  agent owns herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent 1 owns agent 2 and agent 3 owns herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' One agent owns all three  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' To show that the third case must hold we rule out the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: Each agent owns herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The classic example of a roommate problems without a  stable matching serves to obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Define ≿ as follows:  ≿1: 2, 3, 1  ≿2: 3, 1, 2  ≿3: 1, 2, 3  Since f is efficient at least two agents must get matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Assume that f(≿)(1) = 2 and  consider the deviation to ≿′  2: 3, 2, 1 and ≿′  3: 2, 3, 1 for agents 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since each agent owns herself,  f(≿1, ≿′  −1)(2) ≿2 2 and f(≿1, ≿′  −1)(3) ≿3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So f(≿1, ≿′  −1) either keeps all agents single or pairs up  agents 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since the latter Pareto dominates the former, f(≿1, ≿′  −1)(2) = 3 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  f(≿1, ≿′  −1)(2) = 3 ≻2 1 = f(≿)(2) and f(≿1, ≿′  −1)(3) = 2 ≻3 3 = f(≿)(3) a contradiction to the  group strategy-proofness of f results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Mutatis mutandis the same arguments rule out the remaining  two matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So it cannot be that each agent owns herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: Agent 1 owns agent 2 and agent 3 owns herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Transform the profile ≿ in two  steps to (≿2, ≿′  −2) where  ≿1: 3, 2, 1  ≿′  1: 3, 1, 2  ≿2: 3, 2, 1  ≿3: 2, 1, 3  ≿′  3: 2, 3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since agent 1 owns agent 2, f(≿)(1) equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The latter must hold since the matching {(1, 2), ()}  is (at ≿) Pareto dominated by {(1, 3)(2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness f(≿) = f(≿′  1, ≿−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now  swap agent 3’s preference to ≿′  3: By strategy-proofness and since agent 3 owns herself 2 ≻3 f(≿2, ≿′  −2  )(3) ≿′  3 3, so that f(≿2, ≿′  −2)(3) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Conditioning on agent 3 staying single, agents 1 and 2 must, by  21  efficiency, also stay single at f(≿2, ≿′  −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since matching agents 2 and 3 Pareto  dominates f(≿2, ≿′  −2) at (≿2, ≿′  −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='5  The case of four agents  In the present subsection assume that f : Ω1 → Σ1 is a group strategy-proof and efficient mechanism  for four agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For each profile of preferences ≿ define ≿ so that each agent i ranks being single at the  bottom keeping all other rankings identical to ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say Ω  1 is the subdomain of such profiles where each  agent ranks being single at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The set of matchings where no agent is single is Σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' There  are exactly four such matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The restriction f of f to Ω  1 is a dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f is group strategy-proof and efficient, its restriction to Ω  1 is so too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f is efficient  and since each agent ranks being single at the bottom no agent stays single according to f(≿) for any  ≿∈ Ω  1 and we can represent f as a mechanism mapping Ω  1 to Σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since Σ  1 contains three matchings,  each of which is fully determined by the match of a single agent, we are facing a classic social choice  problem with three options where four agents may hold any preferences over these three options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By  the Gibbard Satterthwaite theorem f : Ω  1 → Σ  1 must be a dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For the reminder of the present section say agent 1 is the dictator in f : Ω  1 → Σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For each i ∈ {1, 2, 3, 4}, f −i is a sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 6 f −i is efficient and group strategy-proof for three agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By the preceding section  f −i is a sequential dictatorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For the remainder of the section assume that agent 2 is the dictator in f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' An agent i∗ ∈ {1, 2} is the dictator in both f −3 and f −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' First fix a profile ≿ such that  ≿1:  1, ·, ·, ·  ≿2:  2, ·, ·, ·  ≿3:  2, 3, 4, 1  ≿4:  2, 4, 3, 1  The efficiency of f implies f(≿)(1) = 1, and Lemma 6 then implies f −1(≿) ⊂ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  top(≿2) = 2 and since agent 2 is the dictator in f −1, f(≿)(2) = f −1(≿)(2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By efficiency agents 3  and 4 also remain single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 6 f(≿) must then be consistent with f −3(≿) and f −4(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  f(≿)(4) = 4 and since 2 ≿4 4, agent 4 cannot be the dictator in f −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Mutatis mutandis we see that  agent 3 cannot be the dictator in f −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  22  To see that f −3 and f −4 must have the same dictator fix ≿ such that  ≿1:  1, 2, ·, ·,  ≿2:  1, 2, ·, ·,  ≿3:  3, ·, ·, ·,  ≿4:  4, ·, ·, ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since f is efficient, f(≿)(3) = 3 and f(≿)(4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 6, f(≿)(1) = f −3(≿)(1) = f −4(≿)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since top(≿1) = top(≿2) = 1, we then get f −3(≿)(1) = f −4(≿)(1) = i∗ so f −3 and f −4 have the same  dictator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent 1 is the dictator in f −j for j = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix ≿ and the deviations ≿′  ≿1: 3, 1, 2, 4  ≿′  1: 3, 4, 1, 2  ≿2: 3, 2, 1, 4  ≿3: 2, 1, 3, 4  ≿4: 4, ·, ·, ·  ≿′  4: 1, 4, ·, ·  By efficiency f(≿)(4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 6 f −4(≿) ⊂ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose 1 is not the dictator in f −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So 2 or 3  must be the dictator in f −4 and we get f(≿) = {{1}, {2, 3}, {4}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Changing agent 1 and 4’s preferences  to ≿′  1,4, group strategy-proofness implies f(≿′  1,4, ≿2,3)(1) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By efficiency f(≿′  1,4, ≿2,3)(2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now  define ≿′′ to be identical to (≿′  1,4, ≿2,3) except that each agent ranks being single at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By  group strategy-proofness f(≿′  1,4, ≿2,3) = f(≿′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since ≿′′∈ Σ  1 and agent 1 is  by assumption the dictator for problems where all agents rank being single at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So agent 1  must be the dictator in f −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 13 agent 1 is also the dictator in f −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now define ≿  ≿1:  1, 3, ·, ·,  ≿2:  2, ·, ·, ·  ≿3:  1, 3, ·, ·,  ≿4:  4, ·, ·, ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By efficiency f(≿)(2) = 2 and f(≿)(4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 6 f(≿) is consistent with f −2(≿) and  f −4(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agent 1 is the dictator in f −4, f −4(≿)(1) = 1 = f(≿)(1) = f −2(≿)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The latter  implies that agent 3 cannot be the dictator in f −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Mutatis mutandis the dictator of f −2 cannot be  agent 4 either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So the dictator of f −2 must be agent 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent 1 is the dictator in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  23  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We have to show that f(≿)(1) = top(≿1) for all ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix an arbitrary ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Assume without loss  of generality that top(≿1) ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If f(≿)(k) = k for k = 3 or k = 4, then f −k(≿) ⊂ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since 1  is the dictator in f −k for k = 3, 4 we get that f −k(≿)(1) = f(≿)(1) = top(≿1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So for the remainder  assume that (3, 4) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: top(≿1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose we have f(≿)(1) ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since (3, 4) ∈ f(≿), f(≿) must then equal {(1), (2), (3, 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness we can  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g assume that  ≿1:  2, 1, 3, ·,  ≿2:  2, ·, ·, ·  ≿3:  4, ·, ·, ·,  ≿4:  3, 4·, ·, ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By strategy-proofness f(≿′  1, ≿−1)(1) ∈ {1, 3} for ≿′  1: 2, 3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If f(≿′  1, ≿−1)(1) = 1, then group  strategy-proofness implies f(≿′  1, ≿−1) = f(≿), and f(≿′  1, ≿−1) is consistent with f −2(≿′  1, ≿−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A  contradiction arises since agent 1, would as the dictator in f −2 choose agent 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So we must have  f(≿′  1, ≿−1)(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Efficiency then implies f(≿′  1, ≿−1)(4) = 4 so that f(≿′  1, ≿−1) is consistent with  f −4(≿′  1, ≿−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since agent 1, would as the dictator in f −4 choose agent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: top(≿1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since (3, 4) ∈ f(≿), f(≿) must then equal {(1, 2), (3, 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group  strategy-proofness assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' that ≿ is given by  ≿1:  1, 2, 3, ·,  ≿2:  1, 2·, ·,  ≿3:  4, ·, ·, ·,  ≿4:  3, 4·, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now swap agents 2 and 3 in agent 1’s ranking, so that ≿′  1: 1, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By strategy-proofness, f(≿′  1, ≿−1  )(1) ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose f(≿′  1, ≿−1) = 2, then we have f(≿′  1, ≿−1) = f(≿) by group strategy-proofness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now define ≿′′ to be identical to (≿′  1, ≿−1) except that each agent drops being single to the bottom of  their ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness f(≿′′) = f(≿′  1, ≿−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since ≿′′∈ Σ  1,  but f(≿′′)(1) = 2 ̸= top(≿′′)(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So we must have f(≿′  1, ≿−1)(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By efficiency agents 2 and 4  must then stay single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since f(≿′  1, ≿−1)(1) must then equal f −2(≿′  1, ≿−1)(1) = 1  since agent 1 is the dictator in f −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  24  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='6  The case of n ≥ 4 agents  For this section fix a group strategy-proof and efficient mechanism f for n + 1 agents and assume that  all such mechanisms for n or fewer agents are sequential dictatorships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If agent 1 is the dictator in f −2, then some agent j ∈ {1, 2} is the dictator in f −k for  all k /∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose agent 4 was the dictator in f −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix ≿ such that each agent i ̸= 3 ranks agents 1  and 3 respectively at the top and at the bottom, agents i ̸= 1, 3 rank being single in second place and  top(≿3) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  ≿1:  1, ·, ·, · · · , 3  ≿2:  1, 2, ·, · · · , 3  ≿3:  3, ·, ·, · · ·  ≿4:  1, 4, ·, · · · , 3 By efficiency f(≿) is consistent with f −3(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agent 4 is the dictator in f −3, and since  top(≿4) = 1, we have f(≿)(4) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By efficiency f(≿)(i) = i for all i /∈ {1, 4}, in particular f(≿  )(2) = 2, so that f −2(≿) is consistent with f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agent 1 is the dictator in f −2, we obtain the  contradiction that f(≿)(1) = f −2(≿)(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agents 3 and 4 were chosen arbitrarily in N \\{1, 2}  the dictator in f −k is for each k /∈ {1, 2} is either agent 1 or agent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To see that f −k for each k /∈ {1, 2} must have the same dictator consider the profile ≿ where  agents 1 and 2 both rank agents 1 and 2 in first and second place and where any agent i /∈ {1, 2} top  ranks being single, so  ≿1:  1, 2, ·, ·,  ≿2:  1, 2, ·, ·, ·,  ≿3:  3, ·, ·, ·,  ≿4:  4, ·, ·, ·, By efficiency, f(≿)(i) = i for each i /∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So f(≿) is consistent with f −i(≿) for each  i /∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By the preceding paragraph some agent j ∈ {1, 2} is the dictator in j−3, so f(≿)(j) =  f −3(≿)(j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now consider any k /∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since f(≿) is consistent with f −k, agent j must also  be the dictator in f −k  25  By Lemma 16 one agent j is the dictator in all submechanisms f −k with j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For the  remainder assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g that agent 1 is this agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent 1 is the dictator in f −j,k for any two agents j, k such that 1 /∈ {j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix ≿ such that two agents other than agent 1, say agents 2 and 3 top rank each other while a  third agent, say agent 4, top ranks himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' All remaining agents top rank agent 1 and rank agents 2,  3 and 4 at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  ≿1:  1, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , 2, 3, 4  ≿2:  3, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ,  ≿3:  2, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ,  ≿4:  4, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' ,  ≿5:  1, ·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' , 2, 3, 4 By efficiency f(≿) matches agent 4 with himself and agents 2 and 3 with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So f(≿)  is consistent with f −4(≿) and f −2,3(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agent 1 is the dictator in f −4(≿), f(≿)(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  f −2,3(≿) is consistent with f(≿), f −2,3(≿)(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since all agents but agent 4 top rank agent 1 in ≿,  the dictator in f −2,3 is either agent 1 or agent 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Repeating the same arguments with swapping agent  4 and 5, we see that either agent 1 or agent 5 is the dictator in f −2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In sum, agent 1 must be the  dictator in f −2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since agents 2 and 3 were chosen arbitrarily, agent 1 is the dictator in any f −j,k for  1 /∈ {j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Agent 1 is the dictator in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix a profile ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose f(≿)(1) ̸= top(≿1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since n ≥ 5, we can fix an agent j such that  {j, f(≿)(j)} ∩ {1, f(≿)(1), top(≿1)} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If j = f(≿)(j), then f(≿) is consistent with f −j(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  agent 1 is by Lemma 16 the dictator in f −j and since top(≿1) ̸= j we then obtain f(≿)(1) = f −j(≿  )(1) = top(≿1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If j ̸= f(≿)(j), define f(≿)(j) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In that case f(≿) is consistent with f −j,k(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since agent 1 is by Lemma 17 the dictator in f −j,k and since top(≿1) /∈ {j, k} we can conclude as  above that f(≿)(1) = f j,k(≿)(1) = top(≿1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  D  Proof of Theorem 1  Fix a mechanism f : Ω2 → Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Restricted to the domain Ω2  symm the mechanism f is by Lemma 2 and  Theorem 2 a serial dictatorship in which one couple can choose to either stay alone (marry each other)  or pair up with a different couple (swap partners).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Recall that f is assumed to be weakly gender-  neutral with respect to σ that is defined such that σ(mi) = wi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Without loss of generality say  the couple (m1, w1) is the dictator in the embedded roommates mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We call this the “royal  26  couple” and all other agents “commoners.” The rest of this proof is dedicated to showing that the  royal couples’ powers also apply to symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' At the same time we must show that any  conflict between the royal couples interests (when one wants the royals to marry and the other does  not) must be mediated using either the matched-by-default or the unmatched-by-default protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For Lemmas 19, 20, and 21 fix a symmetric profile ≿sym  −m1,w1 for all commoners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say Ωl,k  m1,w1 is the set of all preferences of m1 and w1 that respectively top rank wl and  mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then either  f(≿m1,w1, ≿sym  −m1,w1)(m1) = wl and f(≿m1,w1, ≿sym  −m1,w1)(w1) = mk for all ≿m1,w1∈ Ωl,k  m1,w1, or  f(≿m1,w1, ≿sym  −m1,w1)(m1) ̸= wl and f(≿m1,w1, ≿sym  −m1,w1)(w1) = mk for all ≿m1,w1∈ Ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='k  m1,w1 or  f(≿m1,w1, ≿sym  −m1,w1)(m1) = wl and f(≿m1,w1, ≿sym  −m1,w1)(w1) ̸= mk for all ≿m1,w1∈ Ωl,k  m1,w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Pick an arbitrary ≿m1,w1∈ Ωl,k  m1,w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: k = l Let ≿′  m1= σ(≿w1) so that Theorem 2 implies (m1, wk), (w1, mk) ∈ f(≿′  m1, ≿w1  , ≿sym  −m1,w1) Since f(≿′  m1, ≿w1, ≿sym  −m1,w1)(m1) = top(≿′  m1) = top(≿m1), group strategy-proofness then  implies that (m1, wk), (w1, mk) ∈ f(≿m1,w1, ≿sym  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: k ̸= l  Let ≿′  m1: wl, wk and ≿′  w1: mk, ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness and Case 1, f(≿′  m1,w1, ≿sym  −m1,w1)  either marries the royal couple with their most preferred partners (Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1), or with ml and wl  (Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2) or with mk and wk (Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1 the group strategy-proofness of f implies  f(≿m1,w1, ≿sym  −m1,w1) matches each royal with their top choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  For Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 Strategy-proofness implies f(≿w1, ≿′  m1, ≿sym  −m1,m1)(w1) ̸= mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now since m1 could  swap to make an announcement symmetric to ≿w1, strategy-proofness and Case 1 imply that f(≿w1  , ≿′  m1, ≿sym  −m1,m1)(m1) ≿′  m1 wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However, if f(≿w1, ≿′  m1, ≿sym  −m1,m1)(m1) = wk, we obtain a violation of  group strategy-proofness since if w1 announces σ(≿m1) she gets the ≿w1-preferred mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So f(≿w1, ≿′  m1  , ≿sym  −m1,m1)(m1) = wl must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Finally, by strategy-proofness for m1 we have f(≿w1, ≿′  m1, ≿sym  −m1,m1  )(m1) = f(≿m1,w1, ≿sym  −m1,m1)(m1) = wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Group strategy-proofness then implies f(≿w1, ≿′  m1, ≿sym  −m1,m1  ) = f(≿m1,w1, ≿sym  −m1,m1), so that  f(≿w1, ≿′  m1, ≿sym  −m1,m1 (m1) = f(≿m1,w1, ≿sym  −m1,m1)(w1) ̸= mk  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Mutatis mutandis the arguments of Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 apply to Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say Ω∗  m1,w1 is the set of preferences for the royals where exactly one royal top ranks the  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then either a) or b) holds for all ≿m1,w1∈ Ω∗  m1,w1  a) (m1, w1) ∈ f(≿m1,w1, ≿sym  −m1,m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  b) f(≿m1,w1, ≿sym  −m1,m1) matches the royal who top ranks a commoner with that commoner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Let ≿′  m1: w1, wk and ≿′  w1: mk, m1 for k ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since it is not possible to match both royals with  their top-ranked partners, Lemma 19 yields that one of the royals gets their top partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  27  Case 1: (m1, w1) ∈ f(≿′  m1,w1, ≿sym  −m1,m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then Lemma 19 implies that (m1, w1) ∈ f(≿′′  m1,w1  , ≿sym  −m1,w1) for any ≿′′  m1,w1 with top(≿′′  m1) = w1 and top(≿′′  w1) = mk (including ≿′  w1 that rank m1  last).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then group strategy-proofness implies that (m1, w1) ∈ f(≿m1,w1, ≿sym  −m1,w1) for any ≿m1,w1 with  top(≿m1) = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By gender-neutrality we also get that (m1, w1) ∈ f(≿m1,w1, ≿sym  −m1,w1) if top(≿w1) =  m1 and if ≿m1 is any preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: f(≿′  m1,w1, ≿sym  −m1,m1)(w1) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If (m1, w1) ∈ f(≿′  m1,w1, ≿sym  −m1,m1) held for some  ≿m1,w1 with top(≿m1) = w1 and top(≿w1) = ml for l ̸= 1 case 1 would imply the contradiction that  (m1, w1) ∈ f(≿′  m1,w1, ≿sym  −m1,m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Hence by Lemma 19 we have (w1, ml) ∈ f(≿(m1,w1), ≿sym  −m1,w1)(w1) =  ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' gender-neutrality then implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say top(≿m1) = wl and top(≿w1) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If k = 1 = l or if k ̸= 1 ̸= l, f(≿m1,w1, ≿sym  −m1,w1)  matches the royal couple with their most preferred partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If k = 1 = l the claim follows from first part of the proof of Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So let k ̸= 1 ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' From  Lemma 20 there are two cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: If exactly one royal top-ranks the other, the royals are matched Consider  ≿′  m1: wl, w1 and ≿′  w1: mk, m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By Lemma 19 at least one royal must get their top partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By  strategy-proofness, neither royal can do worse than the other royal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since m1 is only matched with  w1 if w1 is matched with m1 we then get (w1, mk), (m1, wl) ∈ f(≿′  m1,w1, ≿sym  −m1,w1) and by group  strategy-proofness f(≿m1,w1, ≿sym  −m1,w1) = f(≿′  m1,w1, ≿sym  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: If exactly one royal top-ranks the other, the royal top-ranking a commoner  gets their top match Consider the preferences  ≿′  m1: wl, w1,  ≿′′  m1: w1, wl  ≿′  w1: mk, m1,  ≿′′  w1: m1, mk  derived from ≿m1 and ≿w1 keeping all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since we are in Case 2, we have f(≿′  m1, ≿′′  w1, ≿sym  −m1,w1)(m1) = wl and f(≿′  w1, ≿′′  m1, ≿sym  −m1,w1  )(w1) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness we then get that (m1, wl), (w1, mk) ∈ f(≿′  m1, ≿′  w1, ≿sym  −m1,w1  )(m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Applying group strategy-proofness once again to drop each other in their royals’ rankings we  get (m1, wl), (w1, mk) ∈ f(≿m1,w1, ≿sym  −m1,w1) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The preceding three Lemmas referred to an arbitrarily fixed symmetric profile for all commoners  ≿sym  −m1,w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We showed that for any such fixed profile, the royals must be matched according to a  matched-by-default or a unmatched-by-default protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The next Lemma shows that for a vast set of  profiles for the commoners the royals get their top choices, if these top choices do not stand in conflict  (so if they both want to marry commoners or want to marry each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=')  Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix ≿ such that a subset of all pairs of commoners have symmetric preferences, while all  remaining commoners bottom-rank the royals, but do not necessarily have symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say  top(≿w1) = mk and top(≿m1) = wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If l = k = 1 or l ̸= 1 ̸= k, then (m1, wl), (w1, mk) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  28  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We use induction over the number m of pairs who do not have symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Start of the induction: m = 0 so that ≿−m1,w1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' In this case the claim holds  by Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Induction step: Suppose the claim holds up to some m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix an arbitrary profile ≿−m1,w1  such that m + 1 pairs have (potentially) asymmetric preferences, ranking the royals at the bottom,  and such that the remaining n − m − 1 pairs have symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Without loss, suppose that  (m2, w2) are a type that do not have symmetric preferences and consider ≿′  w1: mk, m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1: k ̸= 1 ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose by way of contradiction that f(≿)(w1) ̸= mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1: Couple (mk, wk) does not have symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For ≿′  mk= σ(≿wk), bottom(≿wk  ) = m1 implies bottom(≿′  mk) = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since (≿′  mk, ≿−mk) is covered by the hypothesis of the induc-  tion f(≿′  mk, ≿−mk)(w1) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We then obtain a contradiction to strategy-proofness since w1 ̸= f(≿  )(mk) ≻′  mk f(≿′  mk, ≿−mk)(mk) = w1 = bottom(≿′  mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 Couple (mk, wk) does have symmetric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By strategy-proofness and the as-  sumption that f(≿)(w1) ̸= mk, we have f(≿′  w1, ≿−w1)(w1) ̸= mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1, w1 would get matched  with m2 if she were to top rank m2 (since we’ve assumed m2 and w2 announce asymmetric prefer-  ences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So by strategy-proofness f(≿′  w1, ≿−w1)(w1) = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now consider ≿′  m2= σ(≿w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since couple  (m2, w2) does not have symmetric preferences bottom(≿m2) = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' We then obtain a contradiction  to strategy-proofness since (≿′  w1,m2, ≿−w1,m2) is covered by the hypothesis of the induction so that  f(≿′  w1,m2, ≿−w1,m2)(w1) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The latter implies that m2 can improve his match by switching his  preference from ≿m2 to ≿′  m2 to avoid being matched with w1 = bottom(≿m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Cases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 proves that (w1, mk) ∈ f(≿) holds in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By gender-neutrality (m1, wl) ∈  f(≿) also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Case 2: k=l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose (m1, w1) /∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' As in the proof of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='2 f(≿′  w1, ≿−w1)(w1) =  m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Just as in that proof we obtain a contradiction to strategy-proofness since m2 would be better-off  if he symmetrized his preference with w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The next two Lemmas pertain to the case that the royals top rank a symmetric pair (mi, wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In this case the royals are matched with their top choices - for any profile of commoners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix ≿ so that the royals top rank each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Then f(≿) marries the royals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Define ≿′  −m1,w1 as a modification of ≿−m1,w1 where m1 and w1 are moved to  the top of all agents preferences, but are otherwise unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Sequentially swap each commoner  i’s preference from ≿i to ≿′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness, with each such swap the matching either  stays constant or commoner i marries a royal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Therefore (m1, w1) /∈ f(≿m1,w1, ≿′  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose  (m1, wi), (w1, mj) ∈ f(≿m1,w1, ≿′  −m1,w1), with i ̸= 1 ̸= j and i = j permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Start with ≿′  −m1,w1,  to define a new profile ≿′′  −m1,w1 by dropping the royals to the bottom of all commoners’ rankings all  commoners other than in wi, mi, wj and mj rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Moreover let ≿′  wi :  = σ(≿mi) and if i ̸= j  also ≿′  mj : = σ(≿wj) so that the couples (mi, wi) as well as (mj, wj) (if different) have symmetric  preferences where the agents who are matched with the royal couples by f(≿m1,w1, ≿′  −m1,w1) top  rank the royal couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness none of these changes affect the match so that  29  f(≿m1,w1, ≿′  −m1,w1) = f(≿m1,w1, ≿′′  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' However, a contradiction arises since ≿′′  −m1,w1 is covered  by Lemma 22, so (m1, w1) ∈ f(≿m1,w1, ≿′′  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If top(≿m1) = wl and top(≿w1) = ml, then (m1, wl), (w1, ml) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix the royal’s preferences such that they rank each other in second place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose f(≿) did  not match both royals with their top ranked partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Observation 1: One royal must get their most preferred partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By the preceding Lemma and group strategy-proofness the royals cannot both prefer each other  to their matches f(≿)(m1) and f(≿)(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now suppose we had f(≿)(w1) = m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-  proofness f(≿) = f(≿m1,w1, ≿′  −m1,w1) where the commoners’ preferences ≿′  −m1,w1 are derived from  ≿−m1,w1 by dropping the royals to the bottom of each commoner’s preference keeping all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A  contradiction arises since ≿′  −m1,w1 is covered by Lemma 22, so that (m1, w1) ∈ f(≿m1,w1, ≿′  −m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In the only remaining case one royal gets their most preferred partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Observation 2: We may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g assume that bottom(≿ml) = w1 and bottom(≿wl) = m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Suppose this did not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For concreteness assume that (m1, wl) ∈ f(≿), noting that the  following arguments apply mutatis mutandis to the alternative case where (w1, ml) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since  (w1, ml) /∈ f(≿), we may by group strategy-proofness w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g assume that bottom(≿ml) = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now  define ≿′  wl by dropping m1 to the bottom of wl’s ranking keeping all else equal to ≿wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If (m1, wl) /∈  f(≿′  wl, ≿−wl), then Observation 1 implies that (w1, ml) ∈ f(≿′  wl, ≿−wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By Observations 1 and 2 and group strategy-proofness we may assume that (m1, wl)(w1, mj) ∈  f(≿) for some j ̸= l, top(≿mj) = w1, bottom(≿mi) = w1 for all i ̸= j, 1 and bottom(≿wi) = m1 for all  i ̸= 1  Now lift m1 in wj’s ranking so that top(≿′  wj) = m1 keeping all else equal to ≿wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group  strategy-proofness we either have f(≿) = f(≿′  wj, ≿−wj) or (m1, wj) ∈ f(≿′  wj, ≿−wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now define  ≿′′  mj,wj as two gender-neutral preferences ≿′′  mj= σ(≿′′  wj) so that an agent in {mj, wj} who is matched  with a commoner keeps their preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By group strategy-proofness f(≿) = f(≿′  wj, ≿−wj) =  f(≿′′  wj,mj, ≿−mj,wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction then arises since (≿′′  wj,mj, ≿−mj,wj) is covered by Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  And we can conclude that (m1, wl), (w1, ml) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  To conclude the proof, note that the group strategy-proofness of f together with (m1, wl), (w1, ml) ∈  f(≿) imply that the royals keep their most preferred partners if they drop each other to any place in  their rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' If top(≿m1) = wl, top(≿w1) = mk, and k ̸= 1 ̸= l, then (m1, wl), (w1, mk) ∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Define ≿′  w1: mk, ml, ≿′  m1: wl, wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Keeping all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By the Lemma 24 and group strategy-  proofness the royals either get their top choices or they marry mk and wk or ml and wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So suppose that  (m1, wk), (w1, mk) ∈ f(≿′  m1,w1, ≿−m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Define ≿′  −m1,w1 for the commoners so that top(≿′  mk) = w1,  ≿′  wk= σ(≿′  mk), bottom(≿′  mi) = w1 and bottom(≿′  wi) = m1 for all i ̸= k keeping all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By  group strategy-proofness we have f(≿′  m1,w1, ≿−m1,w1) = f(≿′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' A contradiction arises since ≿′  −m1,w1 is  30  covered by Lemma 22 so that the royals must get their top choices in f(≿′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Mutatis mutandis the same  arguments rule out the case that (m1, wl), (w1, ml) ∈ f(≿′  m1,w1, ≿−m1,w1) and f(≿′  m1,w1, ≿−m1,w1)  must match the royals with their top choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Dropping the less liked partners ml and wk in their  rankings, group strategy-proofness yields f(≿′  m1,w1, ≿−m1,w1) = f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For any fixed ≿−m1,w1 the royals are either  matched-by-default  unmatched-by-default  choosing sequentially dictatorially with m1 going first  choosing sequentially dictatorially with w1 going first  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since all four regimens find the same choices when the royals either top rank each other or when  they both top rank commoners, we focus on the case where exactly one royal wants to marry the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In particular we assume top(≿w1) = mk and ≿m1: w1, wl for k ̸= 1 ̸= l and note that the arguments  apply by gender-neutrality to the case where w1 wants to marry m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose that (m1, w1) /∈ f(≿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By Lemmas 24 and 25 and group strategy-proofness f(≿)(m1) ≿m1 wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Since m1 is by assumption  not matched with the only partner he prefers to wl we have f(≿)(m1) = wl and f(≿)(w1) = mk  Fix any ≿′  m1,w1 with ≿′  w1= mk′, mk and ≿′  m1: w1, wl′ for l′ ̸= 1 ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-  proofness we may in the preceding paragraph assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g that ≿w1: mk, mk′ and ≿m1: w1, wl, wl′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  By strategy-proofness f(≿′  m1, ≿−m1)(m1) ∈ {wl′, wl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By the above arguments, (m1, wl′), (w1, mk) ∈  f(≿′  m1, ≿−m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Now swap mk and mk′ in w1’s ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By strategy-proofness f(≿′  m1,w1, ≿−m1,w1  )(w1) ∈ {mk, m′  k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemmas 24 and 25 and group strategy-proofness (m1, wl′), (w1, mk′) ∈ f(≿′  m1,w1  , ≿−m1,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Dropping mk in woman w1’s ranking we see that woman w1 always gets her will (given  ≿−m1,w1) to marry a commoner if she once gets her will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By symmetry, she would always get her will  to marry m1 if she once gets her will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So the only possible regimes at and ≿−m1,w1 are the two serial  dictatorships as well as matched and unmatched-by-default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The royals are either matched-by-default or unmatched-by-default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix an arbitrary ≿−m1,w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By Lemma 26 the royals are either matched or unmatched-by-default  or engaged in a serial dictatorship with one of the royals choosing first, the other second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Suppose that  different profiles for the commoners were governed by different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Fix two such profiles where  the regimes change with the preference of one agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Say w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g that this agent is m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So the picking  regime for w1 and m1 is governed by different rules at ≿−m1,w1 and at (≿′  w2, ≿−m1,w1,m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  The outcomes of all regimes are identical when the two royals either top rank each other or  when they both rank commoners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  So we fix a profile ≿m1,w1 where exactly one royal top ranks  a commoner and say that at ≿−m1,w1 the royal who wants to marry the other royal wins, while  at (≿′  w2, ≿−m1,w1,m2) the royals marry commoners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' For concreteness assume that ≿m1: w1, wl and  ≿w1: mk, so that the regime change from ≿−m1,w1 to (≿′  w2, ≿−m1,w1,m2) is one from matched-by-  default or serial dictatorship with m1 going first to unmatched-by-default or serial dictatorship with  w1 going first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Mutatis mutandis the same arguments apply when ≿m1,w1 is such that w1 wants to  31  marry m1 who in turn wants to marry a commoner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  In sum we have that (m1, w1) ∈ f(≿) and  (w1, mk), (m1, wl) ∈ f(≿′  m2, ≿−m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Since the regime stays fixed as long as the commoners’ preferences stay fixed we may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='g  assume that mk = m2 so that f(≿′  w2, ≿−m1,w1,m2)(m2) = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By group strategy-proofness f(≿) =  f(≿∗  m2, ≿−m2) for any ≿∗  m2 that top ranks f(≿)(m2) and f(≿′′  m2, ≿−m1,w1,m2) = f(≿′  w2, ≿−m1,w1,m2)  for any ≿′′  m2: w1, f(≿)(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  Now change the royal couples preferences to ≿′  m1: w1, f(≿)(m2) and  ≿′  w1: m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Give that the regimes stay fixed at (≿∗  m2, ≿−m1,w1,m2) and (≿′′  m2, ≿−m1,w1,m2) we get  f(≿′  m1,w1, ≿∗  m2, ≿−m1,w1,m2) = f(≿)  (m1, f(≿)(m2)), (w1, m3) ∈ f(≿′  m1,w1, ≿′′  m2, ≿−m1,w1,m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  A contradiction to strategy-proofness results since f(≿′  m1,w1, ≿∗  m2, ≿−m1,w1,m2) matches m2  with f(≿)(m2) which is according to ≿′′  m2, m2’s second favorite wife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Conversely, since  (m1, f(≿)(m2)), (w1, m3) ∈ f(≿′  m1,w1, ≿′′  m2, ≿−m1,w1,m2)  the latter neither matches m2 with his ≿′′  m2-favorite wife w1 nor with his second favorite f(≿)(w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We in sum get that the regime with which the royals m1, w1 choose partners stays fixed for  all ≿−m1,w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' By gender-neutrality the royals must use a symmetric regime when ≿−m1,w1 is gender-  neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' So neither of the two serial dictatorships can govern the choices by the royals and we must  have that the regime is either matched-by-default or unmatched-by-default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  We have shown that there is a single royal couple who are either matched-by default, or un-  matched by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' The notion of gender-neutrality implies that, given the matches of the royal couple,  the remaining agents are engaged in a continuation mechanism which is required to be gender-neutral  with respect to some symmetry of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Applying the same arguments to these submechanisms  gives a second royal couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content=' Continuing in this way we get a sequence of royal couples until just four  agents remain at which point our arguments break down, and any one of the four-agent mechanisms  described in section A can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
+page_content='  32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFPT4oBgHgl3EQfPTQG/content/2301.13037v1.pdf'}
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+Scene Synthesis from Human Motion
+Sifan Ye∗
+Stanford University
+United States of America
+sifan.ye@cs.stanford.edu
+Yixing Wang∗
+Stanford University
+United States of America
+yiw998@stanford.edu
+Jiaman Li
+Stanford University
+United States of America
+jiamanli@stanford.edu
+Dennis Park
+Toyota Research Institute
+United States of America
+dennis.park@tri.global
+C. Karen Liu
+Stanford University
+United States of America
+karenliu@cs.stanford.edu
+Huazhe Xu†
+Stanford University
+United States of America
+huazhexu@stanford.edu
+Jiajun Wu†
+Stanford University
+United States of America
+jiajunwu@cs.stanford.edu
+(a) Human Motion Input
+(b) Synthesized Scene with Semantic Labels
+(c) Synthesized Scene with Textures
+Figure 1: From a human motion sequence, SUMMON synthesizes physically plausible and semantically reasonable objects.
+ABSTRACT
+Large-scale capture of human motion with diverse, complex scenes,
+while immensely useful, is often considered prohibitively costly.
+Meanwhile, human motion alone contains rich information about
+the scene they reside in and interact with. For example, a sitting
+human suggests the existence of a chair, and their leg position
+further implies the chair’s pose. In this paper, we propose to syn-
+thesize diverse, semantically reasonable, and physically plausible
+scenes based on human motion. Our framework, Scene Synthesis
+from HUMan MotiON (SUMMON), includes two steps. It first uses
+ContactFormer, our newly introduced contact predictor, to obtain
+temporally consistent contact labels from human motion. Based
+on these predictions, SUMMON then chooses interacting objects
+and optimizes physical plausibility losses; it further populates the
+scene with objects that do not interact with humans. Experimental
+results demonstrate that SUMMON synthesizes feasible, plausible,
+and diverse scenes and has the potential to generate extensive
+human-scene interaction data for the community.
+∗ and † indicate equal contribution. https://sites.google.com/stanford.edu/summon
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+© 2022 Copyright held by the owner/author(s).
+ACM ISBN 978-1-4503-9470-3/22/12.
+https://doi.org/10.1145/3550469.3555426
+CCS CONCEPTS
+• Computing methodologies → Motion processing; Shape in-
+ference; Scene understanding.
+KEYWORDS
+Scene synthesis, motion analysis, activity understanding
+ACM Reference Format:
+Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C. Karen Liu, Huazhe Xu,
+and Jiajun Wu. 2022. Scene Synthesis from Human Motion. In SIGGRAPH
+Asia 2022 Conference Papers (SA ’22 Conference Papers), December 6–9, 2022,
+Daegu, Republic of Korea. ACM, New York, NY, USA, 9 pages. https://doi.
+org/10.1145/3550469.3555426
+1
+INTRODUCTION
+Capturing, modeling, and synthesizing realistic human motion in
+3D scenes is crucial in a spectrum of applications such as virtual
+reality, game character animation, and human-robot interaction. To
+facilitate research in this area, a plethora of datasets [Hassan et al.
+2019; Mahmood et al. 2019; Savva et al. 2016] have been curated
+to capture human motion. For example, Bhatnagar et al. [2022]
+collected trajectories of humans manipulating objects. The PROX-E
+dataset [Zhang et al. 2020a] contains human contact with a scene
+mesh. However, building high-quality, large-scale datasets anno-
+tated with both diverse human motions and rich 3D scenes remains
+challenging. This is mainly because current data capture pipelines
+depend on costly devices, such as MoCap systems, structure cam-
+eras, and 3D scanners, and therefore can only be conducted in
+laboratory settings, which entails limited physical space and scene
+arXiv:2301.01424v1  [cs.GR]  4 Jan 2023
+
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C. Karen Liu, Huazhe Xu, and Jiajun Wu
+diversity. Inspired by recent advances in modeling 3D human poses
+and their contact with environments, we aim to address these chal-
+lenges by exploring a new possibility: can we learn to synthesize the
+scenes only from human motion? If successful, our system will also
+have many potential applications beyond data collection, such as
+providing suggestions during the creation of virtual environments
+based on artists’ motions in VR.
+Recent works have proposed to estimate room layouts based
+on human trajectories and learned room priors [Nie et al. 2022].
+However, only semantics, not affordances, was considered in the
+reconstructed layouts. Yi et al. [2022] proposed to reconstruct scene
+objects from visual inputs and then use Human-Scene Interactions
+(HSIs) to further improve the feasibility. While such a method pro-
+duces physically plausible reconstructions, it requires additional
+visual inputs so that the reconstructed scenes are restricted.
+We propose Scene Synthesis from HUMan MotiON (SUMMON),
+a method that predicts feasible object placements in a scene based
+solely on 3D human pose trajectories, as shown in Figure 1. SUM-
+MON consists of two modules: a human-scene contact prediction
+module and a scene synthesis module. The human-scene contact
+prediction module, named ContactFormer, leverages existing HSI
+data to learn a mapping from human body vertices to the semantic
+label of the objects that are in contact. ContactFormer advances
+previous methods [Hassan et al. 2021b] by incorporating temporal
+cues to enhance the consistency in label prediction in time. Given
+the estimated semantic contact points, the scene synthesis module
+first searches for objects that fit the contact points in terms of se-
+mantics and physical affordances to the agent; it then populates the
+scene with other objects that have no contact with humans, based
+on human motion and objects inferred from previous steps.
+We conduct our experiments using the PROXD [Hassan et al.
+2019] and the GIMO [Zheng et al. 2022] datasets. In terms of con-
+tact estimation, ContactFormer outperforms previous single-frame
+contact prediction methods [Hassan et al. 2021b]. In terms of scene
+synthesis, our proposed system shows more realistic, physically
+plausible, and diverse scenes than baselines, using various metrics
+and human evaluation.
+Our contributions are threefold. First, we propose SUMMON, a
+system that synthesizes semantically reasonable, physically plausi-
+ble, and diverse scenes based only on human motion trajectories.
+Second, as a part of SUMMON, we propose a contact prediction mod-
+ule ContactFormer that outperforms existing methods by modeling
+the temporal consistency in semantic labels. Third, we demonstrate
+that the scenes synthesized by SUMMON consistently outperform
+existing methods both qualitatively and quantitatively.
+2
+RELATED WORKS
+Scene affordance learning. Learning affordance from human-scene
+interaction has caught much attention recently [Chen et al. 2019;
+Chuang et al. 2018; Delaitre et al. 2012; Fouhey et al. 2012; Gupta
+et al. 2011; Wang et al. 2019a; Zhu et al. 2014]. In the literature,
+researchers study how to put human skeletons in a scene. For ex-
+ample, Wang et al. [2017] proposed to learn the affordance from
+sitcom videos for positioning skeletons in a static image. Li et al.
+[2019a] introduced a generative model of 3D poses to predict plau-
+sible human poses in a scene. Along with developing better human
+body representations, there have been methods that try to put a
+3D human body into the scene [Zhang et al. 2020a]. More recently,
+POSA [Hassan et al. 2021b] learns a model that augments a SMPL-X
+human body model vertices with contact probability and semantic
+labels to place human poses in a 3D scene mesh. Blinn et al. [2021]
+proposed a fitting and comfort-based loss to train an affordance-
+aware model to generate chairs that fit a human body pose. Several
+works also try to collect or generate data that involve human-scene
+interactions. For example, VirtualHome [Puig et al. 2018] provides
+a simulated 3D environment where humanoid agents can interact
+with 3D objects. BEHAVE [Bhatnagar et al. 2022] provides a dataset
+of real full-body human parameterized using SMPL interacting and
+manipulating objects in 3D with contact points. Our work takes an
+additional step from the affordance learning works: we first learn to
+understand the affordance, then leverage them to synthesize scenes
+that can be used for other related tasks.
+Human motion synthesis. Motion synthesis is a long-standing
+problem in computer graphics and vision [Brand and Hertzmann
+2000; Holden et al. 2016; Kovar and Gleicher 2003; Park et al. 2002;
+Spallone 2015]. Xu et al. [2020] proposed a hierarchical way to
+generate long-horizon motion by using a memory bank to retrieve
+short-horizon reference clips. Harvey et al. [2020] proposed to pre-
+dict motion robustly with additional embeddings. Recently, many
+works also take the environment into consideration [Hassan et al.
+2021a; Rempe et al. 2021; Wang et al. 2021a]. For example, Wang
+et al. [2021a] combined long-term human motion synthesis condi-
+tioned on a scene mesh with affordance optimization to generate
+realistic human trajectories. SAMP [Hassan et al. 2021a] learns
+generalized interaction for object classes across different instances
+of that class. Our work is trying to solve the inverse problem that
+generates plausible scenes given human motion trajectories.
+Scene synthesis. Our work is also closely related to synthesizing
+plausible 3D scenes and room layout [Li et al. 2019b; Luo et al. 2020;
+Purkait et al. 2020; Ritchie et al. 2019; Wang et al. 2019b, 2021b;
+Zhang et al. 2020b; Zhou et al. 2019]. For example, ATISS [Paschali-
+dou et al. 2021] learns an autoregressive generative model for furni-
+ture placement. It can be used for generating plausible novel room
+layouts, completing a scene given existing objects, and suggest-
+ing possible placements given spatial constraints. Another work,
+Pose2Room [Nie et al. 2022], predicts bounding boxes of objects
+conditioned on 3D human pose trajectory. MOVER [Yi et al. 2022]
+reconstructs 3D objects constrained by 3D human body predictions
+from monocular RGB videos. Unlike these prior methods, our model
+generates not only layouts but also affordable objects with only
+human trajectories.
+3
+METHOD
+We aim to predict a set of furniture objects and a physically plausible
+3D configuration of them only from human motion sequences.
+We first introduce the human body and contact representation
+in Sec. 3.1. SUMMON generates a temporally consistent contact
+semantic estimation for each vertex of the human body to retrieve
+suitable objects (Sec. 3.2). Then we optimize object placement based
+on the contact locations and physical plausibility (Sec. 3.3). An
+illustration of our method is shown in Figure 2.
+
+Scene Synthesis from Human Motion
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Objects 
+in Interaction
+Per-frame Contact 
+Predictions
+Accumulated 
+Contact Points
+Object
+Optimization
+Input Human
+Meshes
+ContactFormer
+Chair
+Partial
+Contact Points
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 2: The overview of SUMMON: (a) an input sequence of human body meshes interacting with a scene, (b) the Contact-
+Former that predicts per-frame contact labels, (c) per-frame contact predictions, (d) estimated contact points, (e) synthesized
+objects, and (f) objects in interaction.
+3.1
+Human Body and Contact Representation
+We use a modified version of SMPL-X [Pavlakos et al. 2019] as the
+representation of human body poses. Specifically, we parameterize
+the human body with 𝑀(𝜃, 𝛽) : R|𝜃 |×|𝛽 | → R3𝑁 , where 𝜃 denotes
+pose parameters, 𝛽 denotes coefficients in a learned shape space,
+and 𝑁 is the number of vertices in a SMPL-X body mesh. For com-
+putation efficiency, we downsample the vertices from 10,475 to
+655 points, following the prior work by Hassan et al. [2021b].
+We represent contact information by per-vertex features. For
+each vertex 𝑣𝑏 ∈ 𝑉𝑏, where 𝑉𝑏 is all vertices of a human body, we
+use a one-hot vector 𝑓 to represent the contact semantic label for
+that vertex. Each vector 𝑓 has a length of |𝑓 | = 𝐶 + 1, where 𝐶 is
+the number of object classes. We introduce an extra “void” class to
+represent vertices without contact. We use 𝐹 to denote the contact
+semantic labels for all vertices in a body pose.
+3.2
+Human-Scene Contact Prediction
+Our dataset consists of a sequence of paired vertices and contact
+semantic labels {(𝑉 1
+𝑏 , 𝐹 1), (𝑉 2
+𝑏 , 𝐹 2), ..., (𝑉 𝑛
+𝑏 , 𝐹𝑛)}, where 𝑉 𝑖
+𝑏 repre-
+sents the human body vertices (Figure 2(a)), 𝐹𝑖 represents the con-
+tact semantic labels for frame 𝑖, and 𝑛 is the varied sequence length.
+We first train a conditional Variational Autoencoder (cVAE) to learn
+a probabilistic model of contact semantic labels conditioned on ver-
+tex positions. Then we deploy transformer layers on top of the cVAE
+to improve temporal consistency. We refer to this framework as
+ContactFormer. An illustration of the overall network architecture
+is shown in Figure 3.
+Contact semantics prediction. We first train a model to predict
+contact semantic labels for each individual pose. Given a pair of
+body vertices and contact semantic labels (𝑉𝑏, 𝐹), we first fuse these
+two components: 𝐼𝑒 = Concat(𝑉𝑏, 𝐹). We feed 𝐼𝑒 into a graph neural
+network (GNN) encoder 𝐺𝐸𝑛𝑐 to get a latent Gaussian space with
+the mean 𝐻𝜇 and the standard deviation 𝐻𝜎. Then we sample a
+latent vector 𝑧 from the latent Gaussian space and concatenate it
+with each vertex position: 𝐼𝑑 = Concat(𝑉𝑏,𝑧). We feed 𝐼𝑑 into a
+GNN
+Encoder
+GNN
+Decoder
+������������
+Positional
+Embedding
+Transformer
+MLP
+�������������1, ⋯ , �������������������������
+������������1, ⋯ , ������������������������
+������������������������
+1, ⋯ , ������������������������
+������������
+������������������������1, ⋯ , ������������������������������������
+(������������������������
+1, ������������1), ⋯ , (������������������������
+������������, ������������������������)
+For frame i,
+������������������������: Ground-truth semantic labels
+������������������������������������: Initial predicted semantic labels
+������������������������: Hidden feature
+�������������������������: Final predicted semantic labels
+z : Latent vector
+Figure 3: The architecture of ContactFormer. We first use a
+GNN-based variational autoencoder to encode the contact
+points. Then a transformer is applied to improve the tempo-
+ral information fusion. We also add a sinusoidal positional
+encoding to the output of the GNN decoder.
+GNN decoder 𝐺𝐷𝑒𝑐 to predict the reconstructed contact semantic
+labels 𝐹𝑝. Note that both GNNs in the encoder and the decoder
+share the same structure as in Hassan et al. [2021b]. Each vertex
+feature ℎ𝑘𝑥 for vertex 𝑥 at layer 𝑘 is updated by
+ℎ𝑘
+𝑥 = Linear(Concat({ℎ𝑘−1
+𝑥′
+: 𝑥 ′ ∈ 𝑁 (𝑥)})),
+(1)
+where 𝑁 (𝑥) is defined as the 𝑚-nearest neighbor vertices of 𝑥 in a
+spiral-ordered sequence, as proposed by Gong et al. [2019].
+ContactFormer: We train a transformer to extract temporal infor-
+mation from a pose sequence to enhance prediction consistency, as
+shown in Figure 3. Specifically, given a sequence of pose and con-
+tact semantic labels {(𝑉 1
+𝑏 , 𝐹 1), ..., (𝑉 𝑛
+𝑏 , 𝐹𝑛)} from frame 1 to 𝑛, we
+first use the previous model to reconstruct contact semantic labels
+𝐹𝑖𝑝 independently for each frame 𝑖. We then embed each 𝐹𝑖𝑝 into a
+hidden feature space, augmenting it with a sinusoidal positional
+
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C. Karen Liu, Huazhe Xu, and Jiajun Wu
+embedding before feeding it to the transformer module. The output
+of the transformer module is a sequence of n vectors {𝐻1, ..., 𝐻𝑛}.
+For each frame 𝑖, we concatenate 𝐻𝑖 with the initial prediction 𝐹𝑖𝑝
+and use a multi-layer perceptron (MLP) to get a final prediction ^𝐹𝑖.
+The final prediction is shown in Figure 2(c).
+Training: We optimize the model’s parameters by the following
+loss function:
+L = L𝑟𝑒𝑐 + 𝛼 · L𝐾𝐿,
+(2)
+where L𝑟𝑒𝑐 is the sum of the categorical cross entropy (CCE) loss be-
+tween the ground truth semantic label 𝐹𝑖 and the model prediction
+^𝐹𝑖 for any frame 𝑖:
+L𝑟𝑒𝑐 =
+∑︁
+𝑖
+CCE(𝐹𝑖, ^𝐹𝑖),
+(3)
+and L𝐾𝐿 is the Kullback-Leibler divergence loss between the latent
+Gaussian space and the normal distribution N:
+L𝐾𝐿 = 𝐾𝐿(𝑄(𝑧|𝐹,𝑉𝑏)||N).
+(4)
+Here we use 𝑄 to represent the encoder network in our cVAE
+combined with the sampling process with the reparameterization
+trick. Inspired by Higgins et al. [2016], we also multiply L𝐾𝐿 with a
+weight𝛼 to control the balance between the reconstruction accuracy
+and diversity.
+3.3
+Scene Synthesis
+Contact Object Recovery. Given the accumulated contact points
+from each frame predicted by ContactFormer (Figure 2(d)), we
+further reduce spatial prediction noise by performing a local object
+class majority voting as shown in Figure 4. Then, the vertices of each
+predicted object class are clustered into possible contact instances
+𝑉𝑐, using the shortest length of all object edges in that class as 𝜖 for
+clustering. In practice, we downsample the contact vertices to keep
+later computations tractable.
+We then optimize the poses of the object point cloud 𝑉𝑜 by
+minimizing the following losses:
+L(𝑉𝑐,𝑉𝑜) = L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 + L𝑝𝑒𝑛.
+(5)
+The contact loss L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is defined as
+L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡
+1
+|𝑉𝑐 |
+∑︁
+𝑣𝑐 ∈𝑉𝑐
+min
+𝑣𝑜 ∈𝑉𝑜
+||𝑣𝑐 − 𝑣𝑜 ||2
+2,
+(6)
+where 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is a tunable hyperparameter. This loss encourages
+the object to be in contact with the predicted human vertices. The
+penetration loss L𝑝𝑒𝑛 is defined as:
+L𝑝𝑒𝑛 = 𝜆𝑝𝑒𝑛
+∑︁
+𝑑𝑖𝑐<𝑡
+𝑑𝑖
+𝑐
+2,
+(7)
+where 𝑑𝑖𝑐 are signed distances between the object and the human
+body sequence, 𝑡 is the penetration distance threshold. This loss
+prevents the object from penetrating the human body sequence.
+Intuitively, these losses encourage objects to be in contact with
+human meshes, but not penetrate them. An illustration of the opti-
+mized object placement is shown in Figure 2(e). To improve compu-
+tation efficiency, we choose to compute human SDF from merged
+human meshes of the motion sequence. To have a consistent scale
+of loss across different objects, we choose the number of sampled
+points according to the size of the object.
+Inconsistencies
+After Voting
+table
+chair
+sofa
+bed
+Figure 4: Illustration of the local majority voting. From the
+zoomed-in box, there are multiple inconsistent points in the
+original contact points. The pink points represent the se-
+mantic label bed, and the green points represent the label
+sofa. We alleviate this issue by adding majority voting.
+Constrained Scene Completion. To obtain a complete scene, we
+also predict non-contact objects as a scene completion task con-
+strained by 3D human trajectories and existing in-contact objects.
+The floor is divided into a grid, and each cell is labeled as occupied
+if feet vertices or object vertices are in close proximity. Consider-
+ing the furniture categories in a room as a sequence, we train an
+autoregressive transformer model on the 3D-FRONT dataset [Fu
+et al. 2021a]. The model takes as input the categories of existing
+objects and returns a probability distribution of the next potential
+object category. We sample a category from the distribution and
+randomly place an object of that category onto an unoccupied floor
+grid. To prevent the sampled object from penetrating the human
+body sequence, we further optimize the object’s translation and
+rotation using our L𝑝𝑒𝑛 (see Equation 7).
+4
+EXPERIMENT SETUP
+In this section, we introduce the datasets and implementation details
+for the whole SUMMON framework.
+4.1
+Datasets
+We use the PROXD [Hassan et al. 2019] dataset for training our
+ContactFormer. PROXD uses RGB-D cameras to capture 20 human
+subjects interacting with 12 scenes. We represent human poses
+using the SMPL-X format to reconstruct human body meshes. The
+pose sequences in PROXD are estimated using SMPLify [Bogo
+et al. 2016] and contain many jitters. We apply LEMO [Zhang et al.
+2021], a learned temporal motion smoothness prior, to produce
+smooth human motion as training data. Our ground truth per vertex
+contact semantic labels are generated using scene SDF with contact
+semantic labels from PROX-E [Zhang et al. 2020a], which extends
+PROXD by manually annotating the scene meshes with predefined
+object categories. We define human-scene contact as the signed
+distance between a human vertex and the scene to be less than
+0.05.
+We select objects from 3D-FUTURE [Fu et al. 2021b] to be placed
+into the scenes. 3D-FUTURE is a dataset of categorized 3D models
+of furniture with their original sizes. We use a selected subset of
+3D-FUTURE to reduce candidate search time. To simplify contact
+estimation and limit predicted object classes to the available ones
+
+Scene Synthesis from Human Motion
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+in our object dataset, we reduce the contact object categories in the
+PROX-E dataset from 42 to 8.
+We use the GIMO dataset [Zheng et al. 2022] as another test
+dataset for evaluating the generalization ability of the proposed
+method on out-of-distribution data.
+4.2
+Implementation
+ContactFormer. For the encoder and decoder GNNs, we choose
+the number of hidden layers to be 3. The dimension for each hidden
+vertex feature in the GNNs is 64. In the GNN encoder, we downsam-
+ple the body vertices after each hidden layer by a factor of 4. We
+deploy a similar architecture for the transformer layers as used in
+the previous work [Vaswani et al. 2017]. We provide training details
+and hyperparameter choices in the supplementary materials.
+To compare different architectures’ capacities for extracting tem-
+poral information, we also implement models that use MLP and
+LSTM [Greff et al. 2016] modules as the final block on top of the
+GNN decoder. For the model that uses the MLP module, we deploy
+a max pooling layer to the output of the GNN decoder along the
+dimension of vertices. Then we feed it to an MLP block to get the
+embedding for the whole sequence. The sequence embedding is
+then fused with the output of the GNN decoder to get the final
+prediction via a linear projection. For the model that uses the LSTM
+module, we linearly project the outputs from the GNN decoder into
+a higher dimensional embedding space and feed them to a bidirec-
+tional LSTM layer to extract features for each frame. Frame features
+are then concatenated with the output from the GNN decoder to
+obtain final semantic labels.
+Contact Object Recovery. To reduce noise in contact semantic
+estimation, we use majority semantic voting in point cloud clusters
+with 𝜖 = 0.1 and 𝑚𝑖𝑛𝑃𝑡𝑠 = 10. In point cloud clustering for object
+instance fitting, we used different values for 𝜖 for different classes
+due to their different sizes.
+To place objects into the scene at an appropriate height, we first
+cluster all the human body vertices that are in contact with the floor.
+We then take the minimum medians of all clusters as the estimated
+floor height. Next, we translate the object to place its lowest vertex
+on the floor.
+To avoid local minima, we perform a grid search for translation
+along the floor plane and rotation around the up axis to warm-
+start the initial transformation. We then optimize for the same
+transformation parameters on top of the results from the grid search.
+In both cases, We use different 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡, 𝜆𝑝𝑒𝑛 and 𝑡 to accommodate
+for different properties of object classes. We keep the transformation
+that achieves the lowest loss as the optimization result.
+To achieve scene diversity, we consider inter-class and intra-class
+diversity. Inter-class diversity is when a human motion is likely
+to interact with different classes of objects. For example, sitting
+down can be performed on a chair, a bed, or a sofa. To achieve this,
+we first sample per-vertex contact semantics based on the contact
+probability distribution predicted by ContactFormer. During local
+clustering of contact object recovery, we consider class labels in
+local clusters as a probability distribution and sample the cluster
+contact class. Intra-class diversity is when a human motion is likely
+to interact with different instances of the same object class. To
+Table 1: Results of contact prediction. We use the reconstruc-
+tion accuracy and the consistency score as metrics. Our Con-
+tactFormer clearly outperforms the baselines.
+Models
+Reconstruction Acc. ↑
+Consistency Score ↑
+MLP Predictor
+0.9082
+0.8922
+LSTM Predictor
+0.9087
+0.9209
+POSA
+0.9106
+0.8816
+ContactFormer (ours)
+0.9120
+0.9518
+First time on the bed
+Second time on the bed
+POSA
+ContactFormer
+Bed
+Sofa
+Figure 5: Visualizations of the contact prediction results
+of POSA and our ContactFormer. Left: Contact predictions
+from POSA and ContactFormer when the person lies on
+the bed. Right: Contact predictions from POSA and Contact-
+Former when the person lies on the bed again after walking
+around. ContactFormer has better consistency when the per-
+son lies in bed for the second time.
+achieve this, we perform grid search and optimization on all the
+instances from the object class.
+5
+EVALUATIONS
+In this section, we introduce evaluation metrics, baselines, and
+results on contact prediction and scene synthesis. We encourage
+the readers to watch the video in the supplementary materials.
+5.1
+Contact Semantic Prediction
+Baselines. We compare with three baselines, including POSA [Has-
+san et al. 2021b], an architectural variant that uses a multi-layer
+perceptron (MLP) based predictor, and a temporal information fu-
+sion variant that uses a bidirectional LSTM [Greff et al. 2016].
+Metrics. We use two metrics for evaluating the contact semantic
+prediction: reconstruction accuracy and consistency score. The re-
+construction accuracy is computed as the average correctness of
+the predicted label compared with the ground-truth label for each
+vertex. The consistency score is designed following this intuition: if
+we accumulate predicted contact points from each frame, close con-
+tact points should have consistent contact semantic labels. Hence,
+this loss is computed as follows: Given a pose sequence and the ac-
+cumulated contact points, for each point, we compare its predicted
+contact label with the contact labels of its neighboring points to
+see if the prediction agrees with the majority of the neighboring
+contact labels (i.e., a high consistency score).
+Results. Table 1 shows the reconstruction accuracy and the con-
+sistency score of all methods on the validation set of PROXD. We
+find that ContactFormer achieves competitive performance in terms
+
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C. Karen Liu, Huazhe Xu, and Jiajun Wu
+PROXD
+PROXD
+GIMO
+Human Meshes
+ContactICP
+Pose Priors
+Ours Snapshot 1
+Ours Snapshot 2
+Ours Snapshot 3
+Figure 6: Visualizations of the generated objects based on human trajectories. The human trajectories are from the PROXD
+dataset and the unseen GIMO dataset. The first column shows the trajectory. The second column shows the results from
+the ContactICP baseline. The third column shows the results from the Pose Prior baseline. The fourth to sixth columns are
+snapshots of results generated by SUMMON.
+of reconstruction accuracies and significantly outperforms all the
+baselines in consistency scores. This demonstrates the superior-
+ity of the transformer-based architecture in predicting temporally
+consistent yet accurate contact labels.
+Figure 5 visualizes the output contact labels from ContactFormer
+and POSA. We notice that ContactFormer predicts consistent labels
+the second time the human tries to lie on the bed, while POSA, due
+to its lack of temporal information, predicts a different label.
+User study. We conduct a user study to evaluate the quality of
+the contact semantic label predictions, where we compare Con-
+tactFormer with POSA. For each pose sequence in the validation
+dataset, we render a video showing the human motion, predicted
+contact semantic labels, and the ground truth scene. We show the
+predicted contact semantic labels by rendering small areas around
+body vertices in different colors depending on their labels. Each
+video is rendered from a camera angle that can clearly capture
+human motion and semantic labels. For each pose sequence, we ask
+the human subjects the following question: "Which video seems to
+have a more reasonable contact label prediction?". Among 22 users,
+78.12% of the users choose ContactFormer over POSA, believing
+ContactFormer provides more reasonable and convincing results.
+This result echoes the quantitative results in Table 1.
+5.2
+Contact Object Recovery
+Baselines. Since our problem is novel and there are no baselines,
+we devise two reasonable baselines ourselves: contact-informed
+point cloud registration (ContactICP) based on point-to-point ICP [Besl
+and McKay 1992] and object alignment with pose priors (Pose Pri-
+ors) based on the orientation of the hip. We provide the details of
+those methods in the supplementary materials.
+Metrics. We use the non-collision score proposed by Zhang et al.
+[2020a]. This score estimates the collision ratio between human
+body mesh and scene objects. Since all the methods, including
+Table 2: Non-collision scores for contact object recovery on
+the smoothed PROXD and the unseen GIMO dataset. For
+each sequence, the score is computed to be the mean of all
+possible generated scenes. Higher scores are better.
+Method
+PROXD
+GIMO
+ContactICP
+0.654
+0.820
+Pose Priors
+0.703
+0.798
+SUMMON w/o optimization
+0.815
+0.937
+SUMMON (ours)
+0.851
+0.951
+SUMMON, first align the object to the centroid of contact points,
+contact constraints are naturally satisfied.
+Results. For each sequence, we compute the mean of the non-
+collision scores for all the objects in the scene. In Table 2, we compare
+the mean non-collision scores on the smoothed PROXD dataset [Zhang
+et al. 2021], which was used during training, and the unseen GIMO
+dataset [Zheng et al. 2022], which also provides SMPL-X parameters
+for humans interacting with scenes.
+We visualize comparisons between our method and the base-
+lines in Figure 6. We find that SUMMON can synthesize objects
+that are physically plausible and semantically reasonable. Contac-
+tICP usually suffers from large penetrations because the contact
+points might be sparse for registration. While Pose Priors can have
+seemingly correct object locations and orientations, it often fails to
+consider physical constraints.
+Figure 7 demonstrates various possible scenes generated from
+the same human motion trajectory by SUMMON. We find that
+SUMMON can generalize intra-class (e.g., chairs with different ap-
+pearances) and inter-class (e.g., sofa to a bed). We provide additional
+examples in the supplementary materials.
+Human user study. We follow the same procedure as in Sec-
+tion 5.1. Instead of contact prediction, we present the users with
+
+灵AScene Synthesis from Human Motion
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Human Meshes
+Possible Scene 1
+Possible Scene 2
+Possible Scene 3
+Figure 7: Visualizations of possible alternative object placements generated by SUMMON based on the same human trajectories.
+In this example, an in-contact object can be a chair, a sofa, or a bed, as long as it does not violate physical constraints. SUMMON
+can also generate different instances (e.g., chairs) within the same category.
+Human Meshes
+Object Recovery
+Synthesized Scene 1
+Synthesized Scene 2
+Figure 8: Visualizations of scene completion. Based on all the in-contact objects and human motion trajectories, SUMMON
+now generates the objects that are not in contact with human meshes. While there is no contact, it makes the scene more
+complete and introduces the potential for future synthesized human motion sequences to interact with additional objects.
+the animated human motion sequences and the predicted objects in
+the scene, and ask them to choose the most plausible and realistic
+placement. From the statistics, we find that 74.5% of the users se-
+lect SUMMON over ContactICP and Pose Priors. We find that Pose
+Priors has a 23.5% user selection rate, showing that it can produce
+reasonable results in some cases.
+Ablations. We also perform ablation on the optimization objec-
+tives. Table 3 shows that both the penetration loss and the contact
+loss are important for SUMMON. Intuitively, the penetration loss
+helps the object to avoid a collision, while the contact loss helps
+to keep the object close to humans. We use both the non-collision
+score and the contact score. The contact score is computed as the
+fraction of objects in the scenes that are in contact with the human
+trajectory Zhang et al. [2020a].
+5.3
+Scene completion
+To generate a full-fledged scene, we train another object generation
+model following Paschalidou et al. [2021] as in Section 3.3. The
+model outputs a family of possible objects that does not contact or
+penetrate human meshes. Using this model, we generate a fuller
+scene with both in-contact and no-contact objects. Visualized re-
+sults are in Figure 8. The completed scenes have additional objects,
+such as a TV stand or a coffee table. While there is no contact be-
+tween these objects and the human meshes, they make the scene
+semantically more realistic.
+Table 3: Ablation study on the losses. The penetration loss
+and the contact loss are ablated. We use the non-collision
+score and the contact score as metrics.
+Method
+non-collision score ↑
+contact score ↑
+SUMMON
+0.894
+1
+w/o penetration loss
+0.656
+1
+w/o contact loss
+0.995
+0.194
+6
+CONCLUSION
+We propose Scene Synthesis from HUMan MotiON (SUMMON), a
+framework that generates multi-object scenes from a sequence of
+human interaction. SUMMON leverages human contact estimations
+and scene priors to produce scenes that realistically support the
+interaction and the semantic context. The flexibility of SUMMON
+also enables the synthesis of diverse scenes from a single motion
+sequence. We hope this can also shed light on generating inexpen-
+sive diverse human-scene interaction datasets. In the future, we are
+interested in exploring the following directions. Since PROXD does
+not consider soft-body interactions, a potential direction would
+be considering soft-body deformation of objects such as beds and
+sofas. Our method considers synthesized scenes to be stationary,
+hence future works can include movement and rearrangement of
+furniture during human-scene interaction. As PROXD categorizes
+all the smaller interaction objects such as books, cups, or TV re-
+motes into a single category, one potential extension to our method
+would be to include interactions with more specific small objects.
+
+SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea
+Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C. Karen Liu, Huazhe Xu, and Jiajun Wu
+ACKNOWLEDGMENTS
+This work is in part supported by the Stanford Human-Centered
+AI Institute (HAI), the Toyota Research Institute (TRI), Innopeak,
+Meta, Bosch, and Samsung.
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+
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+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='edu  Huazhe Xu†  Stanford University  United States of America  huazhexu@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='edu  Jiajun Wu†  Stanford University  United States of America  jiajunwu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='edu  (a) Human Motion Input  (b) Synthesized Scene with Semantic Labels  (c) Synthesized Scene with Textures  Figure 1: From a human motion sequence, SUMMON synthesizes physically plausible and semantically reasonable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  ABSTRACT  Large-scale capture of human motion with diverse, complex scenes,  while immensely useful, is often considered prohibitively costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Meanwhile, human motion alone contains rich information about  the scene they reside in and interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, a sitting  human suggests the existence of a chair, and their leg position  further implies the chair’s pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In this paper, we propose to syn-  thesize diverse, semantically reasonable, and physically plausible  scenes based on human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our framework, Scene Synthesis  from HUMan MotiON (SUMMON), includes two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' It first uses  ContactFormer, our newly introduced contact predictor, to obtain  temporally consistent contact labels from human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Based  on these predictions, SUMMON then chooses interacting objects  and optimizes physical plausibility losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' it further populates the  scene with objects that do not interact with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Experimental  results demonstrate that SUMMON synthesizes feasible, plausible,  and diverse scenes and has the potential to generate extensive  human-scene interaction data for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  ∗ and † indicate equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='com/stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='edu/summon  Permission to make digital or hard copies of part or all of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  © 2022 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  ACM ISBN 978-1-4503-9470-3/22/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1145/3550469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3555426  CCS CONCEPTS  Computing methodologies → Motion processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Shape in-  ference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  KEYWORDS  Scene synthesis, motion analysis, activity understanding  ACM Reference Format:  Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Karen Liu, Huazhe Xu,  and Jiajun Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Scene Synthesis from Human Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In SIGGRAPH  Asia 2022 Conference Papers (SA ’22 Conference Papers), December 6–9, 2022,  Daegu, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1145/3550469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3555426  1  INTRODUCTION  Capturing, modeling, and synthesizing realistic human motion in  3D scenes is crucial in a spectrum of applications such as virtual  reality, game character animation, and human-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To  facilitate research in this area, a plethora of datasets [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Savva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016] have been curated  to capture human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2022]  collected trajectories of humans manipulating objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The PROX-E  dataset [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020a] contains human contact with a scene  mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' However, building high-quality, large-scale datasets anno-  tated with both diverse human motions and rich 3D scenes remains  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' This is mainly because current data capture pipelines  depend on costly devices, such as MoCap systems, structure cam-  eras, and 3D scanners, and therefore can only be conducted in  laboratory settings, which entails limited physical space and scene  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='01424v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='GR]  4 Jan 2023  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Karen Liu, Huazhe Xu, and Jiajun Wu  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Inspired by recent advances in modeling 3D human poses  and their contact with environments, we aim to address these chal-  lenges by exploring a new possibility: can we learn to synthesize the  scenes only from human motion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' If successful, our system will also  have many potential applications beyond data collection, such as  providing suggestions during the creation of virtual environments  based on artists’ motions in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Recent works have proposed to estimate room layouts based  on human trajectories and learned room priors [Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  However, only semantics, not affordances, was considered in the  reconstructed layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2022] proposed to reconstruct scene  objects from visual inputs and then use Human-Scene Interactions  (HSIs) to further improve the feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' While such a method pro-  duces physically plausible reconstructions, it requires additional  visual inputs so that the reconstructed scenes are restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We propose Scene Synthesis from HUMan MotiON (SUMMON),  a method that predicts feasible object placements in a scene based  solely on 3D human pose trajectories, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' SUM-  MON consists of two modules: a human-scene contact prediction  module and a scene synthesis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The human-scene contact  prediction module, named ContactFormer, leverages existing HSI  data to learn a mapping from human body vertices to the semantic  label of the objects that are in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ContactFormer advances  previous methods [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b] by incorporating temporal  cues to enhance the consistency in label prediction in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Given  the estimated semantic contact points, the scene synthesis module  first searches for objects that fit the contact points in terms of se-  mantics and physical affordances to the agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' it then populates the  scene with other objects that have no contact with humans, based  on human motion and objects inferred from previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We conduct our experiments using the PROXD [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2019] and the GIMO [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In terms of con-  tact estimation, ContactFormer outperforms previous single-frame  contact prediction methods [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In terms of scene  synthesis, our proposed system shows more realistic, physically  plausible, and diverse scenes than baselines, using various metrics  and human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Our contributions are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' First, we propose SUMMON, a  system that synthesizes semantically reasonable, physically plausi-  ble, and diverse scenes based only on human motion trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Second, as a part of SUMMON, we propose a contact prediction mod-  ule ContactFormer that outperforms existing methods by modeling  the temporal consistency in semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Third, we demonstrate  that the scenes synthesized by SUMMON consistently outperform  existing methods both qualitatively and quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2  RELATED WORKS  Scene affordance learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Learning affordance from human-scene  interaction has caught much attention recently [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Delaitre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Fouhey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Gupta  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In the literature,  researchers study how to put human skeletons in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For ex-  ample, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2017] proposed to learn the affordance from  sitcom videos for positioning skeletons in a static image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  [2019a] introduced a generative model of 3D poses to predict plau-  sible human poses in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Along with developing better human  body representations, there have been methods that try to put a  3D human body into the scene [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' More recently,  POSA [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b] learns a model that augments a SMPL-X  human body model vertices with contact probability and semantic  labels to place human poses in a 3D scene mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Blinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2021]  proposed a fitting and comfort-based loss to train an affordance-  aware model to generate chairs that fit a human body pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Several  works also try to collect or generate data that involve human-scene  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, VirtualHome [Puig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2018] provides  a simulated 3D environment where humanoid agents can interact  with 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' BEHAVE [Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022] provides a dataset  of real full-body human parameterized using SMPL interacting and  manipulating objects in 3D with contact points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our work takes an  additional step from the affordance learning works: we first learn to  understand the affordance, then leverage them to synthesize scenes  that can be used for other related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Human motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Motion synthesis is a long-standing  problem in computer graphics and vision [Brand and Hertzmann  2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Holden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Kovar and Gleicher 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Spallone 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2020] proposed a hierarchical way to  generate long-horizon motion by using a memory bank to retrieve  short-horizon reference clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2020] proposed to pre-  dict motion robustly with additional embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Recently, many  works also take the environment into consideration [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Rempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2021a] combined long-term human motion synthesis condi-  tioned on a scene mesh with affordance optimization to generate  realistic human trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' SAMP [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a] learns  generalized interaction for object classes across different instances  of that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our work is trying to solve the inverse problem that  generates plausible scenes given human motion trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Scene synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our work is also closely related to synthesizing  plausible 3D scenes and room layout [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Purkait et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019b, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, ATISS [Paschali-  dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021] learns an autoregressive generative model for furni-  ture placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' It can be used for generating plausible novel room  layouts, completing a scene given existing objects, and suggest-  ing possible placements given spatial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Another work,  Pose2Room [Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022], predicts bounding boxes of objects  conditioned on 3D human pose trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' MOVER [Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022]  reconstructs 3D objects constrained by 3D human body predictions  from monocular RGB videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Unlike these prior methods, our model  generates not only layouts but also affordable objects with only  human trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  3  METHOD  We aim to predict a set of furniture objects and a physically plausible  3D configuration of them only from human motion sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We first introduce the human body and contact representation  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' SUMMON generates a temporally consistent contact  semantic estimation for each vertex of the human body to retrieve  suitable objects (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then we optimize object placement based  on the contact locations and physical plausibility (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' An  illustration of our method is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Scene Synthesis from Human Motion  SA ’22 Conference Papers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' December 6–9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Daegu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Republic of Korea  Objects  in Interaction  Per-frame Contact  Predictions  Accumulated  Contact Points  Object  Optimization  Input Human  Meshes  ContactFormer  Chair  Partial  Contact Points  (a)  (b)  (c)  (d)  (e)  (f)  Figure 2: The overview of SUMMON: (a) an input sequence of human body meshes interacting with a scene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' (b) the Contact-  Former that predicts per-frame contact labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' (c) per-frame contact predictions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' (d) estimated contact points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' (e) synthesized  objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' and (f) objects in interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1  Human Body and Contact Representation  We use a modified version of SMPL-X [Pavlakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019] as the  representation of human body poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Specifically, we parameterize  the human body with 𝑀(𝜃, 𝛽) : R|𝜃 |×|𝛽 | → R3𝑁 , where 𝜃 denotes  pose parameters, 𝛽 denotes coefficients in a learned shape space,  and 𝑁 is the number of vertices in a SMPL-X body mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For com-  putation efficiency, we downsample the vertices from 10,475 to  655 points, following the prior work by Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We represent contact information by per-vertex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For  each vertex 𝑣𝑏 ∈ 𝑉𝑏, where 𝑉𝑏 is all vertices of a human body, we  use a one-hot vector 𝑓 to represent the contact semantic label for  that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Each vector 𝑓 has a length of |𝑓 | = 𝐶 + 1, where 𝐶 is  the number of object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We introduce an extra “void” class to  represent vertices without contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use 𝐹 to denote the contact  semantic labels for all vertices in a body pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='2  Human-Scene Contact Prediction  Our dataset consists of a sequence of paired vertices and contact  semantic labels {(𝑉 1  𝑏 , 𝐹 1), (𝑉 2  𝑏 , 𝐹 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', (𝑉 𝑛  𝑏 , 𝐹𝑛)}, where 𝑉 𝑖  𝑏 repre-  sents the human body vertices (Figure 2(a)), 𝐹𝑖 represents the con-  tact semantic labels for frame 𝑖, and 𝑛 is the varied sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We first train a conditional Variational Autoencoder (cVAE) to learn  a probabilistic model of contact semantic labels conditioned on ver-  tex positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then we deploy transformer layers on top of the cVAE  to improve temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We refer to this framework as  ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' An illustration of the overall network architecture  is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Contact semantics prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We first train a model to predict  contact semantic labels for each individual pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Given a pair of  body vertices and contact semantic labels (𝑉𝑏, 𝐹), we first fuse these  two components: 𝐼𝑒 = Concat(𝑉𝑏, 𝐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We feed 𝐼𝑒 into a graph neural  network (GNN) encoder 𝐺𝐸𝑛𝑐 to get a latent Gaussian space with  the mean 𝐻𝜇 and the standard deviation 𝐻𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then we sample a  latent vector 𝑧 from the latent Gaussian space and concatenate it  with each vertex position: 𝐼𝑑 = Concat(𝑉𝑏,𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We feed 𝐼𝑑 into a  GNN  Encoder  GNN  Decoder  ������������  Positional  Embedding  Transformer  MLP  �������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' �������������������������  ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ������������������������  ������������������������  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ������������������������  ������������  ������������������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ������������������������������������  (������������������������  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ������������1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' (������������������������  ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ������������������������)  For frame i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  ������������������������: Ground-truth semantic labels  ������������������������������������: Initial predicted semantic labels  ������������������������: Hidden feature  �������������������������: Final predicted semantic labels  z : Latent vector  Figure 3: The architecture of ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We first use a  GNN-based variational autoencoder to encode the contact  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then a transformer is applied to improve the tempo-  ral information fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We also add a sinusoidal positional  encoding to the output of the GNN decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  GNN decoder 𝐺𝐷𝑒𝑐 to predict the reconstructed contact semantic  labels 𝐹𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Note that both GNNs in the encoder and the decoder  share the same structure as in Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Each vertex  feature ℎ𝑘𝑥 for vertex 𝑥 at layer 𝑘 is updated by  ℎ𝑘  𝑥 = Linear(Concat({ℎ𝑘−1  𝑥′  : 𝑥 ′ ∈ 𝑁 (𝑥)})),  (1)  where 𝑁 (𝑥) is defined as the 𝑚-nearest neighbor vertices of 𝑥 in a  spiral-ordered sequence, as proposed by Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  ContactFormer: We train a transformer to extract temporal infor-  mation from a pose sequence to enhance prediction consistency, as  shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Specifically, given a sequence of pose and con-  tact semantic labels {(𝑉 1  𝑏 , 𝐹 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', (𝑉 𝑛  𝑏 , 𝐹𝑛)} from frame 1 to 𝑛, we  first use the previous model to reconstruct contact semantic labels  𝐹𝑖𝑝 independently for each frame 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We then embed each 𝐹𝑖𝑝 into a  hidden feature space, augmenting it with a sinusoidal positional  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Karen Liu, Huazhe Xu, and Jiajun Wu  embedding before feeding it to the transformer module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The output  of the transformer module is a sequence of n vectors {𝐻1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', 𝐻𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  For each frame 𝑖, we concatenate 𝐻𝑖 with the initial prediction 𝐹𝑖𝑝  and use a multi-layer perceptron (MLP) to get a final prediction ^𝐹𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  The final prediction is shown in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Training: We optimize the model’s parameters by the following  loss function:  L = L𝑟𝑒𝑐 + 𝛼 · L𝐾𝐿,  (2)  where L𝑟𝑒𝑐 is the sum of the categorical cross entropy (CCE) loss be-  tween the ground truth semantic label 𝐹𝑖 and the model prediction  ^𝐹𝑖 for any frame 𝑖:  L𝑟𝑒𝑐 =  ∑︁  𝑖  CCE(𝐹𝑖, ^𝐹𝑖),  (3)  and L𝐾𝐿 is the Kullback-Leibler divergence loss between the latent  Gaussian space and the normal distribution N:  L𝐾𝐿 = 𝐾𝐿(𝑄(𝑧|𝐹,𝑉𝑏)||N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  (4)  Here we use 𝑄 to represent the encoder network in our cVAE  combined with the sampling process with the reparameterization  trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Inspired by Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2016], we also multiply L𝐾𝐿 with a  weight𝛼 to control the balance between the reconstruction accuracy  and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3  Scene Synthesis  Contact Object Recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Given the accumulated contact points  from each frame predicted by ContactFormer (Figure 2(d)), we  further reduce spatial prediction noise by performing a local object  class majority voting as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then, the vertices of each  predicted object class are clustered into possible contact instances  𝑉𝑐, using the shortest length of all object edges in that class as 𝜖 for  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In practice, we downsample the contact vertices to keep  later computations tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We then optimize the poses of the object point cloud 𝑉𝑜 by  minimizing the following losses:  L(𝑉𝑐,𝑉𝑜) = L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 + L𝑝𝑒𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  (5)  The contact loss L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is defined as  L𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡  1  |𝑉𝑐 |  ∑︁  𝑣𝑐 ∈𝑉𝑐  min  𝑣𝑜 ∈𝑉𝑜  ||𝑣𝑐 − 𝑣𝑜 ||2  2,  (6)  where 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is a tunable hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' This loss encourages  the object to be in contact with the predicted human vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The  penetration loss L𝑝𝑒𝑛 is defined as:  L𝑝𝑒𝑛 = 𝜆𝑝𝑒𝑛  ∑︁  𝑑𝑖𝑐<𝑡  𝑑𝑖  𝑐  2,  (7)  where 𝑑𝑖𝑐 are signed distances between the object and the human  body sequence, 𝑡 is the penetration distance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' This loss  prevents the object from penetrating the human body sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Intuitively, these losses encourage objects to be in contact with  human meshes, but not penetrate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' An illustration of the opti-  mized object placement is shown in Figure 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To improve compu-  tation efficiency, we choose to compute human SDF from merged  human meshes of the motion sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To have a consistent scale  of loss across different objects, we choose the number of sampled  points according to the size of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Inconsistencies  After Voting  table  chair  sofa  bed  Figure 4: Illustration of the local majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' From the  zoomed-in box, there are multiple inconsistent points in the  original contact points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The pink points represent the se-  mantic label bed, and the green points represent the label  sofa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We alleviate this issue by adding majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Constrained Scene Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To obtain a complete scene, we  also predict non-contact objects as a scene completion task con-  strained by 3D human trajectories and existing in-contact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  The floor is divided into a grid, and each cell is labeled as occupied  if feet vertices or object vertices are in close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Consider-  ing the furniture categories in a room as a sequence, we train an  autoregressive transformer model on the 3D-FRONT dataset [Fu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The model takes as input the categories of existing  objects and returns a probability distribution of the next potential  object category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We sample a category from the distribution and  randomly place an object of that category onto an unoccupied floor  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To prevent the sampled object from penetrating the human  body sequence, we further optimize the object’s translation and  rotation using our L𝑝𝑒𝑛 (see Equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  4  EXPERIMENT SETUP  In this section, we introduce the datasets and implementation details  for the whole SUMMON framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1  Datasets  We use the PROXD [Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019] dataset for training our  ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' PROXD uses RGB-D cameras to capture 20 human  subjects interacting with 12 scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We represent human poses  using the SMPL-X format to reconstruct human body meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The  pose sequences in PROXD are estimated using SMPLify [Bogo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016] and contain many jitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We apply LEMO [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2021], a learned temporal motion smoothness prior, to produce  smooth human motion as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our ground truth per vertex  contact semantic labels are generated using scene SDF with contact  semantic labels from PROX-E [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020a], which extends  PROXD by manually annotating the scene meshes with predefined  object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We define human-scene contact as the signed  distance between a human vertex and the scene to be less than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We select objects from 3D-FUTURE [Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b] to be placed  into the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3D-FUTURE is a dataset of categorized 3D models  of furniture with their original sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use a selected subset of  3D-FUTURE to reduce candidate search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To simplify contact  estimation and limit predicted object classes to the available ones  Scene Synthesis from Human Motion  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  in our object dataset, we reduce the contact object categories in the  PROX-E dataset from 42 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We use the GIMO dataset [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022] as another test  dataset for evaluating the generalization ability of the proposed  method on out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='2  Implementation  ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For the encoder and decoder GNNs, we choose  the number of hidden layers to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The dimension for each hidden  vertex feature in the GNNs is 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In the GNN encoder, we downsam-  ple the body vertices after each hidden layer by a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We  deploy a similar architecture for the transformer layers as used in  the previous work [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We provide training details  and hyperparameter choices in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  To compare different architectures’ capacities for extracting tem-  poral information, we also implement models that use MLP and  LSTM [Greff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016] modules as the final block on top of the  GNN decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For the model that uses the MLP module, we deploy  a max pooling layer to the output of the GNN decoder along the  dimension of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Then we feed it to an MLP block to get the  embedding for the whole sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The sequence embedding is  then fused with the output of the GNN decoder to get the final  prediction via a linear projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For the model that uses the LSTM  module, we linearly project the outputs from the GNN decoder into  a higher dimensional embedding space and feed them to a bidirec-  tional LSTM layer to extract features for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Frame features  are then concatenated with the output from the GNN decoder to  obtain final semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Contact Object Recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To reduce noise in contact semantic  estimation, we use majority semantic voting in point cloud clusters  with 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1 and 𝑚𝑖𝑛𝑃𝑡𝑠 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In point cloud clustering for object  instance fitting, we used different values for 𝜖 for different classes  due to their different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  To place objects into the scene at an appropriate height, we first  cluster all the human body vertices that are in contact with the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We then take the minimum medians of all clusters as the estimated  floor height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Next, we translate the object to place its lowest vertex  on the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  To avoid local minima, we perform a grid search for translation  along the floor plane and rotation around the up axis to warm-  start the initial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We then optimize for the same  transformation parameters on top of the results from the grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  In both cases, We use different 𝜆𝑐𝑜𝑛𝑡𝑎𝑐𝑡, 𝜆𝑝𝑒𝑛 and 𝑡 to accommodate  for different properties of object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We keep the transformation  that achieves the lowest loss as the optimization result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  To achieve scene diversity, we consider inter-class and intra-class  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Inter-class diversity is when a human motion is likely  to interact with different classes of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For example, sitting  down can be performed on a chair, a bed, or a sofa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To achieve this,  we first sample per-vertex contact semantics based on the contact  probability distribution predicted by ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' During local  clustering of contact object recovery, we consider class labels in  local clusters as a probability distribution and sample the cluster  contact class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Intra-class diversity is when a human motion is likely  to interact with different instances of the same object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' To  Table 1: Results of contact prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use the reconstruc-  tion accuracy and the consistency score as metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our Con-  tactFormer clearly outperforms the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Models  Reconstruction Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ↑  Consistency Score ↑  MLP Predictor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='8922  LSTM Predictor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9087  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9209  POSA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='8816  ContactFormer (ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='9518  First time on the bed  Second time on the bed  POSA  ContactFormer  Bed  Sofa  Figure 5: Visualizations of the contact prediction results  of POSA and our ContactFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Left: Contact predictions  from POSA and ContactFormer when the person lies on  the bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Right: Contact predictions from POSA and Contact-  Former when the person lies on the bed again after walking  around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ContactFormer has better consistency when the per-  son lies in bed for the second time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  achieve this, we perform grid search and optimization on all the  instances from the object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  5  EVALUATIONS  In this section, we introduce evaluation metrics, baselines, and  results on contact prediction and scene synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We encourage  the readers to watch the video in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1  Contact Semantic Prediction  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We compare with three baselines, including POSA [Has-  san et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b], an architectural variant that uses a multi-layer  perceptron (MLP) based predictor, and a temporal information fu-  sion variant that uses a bidirectional LSTM [Greff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use two metrics for evaluating the contact semantic  prediction: reconstruction accuracy and consistency score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The re-  construction accuracy is computed as the average correctness of  the predicted label compared with the ground-truth label for each  vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The consistency score is designed following this intuition: if  we accumulate predicted contact points from each frame, close con-  tact points should have consistent contact semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Hence,  this loss is computed as follows: Given a pose sequence and the ac-  cumulated contact points, for each point, we compare its predicted  contact label with the contact labels of its neighboring points to  see if the prediction agrees with the majority of the neighboring  contact labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', a high consistency score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Table 1 shows the reconstruction accuracy and the con-  sistency score of all methods on the validation set of PROXD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We  find that ContactFormer achieves competitive performance in terms  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Karen Liu, Huazhe Xu, and Jiajun Wu  PROXD  PROXD  GIMO  Human Meshes  ContactICP  Pose Priors  Ours Snapshot 1  Ours Snapshot 2  Ours Snapshot 3  Figure 6: Visualizations of the generated objects based on human trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The human trajectories are from the PROXD  dataset and the unseen GIMO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The first column shows the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The second column shows the results from  the ContactICP baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The third column shows the results from the Pose Prior baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The fourth to sixth columns are  snapshots of results generated by SUMMON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  of reconstruction accuracies and significantly outperforms all the  baselines in consistency scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' This demonstrates the superior-  ity of the transformer-based architecture in predicting temporally  consistent yet accurate contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Figure 5 visualizes the output contact labels from ContactFormer  and POSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We notice that ContactFormer predicts consistent labels  the second time the human tries to lie on the bed, while POSA, due  to its lack of temporal information, predicts a different label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  User study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We conduct a user study to evaluate the quality of  the contact semantic label predictions, where we compare Con-  tactFormer with POSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For each pose sequence in the validation  dataset, we render a video showing the human motion, predicted  contact semantic labels, and the ground truth scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We show the  predicted contact semantic labels by rendering small areas around  body vertices in different colors depending on their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Each  video is rendered from a camera angle that can clearly capture  human motion and semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For each pose sequence, we ask  the human subjects the following question: "Which video seems to  have a more reasonable contact label prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Among 22 users,  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='12% of the users choose ContactFormer over POSA, believing  ContactFormer provides more reasonable and convincing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  This result echoes the quantitative results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='2  Contact Object Recovery  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Since our problem is novel and there are no baselines,  we devise two reasonable baselines ourselves: contact-informed  point cloud registration (ContactICP) based on point-to-point ICP [Besl  and McKay 1992] and object alignment with pose priors (Pose Pri-  ors) based on the orientation of the hip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We provide the details of  those methods in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use the non-collision score proposed by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' This score estimates the collision ratio between human  body mesh and scene objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Since all the methods, including  Table 2: Non-collision scores for contact object recovery on  the smoothed PROXD and the unseen GIMO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For  each sequence, the score is computed to be the mean of all  possible generated scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Higher scores are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Method  PROXD  GIMO  ContactICP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='654  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='820  Pose Priors  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='703  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='798  SUMMON w/o optimization  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='815  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='937  SUMMON (ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='851  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='951  SUMMON, first align the object to the centroid of contact points,  contact constraints are naturally satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' For each sequence, we compute the mean of the non-  collision scores for all the objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Table 2, we compare  the mean non-collision scores on the smoothed PROXD dataset [Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021], which was used during training, and the unseen GIMO  dataset [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022], which also provides SMPL-X parameters  for humans interacting with scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  We visualize comparisons between our method and the base-  lines in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We find that SUMMON can synthesize objects  that are physically plausible and semantically reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Contac-  tICP usually suffers from large penetrations because the contact  points might be sparse for registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' While Pose Priors can have  seemingly correct object locations and orientations, it often fails to  consider physical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Figure 7 demonstrates various possible scenes generated from  the same human motion trajectory by SUMMON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We find that  SUMMON can generalize intra-class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', chairs with different ap-  pearances) and inter-class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', sofa to a bed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We provide additional  examples in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Human user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We follow the same procedure as in Sec-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Instead of contact prediction, we present the users with  灵AScene Synthesis from Human Motion  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Human Meshes  Possible Scene 1  Possible Scene 2  Possible Scene 3  Figure 7: Visualizations of possible alternative object placements generated by SUMMON based on the same human trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  In this example, an in-contact object can be a chair, a sofa, or a bed, as long as it does not violate physical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' SUMMON  can also generate different instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=', chairs) within the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Human Meshes  Object Recovery  Synthesized Scene 1  Synthesized Scene 2  Figure 8: Visualizations of scene completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Based on all the in-contact objects and human motion trajectories, SUMMON  now generates the objects that are not in contact with human meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' While there is no contact, it makes the scene more  complete and introduces the potential for future synthesized human motion sequences to interact with additional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  the animated human motion sequences and the predicted objects in  the scene, and ask them to choose the most plausible and realistic  placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' From the statistics, we find that 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='5% of the users se-  lect SUMMON over ContactICP and Pose Priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We find that Pose  Priors has a 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='5% user selection rate, showing that it can produce  reasonable results in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We also perform ablation on the optimization objec-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Table 3 shows that both the penetration loss and the contact  loss are important for SUMMON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Intuitively, the penetration loss  helps the object to avoid a collision, while the contact loss helps  to keep the object close to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use both the non-collision  score and the contact score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The contact score is computed as the  fraction of objects in the scenes that are in contact with the human  trajectory Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3  Scene completion  To generate a full-fledged scene, we train another object generation  model following Paschalidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' [2021] as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The  model outputs a family of possible objects that does not contact or  penetrate human meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Using this model, we generate a fuller  scene with both in-contact and no-contact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Visualized re-  sults are in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The completed scenes have additional objects,  such as a TV stand or a coffee table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' While there is no contact be-  tween these objects and the human meshes, they make the scene  semantically more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Table 3: Ablation study on the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The penetration loss  and the contact loss are ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We use the non-collision  score and the contact score as metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Method  non-collision score ↑  contact score ↑  SUMMON  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='894  1  w/o penetration loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='656  1  w/o contact loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='995  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='194  6  CONCLUSION  We propose Scene Synthesis from HUMan MotiON (SUMMON), a  framework that generates multi-object scenes from a sequence of  human interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' SUMMON leverages human contact estimations  and scene priors to produce scenes that realistically support the  interaction and the semantic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' The flexibility of SUMMON  also enables the synthesis of diverse scenes from a single motion  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' We hope this can also shed light on generating inexpen-  sive diverse human-scene interaction datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In the future, we are  interested in exploring the following directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Since PROXD does  not consider soft-body interactions, a potential direction would  be considering soft-body deformation of objects such as beds and  sofas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Our method considers synthesized scenes to be stationary,  hence future works can include movement and rearrangement of  furniture during human-scene interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' As PROXD categorizes  all the smaller interaction objects such as books, cups, or TV re-  motes into a single category, one potential extension to our method  would be to include interactions with more specific small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Sifan Ye, Yixing Wang, Jiaman Li, Dennis Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Karen Liu, Huazhe Xu, and Jiajun Wu  ACKNOWLEDGMENTS  This work is in part supported by the Stanford Human-Centered  AI Institute (HAI), the Toyota Research Institute (TRI), Innopeak,  Meta, Bosch, and Samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
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+page_content='  Huan Fu, Bowen Cai, Lin Gao, Ling-Xiao Zhang, Jiaming Wang, Cao Li, Qixun Zeng,  Chengyue Sun, Rongfei Jia, Binqiang Zhao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3d-front: 3d furnished  rooms with layouts and semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International Conference on Computer Vision  (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 10933–10942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Huan Fu, Rongfei Jia, Lin Gao, Mingming Gong, Binqiang Zhao, Steve Maybank, and  Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3d-future: 3d furniture shape with texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' International  Journal of Computer Vision (IJCV) 129, 12 (2021), 3313–3337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Shunwang Gong, Lei Chen, Michael Bronstein, and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Spi-  ralnet++: A fast and highly efficient mesh convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International  Conference on Computer Vision Workshops (ICCVW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 4141–4148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Klaus Greff, Rupesh K Srivastava, Jan Koutník, Bas R Steunebrink, and Jürgen Schmid-  huber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' LSTM: A search space odyssey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' IEEE Transactions on Neural Networks  and Learning Systems 28, 10 (2016), 2222–2232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Abhinav Gupta, Scott Satkin, Alexei A Efros, and Martial Hebert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' From 3d  scene geometry to human workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision and Pattern  Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 1961–1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Félix G Harvey, Mike Yurick, Derek Nowrouzezahrai, and Christopher Pal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Robust  motion in-betweening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM Transactions on Graphics (TOG) 39, 4 (2020), 60–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Mohamed Hassan, Duygu Ceylan, Ruben Villegas, Jun Saito, Jimei Yang, Yi Zhou, and  Michael Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Stochastic Scene-Aware Motion Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International  Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 11354–11364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Mohamed Hassan, Vasileios Choutas, Dimitrios Tzionas, and Michael J Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Resolving 3D human pose ambiguities with 3D scene constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International  Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2282–2292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Mohamed Hassan, Partha Ghosh, Joachim Tesch, Dimitrios Tzionas, and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Populating 3D Scenes by Learning Human-Scene Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In  Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 14708–14718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew  Botvinick, Shakir Mohamed, and Alexander Lerchner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' beta-vae: Learning  basic visual concepts with a constrained variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International  Conference on Learning Representation (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Daniel Holden, Jun Saito, and Taku Komura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' A deep learning framework for  character motion synthesis and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM Transactions on Graphics (TOG) 35, 4  (2016), 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Lucas Kovar and Michael Gleicher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Flexible automatic motion blending with  registration curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Symposium on Computer Animation (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 214–224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Manyi Li, Akshay Gadi Patil, Kai Xu, Siddhartha Chaudhuri, Owais Khan, Ariel Shamir,  Changhe Tu, Baoquan Chen, Daniel Cohen-Or, and Hao Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Grains:  Generative recursive autoencoders for indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM Transactions on Graphics  (TOG) 38, 2 (2019), 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Xueting Li, Sifei Liu, Kihwan Kim, Xiaolong Wang, Ming-Hsuan Yang, and Jan Kautz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Putting Humans in a Scene: Learning Affordance in 3D Indoor Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  In Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 12360–12368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Andrew Luo, Zhoutong Zhang, Jiajun Wu, and Joshua B Tenenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' End-to-  end optimization of scene layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision and Pattern  Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3754–3763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Naureen Mahmood, Nima Ghorbani, Nikolaus F Troje, Gerard Pons-Moll, and Michael J  Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' AMASS: Archive of motion capture as surface shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International  Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 5442–5451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Yinyu Nie, Angela Dai, Xiaoguang Han, and Matthias Nießner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Pose2Room: Un-  derstanding 3D Scenes from Human Activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In European Conference on Computer  Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Sang Il Park, Hyun Joon Shin, and Sung Yong Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' On-line locomotion generation  based on motion blending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Symposium on Computer Animation (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 105–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Despoina Paschalidou, Amlan Kar, Maria Shugrina, Karsten Kreis, Andreas Geiger, and  Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ATISS: Autoregressive Transformers for Indoor Scene Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  In Advances in Neural Information Processing Systems (NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 12013–12026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Georgios Pavlakos, Vasileios Choutas, Nima Ghorbani, Timo Bolkart, Ahmed A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Osman, Dimitrios Tzionas, and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Expressive Body Capture:  3D Hands, Face, and Body From a Single Image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision  and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 10975–10985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Xavier Puig, Kevin Ra, Marko Boben, Jiaman Li, Tingwu Wang, Sanja Fidler, and  Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Virtualhome: Simulating household activities via programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  In Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 8494–8502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Pulak Purkait, Christopher Zach, and Ian Reid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Sg-vae: Scene grammar variational  autoencoder to generate new indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In European Conference on Computer  Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 155–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Davis Rempe, Tolga Birdal, Aaron Hertzmann, Jimei Yang, Srinath Sridhar, and  Leonidas J Guibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Humor: 3d human motion model for robust pose es-  timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 11488–11499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Daniel Ritchie, Kai Wang, and Yu-an Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Fast and flexible indoor scene synthesis  via deep convolutional generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision and  Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 6182–6190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Manolis Savva, Angel X Chang, Pat Hanrahan, Matthew Fisher, and Matthias Nießner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Pigraphs: learning interaction snapshots from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM Transactions  on Graphics (TOG) 35, 4 (2016), 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Roberta Spallone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Digital reconstruction of demolished architectural master-  pieces, 3D modeling, and animation: the case study of Turin Horse Racing by  Mollino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Handbook of research on emerging digital tools for architectural surveying,  modeling, and representation (2015), 476–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N  Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In  Advances in Neural Information Processing Systems (NIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Jiashun Wang, Huazhe Xu, Jingwei Xu, Sifei Liu, and Xiaolong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Synthe-  sizing long-term 3d human motion and interaction in 3d scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on  Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 9401–9411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Kai Wang, Yu-An Lin, Ben Weissmann, Manolis Savva, Angel X Chang, and Daniel  Ritchie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Planit: Planning and instantiating indoor scenes with relation graph  and spatial prior networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Xiaolong Wang, Rohit Girdhar, and Abhinav Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Binge watching: Scaling  affordance learning from sitcoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision and Pattern  Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2596–2605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Xinpeng Wang, Chandan Yeshwanth, and Matthias Nießner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Sceneformer:  Indoor scene generation with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International Conference on 3D Vision  (3DV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 106–115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Zhe Wang, Liyan Chen, Shaurya Rathore, Daeyun Shin, and Charless Fowlkes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Geometric pose affordance: 3d human pose with scene constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' arXiv preprint  arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='07718 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Jingwei Xu, Huazhe Xu, Bingbing Ni, Xiaokang Yang, Xiaolong Wang, and Trevor  Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Hierarchical Style-based Networks for Motion Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In European  Conference on Computer Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 178–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Hongwei Yi, Chun-Hao P Huang, Dimitrios Tzionas, Muhammed Kocabas, Mohamed  Hassan, Siyu Tang, Justus Thies, and Michael J Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Human-aware object  placement for visual environment reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision  and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 3959–3970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Siwei Zhang, Yan Zhang, Federica Bogo, Marc Pollefeys, and Siyu Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Learning  motion priors for 4d human body capture in 3d scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International Conference  on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 11343–11353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Song-Hai Zhang, Shao-Kui Zhang, Wei-Yu Xie, Cheng-Yang Luo, and Hong-Bo Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Fast 3d indoor scene synthesis with discrete and exact layout pattern  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='00328 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Yan Zhang, Mohamed Hassan, Heiko Neumann, Michael J Black, and Siyu Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Generating 3d people in scenes without people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In Conference on Computer Vision  and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 6194–6204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Yang Zheng, Yanchao Yang, Kaichun Mo, Jiaman Li, Tao Yu, Yebin Liu, Karen Liu,  and Leonidas Guibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' GIMO: Gaze-Informed Human Motion Prediction in  Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In European Conference on Computer Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Scene Synthesis from Human Motion  SA ’22 Conference Papers, December 6–9, 2022, Daegu, Republic of Korea  Yang Zhou, Zachary While, and Evangelos Kalogerakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Scenegraphnet: Neural  message passing for 3d indoor scene augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In International Conference on  Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 7384–7392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  Yuke Zhu, Alireza Fathi, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' Reasoning about object affordances in a  knowledge base representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content=' In European Conference on Computer Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
+page_content='  408–424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAzT4oBgHgl3EQfdvwv/content/2301.01424v1.pdf'}
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